摘要:

C o n t e n t sVol. 4, No. 2. October 1971Computer simulation of some physicochemical problemsMoti La1Lord Ernest Rutherford of Nelson (1 871-1937)R. H. CraggRecent advances in the chemistry of noble gas elementsN. K. JhaGraph theory in chemistryD. H. RouvrayCumulative index971291471731970 The Royal Institute of Chemistry30 Russell Square, London WCI B 5D

ISSN:0035-8940    

DOI:10.1039/RR97104FP003

出版商:RSC    

年代:1971

数据来源: RSC

摘要:

Chemical Society ReviewsAs from January 1972, arising from the amalgamation of The Societies, thereview periodicals Quarterly Reviews and R.I.C. Reviews will be replaced byone new review publication, Chemical Society Reviews.The new periodical will not be simply a mixture of its two predecessors.It will, however, constitute a logical continuation of both, and will aim toprovide articles of interest to a wide readership. The articles will rangeover the whole of chemistry and its interfaces with other disciplines, andthere will be a conscious attempt to provide consistency of depth of treatment,so that each article is of interest to chemists in general, and not merely tothose with a specialist interest in the subject being reviewed.Chemical Society Reviews will appear quarterly, and should comprise approxi-mately 30 articles (600 pp) pa. Although the majority of articles will bespecially commissioned, the Society will always be prepared to consideroffers of articles for publication. In such cases a short synopsis should besubmitted rather than the completed article.Members of The Chemical Society will have an opportunity to subscribeto Chemical Society Reviews at E2.00 pa; they should place their orders ontheir Annual Subscription renewal forms in the usual way. Non-membersmay order Chemical Society Reviews (E8.00 pa; remittance with order)from The Publications Sales Office, The Chemical Society, BlackhorseRoad, Letchworth, Herts, SG6 IHN

ISSN:0035-8940    

DOI:10.1039/RR97104FX005

出版商:RSC    

年代:1971

数据来源: RSC

摘要:

COMPUTER SIMULATION OF SOME PHYSICOCHEMICAL PROBLEMS Moti La1 . . . . 97 98 . . . . . . . I . . . . . . . . . . . . . . . . . . . . 102 103 . . .. . . . . . . . . . . . . . . . . . * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical theory of fluids, 103 Computer studies, 105 Simple chain molecules . . . . Biological macromolecules Adsorption Reaction kinematics Systems corresponding to the Ising model Applications of the analog computer . . . . . . ‘ . . . . . . . . . . . . . . . . . . . . . * . 102 . . 111 113 117 118 120 122 124 125 .. . . Conclusion. . .. . _ . . .. References . . . ... . . . . . . .. . . . . .. Unilever Research, Port Sunlight Laboratory, Port Sunlight, Wirral, Cheshire 162 4XN * ’ . Introduction RCsumC of statistical mechanics. . ‘Monte Carlo’ method . . . . The method of molecular dynamics The fluid state . . . . .. INTRODUCTION . . .. . . Undoubtedly the electronic computer is one of the most powerful tools in the hands of modern scientists and technologists. Apart from taking the drudgery of manual computation out of scientific and technical work, it has allowed us to tackle many problems pursuing hitherto unexplorable directions. This led to the evolution of completely new systems for informa- tion storage and retrieval, cataloguing, accountancy, etc., and a great extension of the scope of investigation into engineering designs and chemical engineering operations. The recent flourishing of such subjects as operational research and cybernetics is a consequence of the advent of high-speed computers.In science, an area where computers have proved of tremendous use is model building and simulation. Such studies have allowed a deeper insight and, often, a growing understanding of a situation or a phenomenon in question to be gained. Earlier work on computer simulation of the random neutron diffusion process in fissile material by von Neumann and Ulam provides an illuminating example of the success of computers even in the early stages of their development.1 Modern computers are capable of handling more complicated simulations : simulations of simple fluids,2a,ZbJ solids and alloys4~5 have successfully been achieved-the behaviour of the computer models has closely approximated that of the real systems; extensive studies on the computer simulation of chain molecules have been made;6,79* certain chemical kinetic processes have been simulated with the object of elucidating reaction mechanisms ; 9 simulation of the processes of adsorption,loJl La1 97 diffusion12 and percolation13 has been done with some success; computer simulation techniques have entered the domain of biochemistry.l4 This article intends to give a detailed account of this mode of computer application in the above spectrum of physicochemical areas, and to assess, where possible and appropriate, the values of such studies as aids to verifying the correspond- ing analytical theories and promoting further comprehension of the basis of the subjects concerned.An article dealing with a similar topic appeared in the literature a decade ag0.15 Advances of much significance made in the applications of the computer methods and their extension to several new areas since then, call for a new review of the subject. Most of the aforementioned computer simulation studies aim to correlate the molecular behaviour of the systems with their phenomenological proper- ties ; this results in their deep involvement with statistical mechanics. Therefore a brief outline of relevant basic statistical mechanics is given before describing simulation techniques and the behaviour of the simulated systems.RE SUM^ OF STATISTICAL MECHANICS We consider a system of N particles that is able to conform to various con- figurations, say 1, 2, 3, etc., with the corresponding energies 2 1 , ZZ, .%3, etc. Statistical mechanics provides the following expression for the canonical ensemble average, the equilibrium value, P, of a property P of the system: 1 2 C Pi exp ( - - Z i / k T ) where Pi and Hi are the values of the property P and energy, respectively, corresponding to a configuration i; k is the Boltzmann constant, and T is the temperature. The summations in 1 are carried over all the possible configurations that the system can assume. The denominator of the right-hand side of 1 is called the partition function of the system and is usually denoted by 2; thus i Z = C exp ( - 2 i / k T ) If the 'internal' energy of the system (the energy due to the internal degrees of freedom, e.g. internal vibrations, rotations, etc., of the particles) is scaled at zero level, and assumed to be invariant under ordinary thermodynamic conditions, Zi is then 3 The two sets of vectors PI, Pz, .. . PN and q1, 92, . . . qN represent, respec- tively, the momenta and the positions of the particles in the configuration i; the subscripts indicate the corresponding particles, and s is the mass of each particle. The first term in 3 is the kinetic energy of the system, and is a function of temperature only; while U is the potential energy, and depends only on the positions of the particles.R. I. C. Reviews 98 From 2 and 3, 4 In the classical limit, for a system of non-localized particles, 4 assumes the form dP2 . . . dPN dql .dqz . . . . dqiv dPz . . . ~ P N 1 . . . 1 exp (- U/kT) dgi .dqz . . . . dq,v where I stands for the integral 5 6 ii 1 . . . 1 exp (-U/kT) dql .dq2 . . . . dqN called the configurational partition function or the configurationa 111 tegral of the system. The partition function is related to the thermodynamic properties of the system through the equation A = - kT In 2; A is the free energy of the system 7 The expressions relating the partition function to various other thermo- dynamic functions can be readily deduced from 7 by making use of standard thermodynamic formulae which relate free energy to the thermodynamic functions in question.Thus we obtain S (entropy) = k In Z + (kT/Z) aT p (pressure) = ~- a l n z Cv (heat capacity at constant volume) = - kT Another approach to the classical thermodynamics from the molecular state of the system is provided by the molecular distribution functions. A molecular distribution function nnz is defined as the probability of the occurrence of a configuration in which each position of a set of m positions La1 99 ql, q2 . . . , qm is occupied by a particle. It follows from the above definition that ~- ~- 12b r is the distance between ql and 92; nl(q1) is the probability of finding a particle at ql, and n2(qz) is that of finding a particle at 9 2 . If we assume that the potential energy, U, of the system can be expressed as a sum of the pair interaction energies of the constituent particles, i.e.13 where zrij(r) is the interaction energy between two particles i and j situated at a distance r from each other (the notation i < j assures that the pairs ij and j i are equivalent and hence are not counted twice), g is related to U as 1 . . . j e x p (- U / k T ) dqm+l. 4 m + 2 . . . dqN N ! nm = - 1 . . . I e x p (- U/kT) dql.dq2. . . . dqN (N - rn)! 9 In the theory of fluids the most significant distribution function is the pair distribution function nz(q1, q2), the probability of finding any two particles each at the positions q1 and qz simultaneously; which is simply equal to 1 1 We define a function g , called the radial distribution function, as 12a Aq1, (Sl - ad1 = nz(q1, sd/nl(al> x nz(q2) In an isotropic fluid, the above reduces to gw = n2(r)/(N/V)2 n a, U = (N2/2 V ) J u(r).g(r) .4m2 dr 0 virial theorem, yields the following equation of state p = (NkTJV) - 2z<: 1 T ) . g ( r ) . r 3 dr Hence, 14 15 where (3/2)NkT is the kinetic energy of the system (assuming that the con- stituent particles are spherical). Equation 15, in conjunction with Clausius’ 16 3 v 0 R. I. C. Reviews 100 Equations 15 and 16 lead to the expressions relating the radial distribution function to various other thermodynamic functions of the system. Recently, the direct correlation function, c(r), defined by the following equation 17, the Ornstein-Zernike equation, has come into much prominence due to its appearance in the advanced theories of fluids 17 0 rij represents the distance between a particle i and a particle j ; and h(r) = g(r) - 1. It emerges from the preceding account that the essential requisite for obtain- ing explicit relations between the molecular properties and bulk thermo- dynamic functions is the solution of the equations 15 and 16 which in turn demand the knowledge of the quantitative nature of the functions u(r) and g(r). Through the combination of experimental and theoretical considera- tions, several approximate but convenient forms of u(r) have been deduced.These potential functions may be considered adequate for the present pur- poses. Although g(r) can be determined from x-ray and neutron diffraction or scattering studies,l6J7 H and p , according to the equations 15 and 16, are so sensitive to g(r) that a very high accuracy, unattainable by the existing experimental techniques, in the values of g(r) is required for their (H and p ) calculation. One has, therefore, to resort to theoretical methods for evaluating radial distribution functions.Complete solutions of the configurational integral and radial distribution function for realistic systems defy analytical methods unless certain simpli- fying assumptions are introduced. There might not be valid justifications for presuming that such assumptions, mainly dictated by mathematical con- siderations, would be compatible with the physical characteristics of the system.Alternatively, in the event of an adequate knowledge about the physical picture of the system, its mathematical simulation can be evolved which may then be employed for the direct evaluation of the partition func- tion and g(r). But for certain statistical considerations, the prohibitive amount of calculations involved would have denied the adoption of such a procedure. The ‘Monte Carlo’ method involves the use of such statistical considerations for reducing the calculations to an amount that can be handled adequately by modern computers.18 Another technique which is equally extensively employed in simulating statistical mechanical systems by computer is the method of molecular dynamics.19 While the ‘Monte Carlo’ method gives ensemble averages of the properties of the system under study, the method of molecular dynamics follows the trajectories of the system in time, thus yielding time-averages of the properties.The results obtained from the two approaches would, however, be identical at equilibrium. In the next two sections some basic aspects of the two methods are dealt with. La1 101 ‘MONTE CARLO’ METHOD ‘Monte Carlo’ method is a statistical technique involving random sampling. Its use for treating probabilistic problems-problems posed by the situations governed by stochastic processes, e.g., diffusion of neutrons in fissile materials-is obvious and has a long history. The great utility of the method, as a numerical technique, in the solution of deterministic mathematical problems that are intractable by analytical methods has, however, been realized relatively recently.20 Multiple integral expressions representing the partition functions of the systems of interacting particles typify such prob- lems.The following illustrates the application of the ‘Monte Carlo’ approach to statistical mechanical problems. Consider a fluid system of N particles. The simplest way of determining the statistical thermodynamic properties of the system would be to compute the Boltzmann factors corre- sponding to all the configurations accessible to the system, and then obtain the partition function by summing them. But the impracticability of carrying out such an operation is obvious. However, instead of attempting to consider all the configurations the system can assume, a random sample from the con- figuration population can be drawn.The properties computed on the basis of the sample thus collected may be assumed to be unbiased estimates of the properties of the configuration population. Estimation procedures designed along these lines are essentially ‘Monte Carlo’ methods. Two considerations account for the magnitude of the sample: the ‘speed’ with which sample properties converge to the population properties, and the desired limits of accuracy in the estimated values. The uncertainty in a statistical estimate is measured by the variance, 02, which is defined as THE METHOD OF MOLECULAR DYNAMICS 18 where Pi is the value of a property, P, corresponding to a configuration i chosen in the sample; is the estimate of p ; and n is the sample size.The magnitude of 02 is determined by the sample size, n, as well as the mode of sampling: two samples of the same size but obtained from different sampling procedures often render different variance. An efficient ‘Monte CaI lo’ scheme is one that would involve a sampling procedure producing small variance. Efficient sampling designing is known as variance-reduction. Several variance-reduction techniques occur in the literature ;21 the choice depends on the nature of the problem under consideration. In ‘Monte Carlo’ calculations related to statistical mechanical problems-which is the main concern in this article-a commonly used sampling procedure is importance sampling. Importance sampling allows the sample to have the highest concentration of its points in the high-probability region of the distribution which the population follows ; the configuration populations of ordinary statistical mechanical systems follow a Boltzmann distribution.Assuming that the constituent particles of the system to be investigated behave classically, Newton’s equations-of-motion for each particle are R. I. C. Reviews 102 numerically integrated to obtain the momenta and the positions as functions of time.” These are continually recorded as the computation progresses. The time-average values of position- and velocity-dependent properties of the system can be calculated from this data. At the initial point (zero time) the particles assume some arbitrary but convenient set of momenta and positions.The temperature of the system at an instant t is given by 19 where q,,, and q,,t represent the positions of a particle n corresponding to the times o and t respectively and 20 s( n=l 5 (v?b,t)2)!3Nk where v,,t is the velocity of a particle n at t, and s is the particle mass. The method of molecular dynamics has a wider scope than the ‘Monte Carlo’ method: while the ‘Monte Carlo’ method can only be applied to study equilibrium situations, the method of molecular dynamics can be used in investigating time-dependent phenomena such as relaxation and non-equi- librium behaviour. Quantitative information for important primary proper- ties such as particle-velocity distribution, collision rate, pair and triplet distribution functions, pressure, and mean potential energy of the particles can be derived from molecular dynamic computation. Other quantities of great interest are the mean square of particle displacement, ((q2))t, and the velocity autocorrelation function, Z ( t ) , which are defined as Z ( t ) = ) c vn,o - vn, t N Y N n=l ((q2))t and Z ( t ) are related to the process of diffusion in the system.Having outlined the necessary basic statistical mechanical formalism and introduced two current computer simulation approaches, ‘Monte Carlo’ method and the method of molecular dynamics, applications in various areas of physicochemical research can now be considered. THE FLUID STATE Statistical theory of fluids In the case of non-interacting particle systems, the partition function and the radial distribution function are subject to analytical solutions.However when a system consists of particles which interact with one another in accord with a realistic law, the above integrals cannot be evaluated as such: the multi-integrals involving the potential energy of the system cannot be separated into parts so that each part would depend only on the coordinates of one particle. The aim of the statistical mechanical theories of fluids is, therefore, to introduce simplifications so as to reduce the configurational integral to the forms amenable to analytical solutions. The important theories * The details of the procedure are set out in: B.J. Alder and T. Wainwright, J. chem. Phys., 1959, 31,459; A. Rahman, Phys. Rev., 1964, 136, A405. La1 103 are : the cell theory,22 Mayer's theory of cluster integrals,23 Bogoliubov- Born - Green - Kirkwood - Yvon (BBGKY) theory,24 hypernetted chain (hnc) theory,25 Percus-Yevik (PY) theory,26 and perturbation theory.104 The cell theory assumes that each particle in a fluid system is confined to a spherical cell made up of the surrounding particles. The configurational energy of a particle can be considered to be composed of two parts : (i) energy, UO, possessed by the particle when positioned at the centre of the cell (lattice energy), (ii) UD, the energy associated with the displacement of the particle from the cell centre.The configurational partition function, q, of the particle is, then 21 23 24 1 exp [-(uo + u ~ ) / k T ] 4 r v ~ dr cell where r is the radius of the cell. The configurational partition function of the system would simply be equal to q N . Mayer's theory, in its essentials, is concerned with a series expansion of the integrand exp - ( U / k T), and the subsequent term-by-term integration of the series. If exp -(uzj/kT) is expressed as (1 + f i j ) , I N \ N =1+cc.h+ N i j i < j c c N c (.fij.hk.f3k + . f i j . f i k + j i j . h k +J;:k.ftk) + * * - 22 i j k i < j < k It may be realized that the second, third, etc., terms in the above series act as successive corrective terms modifying the partition function to take inter- molecular interactions into account.The second term involves molecular pairs (ij), the third term triplets (ijk), the fourth term quadruplets, and so on. The consequences of the work carried out in this direction by Ursell, Mayer, Born, and others provide a sound theoretical foundation to the classical virial equation of state for imperfect gases. The usefulness of this procedure is limited to low-density gases and liquids; with high-density gases and liquids, the above series fails to converge. BBGKY, hnc and PY theories enable one to evaluate g(r), assuming U(Y) is known. The two basic equations of the BBGKY theory are: g123 = 812 -g23 *g13 g12, g23, etc. correspond to the pairs 1-2,2-3, etc. V is the differential operator (a/&, a/ay, a/az).Equation 23 is directly derivable from 10,24 is an approxi- mation-introduced by Kirkwood-which estimates the triplet distribution function g123 in terms of the radial distribution functions g12, g23 and g13. The R. I. C. Reviews 104 substitution of 24 into 23 would lead to an equation involving only gijs and uir). This resulting equation, called the BBGKY equation, can be solved numeri- cally for g. The hnc theory evolved from Montroll and Mayer’s procedure of the summation over a certain class of interaction graphs (in the terminology of Ursell and Mayer);27 while the PY theory originated from an attempt to apply the collective variable method-in a fashion similar to that applied by Bohm and Pines to a system of charged particles28-to non-polar fluids.Though the two theories come from different directions, the structure of the equations (relating c(r) to u(r)) which they yield bear similarity to each other: 25 c(r) = h(r) - In [l + h(r)] - u(r)/kT (hnc) c(r) = [l -+ h(r)][1 - exp (u(r)/kT)] (PY) Computer studies 26 The substitutions of 25 and 26 in the Ornstein-Zernike equation 17 gives the well-known hnc and PY integral equations respectively. The development of the perturbation theory with realistic systems is due to Barker and Henderson.104 The theory treats an intermolecular potential as a modified function of an arbitrary potential. The potential can be defined in terms of three parameters: d, the hard-sphere parameter, a, the inverse steepness parameter for the repulsive region, and y, the depth parameter for the attractive region.a = y = 1 corresponds to the arbitrary potential function, and a = y = 0 describes a hard-sphere potential. The theory is essentially concerned with the expansion of the configurational integral in double Taylor series in terms of a and y about a = y = 0. The expansion has satisfactory convergence under certain realistic conditions. While attempting numerical solutions of the BBGKY equation for the case of hard-sphere systems, Kirkwood, Maun and Alder discovered that the equation does not possess any integrable solution in the density region above (N/V)a3 = 0.95, CT being the molecular diameter (for the close-packed system (N/V)o3 = 1.414).29 This prompted Alder and Wainwright to simu- late the system on computer by adopting the method of molecular dynamics.19 The simulated system underwent solid/fluid transition at approximately (N/ V)a3 = 0.95, the point beyond which conventional numerical procedures cease to yield a solution for the BBGKY equation.The same system was also simulated by Wood and Jacobson employing the ‘Monte Carlo’ method; this study virtually reproduced Alder and Wainwright’s results.ls Because of the rather small number of particles constituting the system (-100-500)-a computer limitation-and because of the apparent artificiality of the simu- lated systems, the validity of the conclusions may be doubtful. But a close agreement between the equation of state, for the fluid regon, derived from the computer studies and that obtained by determining successive virial coeffi- cients (Fig.I ) would certainly dispel such doubts, establishing a firm reliability in ‘Monte Carlo’ and molecular dynamic calculations. Indeed, the computer methods can now be regarded as ‘experimental’ techniques, and hence the results obtained from them should provide a fair basis for assessing the relative merits of various analytical theories. La1 105 of hard spheres; ~ ‘Monte Fig. I . The equation of state for a system derived from molecular dynamic studies;lg --- virial expansion. (Ref. 18.) Carlo’ results of Wood and Jacobson;]* The inference of phenomenological properties from those of the submicro- scopic simulated system utilizes ‘periodic boundary conditions’. These state that a macroscopic system is formed from the successive translational replications of the system under computer study.Therefore, the molecular interactions between the system and its surrounding replicas are to be taken into account. In the molecular dynamic studies, the periodic boundary con- ditions stipulate that if a particle leaves the ‘cell’-the confines of the system- another particle with identical kinetic properties would enter the cell in- stantaneously from the opposite direction, so that in all instances the number of the particles in the cell remains constant. The ‘Monte Carlo’ method in simpler form, viz. collecting a sample of accessible random configurations generated independently of one another, would be extremely inefficient, involving a prohibitive amount of computer time.Therefore it would be necessary to devise an efficient sampling scheme for a successful ‘Monte Carlo’ calculation. A successful, and hence extensively used, variance-reduction scheme was originated by Metropolis and co- w o r k e r ~ . ~ ~ A large sequence of configurations generated by their method tends towards a Boltzmann distribution; hence the simple average of a property over the sample converges to the canonical ensemble average value. The innovation of the Metropolis sampling technique, with the advent of fast electronic computers with sufficient storage capacities, has led to success- R. I. C. Reviews 106 ful ‘Monte Carlo’ studies on systems with realistic intermolecular inter- actions, e.g.the work of Wood and Parker, in 1957, on the computer simulation of argon at 55°C.31 They assumed that the molecules interact in accord with the Lennard-Jones equation, and that the configurational energy is expressible as the sum of the total intermolecular pair interactions. The computed compressibility, energy and heat capacity data were found to agree with the experimental results of Michel and co-workers in the pressure range 15.19875-202.65 MPa; however at high pressure, viz. 202.65-1519.875 MPa the calculated compressibility factors were at variance with Bridgnian’s experi- mental values. The computer study indicated the existence of fluid/solid transitions, but the predicted freezing pressure was found to be lower and the volume change higher than experimental estimates.Wood, later, extended the computations to higher temperatures.2b In a similar study, by McDonald and Singer, the calculations were carried out in the temperature range - 100-150°C.32 Fair agreement between the computed and experi- mental compressibility factors was observed but the discrepancy between the ‘Monte Carlo’ and actual energy values was marked. They attributed this to the inadequacy of the Lennard-Jones potential in representing inter- molecular interactions. ‘Monte Carlo’ work by Verlet and co-workers has been mainly directed to the assessment of the current statistical mechanical theories of fluids by comparing theoretical and ‘Monte Carlo’ computed values of thermodynamic functions.33 Figure 2 shows such a comparison, where the compressibility factor, p/RTp, as a function of density p, calculated from the PY equation (with the first correction term), is represented by the 1.7 1.6 1.5 1.4 1.3 P/p, 1.2 1.1 3.1 ,0.2 ,0.3 ,0.4 10.5 ,0.6 ‘0.7 9.8 ,O.S Fig.2. Compressibility factors, versus density isotherm for argon :or- responding t o the reduced temperature, kT/E*, I .35. I ‘Monte Carlo’ results; calculated from the PY equation with the first correction term; 0 experi- mental values. (Ref. 33.) La1 P 107 solid line; the dots are the experimental points for argon and the vertical lines represent ‘Monte Carlo’ results. Close agreement between ‘Monte Carlo’ and theoretical results is demonstrated up to p = 0.55; beyond this, the deviation of the theoretical line from the computer results is considerable. The experimental results differ substantially from those of ‘Monte Carlo’ and PY.This again shows the invalidity of the L-J potential. Thus if a correct intermolecular potential is found, the PY equation solution should be able to reproduce experimental results in the low density region. Many-body forces cause an additional complication at high densities, their exact treat- ment is beyond present theories. In a subsequent study, computer ‘experi- ments’ were designed to test the equations of state derived from Barker and Henderson’s theory; the theory was in reasonable agreement with the machine results.105 Further progress in ‘Monte Carlo’ applications has been made by Verlet who carried out a successful study on phase transition behaviour of Lennard-Jones fluids.34 Figure 3 shows the experimental and ‘Monte Carlo’ co-existence curves for argon; Fig.4 gives the melting pressure versus temperature curve. Excellent agreement between experimental points and the computed curve has been found for the solid/fluid transition; however the agreement is rather poor for the liquid/gas transition. The ‘Monte Carlo’ melting pressures are somewhat lower than the experimental pressures in the whole of the temperature range studied. Singer35 has extended ‘Monte Carlo’ studies to binary fluid mixtures. Comparison between ‘Monte Carlo’ results and those calculated from various Fig.3. Coexistence diagram for argon in terms of reduced temperature T* and densityp*; - machine computation results; - - - ex p e r i m e n t al c u r ve ;I09 melting data.111 melting data;llO (Ref. 34.) function of temperature; - Monte Fig. 4. Melting pressure of argo; as a Carlo’ results; experimental results.111 (Ref. 34.) ,0.2 ,0.4 ,0.6 p.8 11.0 , R. I, C. Reviews 108 theories of mixtures of non-polar spherical molecules has revealed the superiority of the van der Waals version of average potential model, as given by Rowlinson and co-workers,36 over Prigogine’s theory.37 A paper by Barker and Watts on water structure was perhaps the first on the application of the ‘Monte Carlo’ method to the systems composed of complex Fig.5. Radial distribution function of water (298.15 K). A ‘Monte Carlo’ results based on 54 000 configurations; ‘Monte Carlo’ results based on I 10 000 configurations; - ex- perimental.112 (Ref. 38.) A A 2.5 1.5 1 .o 0.5 1 J 2.0 Distance (A) 6.C .3.0 ,5.0 .4.0 1 I molecules.3* They assumed that the pair potential energy of water molecules is adequately expressible by the expression suggested by Rowlinson. The calculated energy and heat capacity agreed reasonably well with experiment, but the computed radial distribution compared less favourably (Fig. 5). Another interesting class of fluid systems which has been the subject of computer simulation studies is that of charged p a r t i ~ l e s .3 9 ~ ~ 0 ~ ~ ~ ~ 4 2 The most recent example is provided by a ‘Monte Carlo’ study of a neutral system of classical charged particles,43 which was designed to derive the thermo- dynamic as well as structural properties of a simulated potassium chloride system. The internal energy, pressure, heat capacity, thermal pressure coeffi- cients, thermal expansion coefficients, compressibility, entropy, normal melting points and radial distribution functions were computed. The Born-- Mayer-Huggins potential, with coefficients obtained from the crystal proper- t i e ~ , ~ 4 was assumed to represent the inter-particle interactions. The computed values agreed reasonably well with the available experimental data. Molecular dynamic studies on the systems of Lennard-Jones particles have been performed by Verlet.45346 The simulated systems were allowed to assume the same Lennard-Jones parameters as those for argon.Verlet derived energy and Compressibility values from this study and also estimated critical constants of the simulated system, which were substantially smaller than those of argon.45 The molecular dynamic ‘experiments’ also enabled Verlet to obtain molecular distribution functions.46 Figures 6 and 7 illustrate the computed g(r) and h(r); the figures also include those calculated from the PY equation. The agreement between ‘experimental’ and theoretical values of the functions is gratifying. Fig. 7. h(r) as a function of r ; - molecu- lar dynamic results; --- results given Fig. 6. Radial distribution function of a Lennard-Jones fluid (T* = I .4, p* = 0.8) derived from molecular dynamic study, and from the PY equation with the first correc- tion term.(Ref. 46.) 2 c2 by the PY equation. (Ref. 46.) I * r r .5 I - 2 ,I .5 ,0.5 ,2 R.I.C. Reviews 110 The molecular dynamic calculations by Rahman have contributed greatly to the understanding of the mechanism of diffusion in liquids.106 His study assumes Bernal’s description of the liquid state in terms of Voronoi poly- hedra.lo7 He argued that the velocity autocorrelation function, Z(t), is resolvable into two parts : S(t), ascribed to the ‘slip’ of the molecule along the direction of the elongation of the primary polyhedron caused by the fluctua- tions of the neighbouring molecules, and R(t), due to the ‘rattling’ of the molecules.The ‘slipping’ is a consequence of the short-range fluctuations and hence is the essential characteristic of the liquid state. Further, S(t) can be resolved into S+(t) and S-(t); S-(t) accounts for the molecules moving in the direction opposite to that of displacement at the initial point. R(t) and S-(t) describe a sort of oscillatory motion. The most significant contribu- tion to the process of self-diffusion is due to S+(t). SIMPLE CHAIN MOLECULES A simple polymer molecule may be regarded as a chain made up by the consecutive joining of segments which, in turn, are constituted of groups of atoms. An n-alkane molecule epitomizes the above description of poly- mers. The segments in this case are the groups CH3- (end) and -CH2- (middle).The connection between successive segments is provided by the C-C bonds. Thus the succession of C-C links forms the backbone structure of the chains. To a fair approximation, the number of links n, the bond length, I, and the bond angle, 8, of the backbone structure, suffices to charac- terize linear chain molecules. The rotation of the backbone bonds enables the molecule to conform to numerous configurations. The geometrical functions which characterize chain molecular configurations are the magnitude of the end-to-end vector, r, and the radius of gyration, S. For chain models, neglecting intersegmental interactions, the averages of 1-2 and S2 (to be denoted as (r2) and (S2)) in terms of n, I, 8 and C$ (bond rotational angle) are analytically expressible.In the limit of infinite chain length, such expressions reduce to4’ 1 + cos 8 1 + (cos 4 ) 1 - cos 8 1 - (cos 4 ) ~ (b) 27 hence ((r2>/n12) n-t to = ((S2>/nlz) n-+ccI = ~ 1 1 + cos 8 1 + (cos +) 6 1 - cos 8 ’ 1 - {COS 4) But when the intersegmental interactions are taken into account, the problem becomes extremely complex and often cannot be treated analytically. The most important stipulation imposed by the intersegmental interactions is the excluded-volume condition-the chain cannot intersect itself. Some attempts have been made to correlate <r2) and (S2) with chain length for the chain models incorporating the excluded volume condition ;48949p50 however the validity of these treatments is doubtful.51 La1 111 Another approach to deriving the statistical averages of various configura- tional properties of chains is based on the simulation of the configurational behaviour of such chains using computers.The simulation procedure involves collecting a large sample of randomly generated chain configurations ; only those configurations which conform to the excluded-volume condition are included in the sample. The averages of various properties taken over the sample is assumed to converge to the canonical ensemble averages and thus represents the corresponding equilibrium values. Such work has been pioneered by Wall and co-workers who studied chains confined to various types of lattices. Their studies firmly established the following relationships.6 In the limit, n -+ co, __- (s2) = c (constant) 29 <r2> Values of y obtained are approximately 1.50 and 1.20 for two- and three- dimensional chains;8 c assumes the values ca 0.14552@ and cu 0.157,52354 respectively, for the two kinds of chains.Recent work along similar lines on off-lattice chains yielded slightly higher values for y, but the value of c remained virtually ~naltered.5~ Another aspect of chain configuration statistics which has attracted much attention is the distributions associated with chain segments and chain-end separations.56~57~58~59 For chain models, neglecting the excluded-volume condition, such distributions are Gaussian in the limit of infinite chain length.The introduction of excluded volume in the model would pose pro- found mathematical difficulties which have so far prevented the development of an undisputed analytical treatment for such chains. This necessitates resorting to the computer simulation based numerical procedures. The simulation studies of Mazur and co-workers have shown that the inclusion of excluded volume destroys the Gaussian character of the limiting distribu- tion. They found that the following function would adequately describe the distribution W(r) = Aexp-(r/a)m where for three-dimensional chains 171 Y 3.2.56957 In the case of two-dimen- sional chains, the functional form of the distribution function is identical to 29, but the value of m required to reproduce the computed moments is approximately equal to 5.8.60 An interesting application of the ‘Monte Carlo’ technique has been made by Verdier and Stockmayer who simulated the dynamic behaviour of ex- cluded-volume chains in infinitely dilute solutions.61962 A general agreement between the relaxation behaviour of the end-to-end distance of the chains revealed by ‘Monte Carlo’ study and that predicted by the Rouse and Zimm theory provides encouragement for applying the method in exploring the nature of the relaxations associated with such complex but interesting 112 R.I . C. Reviews processes as untying of knots in polymer chains, and first contacts between chain ends. The simple excluded-volume effect in the model chains described above implies a hard-sphere potential for intersegmental interactions.Because of the unrealistic nature of this potential, the correspondence between real chain molecules and the above models would be severely limited. It would be necessary to apply a realistic intersegmental potential to improve such a correspondence. Wall and co-workers, in later ‘Monte Carlo’ calculations, assumed a square-well potential for such interaction~.~8Jj3,6~ The chains obeyed the equations 28 with y and c varying with temperature. Furthermore, this study demonstrated the existence of the Flory temperature 8 (the tempera- ture at which the configurational behaviour of the chains is identical to that of the random walk model; hence at 8, y = 1 , c = 6) for the model con- sidered.Even the square-well potential portrays a very crude picture of the intersegmental interactions. Further refinement of the model would involve using more complex potentials such as those of Lennard-Jones, Buckingham, etc., which describe intermolecular interactions more realistically.65 The ‘Monte Carlo’ calculations described so far have been entirely con- cerned with isolated-chain models. Such models relate to dilute solutions, where interchain interactions are negligible. In crystals and in concentrated solutions surrounding chains would undoubtedly exert a considerable influence on the molecule. Very few ‘Monte Carlo’ studies have been devoted to the investigation of such effects. Whittington and Chapman performed calculations on a very simple model simulating a multi-chain system.66 The end-to-end distance of the chains and the entropy of the system were com- puted as functions of interchain distance.The study indicated a phase- transition in the system. In another investigation, calculations were made for the rotational energies of hydrocarbon chains in crystals employing the ‘Monte Carlo’ method.67 The behaviour of energy as a function of temperature revealed a phase-transition in the chains in conformity with experimental findings. The other computer technique, which has been applied in chain statistical calculations, is the method of exact enumeration. For lattice-confined short chains, it has been possible to generate all the allowable configurations on computer; this would allow calculation of exact values of various chain properties.The values for infinite chain length for properties which converge rapidly to their asymptotic limits as functions of chain lengths, e.g. the ratio (S2)/($>, can be deduced through extrapolation procedures. The exact enumeration method has been the basis of most of the work on chain con- figuration statistics carried out by Domb and co-workers.68 In general, the results obtained from the two methods are in agreement.69 BIOLOGICAL MACROMOLECULES For the past two decades or so, the structure of biological macromolecules, particularly nucleic acids, polypeptides, etc., has been the subject of profound interest to physical scientists. The first major breakthrough was made by Watson and Crick‘s proposal of a double helical structure for the DNA La1 113 Fig.8. A pictorial view of the minimum energy conformation of Gramicidine S. (Ref. 73.) molecule.70 Briefly, the molecule is composed of two right-handed helices wound around a common axis; the two helices are connected at various points through the hydrogen bonds occurring between the bases adenine- thymine, guanine-cytosine, etc. Subsequent x-ray studies confirmed the above structure. It was realized immediately that the interpretation of such a significant process as the perpetuation of the DNA molecules of identical structure in cell plasma lay in the structure.70 Since then, several other key molecular biological phenomena have been accounted for in terms of molecu- lar conformations.Computer simulation techniques have to be used in the elucidation of certain configurational features and associated processes because of the immense complexity of these molecules. Some important illustrations of the role of the computer in this field will be given. Scheraga et al. carried out computer reconstructions of simple polypeptide molecules subject to the constraints necessitated by the geometrical as well as energy considerations.71~72~73 As a result, they arrived at the detailed three-dimensional pictures of the molecules. Figure 8 is a simplified pictorial representation of the minimum-energy conformation of Gramicidine S. The structure, in certain respects, is similar to that proposed by Stern et al.on the basis of their nmr study of the molecule74 but while they suggested the existence of four hydrogen bonds, the minimum-energy conformation does not contain any. More recently, Scheraga and co-workers evolved a more comprehensive computer techni que-s t a tis tical search procedure-and R. I.C. Reviews 114 discovered another low-energy conformation, 477 J mol-l lower than that of the above.73 The stability of the helical structure in polynucleotides depends on suitable physicochemical conditions, alteration of such conditions causes the con-. version of helical into coiled conformations. In the past, several models have been postulated to account for this transformation. Crothers, Kallenback and Zimm14 used computer simulation to assess the validity of such models.They confined their investigation to the ‘zipper’ model in which the helix/coil conversion starts at one end of the molecule and proceeds gradually to the other-as if the molecule is being unzipped. The contributions by the different kinds of base pairs to the stability of the helical structure for a heterogeneous polymer-adouble helixcontainingdifferent kinds of base pairs-were assumed to be different, in consonance with the experimental evidence. It was also assumed that the sequence of base pairs along the chain is random. Then helix/coil transition in DNA chains was simulated. Figure 9 compares the transition curves of the simulated chains with those of T2 DNA molecules determined experimentally. The agreement between the computed curves and the experimental points is reasonable for molecules of shorter lengths but not for the high-molecular weight polymers.Thus the zipper model is valid for shorter nucleic acids but fails to account satisfactorily for the helix coil transitions in the longer molecules. The kinetics of the unwinding of polynucleotide helices has been simulated in a study by Simon.75 His approach uses a difference form of the Langevin equation of motion for simulating molecular movement. The object was to obtain information on the role of van der Waals type inter- Fig. 9. Helix/coil transition curves of D N A molecules. L is the number of base pairs in the molecule; 8 is the fraction of H-bonded base pairs; - transition curves of the computer simulated molecules; 0 experimental results for T2 DNA.(Ref. 14.) 115 La1 strand interactions in determining the unwinding kinetics. The results showed that such interactions are of considerable importance in the process. Further- more, the computer investigation supports the view that unwinding starts at helix ends and progresses inwards. Although the existence of folded structures in certain proteins has been confirmed by experiment769 7 7 an adequate understanding of the mechanism of folding during biosynthesis is lacking. De Coen conducted a computer study78 in which it was assumed that the biosynthesis of the folding chains proceeds via a succession of two types of stages occurring alternately : (a) formation of a sequence of amino-acid residues adhering to a minimum- energy conformation, which ‘freezes’ to act as ‘nucleus’ for further growth, (b) addition of a non-hydrogen bond forming amino-acid residue to the ‘nucleus’.The result is a sequence of minimum-energy regions separated by non H-bond forming spots. The simulated structure was a compact globular shape (folded). X-ray studies by Krimm and Tobolsky show that on stretching keratine fibres, the a-helix conformation of the molecules transforms into a sheet-like ,5’ structure.79 Schor, Haukaas and David’s ‘Monte Carlo’ computer simula- tion study aimed at constructing a satisfactory model for this transforma- tion.80 The simulation involved generating stochastic growth of polypeptide chains, through sequential residue addition, based on the Boltzmann proba- bilities associated with the allowed angle pairs (4, 4).The first three residues of the chain were made to assume an a-helical conformation. The helix axis served as the direction of applied tension as well as of the further growth of the chain. Figure I0 presents the tension/length isotherm of the simulated 3.5 3.0 L 2.5 2.0 116 Tension (dynes) 4 2 3 I Fig. 10. Plot of applied tension versus length for a computer simulated keratine molecule of 200 residues. (Ref. 80.) R.I.C. Reviews chain of 200-residue length at 300 K. This model is able to reproduce the essential features of the isotherm in terms of the geometry and interatomic interactions. ADSORPTION In the past, several theoretical attempts have been made to explore the con- figurational behaviour of chain molecules in the adsorbed state.81~82~83~84~85 A serious shortcoming of these theories is that they either ignore, or fail to incorporate satisfactorily, the intersegmental excluded-volume effect in the model.On the other hand, as is clear from the section on simple chain mole- cules, computer simulation methods are capable of considering such an effect. Thus ‘Monte Carlo’ studies would prove quite apt in disclosing the conforma- tional behaviour of polymers at interfaces. Clayfield and Lumb have described a computer study they carried out on the simulation of isolated chains with one end permanently ‘anchored’ to a surface.86 This system would serve as a model for a polymeric dispersant which would prevent colloidal particles from flocculating or from adhering to surfaces.The chains, of up to 300 links, were generated on a four-choice cubic lattice ; successfully generated configurations were those which obeyed the excluded-volume condition. The computation was directed towards the calculation of the averages of various configurational quantities such as end- to-end distance of the chain, maximum height of the chain above the surface, number of adsorbed segments, etc. However the histogram of the heights of chains above the surface, from which the entropy of compression of terminally adsorbed chains could be calculated, was of prime interest. This led to the evaluation of the free energy of two interacting spheres, and a sphere and a plate, in the presence of polymeric dispersant.‘Monte Carlo’ calculations on an improved model of a terminally anchored chain, in which the energy of adsorption of a segment on the surface, e/kT, is taken into consideration, were performed by McCracken.87-The chains O -50 -150 -300 . -200 OS2 Fig. I I. Variation of the fraction of adsorbed segments as a function of the energy of adsorption per segment for ex- cluded volume chains of various lengths. prediction of Rubin’s Theory88 for a non-excluded volume infinite- length chain. (Ref. 87.) Lal 117 were simulated on a four-choice simple cubic lattice. Information on thick- ness of the adsorbed chains, the fraction of the adsorbed segments, end-to- end distances of the chains, loops off the surface etc., was sought.Figure I I gives the computed results of the fractions of segments in the adsorbed state as a function of -e/kT for several chains length. The line represents the calculations for an infinite-length chain in accordance with Rubin’s theory which neglects the excluded-volume effect. For a given e/kT, the fraction of adsorbed segments goes on decreasing as chain length increases; and, as expected, as - e/kT increases, the number of surface-confined segments increases too. The figure provides an indication that for long excluded-volume chains approaching the limiting behaviour, the fractions of adsorbed seg- ments would be considerably less than those calculated by Rubin (for non- excluded volume chains).s8 Furthermore, this study confirms the existence of a critical energy of adsorption-approximately 0.2 kT-below which the extent of adsorption is negligible; above the critical energy, the fraction of adsorbed segments rises sharply with - e/kT and eventually would approach unity for large energies.The ‘Monte Carlo’ calculations by Bluestone and Cronan are based on a fairly realistic model of the chain interacting with a surface.10 All the segments of the chain, including the terminal ones, were assumed to have equal ad- sorbing susceptibility. The interaction between a segment and the surface was assumed to vary with distance from the surface and the intersegmental interactions were ignored. The calculations were performed for 3 1 -segment linear molecules.For greater segment/surface interaction energies, the calcu- lated fractions of the segments adsorbed on the surface were in reasonable agreement with those given by the theories of Frisch,81 HiguchiB2 and Silberberg. 83 When a low interaction energy was assumed, the molecules remained in the desorbed state. The configurational analysis revealed that in the adsorbed state, a chain configuration consists of trains (arrays of seg- ments in the adsorbed state) and loops (arrays of segments away from the surface) which is in agreement with certain theories of polymer adsorption.83 A simulation study of the monolayer adsorption of monatomic gases on solid surfaces was carried outll in which the solid surface was taken as a heterogeneous, hexagonal close-packed planar lattice.It was assumed that in the adsorbed state the particle interacted with the adsorbing site as well as with the surrounding adsorbed molecules. The results were in qualitative agreement with the adsorption equation derived by Hill based on the quasi- chemical approximation. REACTION KINEMATICS The absolute reaction rate theory, embodied in the ‘semi-empirical method’, was developed by Eyring, Polanyi, Hirschfelder, and Wigner. 89 Further progress in the theory has been negligible because of severe mathematical difficulties which have proved insurmountable except, perhaps, for ex- tremely simple cases. A direct approach to the solution for a system of reacting molecules is to simulate the molecular dynamics of such a system on a computer, obtaining the numerical solutions of the classical Harniltonian R.I. C. Reviews 118 differential equations of motion of the particles (constituting the system). For a given value of the Hamiltonian, H, these solutions yield the trajectories of the particles as functions of time. These trajectories contain direct informa- tion concerning the occurrence or non-occurrence of a particular reaction. H2 + H -+ H + Hz-a triatomic system.g0 The potential energy of the Wall and co-workers successfully applied this method to system was calculated from the London-Eyring-Polanyi expression, which involves the interatomic pair potential energies 2.412, 2423 and 2413; uijs are adequately expressed by the Morse equation.The molecular dynamics of the system were worked out for several values of H assuming various initial conditions. Initially, only collinear collisions were considered, but later the computation was carried out for two-dimensional non-linear collisions as well. The computation revealed the following: no reaction would occur when the total energy of the system was below the translational activation energy; the inclusion of vibrational energy in the Hamiltonian reduced the transla- tional activation energy-approximately one-sixth of the vibrational energy could contribute to the activation energy; the existence of a large rotational energy would considerably diminish the probability of the occurrence of a reaction.The above type of calculations were subsequently extended to the systems (linear triatomic) confined to the potential energy surfaces whose saddle points differed in curvature and location. 9 l The computation showed that the reaction would occur only if the energy lay within a certain bounded range, the bounds of the energy range depending upon the position of the saddle point and the relative masses of the reacting particles. Also, the vibrational energy of the product molecule was found to be dependent on the location of the saddle point. of the type: A + B-C = A-C + B have been carried out by Blais and Extensive ‘Monte Carlo’ investigations of exothermic triatomic reactions Bunker92 and by Raff and K a r p l ~ s , ~ ~ ? g particularly the reaction: K + CH3-I = K-I + CH3.Blais and Bunker selected a potential energy surface for the reacting particles that assumed the applicability of Morse potentials for stable molecules B-C and A-C, and fulfilled the criterion that the energy of the reaction is released during the approach of A towards B-C. Various modified versions of Blais and Bunker potential energy functions, used by Raff and Karplus in their study, yielded values of reaction cross-sections in reasonable agreement with experimental results. The studies were aimed at the determination of various reaction attributes as functions of initial conditions, masses of participating particles and nature of the potential energy surfaces. These investigations disclosed that a very large part (ca 90 per cent) of the energy released in the reaction appeared in the vibrational and rotational modes of the product molecule, thus the vibrational and rotational states of newly produced A-C are at highly excited levels, and the total angular momentum of A-C finds its largest contribution from molecular rotation.Calculated laboratory differential cross-sections of the reactions were in good agreement with experimental values. Examination of particle trajectories in the event of a reacting collision (Fig. 12) suggests a simple collision mechanism for the reaction in which the reacting particles interact for a very short period. La1 119 Distance /?,(A) Fig. 12. Particle trajectories in the event of a reactive collision for the simu- lated system K + CH3-I + KI + CH3.--- distance between K and I; distance between I and CH3; distance between K and CH3. -.-.- 20 0 / ,l60 200 Time (0.54 x 10- l4 sec) J20 80 40 (Ref. 9.) The application of the ‘Monte Carlo’ method in the analysis of four-body -+ KI + (CH3-CH2) was reactions of the type: A + B-C-D + A-B + C-D with particular reference to the reaction K + CH3-CH2-I made by Raff.94 The calculations were carried out for the determination of the total reaction cross-section, the differential reaction cross-section, and the distribution of the reaction energy among the available degrees of free- those for the reaction A + B-C -+ A-C + B, were made of the reaction dom of the product molecules (A-D and B-C).Observations, similar to energy distribution. The calculation also showed that the C-C bond in the ethyl group absorbed about 14 per cent of the available energy through its vibrational mode. The occurrence of the reaction assumes two modes, direct interaction and complex formation. Another interesting reaction which has been the subject of ‘Monte Carlo’ calculations is that between an alkali metal atom and a halogen molecule.95 The reaction’s interest lies in its proposed mechanism which involves an electron jump or ‘harpooning’. SYSTEMS CORRESPONDING TO THE ISING MODEL The Ising model, which was originally used to interpret ferro- and anti- ferro-magnetic behaviour, corresponds to a lattice whose sites can assume only two states.In the case of magnetic materials the two states are positive and negative spins which can be assigned the values + 1 and -1 respectively. Let p(i) denote the state of a lattice site i, then a vector assume only the values + 1 and - 1. It is assumed that a site interacts only would completely describe a configuration k of an N-site lattice; p(i) can with its nearest neighbours and with an external field. Thus the total con- R.I.C. Reviews 120 figurational energy of an Ising lattice corresponding to configuration k is given as N N 30 The first term on the rhs of equation 30 gives the nearest neighbour pair interaction energy; the second term gives total field/spin coupling energy of the system ( H is the coupling energy per spin).2J is equal to the energy of transformation of a pair from pzrallel to antiparallel state. An important feature of the Ising lattice is that it exhibits phase transition. Below the phase-transition temperature-the critical temperature, Tc- long-range correlations between the spins exist, above Tc the long-range correlations disappear but short-range correlations persist. The parameters f ( n ) and S, order parameters, describe the short- and long-range correlations in the lattice respectively. They are defined as 1 N 31 and 32 a(n) is the number of the neighbours per lattice site; and sites i and j form an n-th neighbour pair in a configuration k. The equilibrium values of f ( n ) and S would be 33 34 The Ising model can also be used for superlattice transitions in alloys, cooperative transitions which occur in many solid substances, helix/coil transitions in certain biological macromolecules, lattice-bound fluid systems, etc. Solutions of the Ising model may provide an adequate interpretation of the above phenomena.For two-dimensional Ising lattices, in the absence of external field, Onsager, and others, were able to derive exact analytical solutions using highly sophisticated mathematical techniques.96 No such solutions have yet been found for three-dimensional lattices. Therefore numerical techniques such as the ‘Monte Carlo’ method have been employed to tackle the problem. Pioneering work in this direction has been carried out by Fosdick and c o - ~ o r k e r s .~ The ‘Monte Carlo’ method used was based on a Metropolis sampling technique. Initially the computation was performed on square lattice with H = 0. Agreement between the computed f(1) values and those provided by analytical theory established the feasibility of the 121 La1 APPLICATIONS OF THE ANALOG COMPUTER Fig. 13. The long-range parameter, 5 , for a simple cubic lsing lattice as a function of jjkT(K) for various values of H/kT(L). (Ref. I 13.) ‘Monte Carlo’ technique for tackling Ising systems. Subsequently, the calculations were extended to three-dimensional lattices. The purpose of this computation was to evaluate the parameters S , f ( 1) andf(2) assuming various values of J/kT and H/kT. The behaviour of S, as J/kT varies, would indicate the nature of phase-transition in the system.S as a function of J/kT, corre- sponding to several values of H/kT, is plotted in Fig. 13. I t is clear that for H/kT = 0 the system undergoes a sharp transition, the sharpness gradually fades as H/kT increases and eventually disappears for H/kT = 1. For simple cubic and body-centred cubic lattices critical values of J/kT, at H/kT = 0, were approximately estimated by Fosdick and co-workers to be 0.2275 and 0.16 16 respectively. Close correspondence between the Ising model and binary substitution alloys led Gutman to carry out a ‘Monte Carlo’ simulation of such alloys on the basis of the body-centred cubic Ising lattice.5 Besides order parameters, the computation included the evaluation of the entropy and heat capacity of the alloy at several compositions.Figure 14 gives experimental and computed heat capacities of 1 :I P-CuZn alloy as a function of temperature. Although quantitative agreement between the experimental and ‘Monte Carlo’ results is lacking, the qualitative features of the two curves are remarkably similar. Theory confirms the existence of a singularity in the heat capacity versus temperature curve of an infinite two-dimensional Ising lattice. It is con- jectured that heat capacities of the three-dimensional lattices would also follow a similar behaviour.97 ‘Monte Carlo’ studies cannot confirm this because of the finite size of the lattices considered-discontinuities can only be exhibited by infinite crystals. Nevertheless these calculations have been able to give an approximate indication of critical points in such systems.Simulations discussed in the foregoing part of this article would normally require the use of a digital computer. However, for the simulation of certain other processes, particularly those involving the solutions of differential equations and which do not warrant the use of extremely speedy computa- tional equipment, the use of analog computers may prove convenient as well R.I.C. Reviews 122 ? I I I 4 z ll + - Observed p-CuZn - Calculated -3 d I’ r l versus temperature plot for I:I fl-CuZn alloy. C m - 2 Fig. 14. Reduced heat capacity, (Ref. 5.) as advantageous. A great advantage of analog computers is that they are able to perform integration on a continuous basis and are thus able to yield exact solutions of linear as well as non-linear differential equations.98 Further- more, the output from an analog computer can be easily displayed on an oscilloscope ; this greatly facilitates a systematic inspection of variations in output when the parameters which characterize the system under simulation are varied. Analog computers, however, suffer from the disadvantage that they do not possess any ‘memory’. The greatest use of analog computers is in the dynamic analysis of chemical engineering processes giving information valuable for designing reactors. In chemical investigations too, their use-though not so extensive-has proved of benefit. This is exemplified by a recent simulation study of adsorption calorimetry.99 An analog computer was employed to estimate the calorimetric curves that would be obtained when the evolution of the heat of adsorption La1 123 of a gas on a solid was studied using a diathermal calorimeter.The effects of varying the parameters characterizing the design of the calorimeter as well as those determining the kinetics of the adsorption on the shapes of the curves were determined. Inferences from this study would be immensely helpful in evolving a successful design for a calorimeter for studying gas/solid adsorption. Another area of chemical research where analog simulation would find considerable scope is the kinetics of chemical reactions. CONCLUSION It is evident from the foregoing account that ‘Monte Carlo’ and related methods have firmly established themselves as extremely useful techniques in the investigation of a variety of physicochemical problems.These methods have perhaps made their greatest impact in the field of statistical mechanics of fluid systems. ‘Monte Carlo’ and molecular dynamic calculations, based on models that are believed to correspond very closely to real fluids, have produced highly accurate results. It has therefore become quite usual to test the worth of the statistical mechanical theories against the machine computa- tion results. Further, simulation techniques have been able to bring out certain theoretically unpredictable features of the models, e.g. the discovery of solid/fluid transition in hard-sphere systems.l8Jg Molecular dynamic computation shows promise in determining accurate values of various correlation functions such as those of angular and linear velocity, dipole moment, bond forces, etc.100J08 A precise knowledge of these correlation functions would provide a sound basis for the interpretation of infrared and Raman rotational and rovibrational line shapes, nmr quadrupole, magnetic dipole-dipole and spin rotation relaxation times, classical vibration relaxation, etc.Aspects of the microscopic structure of liquids relevant to their role as solvents for chemical processes have largely remained unexplored. In terms of such characteristic microscopic properties of solvents, the ‘solvent cage effect’, solute molecular encounters, etc. can be explained.The molecular dynamic approach could prove suitable for such studies.lo1 In the past, the bulk of ‘Monte Carlo’ calculations carried out on chain molecular systems were based on the idealization of polymer molecules by self-avoiding random walks on or off lattices. Such studies are helpful for investigating chain properties that are insensitive to the ‘microscopic’ details of the model. However, certain other aspects of chain behaviour would in- volve such details as bond angles, nature of bond rotation, long-range intersegmental interactions, etc. Lack of ‘Monte Carlo’ studies which take into consideration these features of real molecules has been due partly to the unrealistic computing time that would be involved in tackling complex models102 and partly to an ignorance of the nature of short- and long-range interactions in polymer molecules.The former impediment seems to be overcome by the availability of such extremely fast and large storage-capacity computers as IBM 360/195, CDC 6600, CDC 7600, etc. Calculations carried out by Flory, and others, on bond rotational energies in chain molecules give an adequate quantitative account of the short-range interactions.103 An approximate quantitative assessment of long-range interactions is also R.Z.C. Reviews 124 possible. A model containing the essential features of the mode of bond rotation in chains, as revealed by Flory’s calculations, as well as allowing long-range interactions, can be evolved for the purpose of simulating the behaviour of real chain molecules on a computer.Successful application of the ‘Monte Carlo’ method to real systems can be expected to lead to serious investigations on chain folding, interchain interactions in vacuum as well as in solvents, interaction of chain molecules with surfaces, etc. Another interest- ing extension of ‘Monte Carlo’ calculations would be to cyclic polymers. Perhaps the greatest challenge to today’s physical scientists is offered by the complexities of biological macromolecules. I hope that computers will prove an invaluable aid to human insight and imagination in discovering those, still unknown, molecular processes that are of profound significance to life. ACKNOWLEDGMENTS Part of this article was written while the author was visiting Research Associate in the Chemistry Department, Dartmouth College, Hanover, NH, US.The author takes this opportunity to express his sincere gratitude to Prof. W. H. Stockmayer for his kind hospitality and is grateful for various discussions that led to the improvement of this article. Thanks are also due to Prof. P. J. Gans, New York University, Dr K. S61c, Midland Laboratory for Macromolecular Science, Dr E. B. Smith, Physical Chemistry Laboratory, Oxford University, and Dr P. J. Anderson, Dr R. V. Scowen and Mr G. C. Peterson of this Laboratory for critical readings of the manuscript. Helpful criticism by the referee is also greatly appreciated. 5 L. Guttman, J. chem. Phys., 1961,34, 1024.REFERENCES 1 J. H. Curtiss et al., Monte Carlo method. Washington: Natn. Bur. Stand. Applied Mathematics Series, 12. 2 (a) B. J. Alder and W. G. Hoover, Physics of simple liquids, ch. 4 (eds H. N. V. Temperley, J. S. 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La1 9 COMPUTER SIMULATION OF SOMEPHYSICOCHEMICAL PROBLEMSMoti La1Unilever Research, Port Sunlight Laboratory, Port Sunlight, Wirral, Cheshire 162 4XN* ’ . Introduction . . . .RCsumC of statistical mechanics. .‘Monte Carlo’ method . . . .The method of molecular dynamicsThe fluid state . . . . ..Statistical theory of fluids, 103Computer studies, 105Simple chain molecules . . . .Biological macromolecules . .Adsorption . . . . . .Reaction kinematics . . . .. .. .. .. .. .. .. .. .. .Systems corresponding to the Ising modelApplications of the analog computer . .Conclusion. . .. . _ . . ..References . . . . .. . . . .. .. .. .. .. .. *. .. .. .. ...... .. .. .. .. .. .. .. .. .. .. .. .... .. .. .. .. I . .. .‘ .. .. .. .. .. .... .. .* .. .. .. .... .. .. .. .... .9798102102103111113117118120122124125INTRODUCTIONUndoubtedly the electronic computer is one of the most powerful tools inthe hands of modern scientists and technologists. Apart from taking thedrudgery of manual computation out of scientific and technical work, ithas allowed us to tackle many problems pursuing hitherto unexplorabledirections. This led to the evolution of completely new systems for informa-tion storage and retrieval, cataloguing, accountancy, etc., and a great extensionof the scope of investigation into engineering designs and chemical engineeringoperations.The recent flourishing of such subjects as operational researchand cybernetics is a consequence of the advent of high-speed computers.In science, an area where computers have proved of tremendous use ismodel building and simulation. Such studies have allowed a deeper insightand, often, a growing understanding of a situation or a phenomenon in questionto be gained. Earlier work on computer simulation of the random neutrondiffusion process in fissile material by von Neumann and Ulam providesan illuminating example of the success of computers even in the early stagesof their development.1 Modern computers are capable of handling morecomplicated simulations : simulations of simple fluids,2a,ZbJ solids andalloys4~5 have successfully been achieved-the behaviour of the computermodels has closely approximated that of the real systems; extensive studieson the computer simulation of chain molecules have been made;6,79* certainchemical kinetic processes have been simulated with the object of elucidatingreaction mechanisms ; 9 simulation of the processes of adsorption,loJlLa1 9diffusion12 and percolation13 has been done with some success; computersimulation techniques have entered the domain of biochemistry.l4 Thisarticle intends to give a detailed account of this mode of computer applicationin the above spectrum of physicochemical areas, and to assess, where possibleand appropriate, the values of such studies as aids to verifying the correspond-ing analytical theories and promoting further comprehension of the basisof the subjects concerned.An article dealing with a similar topic appearedin the literature a decade ag0.15 Advances of much significance made in theapplications of the computer methods and their extension to several newareas since then, call for a new review of the subject.Most of the aforementioned computer simulation studies aim to correlatethe molecular behaviour of the systems with their phenomenological proper-ties ; this results in their deep involvement with statistical mechanics. Thereforea brief outline of relevant basic statistical mechanics is given before describingsimulation techniques and the behaviour of the simulated systems.RE SUM^ OF STATISTICAL MECHANICSWe consider a system of N particles that is able to conform to various con-figurations, say 1, 2, 3, etc., with the corresponding energies 2 1 , ZZ, .%3,etc.Statistical mechanics provides the following expression for the canonicalensemble average, the equilibrium value, P, of a property P of the system:C Pi exp ( - - Z i / k T )1where Pi and Hi are the values of the property P and energy, respectively,corresponding to a configuration i; k is the Boltzmann constant, and T isthe temperature. The summations in 1 are carried over all the possibleconfigurations that the system can assume.The denominator of the right-hand side of 1 is called the partition functionof the system and is usually denoted by 2; thus2If the 'internal' energy of the system (the energy due to the internal degreesof freedom, e.g.internal vibrations, rotations, etc., of the particles) is scaledat zero level, and assumed to be invariant under ordinary thermodynamicconditions, Zi is thenZ = C exp ( - 2 i / k T )i3The two sets of vectors PI, Pz, . . . PN and q1, 92, . . . qN represent, respec-tively, the momenta and the positions of the particles in the configuration i;the subscripts indicate the corresponding particles, and s is the mass of eachparticle. The first term in 3 is the kinetic energy of the system, and is a functionof temperature only; while U is the potential energy, and depends only onthe positions of the particles.98 R. I. C. ReviewFrom 2 and 3,4In the classical limit, for a system of non-localized particles, 4 assumes theformdP2 .. . dPN dql .dqz . . . . dqivdPz . . . ~ P N 1 . . . 1 exp (- U/kT) dgi .dqz . . . . dq,vwhere I stands for the integralii 1 . . . 1 exp (-U/kT) dql .dq2 . . . . dqNcalled the configurational partition function or the configurationaof the system.56111 tegralThe partition function is related to the thermodynamic properties of thesystem through the equationA = - kT In 2; A is the free energy of the system 7The expressions relating the partition function to various other thermo-dynamic functions can be readily deduced from 7 by making use of standardthermodynamic formulae which relate free energy to the thermodynamicfunctions in question. Thus we obtainS (entropy) = k In Z + (kT/Z) aTp (pressure) = ~-a l n z Cv (heat capacity at constant volume) = - kTAnother approach to the classical thermodynamics from the molecularstate of the system is provided by the molecular distribution functions.A molecular distribution function nnz is defined as the probability of theoccurrence of a configuration in which each position of a set of m positionsLa1 9ql, q2 .. . , qm is occupied by a particle. It follows from the above definitionthat9 1 . . . j e x p (- U / k T ) dqm+l. 4 m + 2 . . . dqN N ! ~- ~- nm = - (N - rn)! 1 . . . I e x p (- U/kT) dql.dq2. . . . dqNIn the theory of fluids the most significant distribution function is the pairdistribution function nz(q1, q2), the probability of finding any two particleseach at the positions q1 and qz simultaneously; which is simply equal to1 1We define a function g , called the radial distribution function, asAq1, (Sl - ad1 = nz(q1, sd/nl(al> x nz(q2) 12aIn an isotropic fluid, the above reduces togw = n2(r)/(N/V)2 12br is the distance between ql and 92; nl(q1) is the probability of finding aparticle at ql, and n2(qz) is that of finding a particle at 9 2 .If we assume thatthe potential energy, U, of the system can be expressed as a sum of the pairinteraction energies of the constituent particles, i.e.13where zrij(r) is the interaction energy between two particles i and j situatedat a distance r from each other (the notation i < j assures that the pairs ijand j i are equivalent and hence are not counted twice), g is related to U asa,nU = (N2/2 V ) J u(r). g(r) .4m2 dr0Hence,1415where (3/2)NkT is the kinetic energy of the system (assuming that the con-stituent particles are spherical).Equation 15, in conjunction with Clausius’virial theorem, yields the following equation of statep = (NkTJV) - 2z<: 1 T ) . g ( r ) . r 3 dr 3 v016R. I. C. Reviews 10Equations 15 and 16 lead to the expressions relating the radial distributionfunction to various other thermodynamic functions of the system.Recently, the direct correlation function, c(r), defined by the followingequation 17, the Ornstein-Zernike equation, has come into much prominencedue to its appearance in the advanced theories of fluids170rij represents the distance between a particle i and a particle j ; andh(r) = g(r) - 1.It emerges from the preceding account that the essential requisite for obtain-ing explicit relations between the molecular properties and bulk thermo-dynamic functions is the solution of the equations 15 and 16 which in turndemand the knowledge of the quantitative nature of the functions u(r) andg(r).Through the combination of experimental and theoretical considera-tions, several approximate but convenient forms of u(r) have been deduced.These potential functions may be considered adequate for the present pur-poses. Although g(r) can be determined from x-ray and neutron diffractionor scattering studies,l6J7 H and p , according to the equations 15 and 16,are so sensitive to g(r) that a very high accuracy, unattainable by the existingexperimental techniques, in the values of g(r) is required for their (H and p )calculation.One has, therefore, to resort to theoretical methods for evaluatingradial distribution functions.Complete solutions of the configurational integral and radial distributionfunction for realistic systems defy analytical methods unless certain simpli-fying assumptions are introduced. There might not be valid justifications forpresuming that such assumptions, mainly dictated by mathematical con-siderations, would be compatible with the physical characteristics of thesystem. Alternatively, in the event of an adequate knowledge about thephysical picture of the system, its mathematical simulation can be evolvedwhich may then be employed for the direct evaluation of the partition func-tion and g(r).But for certain statistical considerations, the prohibitive amountof calculations involved would have denied the adoption of such a procedure.The ‘Monte Carlo’ method involves the use of such statistical considerationsfor reducing the calculations to an amount that can be handled adequatelyby modern computers.18 Another technique which is equally extensivelyemployed in simulating statistical mechanical systems by computer is themethod of molecular dynamics.19 While the ‘Monte Carlo’ method givesensemble averages of the properties of the system under study, the methodof molecular dynamics follows the trajectories of the system in time, thusyielding time-averages of the properties.The results obtained from the twoapproaches would, however, be identical at equilibrium. In the next twosections some basic aspects of the two methods are dealt with.La1 10‘MONTE CARLO’ METHOD‘Monte Carlo’ method is a statistical technique involving random sampling.Its use for treating probabilistic problems-problems posed by the situationsgoverned by stochastic processes, e.g., diffusion of neutrons in fissilematerials-is obvious and has a long history. The great utility of the method,as a numerical technique, in the solution of deterministic mathematicalproblems that are intractable by analytical methods has, however, beenrealized relatively recently.20 Multiple integral expressions representing thepartition functions of the systems of interacting particles typify such prob-lems.The following illustrates the application of the ‘Monte Carlo’approach to statistical mechanical problems. Consider a fluid systemof N particles. The simplest way of determining the statistical thermodynamicproperties of the system would be to compute the Boltzmann factors corre-sponding to all the configurations accessible to the system, and then obtainthe partition function by summing them. But the impracticability of carryingout such an operation is obvious. However, instead of attempting to considerall the configurations the system can assume, a random sample from the con-figuration population can be drawn. The properties computed on the basisof the sample thus collected may be assumed to be unbiased estimates ofthe properties of the configuration population.Estimation proceduresdesigned along these lines are essentially ‘Monte Carlo’ methods.Two considerations account for the magnitude of the sample: the ‘speed’with which sample properties converge to the population properties, andthe desired limits of accuracy in the estimated values. The uncertainty in astatistical estimate is measured by the variance, 02, which is defined as18where Pi is the value of a property, P, corresponding to a configuration ichosen in the sample; is the estimate of p ; and n is the sample size. Themagnitude of 02 is determined by the sample size, n, as well as the mode ofsampling: two samples of the same size but obtained from different samplingprocedures often render different variance.An efficient ‘Monte CaI lo’scheme is one that would involve a sampling procedure producing smallvariance. Efficient sampling designing is known as variance-reduction.Several variance-reduction techniques occur in the literature ;21 the choicedepends on the nature of the problem under consideration. In ‘MonteCarlo’ calculations related to statistical mechanical problems-which isthe main concern in this article-a commonly used sampling procedure isimportance sampling. Importance sampling allows the sample to have thehighest concentration of its points in the high-probability region of thedistribution which the population follows ; the configuration populations ofordinary statistical mechanical systems follow a Boltzmann distribution.THE METHOD OF MOLECULAR DYNAMICSAssuming that the constituent particles of the system to be investigatedbehave classically, Newton’s equations-of-motion for each particle are102 R.I. C. Reviewnumerically integrated to obtain the momenta and the positions as functionsof time.” These are continually recorded as the computation progresses.The time-average values of position- and velocity-dependent properties ofthe system can be calculated from this data. At the initial point (zero time)the particles assume some arbitrary but convenient set of momenta andpositions. The temperature of the system at an instant t is given bys( n=l 5 (v?b,t)2)!3Nkwhere v,,t is the velocity of a particle n at t, and s is the particle mass.The method of molecular dynamics has a wider scope than the ‘MonteCarlo’ method: while the ‘Monte Carlo’ method can only be applied to studyequilibrium situations, the method of molecular dynamics can be used ininvestigating time-dependent phenomena such as relaxation and non-equi-librium behaviour.Quantitative information for important primary proper-ties such as particle-velocity distribution, collision rate, pair and tripletdistribution functions, pressure, and mean potential energy of the particlescan be derived from molecular dynamic computation. Other quantities of greatinterest are the mean square of particle displacement, ((q2))t, and the velocityautocorrelation function, Z ( t ) , which are defined as19where q,,, and q,,t represent the positions of a particle n corresponding tothe times o and t respectively and20 ) Z ( t ) = c vn,o - vn, t N Y N n=l((q2))t and Z ( t ) are related to the process of diffusion in the system.Having outlined the necessary basic statistical mechanical formalism andintroduced two current computer simulation approaches, ‘Monte Carlo’method and the method of molecular dynamics, applications in various areasof physicochemical research can now be considered.THE FLUID STATEStatistical theory of fluidsIn the case of non-interacting particle systems, the partition function and theradial distribution function are subject to analytical solutions.Howeverwhen a system consists of particles which interact with one another in accordwith a realistic law, the above integrals cannot be evaluated as such: themulti-integrals involving the potential energy of the system cannot beseparated into parts so that each part would depend only on the coordinatesof one particle.The aim of the statistical mechanical theories of fluids is,therefore, to introduce simplifications so as to reduce the configurationalintegral to the forms amenable to analytical solutions. The important theories* The details of the procedure are set out in: B. J. Alder and T. Wainwright, J. chem.Phys., 1959, 31,459; A. Rahman, Phys. Rev., 1964, 136, A405.La1 10are : the cell theory,22 Mayer's theory of cluster integrals,23 Bogoliubov-Born - Green - Kirkwood - Yvon (BBGKY) theory,24 hypernetted chain (hnc)theory,25 Percus-Yevik (PY) theory,26 and perturbation theory.104The cell theory assumes that each particle in a fluid system is confinedto a spherical cell made up of the surrounding particles.The configurationalenergy of a particle can be considered to be composed of two parts : (i) energy,UO, possessed by the particle when positioned at the centre of the cell (latticeenergy), (ii) UD, the energy associated with the displacement of the particlefrom the cell centre. The configurational partition function, q, of the particleis, then 1 exp [-(uo + u ~ ) / k T ] 4 r v ~ drcellwhere r is the radius of the cell. The configurational partition function of thesystem would simply be equal to q N .Mayer's theory, in its essentials, is concerned with a series expansion ofthe integrand exp - ( U / k T), and the subsequent term-by-term integration ofthe series.If exp -(uzj/kT) is expressed as (1 + f i j ) ,I N \ N21Ni ji < j=1+cc.h+Ni j ki < j < kc c c (.fij.hk.f3k + . f i j . f i k + j i j . h k +J;:k.ftk) + * * - 22It may be realized that the second, third, etc., terms in the above series actas successive corrective terms modifying the partition function to take inter-molecular interactions into account. The second term involves molecularpairs (ij), the third term triplets (ijk), the fourth term quadruplets, and so on.The consequences of the work carried out in this direction by Ursell, Mayer,Born, and others provide a sound theoretical foundation to the classical virialequation of state for imperfect gases.The usefulness of this procedure islimited to low-density gases and liquids; with high-density gases and liquids,the above series fails to converge.BBGKY, hnc and PY theories enable one to evaluate g(r), assuming U(Y)is known. The two basic equations of the BBGKY theory are:23g123 = 812 -g23 *g13 24g12, g23, etc. correspond to the pairs 1-2,2-3, etc. V is the differential operator(a/&, a/ay, a/az). Equation 23 is directly derivable from 10,24 is an approxi-mation-introduced by Kirkwood-which estimates the triplet distributionfunction g123 in terms of the radial distribution functions g12, g23 and g13. The104 R. I. C. Reviewsubstitution of 24 into 23 would lead to an equation involving only gijs anduir).This resulting equation, called the BBGKY equation, can be solved numeri-cally for g. The hnc theory evolved from Montroll and Mayer’s procedureof the summation over a certain class of interaction graphs (in the terminologyof Ursell and Mayer);27 while the PY theory originated from an attempt toapply the collective variable method-in a fashion similar to that appliedby Bohm and Pines to a system of charged particles28-to non-polar fluids.Though the two theories come from different directions, the structure of theequations (relating c(r) to u(r)) which they yield bear similarity to each other:c(r) = h(r) - In [l + h(r)] - u(r)/kT 25c(r) = [l -+ h(r)][1 - exp (u(r)/kT)] (PY) 26The substitutions of 25 and 26 in the Ornstein-Zernike equation 17 givesthe well-known hnc and PY integral equations respectively.The development of the perturbation theory with realistic systems is dueto Barker and Henderson.104 The theory treats an intermolecular potentialas a modified function of an arbitrary potential.The potential can be definedin terms of three parameters: d, the hard-sphere parameter, a, the inversesteepness parameter for the repulsive region, and y, the depth parameterfor the attractive region. a = y = 1 corresponds to the arbitrary potentialfunction, and a = y = 0 describes a hard-sphere potential. The theory isessentially concerned with the expansion of the configurational integral indouble Taylor series in terms of a and y about a = y = 0. The expansionhas satisfactory convergence under certain realistic conditions.(hnc)Computer studiesWhile attempting numerical solutions of the BBGKY equation for the caseof hard-sphere systems, Kirkwood, Maun and Alder discovered that theequation does not possess any integrable solution in the density region above(N/V)a3 = 0.95, CT being the molecular diameter (for the close-packedsystem (N/V)o3 = 1.414).29 This prompted Alder and Wainwright to simu-late the system on computer by adopting the method of molecular dynamics.19The simulated system underwent solid/fluid transition at approximately(N/ V)a3 = 0.95, the point beyond which conventional numerical procedurescease to yield a solution for the BBGKY equation.The same system was alsosimulated by Wood and Jacobson employing the ‘Monte Carlo’ method; thisstudy virtually reproduced Alder and Wainwright’s results.ls Because of therather small number of particles constituting the system (-100-500)-acomputer limitation-and because of the apparent artificiality of the simu-lated systems, the validity of the conclusions may be doubtful. But a closeagreement between the equation of state, for the fluid regon, derived from thecomputer studies and that obtained by determining successive virial coeffi-cients (Fig.I ) would certainly dispel such doubts, establishing a firm reliabilityin ‘Monte Carlo’ and molecular dynamic calculations. Indeed, the computermethods can now be regarded as ‘experimental’ techniques, and hence theresults obtained from them should provide a fair basis for assessing therelative merits of various analytical theories.La1 10Fig. I .The equation of state for a systemof hard spheres; ~ derived frommolecular dynamic studies;lg ‘MonteCarlo’ results of Wood and Jacobson;]* --- virial expansion. (Ref. 18.)The inference of phenomenological properties from those of the submicro-scopic simulated system utilizes ‘periodic boundary conditions’. Thesestate that a macroscopic system is formed from the successive translationalreplications of the system under computer study. Therefore, the molecularinteractions between the system and its surrounding replicas are to be takeninto account. In the molecular dynamic studies, the periodic boundary con-ditions stipulate that if a particle leaves the ‘cell’-the confines of the system-another particle with identical kinetic properties would enter the cell in-stantaneously from the opposite direction, so that in all instances the numberof the particles in the cell remains constant.The ‘Monte Carlo’ method in simpler form, viz.collecting a sample ofaccessible random configurations generated independently of one another,would be extremely inefficient, involving a prohibitive amount of computertime. Therefore it would be necessary to devise an efficient sampling schemefor a successful ‘Monte Carlo’ calculation. A successful, and hence extensivelyused, variance-reduction scheme was originated by Metropolis and co-w o r k e r ~ . ~ ~ A large sequence of configurations generated by their methodtends towards a Boltzmann distribution; hence the simple average of aproperty over the sample converges to the canonical ensemble averagevalue.The innovation of the Metropolis sampling technique, with the advent offast electronic computers with sufficient storage capacities, has led to success-106 R.I. C. Reviewful ‘Monte Carlo’ studies on systems with realistic intermolecular inter-actions, e.g. the work of Wood and Parker, in 1957, on the computersimulation of argon at 55°C.31 They assumed that the molecules interact inaccord with the Lennard-Jones equation, and that the configurational energyis expressible as the sum of the total intermolecular pair interactions. Thecomputed compressibility, energy and heat capacity data were found to agreewith the experimental results of Michel and co-workers in the pressure range15.19875-202.65 MPa; however at high pressure, viz.202.65-1519.875 MPathe calculated compressibility factors were at variance with Bridgnian’s experi-mental values. The computer study indicated the existence of fluid/solidtransitions, but the predicted freezing pressure was found to be lowerand the volume change higher than experimental estimates. Wood, later,extended the computations to higher temperatures.2b In a similar study, byMcDonald and Singer, the calculations were carried out in the temperaturerange - 100-150°C.32 Fair agreement between the computed and experi-mental compressibility factors was observed but the discrepancy betweenthe ‘Monte Carlo’ and actual energy values was marked.They attributedthis to the inadequacy of the Lennard-Jones potential in representing inter-molecular interactions. ‘Monte Carlo’ work by Verlet and co-workershas been mainly directed to the assessment of the current statistical mechanicaltheories of fluids by comparing theoretical and ‘Monte Carlo’ computedvalues of thermodynamic functions.33 Figure 2 shows such a comparison,where the compressibility factor, p/RTp, as a function of density p, calculatedfrom the PY equation (with the first correction term), is represented by theFig. 2. Compressibility factors, P/p,versus density isotherm for argon :or-responding t o the reduced temperature,kT/E*, I .35. I ‘Monte Carlo’ results;calculated from the PY equationwith the first correction term; 0 experi-mental values.(Ref. 33.)La11.71.61.51.41.31.2P 1.13.1 ,0.2 ,0.3 ,0.4 10.5 ,0.6 ‘0.7 9.8 ,O.S10solid line; the dots are the experimental points for argon and the verticallines represent ‘Monte Carlo’ results. Close agreement between ‘MonteCarlo’ and theoretical results is demonstrated up to p = 0.55; beyond this,the deviation of the theoretical line from the computer results is considerable.The experimental results differ substantially from those of ‘Monte Carlo’and PY. This again shows the invalidity of the L-J potential. Thus if a correctintermolecular potential is found, the PY equation solution should be ableto reproduce experimental results in the low density region.Many-bodyforces cause an additional complication at high densities, their exact treat-ment is beyond present theories. In a subsequent study, computer ‘experi-ments’ were designed to test the equations of state derived from Barker andHenderson’s theory; the theory was in reasonable agreement with the machineresults.105 Further progress in ‘Monte Carlo’ applications has been made byVerlet who carried out a successful study on phase transition behaviour ofLennard-Jones fluids.34 Figure 3 shows the experimental and ‘Monte Carlo’co-existence curves for argon; Fig. 4 gives the melting pressure versustemperature curve. Excellent agreement between experimental points andthe computed curve has been found for the solid/fluid transition; however theagreement is rather poor for the liquid/gas transition.The ‘Monte Carlo’melting pressures are somewhat lower than the experimental pressures in thewhole of the temperature range studied.Singer35 has extended ‘Monte Carlo’ studies to binary fluid mixtures.Comparison between ‘Monte Carlo’ results and those calculated from variousFig. 3. Coexistence diagram for argon interms of reduced temperature T* anddensityp*; - machine computationresults; - - - ex p e r i m e n t al c u r ve ;I09melting data;llO melting data.111(Ref. 34.)Fig. 4. Melting pressure of argo; as afunction of temperature; - MonteCarlo’ results; experimental results.111(Ref. 34.),0.2 ,0.4 ,0.6 p.8 11.0 ,108 R. I, C. Reviewtheories of mixtures of non-polar spherical molecules has revealed thesuperiority of the van der Waals version of average potential model, as givenby Rowlinson and co-workers,36 over Prigogine’s theory.37A paper by Barker and Watts on water structure was perhaps the first on theapplication of the ‘Monte Carlo’ method to the systems composed of complex2.0 JFig.5. Radial distribution function of water (298.15 K). A ‘Monte Carlo’ results based on54 000 configurations; ‘Monte Carlo’ results based on I 10 000 configurations; - ex-perimental.112 (Ref. 38.)I 12.5A1.51 .o0.51ADistance (A).3.0 .4.0 ,5.0 6.molecules.3* They assumed that the pair potential energy of water moleculesis adequately expressible by the expression suggested by Rowlinson. Thecalculated energy and heat capacity agreed reasonably well with experiment,but the computed radial distribution compared less favourably (Fig.5).Another interesting class of fluid systems which has been the subject ofcomputer simulation studies is that of charged p a r t i ~ l e s . 3 9 ~ ~ 0 ~ ~ ~ ~ 4 2 Themost recent example is provided by a ‘Monte Carlo’ study of a neutral systemof classical charged particles,43 which was designed to derive the thermo-dynamic as well as structural properties of a simulated potassium chloridesystem. The internal energy, pressure, heat capacity, thermal pressure coeffi-cients, thermal expansion coefficients, compressibility, entropy, normalmelting points and radial distribution functions were computed. The Born--Mayer-Huggins potential, with coefficients obtained from the crystal proper-t i e ~ , ~ 4 was assumed to represent the inter-particle interactions.The computedvalues agreed reasonably well with the available experimental data.Molecular dynamic studies on the systems of Lennard-Jones particleshave been performed by Verlet.45346 The simulated systems were allowed toassume the same Lennard-Jones parameters as those for argon. Verletderived energy and Compressibility values from this study and also estimatedcritical constants of the simulated system, which were substantially smallerthan those of argon.45 The molecular dynamic ‘experiments’ also enabledVerlet to obtain molecular distribution functions.46 Figures 6 and 7 illustratethe computed g(r) and h(r); the figures also include those calculated from thePY equation.The agreement between ‘experimental’ and theoretical valuesof the functions is gratifying.Fig. 6. Radial distribution function of aLennard-Jones fluid (T* = I .4, p* = 0.8)derived from molecular dynamic study, andfrom the PY equation with the first correc-tion term. (Ref. 46.)Fig. 7. h(r) as a function of r ; - molecu-lar dynamic results; --- results givenby the PY equation. (Ref. 46.)2 c2 I*r r,2 - 2 ,0.5 I ,I .5 .5110 R.I.C. ReviewThe molecular dynamic calculations by Rahman have contributed greatlyto the understanding of the mechanism of diffusion in liquids.106 His studyassumes Bernal’s description of the liquid state in terms of Voronoi poly-hedra.lo7 He argued that the velocity autocorrelation function, Z(t), isresolvable into two parts : S(t), ascribed to the ‘slip’ of the molecule along thedirection of the elongation of the primary polyhedron caused by the fluctua-tions of the neighbouring molecules, and R(t), due to the ‘rattling’ of themolecules.The ‘slipping’ is a consequence of the short-range fluctuationsand hence is the essential characteristic of the liquid state. Further, S(t)can be resolved into S+(t) and S-(t); S-(t) accounts for the molecules movingin the direction opposite to that of displacement at the initial point. R(t)and S-(t) describe a sort of oscillatory motion. The most significant contribu-tion to the process of self-diffusion is due to S+(t).SIMPLE CHAIN MOLECULESA simple polymer molecule may be regarded as a chain made up by theconsecutive joining of segments which, in turn, are constituted of groupsof atoms.An n-alkane molecule epitomizes the above description of poly-mers. The segments in this case are the groups CH3- (end) and -CH2-(middle). The connection between successive segments is provided by theC-C bonds. Thus the succession of C-C links forms the backbone structureof the chains. To a fair approximation, the number of links n, the bondlength, I, and the bond angle, 8, of the backbone structure, suffices to charac-terize linear chain molecules. The rotation of the backbone bonds enablesthe molecule to conform to numerous configurations. The geometricalfunctions which characterize chain molecular configurations are the magnitudeof the end-to-end vector, r, and the radius of gyration, S.For chain models, neglecting intersegmental interactions, the averages of1-2 and S2 (to be denoted as (r2) and (S2)) in terms of n, I, 8 and C$ (bondrotational angle) are analytically expressible. In the limit of infinite chainlength, such expressions reduce to4’1 + cos 8 1 + (cos 4 )1 - cos 8 1 - (cos 4 ) ((r2>/n12) n-t to =1 1 + cos 8 1 + (cos +) ((S2>/nlz) n-+ccI = ~ ~ 6 1 - cos 8 ’ 1 - {COS 4) (b) 27henceBut when the intersegmental interactions are taken into account, the problembecomes extremely complex and often cannot be treated analytically.Themost important stipulation imposed by the intersegmental interactions isthe excluded-volume condition-the chain cannot intersect itself.Someattempts have been made to correlate <r2) and (S2) with chain length for thechain models incorporating the excluded volume condition ;48949p50 howeverthe validity of these treatments is doubtful.51La1 11Another approach to deriving the statistical averages of various configura-tional properties of chains is based on the simulation of the configurationalbehaviour of such chains using computers. The simulation procedure involvescollecting a large sample of randomly generated chain configurations ; onlythose configurations which conform to the excluded-volume conditionare included in the sample. The averages of various properties taken over thesample is assumed to converge to the canonical ensemble averages and thusrepresents the corresponding equilibrium values. Such work has beenpioneered by Wall and co-workers who studied chains confined to varioustypes of lattices.Their studies firmly established the following relationships.6In the limit, n -+ co,__- (s2) = c (constant)<r2>Values of y obtained are approximately 1.50 and 1.20 for two- and three-dimensional chains;8 c assumes the values ca 0.14552@ and cu 0.157,52354respectively, for the two kinds of chains. Recent work along similar lines onoff-lattice chains yielded slightly higher values for y, but the value of cremained virtually ~naltered.5~Another aspect of chain configuration statistics which has attracted muchattention is the distributions associated with chain segments and chain-endseparations.56~57~58~59 For chain models, neglecting the excluded-volumecondition, such distributions are Gaussian in the limit of infinite chainlength.The introduction of excluded volume in the model would pose pro-found mathematical difficulties which have so far prevented the developmentof an undisputed analytical treatment for such chains. This necessitatesresorting to the computer simulation based numerical procedures. Thesimulation studies of Mazur and co-workers have shown that the inclusionof excluded volume destroys the Gaussian character of the limiting distribu-tion. They found that the following function would adequately describe thedistributionW(r) = Aexp-(r/a)m 29where for three-dimensional chains 171 Y 3.2.56957 In the case of two-dimen-sional chains, the functional form of the distribution function is identical to29, but the value of m required to reproduce the computed moments isapproximately equal to 5.8.60An interesting application of the ‘Monte Carlo’ technique has been madeby Verdier and Stockmayer who simulated the dynamic behaviour of ex-cluded-volume chains in infinitely dilute solutions.61962 A general agreementbetween the relaxation behaviour of the end-to-end distance of the chainsrevealed by ‘Monte Carlo’ study and that predicted by the Rouse and Zimmtheory provides encouragement for applying the method in exploring thenature of the relaxations associated with such complex but interesting112 R.I . C. Reviewprocesses as untying of knots in polymer chains, and first contacts betweenchain ends.The simple excluded-volume effect in the model chains described aboveimplies a hard-sphere potential for intersegmental interactions.Because ofthe unrealistic nature of this potential, the correspondence between realchain molecules and the above models would be severely limited. It would benecessary to apply a realistic intersegmental potential to improve such acorrespondence. Wall and co-workers, in later ‘Monte Carlo’ calculations,assumed a square-well potential for such interaction~.~8Jj3,6~ The chainsobeyed the equations 28 with y and c varying with temperature. Furthermore,this study demonstrated the existence of the Flory temperature 8 (the tempera-ture at which the configurational behaviour of the chains is identical to thatof the random walk model; hence at 8, y = 1 , c = 6) for the model con-sidered.Even the square-well potential portrays a very crude picture of theintersegmental interactions. Further refinement of the model would involveusing more complex potentials such as those of Lennard-Jones, Buckingham,etc., which describe intermolecular interactions more realistically.65The ‘Monte Carlo’ calculations described so far have been entirely con-cerned with isolated-chain models. Such models relate to dilute solutions,where interchain interactions are negligible. In crystals and in concentratedsolutions surrounding chains would undoubtedly exert a considerableinfluence on the molecule. Very few ‘Monte Carlo’ studies have been devotedto the investigation of such effects.Whittington and Chapman performedcalculations on a very simple model simulating a multi-chain system.66 Theend-to-end distance of the chains and the entropy of the system were com-puted as functions of interchain distance. The study indicated a phase-transition in the system. In another investigation, calculations were made for therotational energies of hydrocarbon chains in crystals employing the ‘MonteCarlo’ method.67 The behaviour of energy as a function of temperaturerevealed a phase-transition in the chains in conformity with experimentalfindings.The other computer technique, which has been applied in chain statisticalcalculations, is the method of exact enumeration. For lattice-confined shortchains, it has been possible to generate all the allowable configurations oncomputer; this would allow calculation of exact values of various chainproperties. The values for infinite chain length for properties which convergerapidly to their asymptotic limits as functions of chain lengths, e.g.the ratio(S2)/($>, can be deduced through extrapolation procedures. The exactenumeration method has been the basis of most of the work on chain con-figuration statistics carried out by Domb and co-workers.68 In general, theresults obtained from the two methods are in agreement.69BIOLOGICAL MACROMOLECULESFor the past two decades or so, the structure of biological macromolecules,particularly nucleic acids, polypeptides, etc., has been the subject of profoundinterest to physical scientists.The first major breakthrough was made byWatson and Crick‘s proposal of a double helical structure for the DNALa1 11Fig. 8. A pictorial view of the minimum energy conformation of Gramicidine S. (Ref. 73.)molecule.70 Briefly, the molecule is composed of two right-handed heliceswound around a common axis; the two helices are connected at variouspoints through the hydrogen bonds occurring between the bases adenine-thymine, guanine-cytosine, etc. Subsequent x-ray studies confirmed the abovestructure. It was realized immediately that the interpretation of such asignificant process as the perpetuation of the DNA molecules of identicalstructure in cell plasma lay in the structure.70 Since then, several other keymolecular biological phenomena have been accounted for in terms of molecu-lar conformations.Computer simulation techniques have to be used in the elucidation ofcertain configurational features and associated processes because of theimmense complexity of these molecules.Some important illustrations of therole of the computer in this field will be given.Scheraga et al. carried out computer reconstructions of simple polypeptidemolecules subject to the constraints necessitated by the geometrical as wellas energy considerations.71~72~73 As a result, they arrived at the detailedthree-dimensional pictures of the molecules. Figure 8 is a simplified pictorialrepresentation of the minimum-energy conformation of Gramicidine S.Thestructure, in certain respects, is similar to that proposed by Stern et al.on the basis of their nmr study of the molecule74 but while they suggestedthe existence of four hydrogen bonds, the minimum-energy conformationdoes not contain any. More recently, Scheraga and co-workers evolved a morecomprehensive computer techni que-s t a tis tical search procedure-and114 R. I.C. Reviewdiscovered another low-energy conformation, 477 J mol-l lower than thatof the above.73The stability of the helical structure in polynucleotides depends on suitablephysicochemical conditions, alteration of such conditions causes the con-.version of helical into coiled conformations. In the past, several models havebeen postulated to account for this transformation.Crothers, Kallenbackand Zimm14 used computer simulation to assess the validity of such models.They confined their investigation to the ‘zipper’ model in which the helix/coilconversion starts at one end of the molecule and proceeds gradually to theother-as if the molecule is being unzipped. The contributions by the differentkinds of base pairs to the stability of the helical structure for a heterogeneouspolymer-adouble helixcontainingdifferent kinds of base pairs-were assumedto be different, in consonance with the experimental evidence. It was alsoassumed that the sequence of base pairs along the chain is random. Thenhelix/coil transition in DNA chains was simulated. Figure 9 compares thetransition curves of the simulated chains with those of T2 DNA moleculesdetermined experimentally.The agreement between the computed curvesand the experimental points is reasonable for molecules of shorter lengthsbut not for the high-molecular weight polymers. Thus the zipper model isvalid for shorter nucleic acids but fails to account satisfactorily for the helixcoil transitions in the longer molecules.The kinetics of the unwinding of polynucleotide helices has been simulatedin a study by Simon.75 His approach uses a difference form of theLangevin equation of motion for simulating molecular movement. Theobject was to obtain information on the role of van der Waals type inter-Fig. 9. Helix/coil transition curves of D N A molecules. L is the number of base pairs in themolecule; 8 is the fraction of H-bonded base pairs; - transition curves of the computersimulated molecules; 0 experimental results for T2 DNA.(Ref. 14.)La1 11strand interactions in determining the unwinding kinetics. The results showedthat such interactions are of considerable importance in the process. Further-more, the computer investigation supports the view that unwinding starts athelix ends and progresses inwards.Although the existence of folded structures in certain proteins has beenconfirmed by experiment769 7 7 an adequate understanding of the mechanismof folding during biosynthesis is lacking. De Coen conducted a computerstudy78 in which it was assumed that the biosynthesis of the folding chainsproceeds via a succession of two types of stages occurring alternately :(a) formation of a sequence of amino-acid residues adhering to a minimum-energy conformation, which ‘freezes’ to act as ‘nucleus’ for further growth,(b) addition of a non-hydrogen bond forming amino-acid residue to the‘nucleus’.The result is a sequence of minimum-energy regions separated bynon H-bond forming spots. The simulated structure was a compact globularshape (folded).X-ray studies by Krimm and Tobolsky show that on stretching keratinefibres, the a-helix conformation of the molecules transforms into a sheet-like,5’ structure.79 Schor, Haukaas and David’s ‘Monte Carlo’ computer simula-tion study aimed at constructing a satisfactory model for this transforma-tion.80 The simulation involved generating stochastic growth of polypeptidechains, through sequential residue addition, based on the Boltzmann proba-bilities associated with the allowed angle pairs (4, 4).The first three residuesof the chain were made to assume an a-helical conformation. The helix axisserved as the direction of applied tension as well as of the further growth ofthe chain. Figure I0 presents the tension/length isotherm of the simulated3.53.0L2.52.0Tension (dynes)2 3 4IFig. 10. Plot of applied tension versuslength for a computer simulated keratinemolecule of 200 residues. (Ref. 80.)116 R.I.C. Reviewchain of 200-residue length at 300 K. This model is able to reproduce theessential features of the isotherm in terms of the geometry and interatomicinteractions.ADSORPTIONIn the past, several theoretical attempts have been made to explore the con-figurational behaviour of chain molecules in the adsorbed state.81~82~83~84~85A serious shortcoming of these theories is that they either ignore, or fail toincorporate satisfactorily, the intersegmental excluded-volume effect in themodel. On the other hand, as is clear from the section on simple chain mole-cules, computer simulation methods are capable of considering such an effect.Thus ‘Monte Carlo’ studies would prove quite apt in disclosing the conforma-tional behaviour of polymers at interfaces.Clayfield and Lumb have described a computer study they carried out onthe simulation of isolated chains with one end permanently ‘anchored’ to asurface.86 This system would serve as a model for a polymeric dispersantwhich would prevent colloidal particles from flocculating or from adhering tosurfaces.The chains, of up to 300 links, were generated on a four-choicecubic lattice ; successfully generated configurations were those which obeyedthe excluded-volume condition. The computation was directed towards thecalculation of the averages of various configurational quantities such as end-to-end distance of the chain, maximum height of the chain above the surface,number of adsorbed segments, etc. However the histogram of the heights ofchains above the surface, from which the entropy of compression of terminallyadsorbed chains could be calculated, was of prime interest. This led to theevaluation of the free energy of two interacting spheres, and a sphere and aplate, in the presence of polymeric dispersant.‘Monte Carlo’ calculations on an improved model of a terminally anchoredchain, in which the energy of adsorption of a segment on the surface, e/kT,is taken into consideration, were performed by McCracken.87-The chainsO -50-150 .-200-300Fig. I I. Variation of the fraction ofadsorbed segments as a function of theenergy of adsorption per segment for ex-cluded volume chains of various lengths.prediction of Rubin’s Theory88for a non-excluded volume infinite-length chain. (Ref. 87.)Lal 117OSwere simulated on a four-choice simple cubic lattice. Information on thick-ness of the adsorbed chains, the fraction of the adsorbed segments, end-to-end distances of the chains, loops off the surface etc., was sought.Figure I Igives the computed results of the fractions of segments in the adsorbed stateas a function of -e/kT for several chains length. The line represents thecalculations for an infinite-length chain in accordance with Rubin’s theorywhich neglects the excluded-volume effect. For a given e/kT, the fraction ofadsorbed segments goes on decreasing as chain length increases; and, asexpected, as - e/kT increases, the number of surface-confined segmentsincreases too. The figure provides an indication that for long excluded-volumechains approaching the limiting behaviour, the fractions of adsorbed seg-ments would be considerably less than those calculated by Rubin (for non-excluded volume chains).s8 Furthermore, this study confirms the existenceof a critical energy of adsorption-approximately 0.2 kT-below which theextent of adsorption is negligible; above the critical energy, the fraction ofadsorbed segments rises sharply with - e/kT and eventually would approachunity for large energies.The ‘Monte Carlo’ calculations by Bluestone and Cronan are based on afairly realistic model of the chain interacting with a surface.10 All the segmentsof the chain, including the terminal ones, were assumed to have equal ad-sorbing susceptibility.The interaction between a segment and the surfacewas assumed to vary with distance from the surface and the intersegmentalinteractions were ignored. The calculations were performed for 3 1 -segmentlinear molecules.For greater segment/surface interaction energies, the calcu-lated fractions of the segments adsorbed on the surface were in reasonableagreement with those given by the theories of Frisch,81 HiguchiB2 andSilberberg. 83 When a low interaction energy was assumed, the moleculesremained in the desorbed state. The configurational analysis revealed that inthe adsorbed state, a chain configuration consists of trains (arrays of seg-ments in the adsorbed state) and loops (arrays of segments away from thesurface) which is in agreement with certain theories of polymer adsorption.83A simulation study of the monolayer adsorption of monatomic gases onsolid surfaces was carried outll in which the solid surface was taken as aheterogeneous, hexagonal close-packed planar lattice.It was assumed thatin the adsorbed state the particle interacted with the adsorbing site as wellas with the surrounding adsorbed molecules. The results were in qualitativeagreement with the adsorption equation derived by Hill based on the quasi-chemical approximation.REACTION KINEMATICSThe absolute reaction rate theory, embodied in the ‘semi-empirical method’,was developed by Eyring, Polanyi, Hirschfelder, and Wigner. 89 Furtherprogress in the theory has been negligible because of severe mathematicaldifficulties which have proved insurmountable except, perhaps, for ex-tremely simple cases. A direct approach to the solution for a system of reactingmolecules is to simulate the molecular dynamics of such a system on acomputer, obtaining the numerical solutions of the classical Harniltonian118 R.I. C. Reviewdifferential equations of motion of the particles (constituting the system).For a given value of the Hamiltonian, H, these solutions yield the trajectoriesof the particles as functions of time. These trajectories contain direct informa-tion concerning the occurrence or non-occurrence of a particular reaction.Wall and co-workers successfully applied this method toH2 + H -+ H + Hz-a triatomic system.g0 The potential energy of thesystem was calculated from the London-Eyring-Polanyi expression, whichinvolves the interatomic pair potential energies 2.412, 2423 and 2413; uijs areadequately expressed by the Morse equation. The molecular dynamics ofthe system were worked out for several values of H assuming various initialconditions.Initially, only collinear collisions were considered, but later thecomputation was carried out for two-dimensional non-linear collisions aswell. The computation revealed the following: no reaction would occur whenthe total energy of the system was below the translational activation energy;the inclusion of vibrational energy in the Hamiltonian reduced the transla-tional activation energy-approximately one-sixth of the vibrational energycould contribute to the activation energy; the existence of a large rotationalenergy would considerably diminish the probability of the occurrence of areaction. The above type of calculations were subsequently extended to thesystems (linear triatomic) confined to the potential energy surfaces whosesaddle points differed in curvature and location.9 l The computation showedthat the reaction would occur only if the energy lay within a certain boundedrange, the bounds of the energy range depending upon the position of thesaddle point and the relative masses of the reacting particles. Also, thevibrational energy of the product molecule was found to be dependent onthe location of the saddle point.Extensive ‘Monte Carlo’ investigations of exothermic triatomic reactionsof the type: A + B-C = A-C + B have been carried out by Blais andBunker92 and by Raff and K a r p l ~ s , ~ ~ ? g particularly the reaction:K + CH3-I = K-I + CH3. Blais and Bunker selected a potentialenergy surface for the reacting particles that assumed the applicability ofMorse potentials for stable molecules B-C and A-C, and fulfilled thecriterion that the energy of the reaction is released during the approach of Atowards B-C. Various modified versions of Blais and Bunker potentialenergy functions, used by Raff and Karplus in their study, yielded values ofreaction cross-sections in reasonable agreement with experimental results.The studies were aimed at the determination of various reaction attributesas functions of initial conditions, masses of participating particles and natureof the potential energy surfaces.These investigations disclosed that a verylarge part (ca 90 per cent) of the energy released in the reaction appeared inthe vibrational and rotational modes of the product molecule, thus thevibrational and rotational states of newly produced A-C are at highlyexcited levels, and the total angular momentum of A-C finds its largestcontribution from molecular rotation. Calculated laboratory differentialcross-sections of the reactions were in good agreement with experimentalvalues.Examination of particle trajectories in the event of a reacting collision(Fig. 12) suggests a simple collision mechanism for the reaction in which thereacting particles interact for a very short period.La1 1120/ Distance /?,(A)Time (0.54 x 10- l4 sec)0 40 80 J20 ,l60 200Fig. 12. Particle trajectories in the eventof a reactive collision for the simu-lated system K + CH3-I + KI + CH3. --- distance between K and I;distance between I and CH3; -.-.- distance between K and CH3.(Ref. 9.)The application of the ‘Monte Carlo’ method in the analysis of four-bodyreactions of the type: A + B-C-D + A-B + C-D with particularreference to the reaction K + CH3-CH2-I -+ KI + (CH3-CH2) wasmade by Raff.94 The calculations were carried out for the determination ofthe total reaction cross-section, the differential reaction cross-section, andthe distribution of the reaction energy among the available degrees of free-dom of the product molecules (A-D and B-C).Observations, similar tothose for the reaction A + B-C -+ A-C + B, were made of the reactionenergy distribution. The calculation also showed that the C-C bond in theethyl group absorbed about 14 per cent of the available energy through itsvibrational mode.The occurrence of the reaction assumes two modes, directinteraction and complex formation.Another interesting reaction which has been the subject of ‘Monte Carlo’calculations is that between an alkali metal atom and a halogen molecule.95The reaction’s interest lies in its proposed mechanism which involves anelectron jump or ‘harpooning’.SYSTEMS CORRESPONDING TO THE ISING MODELThe Ising model, which was originally used to interpret ferro- and anti-ferro-magnetic behaviour, corresponds to a lattice whose sites can assumeonly two states. In the case of magnetic materials the two states are positiveand negative spins which can be assigned the values + 1 and -1 respectively.Let p(i) denote the state of a lattice site i, then a vectorwould completely describe a configuration k of an N-site lattice; p(i) canassume only the values + 1 and - 1.It is assumed that a site interacts onlywith its nearest neighbours and with an external field. Thus the total con-120 R.I.C. Reviewfigurational energy of an Ising lattice corresponding to configuration k isgiven asN N30The first term on the rhs of equation 30 gives the nearest neighbour pairinteraction energy; the second term gives total field/spin coupling energy ofthe system ( H is the coupling energy per spin). 2J is equal to the energy oftransformation of a pair from pzrallel to antiparallel state.An important feature of the Ising lattice is that it exhibits phase transition.Below the phase-transition temperature-the critical temperature, Tc-long-range correlations between the spins exist, above Tc the long-rangecorrelations disappear but short-range correlations persist.The parametersf ( n ) and S, order parameters, describe the short- and long-range correlationsin the lattice respectively. They are defined as1 Nand3132a(n) is the number of the neighbours per lattice site; and sites i and j form ann-th neighbour pair in a configuration k. The equilibrium values of f ( n )and S would be3334The Ising model can also be used for superlattice transitions in alloys,cooperative transitions which occur in many solid substances, helix/coiltransitions in certain biological macromolecules, lattice-bound fluid systems,etc.Solutions of the Ising model may provide an adequate interpretation ofthe above phenomena. For two-dimensional Ising lattices, in the absence ofexternal field, Onsager, and others, were able to derive exact analyticalsolutions using highly sophisticated mathematical techniques.96 No suchsolutions have yet been found for three-dimensional lattices. Thereforenumerical techniques such as the ‘Monte Carlo’ method have been employedto tackle the problem. Pioneering work in this direction has been carried outby Fosdick and c o - ~ o r k e r s . ~ The ‘Monte Carlo’ method used was based on aMetropolis sampling technique. Initially the computation was performed onsquare lattice with H = 0. Agreement between the computed f(1) valuesand those provided by analytical theory established the feasibility of theLa1 12Fig.13. The long-range parameter, 5 , fora simple cubic lsing lattice as a functionof jjkT(K) for various values of H/kT(L).(Ref. I 13.)‘Monte Carlo’ technique for tackling Ising systems. Subsequently, thecalculations were extended to three-dimensional lattices. The purpose of thiscomputation was to evaluate the parameters S , f ( 1) andf(2) assuming variousvalues of J/kT and H/kT. The behaviour of S, as J/kT varies, would indicatethe nature of phase-transition in the system. S as a function of J/kT, corre-sponding to several values of H/kT, is plotted in Fig. 13. I t is clear that forH/kT = 0 the system undergoes a sharp transition, the sharpness graduallyfades as H/kT increases and eventually disappears for H/kT = 1.For simplecubic and body-centred cubic lattices critical values of J/kT, at H/kT = 0,were approximately estimated by Fosdick and co-workers to be 0.2275 and0.16 16 respectively.Close correspondence between the Ising model and binary substitutionalloys led Gutman to carry out a ‘Monte Carlo’ simulation of such alloyson the basis of the body-centred cubic Ising lattice.5 Besides order parameters,the computation included the evaluation of the entropy and heat capacity ofthe alloy at several compositions. Figure 14 gives experimental and computedheat capacities of 1 :I P-CuZn alloy as a function of temperature. Althoughquantitative agreement between the experimental and ‘Monte Carlo’ resultsis lacking, the qualitative features of the two curves are remarkably similar.Theory confirms the existence of a singularity in the heat capacity versustemperature curve of an infinite two-dimensional Ising lattice.It is con-jectured that heat capacities of the three-dimensional lattices would alsofollow a similar behaviour.97 ‘Monte Carlo’ studies cannot confirm thisbecause of the finite size of the lattices considered-discontinuities can onlybe exhibited by infinite crystals. Nevertheless these calculations have beenable to give an approximate indication of critical points in such systems.APPLICATIONS OF THE ANALOG COMPUTERSimulations discussed in the foregoing part of this article would normallyrequire the use of a digital computer.However, for the simulation of certainother processes, particularly those involving the solutions of differentialequations and which do not warrant the use of extremely speedy computa-tional equipment, the use of analog computers may prove convenient as well122 R.I.C. Review4+ - Observed p-CuZn- Calculated-3C m- 2?zI’IIIlldr lFig. 14. Reduced(Ref. 5.)heat capacity, versus temperature plot for I:I fl-CuZn alloy.as advantageous. A great advantage of analog computers is that they areable to perform integration on a continuous basis and are thus able to yieldexact solutions of linear as well as non-linear differential equations. 98 Further-more, the output from an analog computer can be easily displayed on anoscilloscope ; this greatly facilitates a systematic inspection of variations inoutput when the parameters which characterize the system under simulationare varied.Analog computers, however, suffer from the disadvantage thatthey do not possess any ‘memory’.The greatest use of analog computers is in the dynamic analysis of chemicalengineering processes giving information valuable for designing reactors. Inchemical investigations too, their use-though not so extensive-has provedof benefit. This is exemplified by a recent simulation study of adsorptioncalorimetry.99 An analog computer was employed to estimate the calorimetriccurves that would be obtained when the evolution of the heat of adsorptionLa1 12of a gas on a solid was studied using a diathermal calorimeter.The effectsof varying the parameters characterizing the design of the calorimeter aswell as those determining the kinetics of the adsorption on the shapes of thecurves were determined. Inferences from this study would be immenselyhelpful in evolving a successful design for a calorimeter for studying gas/solidadsorption. Another area of chemical research where analog simulationwould find considerable scope is the kinetics of chemical reactions.CONCLUSIONIt is evident from the foregoing account that ‘Monte Carlo’ and relatedmethods have firmly established themselves as extremely useful techniquesin the investigation of a variety of physicochemical problems. These methodshave perhaps made their greatest impact in the field of statistical mechanicsof fluid systems.‘Monte Carlo’ and molecular dynamic calculations, based onmodels that are believed to correspond very closely to real fluids, haveproduced highly accurate results. It has therefore become quite usual to testthe worth of the statistical mechanical theories against the machine computa-tion results. Further, simulation techniques have been able to bring outcertain theoretically unpredictable features of the models, e.g. the discoveryof solid/fluid transition in hard-sphere systems.l8Jg Molecular dynamiccomputation shows promise in determining accurate values of variouscorrelation functions such as those of angular and linear velocity, dipolemoment, bond forces, etc.100J08 A precise knowledge of these correlationfunctions would provide a sound basis for the interpretation of infraredand Raman rotational and rovibrational line shapes, nmr quadrupole,magnetic dipole-dipole and spin rotation relaxation times, classical vibrationrelaxation, etc.Aspects of the microscopic structure of liquids relevant totheir role as solvents for chemical processes have largely remained unexplored.In terms of such characteristic microscopic properties of solvents, the ‘solventcage effect’, solute molecular encounters, etc. can be explained. The moleculardynamic approach could prove suitable for such studies.lo1In the past, the bulk of ‘Monte Carlo’ calculations carried out on chainmolecular systems were based on the idealization of polymer molecules byself-avoiding random walks on or off lattices.Such studies are helpful forinvestigating chain properties that are insensitive to the ‘microscopic’ detailsof the model. However, certain other aspects of chain behaviour would in-volve such details as bond angles, nature of bond rotation, long-rangeintersegmental interactions, etc. Lack of ‘Monte Carlo’ studies which takeinto consideration these features of real molecules has been due partly to theunrealistic computing time that would be involved in tackling complexmodels102 and partly to an ignorance of the nature of short- and long-rangeinteractions in polymer molecules. The former impediment seems to beovercome by the availability of such extremely fast and large storage-capacitycomputers as IBM 360/195, CDC 6600, CDC 7600, etc.Calculations carriedout by Flory, and others, on bond rotational energies in chain moleculesgive an adequate quantitative account of the short-range interactions.103An approximate quantitative assessment of long-range interactions is also124 R.Z.C. Reviewpossible. A model containing the essential features of the mode of bondrotation in chains, as revealed by Flory’s calculations, as well as allowinglong-range interactions, can be evolved for the purpose of simulating thebehaviour of real chain molecules on a computer. Successful application ofthe ‘Monte Carlo’ method to real systems can be expected to lead to seriousinvestigations on chain folding, interchain interactions in vacuum as well asin solvents, interaction of chain molecules with surfaces, etc. Another interest-ing extension of ‘Monte Carlo’ calculations would be to cyclic polymers.Perhaps the greatest challenge to today’s physical scientists is offered bythe complexities of biological macromolecules.I hope that computers willprove an invaluable aid to human insight and imagination in discoveringthose, still unknown, molecular processes that are of profound significanceto life.ACKNOWLEDGMENTSPart of this article was written while the author was visiting ResearchAssociate in the Chemistry Department, Dartmouth College, Hanover, NH,US. The author takes this opportunity to express his sincere gratitude toProf. W. H. Stockmayer for his kind hospitality and is grateful for variousdiscussions that led to the improvement of this article.Thanks are also due toProf. P. J. Gans, New York University, Dr K. S61c, Midland Laboratoryfor Macromolecular Science, Dr E. B. Smith, Physical Chemistry Laboratory,Oxford University, and Dr P. J. Anderson, Dr R. V. Scowen and Mr G. C.Peterson of this Laboratory for critical readings of the manuscript. Helpfulcriticism by the referee is also greatly appreciated.REFERENCES1 J. H. Curtiss et al., Monte Carlo method. Washington: Natn. Bur. Stand. AppliedMathematics Series, 12.2 (a) B. J. Alder and W. G. Hoover, Physics of simple liquids, ch. 4 (eds H. N. V.Temperley, J. S. Rowlinson, and G. S. Rushbrooke) and (6) W. W. Wood, ch. 5.Amsterdam: North Holland Publishing Co., 1968.3 I. R. McDonald and K. Singer, Q.Rev. chern. SOC., 1970, 24, 238.4 L. D. Fosdick, Methods of computational physics, vol. I (eds B. J. Alder and S.Fernbach). New York: Academic Press, 1963.5 L. 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ISSN:0035-8940    

DOI:10.1039/RR9710400097

出版商:RSC    

年代:1971

数据来源: RSC

摘要:

LORD ERNEST RUTHERFORD OF NELSON (1871-1937) R. H. Cragg, BSC, PhD The Chemical Laboratory, University of Kent at Canterbury, Canterbury, Kent Introduction New Zealand 1871-95 . . Cambridge 1895-98 Montreal 1898-1907 Manchester 1907-1919 . . Cambridge 1919-1937 . . References . . INTRODUCTION . . . . . . . . . . . . .. . . 129 . . . . . . . . . . . . . . . . . . . I . . . . . . . . . . . . 137 . . . . . . . . . . . . . . 130 . . . . . . 133 . . . . . . . . . . . . 140 . . . . . . . . . . . . .. .. 145 The centenary of the birth of Lord Rutherford falls this year. By many he is considered the most outstanding scientist since Faraday and equal in impor- tance to Newton. Certainly it would be difficult to envisage the state of science today without the great contributions he made towards its progress, particu- larly his work on the structure of the atom and the transmutation of the elements.Much has been written on Rutherford and for those who wish to pursue the subject further some useful sources of reference can be found at the end of the review. NEW ZEALAND 1871-1 895 The environment in which Rutherford grew up must, doubtless, have been a factor in the development of his character and intelligence. Nineteenth-century New Zealand was a young country, demanding from its people great stamina, and both mental and practical resourcefulness; life was tough but the pioneer- ing spirit was strong and the emergence from such a background of a scientist of Rutherford’s distinction is hardly surprising, Rutherford’s father had emigrated to New Zealand from Scotland with his parents as a child.He had married in 1866 and Ernest, the fourth of 12 children, was born on 30 August 1871 at Spring Grove (later called Bright- water), near Ne1son.l The family moved to Foxhill in 1875, to Havelock in 1882 and finally settled in Pungarehu in 1886. Rutherford’s father was mostly employed in agricultural work ; he eventually concentrated on flax-growing and became something of an expert in developing his own varieties of flax. In addition to his flax-growing, he supplemented the family income by dealing in timber and notably by obtaining a government contract to supply 40000 railway sleepers at 2s. 8d. each for the newly developing rail system. Although everyday life in New Zealand was primitive and spartan in the extreme, the importance of giving children a sound education had been recognized and there was no shortage of good schools.Rutherford attended Crwg 129 129 the state primary school at Brightwater, moving at 11 to Havelock primary school whence, at the age of 15, he won a scholarship to Nelson College. From early childhood Rutherford was a voracious reader, apparently favouring Dickens especially. He was inquisitive and inventive and obviously destined to be a scientist from the day he made his own gunpowder for firing marbles from a toy cannon-having first modified the cannon to increase its range! Later he made his own camera and enjoyed photography as a hobby.Nevertheless, however engrossing the pursuit of his own interests, he was a conscientious and dutiful member of the family, willingly attending to the running and repairing of the water wheels in the mill attached to their home. As one might expect, he loved the countryside and open air and throughout his life maintained that physical fitness and mental alertness were closely allied. He went hop picking in the summer and bathing (once almost fatally for he was unable to swim). Typically, when the news of his scholarship to Canterbury College, Christchurch, came, he was out in the garden digging up pot at oes. So Rutherford’s early life gives some insight into his later development. The stresses of his environment called upon his natural intelligence to assert itself and form a basis for the maturing of his academic capabilities. He entered Canterbury College, Christchurch, in 1889 and, after graduating in 1893, remained to do research work.His laboratory was a cold, draughty cellar which was used as a cloakroom by the students but, despite such incon- venience, he pursued his researches wholeheartedly. Here he developed a new method for the detection of radio waves; having constructed his own detector from fine sewing needles, he initially demonstrated his technique to an audience of professors and students of the University, by transmitting a wireless message through the laboratory. Some time later he was to use his newly developed technique to detect the sparks from an induction coil at a distance of two miles.Success for Rutherford was equated purely with his own personal satisfac- tion and he ignored the commercial possibilities of discoveries such as these. Some years later Marconi made a considerable fortune by perfecting a technique for radio reception. At this time also Rutherford demonstrated that, contrary to the theory of his contemporaries, high frequency currents from a small Tesla coil would strongly magnetize iron. In 1894 he received an MA with a double first in Mathematics and Physics. He was awarded a scholarship by the Commissioners of the 1851 Exhibition which gave him the opportunity of coming to England to work with J. J. Thomson at the Cavendish Laboratory in Cambridge. So he set sail for England in 1895--significantly, the year in which Rontgen discovered x-rays.CAMBRIDGE 1895-1 898 R. I.C. Reviews Rutherford was fortunate in that the regulations at Cambridge had just changed to allow graduates from other Universities to read for a BA degree after completing two years’ residence, and he was the first student to apply under these new regulations. 130 His early impressions of life in England compared to that in New Zealand were given in his letters to his mother. Having met the English agricultural labourers in his walks near Cambridge, he wrote: ‘You can’t imagine how slow- moving, slow-thinking the English villager is. He is very different to anything one gets hold of in the colonies’. An interesting observation which demon- strates how different was the social class of the exploited English farm labourer from that of his respected, often wealthy, New Zealand counterpart.(Cavendish Laboratory Lord Ernest Rutherford of Nelson. Cw?g He observed also, regarding some of the academics he met at dinner, that 131 Rutherford continued his work on the detection of wireless waves since Thomson believed that it was good for a research student to work on a topic of his own choosing. After perfecting his method he was soon able to detect in his lodgings, signals which were transmitted from the laboratory-a distance of half a mile. Thomson gave an account of this work called ‘The Magnetisa- tion of Iron by High-Frequency Discharges and the Investigation of the Effect on Short Steel Needles’, at the Royal Society on 18 June 1896.In 1896 Becquerel discovered that the element uranium could give out a radiation which affected a photographic plate. It was at this time also that Rutherford changed his area of research. He began by studying the ionizing effect of x-rays and showed that x-rayed gases lost their conductivity when passed through porous plugs; this indicated that the ions responsible for the conductivity had been filtered out. After more work he concluded that x-rays produced +vely and -vely charged ions in the gas. He then considered the nature of the rays emitted from uranium. It was found that crystals of potassium uranium sulphate made an impression on a photographic plate both when the experiment was done in the light and in the dark.Rutherford placed some of the uranium crystals between the poles of a magnet and positioned a photographic plate above a slit above the magnet. Rutherford a t Manchester. they were ‘very capable, especially in their conversation, and it is a pity so many of them fossilize as it were in Cambridge, and are not that use in the world they might be’. R.I.C. Reviews 132 On developing the plate and finding that two images appeared Rutherford realized that the emission from the uranium consisted of at least two types- one which had been deflected by the magnet and the other which had only been affected slightly. After further work he showed that they must be particles of matter of different weights.He then concluded that the radiation consisted of a radiation, which was easily stopped by a sheet of thick paper but had a high ionizing effect, and j3 radiation, which was more penetrating but caused less ionization. For his work on uranium, Rutherford was awarded the Coutts-Trotter Scholarship of Trinity College. Early in 1898 the chair in physics at McGill University became vacant and the Director of the MacDonald Physics Laboratory there, John Cox, visited England and approached Thomson for advice. On Thomson’s recommenda- tion Rutherford, still only 28, was appointed professor and set sail for Canada in September 1898. Cox decided to take on the entire administration of the laboratory himself, thus allowing Rutherford to concentrate all his efforts on research.Cox could hardly have realized at the time how important this arrangement was to be, for the success that Rutherford made in Canada was never to be equalled. MONTREAL 1898-1 907 Rutherford found the physics laboratory at McGill University extremely well equipped from a generous endowment given to the University by Sir William MacDonald. Soon after arriving he began researches which were eventually to prove successful. He was the kind of man who was only happy working with a group of people and in his early days in Canada, before he had established such a group, he often became dispirited-on one occasion he almost decided to give up and apply for the Chair of Natural Philosophy which was then vacant at Edinburgh.However, he shortly gathered about him some very able collaborators and during his nine years in Montreal publishedaround 80papers. He was offered chairs at many American universities and also the chair at University College London, but although the salaries they carried were far above the modest one he earned at McGill he turned them down. When at Cambridge Rutherford had discovered that thorium nitrate appeared to be an unusual source of radiation, and in 1898 he set R. B. Owens the problem of studying the radiations from thorium for his research topic. It was found that the radiations appeared to be affected by draughts of air. In 1899 Rutherford found that thorium compounds emitted radioactive particles con- tinuously and that these particles retained their radioactive properties for several minutes.He then studied the properties of the emanation and found that the particles were not charged, not affected in an electric field and could pass through cotton wool. On the other hand the emanated particles could pass through thin layers of metal and ionize the neighbouring gas. Rutherford concluded that the emanation acted as a gas. He had therefore established that radioactive substances could emit not only radiation but also particles, which had the properties of atoms of an ordinary gas. He also demonstrated that, when a solution of a thorium compound was evaporated to dryness on a glass surface, the apparently invisible residue on the glass surface was very radioactive. 133 Rutherford aged 36. In 1900 Rutherford returned to New Zealand to marry his fiancke, Mary Newton. He brought her back to Canada with him and the following year they had a daughter.Frederick Soddy joined Rutherford and investigated the properties of many types of emanation but he was unable to obtain any chemical reactions and concluded, in 1902, that the emanation was a chemically inert gas analogous in nature to the members of the argon family. Further work on thorium showed that new radioactive substances were being produced and also that all different radioactive substances decayed according to the same law and the rate of decay was proportional to the amount of substance. Next it was demonstrated that 01 rays could be deflected by both electric and magnetic fields, that they were positively charged with a velocity of 2% of light and had a mass similar to the hydrogen atom.Rutherford concluded that Q! rays were atomic fragments which were being emitted from exploding atoms and that ,8 rays were of secondary importance. Using a liquid air machine, he and Soddy were able to liquify the emanation and by 1905 Rutherford was convinced that 01 particles were helium atoms containing two positive charges. During this time Rutherford was joined by Hahn who had been working with Ramsey. For some reason Rutherford was rather sceptical of some of Ramsey’s work on radioactivity but Hahn, a very capable person, very quickly showed the quality of Ramsey’s work by demonstrating to Rutherford that radiothorium existed. He then studied the Q! particle emission from radio- 134 R .I. C. Reviews thorium and radioactinium and showed that in his samples of radiothorium there existed a second a-emitting substance called thorium C[iiPo]. As all attempts to separate thorium C were unsuccessful he came to the conclusion that its half life was short. This is not surprising as t+ was found to be 3 x 10-7s. In 1902 Rutherford and Soddy published their theory that radioactivity is a phenomenon accompanying the spontaneous transformation of the atoms of radioactive elements into different kinds of matter. Rutherford’s first book (Radioactivity) was published in 1904 and contained the basis of his theory of radioactivity. His theory was based on the following experimental evidence : (a) Radioactivity is unaffected by any change in external conditions whether by extreme heat or cold, or by the action of any chemical reagent.As it differs from known chemical reactions the process must be atomic in nature. (6) The radioactivity of uranium, thorium and radium is maintained by the production at a constant rate of new kinds of matter which themselves possess temporary radioactivity. The constant activity of the radio elements is due to a state of equilibrium where the rate of production of new matter is equal to the rate of decay of that already produced. (c) In many cases the active products possess well defined chemical pro- perties different from those of the parent elements. ( d ) In some cases the new products, e.g.of thorium and rhodium, have properties of inert gases of high molecular weight. (e) Radioactive change is accompanied by an emission of heat of quite a different order of magnitude from that observed in ordinary chemical reactions. Revolutionary though it was, Rutherford’s theory was accepted with relative ease though naturally there were one or two sceptics who were at first uncon- vinced. One such was Lord Kelvin but as Lord Rayleigh relates, he soon had to admit the soundness of Rutherford ideas : With some lack of proper respect and deference, I asked Lord Kelvin if he would bet with me that within three to six months he would admit that Rutherford was right. Within the time Kelvin- came round, and at a meeting of the British Association he made a public pronouncement in favour of the internal origin of radium energy. The next time I saw him he came up to me and at once said ‘I think I owe you five shillings.Here it is’. By 1906 Rutherford and Boltwood concluded that lead was the end of the radioactive series of uranium. A glimpse of the social side of life at McGill University has been passed on by some of his coworkers and gives an interesting side light. Life at the Canadian university was not always entirely congenial, especially to Otto Hahn and his German colleague Max Levin who found that the meals were insubstantial by German standards and-even worse-that drinking was frowned upon by Rutherford. Consequently, a weekly visit to the local hotel for a large dinner and a bottle of beer was an absolute necessity for them.Fortunately Rutherford was a great pipe smoker, even to the extent of often cragg 135 having to borrow a pipe from Hahn when he mislaid his own, so Hahn was able to indulge that particular weakness in himself without restriction. However, there were occasions when all pipes and cigarettes had to be extinguished and all traces of smoke eliminated; this was an essential prepara- tion for a visit to the laboratory of MacDonald who had given so much of his money made from the tobacco trade to the University but who, ironically, (Cavend ish Laboratory) Rutherford and Ratcliffe. R. I. C. Re views 136 found smoking a detestable habit and would not have been impressed by the pipe-sucking Rutherfordian group.During his time at Montreal Rutherford had many honours bestowed on him. In 1903 he was elected a Fellow of the Royal Society and in 1904 was awarded the Rumford medal. In 1907 he resigned his professorship at McGill and accepted the chair of physics at Manchester University. MANCHESTER 1907-1 9 19 Rutherford’s stay in Manchester was to prove a most happy and scientifically successful period of his life. Professor Schuster, the previous professor of physics, had decided to retire early in order to continue his research indepen- dently and Rutherford was appointed on a salary of El600 per annum. In passing, it is perhaps interesting to note that his nine graduate members of staff shared exactly the same sum p a between them! The laboratory in Manchester had been built just seven years earlier; it had been designed by Schuster and was well equipped, including amongst its apparatus a liquid air machine.Rutherford seemed to attract coworkers of high calibre with great facility; those who collaborated with him in Manchester included men such as Andrade, Bohr, Boltwood, Chadwick, Darwin, Fajans, Geiger, Hevesy, and Marsden. Rutherford was still interested in the nature of the a-particle and one of the first experiments carried out in Manchester, with Royds, was concerned with the identification of the a-particle. The apparatus used consisted of a glass tube blown so thin that &-particles easily penetrated its walls, although the apparatus was gas tight.It was filled with radon and then surrounded by a second glass tube which was highly evacuated. It was shown that helium accumulated in the tube with the passage of time and had therefore been filled as a result of particles entering the tube. In 1908 Rutherford was awarded the Nobel Prize, not for physics but for chemistry. He entitled his lecture, ‘The Chemical Nature of the a-Particles from Radioactive Substances’. Rutherford’s successful relationship with students and collaborators must be ascribed to h s enviable knack of keeping people happy. Certainly his laboratory was a very cheerful place, as may be recognized from the following extract :2 I am sure that the laboratory tea-table, situated in the radioactive training laboratory, was far from being the least important bench in the laboratory. Rutherford provided tea and biscuits every day, and nearly always attended himself, sitting at the table, with the rest of us perched on stools and the neighbouring benches.It was a period of relaxation and general gossip, but the meeting often resolved itself into an informal colloquium, with Ruther- ford taking rather more than a chairman’s part. It was here, too, and in the frequent hospitality enjoyed by the research men in his own house, that Rutherford’s essential friendliness was most apparent. He had in a very high degree that friendly and companionable spirit which is a notable characteristic of people who have spent much of their lives either in New Zealand or Canada.This undoubtedly had its effect in keeping the labora- 131 tory working as a more than commonly united family. Rutherford had, moreover, what is not typical either of England or of any Dominion, a curious insensitivity to differences of academic or social standing, within fairly wide limits, and it probably was very good for the morale of a young research student to see himself treated with as much-or as little-respect as an emeritus professor. During the tea break Rutherford loved chaffing his younger research students. Russell gives one illustration :3 As I was the only Scot amongst them I came in for a good deal of chaff on the real or supposed foibles of Scotsmen. He thought them an over- scholarshipped and over-praised lot.‘You young fellows come down here from across the border with such testimonials written by your Scots professors that, why man alive! if Faraday or Clerk Maxwell were compet- ing against you they wouldn’t even get on to the short list’. The students at Manchester were kept abreast with current developments of science in Friday afternoon physics colloquia.2 The meetings, at which we were joined by large numbers of chemists and mathematicians who came to hear Rutherford, were preceded by an enormous tea-party, generally presided over by Lady Rutherford. Ruther- ford always addressed the first meeting of the session, giving with obvious enjoyment a summary of the main researches carried out in the laboratory during the preceding year.I am fairly certain that this inaugural address was always called the ‘The progress of physics, 1907-1908’ or whatever the session might be, and that the choice of topics was always made in the same way-or, as we should have said a little later, by the same selection principle. I am quite certain that in the atmosphere of sustained and justifiable enthusiasm which Rutherford created at these meetings, no young Man- Chester student could fail to feel that he was a member of a highly privileged community. I think we all felt that we were living very near the centre of the scientific universe, and maturer consideration has only strengthened my conviction that we were right. The sense of privilege naturally grew stronger in those who went on to do research, even in those of us who were doing relatively humble tasks, and not, like Moseley, things that were going to matter fundamentally in science for years to come.Although a relaxed and cheerful working atmosphere helps in research, the final consideration must be the viability of the research problem itself and again Rutherford was unfailingly successful. He once said he had never given a student a dud problem! Napoleon is reported to have once said, ‘There are no bad soldiers, only bad generals’. Rutherford might have adapted this remark to some of his colleagues (and I think he certainly would have, if he had thought of it, for he had a sharp tongue particularly in the dark room while counting scintillations), ‘There are no bad research students, only bad professors’.4 It is no wonder that with the quality of his collaborators the work of the Manchester group became known throughout the world.R. I. C. Re views 138 In 1908 Rutherford redetermined Avogadro’s number by counting the number of a-particles emitted by a known mass of radium in a given time. Although the value he obtained for Avogadro’s number was 40 per cent different from the known value at the time it was very close to the value which is accepted today. Rutherford with Geiger developed a device to measure a-particles. The work was based on an observation Rutherford made whilst in Canada, namely, that during deflection experiments on a-particles the photographic records had sharply defined edges when the containing vessel was highly evacuated but became broader, and the edges more diffuse, in the presence of air.He asked Geiger and Marsden to determine whether a-particles were scattered through large angles. This research was to produce results which gave an important insight into atomic structure. Rutherford records the occasion :2 I agreed with Geiger that young Marsden, whom he had been training in radioactive methds, ought to begin a research. Why not let him see if any a-particles can be scattered through a large angle? I did not believe that they would be.. . . It was quite a most incredible event that has ever happened to me in my life. It was almost as incredible as if you had fired a 15 in shell at a piece of tissue-paper and it came back and hit you.Rutherford concluded that in order to deflect an a-particle through a large angle the positive charge in the atom had to be concentrated in a small space. Thus, in a very simple way Rutherford had laid the foundations of the nuclear theory of the atom. Soon after, Niels Bohr, working with Rutherford in Manchester, applied Plank’s quantum theory to Rutherford’s atom and showed that, with certain assumptions, the spectrum of hydrogen could be calculated. Rutherford’s work in Manchester was much disrupted during the war years and it was not until 1919 that he succeeded in the transmutation of eIements- hitherto the dream of the alchemist. He discovered that some light atomic nuclei could be changed by the impact of energetic a-particles from radio- active substances.His findings were reported in the Philosophical magazine (1919, 37, 581). . . . From the results so far obtained it is difficult to avoid the conclusion that the long-range atoms arising from collision of a-particles with nitrogen are not nitrogen atoms but probably atoms of hydrogen, or atoms of mass two. If this be the case, we must conclude that the nitrogen atom is dis- integrated under the intense forces developed in a close collision with a swift a-particle, and that the hydrogen atom which is liberated formed a constituent part of the nitrogen nucleus. . . . Considering the enormous intensity of the forces brought into play, it is not much a matter of surprise that the nitrogen atom should suffer disintegration as that the a-particle itself escapes disruption into its constituents.The results as a whole support that, if a-particles-or similar projectiles-of still greater energy were avail- able for experiment, we might expect to breakdown the nucleus structure of many of the lighter atoms. cram 139 Whilst at Cambridge, Rutherford again gathered around him a large number of collaborators-people such as Blackett, Chadwick, Cockcroft, Ellis, Kapitza and Oliphant. He continued his researches into the disintegration of light elements and in his Bakerian lecture to the Royal Society (in 1920) announced that he had proved the long range particles to be hydrogen nuclei. In the same year he added the word ‘proton’ to our scientific vocabulary.In 1925 he became president of the Royal Society and was appointed to the Order of Merit. In 1931 as a mark of his outstanding contributions to science he was raised to the peerage, taking the title Baron Rutherford of Nelson. Rutherford’s apparatus t o show the disintegration of nitrogen by a-particles. (Cavendish Laboratory) The results of his experiments can be summed up in the following equation : + :H I$N + $H + ‘:O In April 1919 Rutherford succeeded Thomson as Cavendish Professor of Experimental Physics at Cambridge. CAMBRIDGE 19 19-1 937 His maiden speech given in the House of Lords in May 1931 on the subject, ‘Oil from Coal’, contained much argument that could still be of importance today. It is important to bear in mind that if our oil supplies are at any time sud- denly to be cut short, the greater part of our transport services, the ships etc.would soon become immobile. It is obvious, therefore, that it is of great R.I.C. Reviews 1 40 The Cavendish Laboratory. importance to consider carefully the question of the production of oil in this country so as to be independent of other countries. It is in the national interest that researches into the utilization of coal for the production of oil should be vigorously prosecuted. Rutherford, like any other person of distinction, found his time taken up in many spheres of public life leaving him less than he would like for his researches. Nevertheless his students did not suffer. Although he showed little interest in cvagg 141 the laborious setting up of apparatus with which students began their research course, he always found time to visit them once results arose and discussed their work with genial enthusiasm.He never gave a new student a technically difficult problem as he believed that however capable a person was he should have the incentive of early success. To those students who had to read papers to the Royal Society he gave the following sound advice: Don’t show too many slides. When it is dark in the lecture room some of the audience take the opportunity to leave! He was obviously greatly respected and admired by his students who in their enthusiastic after-dinner gatherings of the Cavendish Society glorified his achievements in songs:5 1 An alpha ray was I, contented with my lot; From Radium C I was set free And outwards I was shot.My speed I quickly reckoned, As I flew through space, Ten thousand miles per second Is not a trifling pace! For an alpha ray Goes a good long way In a short time t, As you easily see; (Cavendish Laboratory) Rutherford’s table at the Cavendish. R.I.C. Reviews 142 Though I don’t know why My speed’s so high, Or why I bear a charge 2e. 2 And in my wild career, as swiftly on I flew, A rarefied gas wouldn’t let me pass, But I pushed my way right through. I had some lively tussles To make it ionize, But I set the small corpuscles A-buzzing round like flies. For an alpha ray Hasn’t time to stay While a trifling mass Of expanded gas, That stands in awe, Of Maxwell’s law, Obstructs the road when I want to pass.3 An electroscope looked on, as I made that gas conduct; Beneath the field the gas did yield And the leaf was greatly ‘bucked’. But in my exultation I lost my senses clean, And I made a scintillation As I struck a zinc blende screen For an alpha ray Makes a weird display With fluorescence green On a zinc blende screen, When the room’s quite dark, You see a spark That marks the spot where I have been. 4 But now I’ve settled down, and move about quite slow: For I alas, am helium gas Since I got that dreadful blow. But though I’m feeling sickly Still no one now denies, That I ran that race so quickly, I’ve won a Nobel Prize.For an alpha ray Is a thing to pay And a Nobel Prize, One cannot despise, And Rutherford Has greatly scored, As all the world now recognize. Although during his time at the Cavendish Laboratory Rutherford was not directly involved with any major discoveries, it was in the Cavendish, in 1932, cram 143 I0 that Cockcroft and Walton transformed lithium atoms into helium by bombarding them with protons accelerated through 600 000 volts. Throughout his life Rutherford wished only to solve the problem because it was there and the attainment of the solution gave him complete satisfaction. The commercial exploitation of his successes was of no interest to him. In connection with Rutherford’s views on industry I remember a conversa- tion I had with him during a high table dinner at Trinity College.. . I was telling my neighbour that every great scientist must be to some extent a madman. Rutherford overheard this conversation and asked me, ‘In your opinion, Kapitza, am I mad too?’ (Royal Institution) A Friday evening discourse at the Royal Institution in 1934. As Kapitza relates :6 ‘Yes, Professor,’ I replied. R.I.C. Reviews 144 LORD ERNEST RUTHERFORD OF NELSON (1871-1937)R. H. Cragg, BSC, PhDThe Chemical Laboratory, University of Kent at Canterbury, Canterbury, KentIntroduction . . . . . . . . . . . . .. . . 129New Zealand 1871-95 . . . . . . . . . . . . . . 129Cambridge 1895-98 . . . . . . . . . . . . . . 130Montreal 1898-1907 . . . . . . . I . . . . . . 133Manchester 1907-1919 .. . . . . . . . . . . . . 137Cambridge 1919-1937 . . . . . . . . . . . . . . 140References . . . . . . . . . . . . . . .. .. 145INTRODUCTIONThe centenary of the birth of Lord Rutherford falls this year. By many he isconsidered the most outstanding scientist since Faraday and equal in impor-tance to Newton. Certainly it would be difficult to envisage the state of sciencetoday without the great contributions he made towards its progress, particu-larly his work on the structure of the atom and the transmutation of theelements. Much has been written on Rutherford and for those who wish topursue the subject further some useful sources of reference can be found at theend of the review.NEW ZEALAND 1871-1 895The environment in which Rutherford grew up must, doubtless, have been afactor in the development of his character and intelligence.Nineteenth-centuryNew Zealand was a young country, demanding from its people great stamina,and both mental and practical resourcefulness; life was tough but the pioneer-ing spirit was strong and the emergence from such a background of a scientistof Rutherford’s distinction is hardly surprising,Rutherford’s father had emigrated to New Zealand from Scotland with hisparents as a child. He had married in 1866 and Ernest, the fourth of 12children, was born on 30 August 1871 at Spring Grove (later called Bright-water), near Ne1son.l The family moved to Foxhill in 1875, to Havelock in 1882and finally settled in Pungarehu in 1886. Rutherford’s father was mostlyemployed in agricultural work ; he eventually concentrated on flax-growingand became something of an expert in developing his own varieties of flax.Inaddition to his flax-growing, he supplemented the family income by dealing intimber and notably by obtaining a government contract to supply 40000railway sleepers at 2s. 8d. each for the newly developing rail system.Although everyday life in New Zealand was primitive and spartan in theextreme, the importance of giving children a sound education had beenrecognized and there was no shortage of good schools. Rutherford attendedCrwg 12the state primary school at Brightwater, moving at 11 to Havelock primaryschool whence, at the age of 15, he won a scholarship to Nelson College.From early childhood Rutherford was a voracious reader, apparentlyfavouring Dickens especially.He was inquisitive and inventive and obviouslydestined to be a scientist from the day he made his own gunpowder for firingmarbles from a toy cannon-having first modified the cannon to increase itsrange! Later he made his own camera and enjoyed photography as a hobby.Nevertheless, however engrossing the pursuit of his own interests, he was aconscientious and dutiful member of the family, willingly attending to therunning and repairing of the water wheels in the mill attached to their home.As one might expect, he loved the countryside and open air and throughouthis life maintained that physical fitness and mental alertness were closelyallied. He went hop picking in the summer and bathing (once almost fatallyfor he was unable to swim).Typically, when the news of his scholarship toCanterbury College, Christchurch, came, he was out in the garden digging uppot at oes.So Rutherford’s early life gives some insight into his later development. Thestresses of his environment called upon his natural intelligence to assert itselfand form a basis for the maturing of his academic capabilities.He entered Canterbury College, Christchurch, in 1889 and, after graduatingin 1893, remained to do research work. His laboratory was a cold, draughtycellar which was used as a cloakroom by the students but, despite such incon-venience, he pursued his researches wholeheartedly. Here he developed a newmethod for the detection of radio waves; having constructed his own detectorfrom fine sewing needles, he initially demonstrated his technique to anaudience of professors and students of the University, by transmitting a wirelessmessage through the laboratory.Some time later he was to use his newlydeveloped technique to detect the sparks from an induction coil at a distance oftwo miles.Success for Rutherford was equated purely with his own personal satisfac-tion and he ignored the commercial possibilities of discoveries such as these.Some years later Marconi made a considerable fortune by perfecting atechnique for radio reception.At this time also Rutherford demonstrated that, contrary to the theory ofhis contemporaries, high frequency currents from a small Tesla coil wouldstrongly magnetize iron.In 1894 he received an MA with a double first in Mathematics and Physics.He was awarded a scholarship by the Commissioners of the 1851 Exhibitionwhich gave him the opportunity of coming to England to work with J.J.Thomson at the Cavendish Laboratory in Cambridge. So he set sail forEngland in 1895--significantly, the year in which Rontgen discovered x-rays.CAMBRIDGE 1895-1 898Rutherford was fortunate in that the regulations at Cambridge had justchanged to allow graduates from other Universities to read for a BA degreeafter completing two years’ residence, and he was the first student to applyunder these new regulations.130 R. I.C. ReviewLord Ernest Rutherford of Nelson. (Cavendish LaboratoryHis early impressions of life in England compared to that in New Zealandwere given in his letters to his mother.Having met the English agriculturallabourers in his walks near Cambridge, he wrote: ‘You can’t imagine how slow-moving, slow-thinking the English villager is. He is very different to anythingone gets hold of in the colonies’. An interesting observation which demon-strates how different was the social class of the exploited English farmlabourer from that of his respected, often wealthy, New Zealand counterpart.He observed also, regarding some of the academics he met at dinner, thatCw?g 13Rutherford a t Manchester.they were ‘very capable, especially in their conversation, and it is a pity somany of them fossilize as it were in Cambridge, and are not that use in theworld they might be’.Rutherford continued his work on the detection of wireless waves sinceThomson believed that it was good for a research student to work on a topicof his own choosing.After perfecting his method he was soon able to detect inhis lodgings, signals which were transmitted from the laboratory-a distanceof half a mile. Thomson gave an account of this work called ‘The Magnetisa-tion of Iron by High-Frequency Discharges and the Investigation of the Effecton Short Steel Needles’, at the Royal Society on 18 June 1896.In 1896 Becquerel discovered that the element uranium could give out aradiation which affected a photographic plate. It was at this time also thatRutherford changed his area of research. He began by studying the ionizingeffect of x-rays and showed that x-rayed gases lost their conductivity whenpassed through porous plugs; this indicated that the ions responsible for theconductivity had been filtered out.After more work he concluded that x-raysproduced +vely and -vely charged ions in the gas.He then considered the nature of the rays emitted from uranium. It wasfound that crystals of potassium uranium sulphate made an impression on aphotographic plate both when the experiment was done in the light and in thedark. Rutherford placed some of the uranium crystals between the poles of amagnet and positioned a photographic plate above a slit above the magnet.132 R.I.C. ReviewOn developing the plate and finding that two images appeared Rutherfordrealized that the emission from the uranium consisted of at least two types-one which had been deflected by the magnet and the other which had onlybeen affected slightly.After further work he showed that they must beparticles of matter of different weights. He then concluded that the radiationconsisted of a radiation, which was easily stopped by a sheet of thick paperbut had a high ionizing effect, and j3 radiation, which was more penetratingbut caused less ionization. For his work on uranium, Rutherford was awardedthe Coutts-Trotter Scholarship of Trinity College.Early in 1898 the chair in physics at McGill University became vacant andthe Director of the MacDonald Physics Laboratory there, John Cox, visitedEngland and approached Thomson for advice.On Thomson’s recommenda-tion Rutherford, still only 28, was appointed professor and set sail for Canadain September 1898. Cox decided to take on the entire administration of thelaboratory himself, thus allowing Rutherford to concentrate all his efforts onresearch. Cox could hardly have realized at the time how important thisarrangement was to be, for the success that Rutherford made in Canada wasnever to be equalled.MONTREAL 1898-1 907Rutherford found the physics laboratory at McGill University extremely wellequipped from a generous endowment given to the University by Sir WilliamMacDonald. Soon after arriving he began researches which were eventually toprove successful. He was the kind of man who was only happy working with agroup of people and in his early days in Canada, before he had establishedsuch a group, he often became dispirited-on one occasion he almost decidedto give up and apply for the Chair of Natural Philosophy which was thenvacant at Edinburgh.However, he shortly gathered about him some very ablecollaborators and during his nine years in Montreal publishedaround 80papers.He was offered chairs at many American universities and also the chair atUniversity College London, but although the salaries they carried were farabove the modest one he earned at McGill he turned them down.When at Cambridge Rutherford had discovered that thorium nitrateappeared to be an unusual source of radiation, and in 1898 he set R. B. Owensthe problem of studying the radiations from thorium for his research topic.Itwas found that the radiations appeared to be affected by draughts of air. In 1899Rutherford found that thorium compounds emitted radioactive particles con-tinuously and that these particles retained their radioactive properties forseveral minutes. He then studied the properties of the emanation and foundthat the particles were not charged, not affected in an electric field andcould pass through cotton wool. On the other hand the emanated particlescould pass through thin layers of metal and ionize the neighbouring gas.Rutherford concluded that the emanation acted as a gas. He had thereforeestablished that radioactive substances could emit not only radiation but alsoparticles, which had the properties of atoms of an ordinary gas.He alsodemonstrated that, when a solution of a thorium compound was evaporatedto dryness on a glass surface, the apparently invisible residue on the glasssurface was very radioactive.13Rutherford aged 36.In 1900 Rutherford returned to New Zealand to marry his fiancke, MaryNewton. He brought her back to Canada with him and the following yearthey had a daughter.Frederick Soddy joined Rutherford and investigated the properties of manytypes of emanation but he was unable to obtain any chemical reactions andconcluded, in 1902, that the emanation was a chemically inert gas analogousin nature to the members of the argon family.Further work on thorium showed that new radioactive substances werebeing produced and also that all different radioactive substances decayedaccording to the same law and the rate of decay was proportional to the amountof substance. Next it was demonstrated that 01 rays could be deflected by bothelectric and magnetic fields, that they were positively charged with a velocity of2% of light and had a mass similar to the hydrogen atom.Rutherford concluded that Q! rays were atomic fragments which were beingemitted from exploding atoms and that ,8 rays were of secondary importance.Using a liquid air machine, he and Soddy were able to liquify the emanationand by 1905 Rutherford was convinced that 01 particles were helium atomscontaining two positive charges.During this time Rutherford was joined by Hahn who had been workingwith Ramsey.For some reason Rutherford was rather sceptical of some ofRamsey’s work on radioactivity but Hahn, a very capable person, very quicklyshowed the quality of Ramsey’s work by demonstrating to Rutherford thatradiothorium existed. He then studied the Q! particle emission from radio-134 R . I. C. Reviewthorium and radioactinium and showed that in his samples of radiothoriumthere existed a second a-emitting substance called thorium C[iiPo]. As allattempts to separate thorium C were unsuccessful he came to the conclusionthat its half life was short. This is not surprising as t+ was found to beIn 1902 Rutherford and Soddy published their theory that radioactivity is aphenomenon accompanying the spontaneous transformation of the atoms ofradioactive elements into different kinds of matter.Rutherford’s first book(Radioactivity) was published in 1904 and contained the basis of his theory ofradioactivity. His theory was based on the following experimental evidence :(a) Radioactivity is unaffected by any change in external conditions whetherby extreme heat or cold, or by the action of any chemical reagent. As itdiffers from known chemical reactions the process must be atomic innature.(6) The radioactivity of uranium, thorium and radium is maintained by theproduction at a constant rate of new kinds of matter which themselvespossess temporary radioactivity. The constant activity of the radio elementsis due to a state of equilibrium where the rate of production of new matter isequal to the rate of decay of that already produced.(c) In many cases the active products possess well defined chemical pro-perties different from those of the parent elements.( d ) In some cases the new products, e.g.of thorium and rhodium, haveproperties of inert gases of high molecular weight.(e) Radioactive change is accompanied by an emission of heat of quite adifferent order of magnitude from that observed in ordinary chemicalreactions.3 x 10-7s.Revolutionary though it was, Rutherford’s theory was accepted with relativeease though naturally there were one or two sceptics who were at first uncon-vinced. One such was Lord Kelvin but as Lord Rayleigh relates, he soon hadto admit the soundness of Rutherford ideas :With some lack of proper respect and deference, I asked Lord Kelvin if hewould bet with me that within three to six months he would admit thatRutherford was right.Within the time Kelvin- came round, and at ameeting of the British Association he made a public pronouncement infavour of the internal origin of radium energy. The next time I saw him hecame up to me and at once said ‘I think I owe you five shillings. Here it is’.By 1906 Rutherford and Boltwood concluded that lead was the end of theradioactive series of uranium.A glimpse of the social side of life at McGill University has been passed onby some of his coworkers and gives an interesting side light. Life at theCanadian university was not always entirely congenial, especially to OttoHahn and his German colleague Max Levin who found that the meals wereinsubstantial by German standards and-even worse-that drinking wasfrowned upon by Rutherford.Consequently, a weekly visit to the local hotelfor a large dinner and a bottle of beer was an absolute necessity for them.Fortunately Rutherford was a great pipe smoker, even to the extent of oftencragg 13Rutherford and Ratcliffe. (Cavend ish Laboratory)having to borrow a pipe from Hahn when he mislaid his own, so Hahn wasable to indulge that particular weakness in himself without restriction.However, there were occasions when all pipes and cigarettes had to beextinguished and all traces of smoke eliminated; this was an essential prepara-tion for a visit to the laboratory of MacDonald who had given so much of hismoney made from the tobacco trade to the University but who, ironically,136 R.I. C. Re viewfound smoking a detestable habit and would not have been impressed by thepipe-sucking Rutherfordian group.During his time at Montreal Rutherford had many honours bestowed onhim. In 1903 he was elected a Fellow of the Royal Society and in 1904 wasawarded the Rumford medal. In 1907 he resigned his professorship at McGilland accepted the chair of physics at Manchester University.MANCHESTER 1907-1 9 19Rutherford’s stay in Manchester was to prove a most happy and scientificallysuccessful period of his life. Professor Schuster, the previous professor ofphysics, had decided to retire early in order to continue his research indepen-dently and Rutherford was appointed on a salary of El600 per annum.Inpassing, it is perhaps interesting to note that his nine graduate members of staffshared exactly the same sum p a between them! The laboratory in Manchesterhad been built just seven years earlier; it had been designed by Schuster andwas well equipped, including amongst its apparatus a liquid air machine.Rutherford seemed to attract coworkers of high calibre with great facility;those who collaborated with him in Manchester included men such asAndrade, Bohr, Boltwood, Chadwick, Darwin, Fajans, Geiger, Hevesy, andMarsden.Rutherford was still interested in the nature of the a-particle and one of thefirst experiments carried out in Manchester, with Royds, was concerned withthe identification of the a-particle.The apparatus used consisted of a glasstube blown so thin that &-particles easily penetrated its walls, although theapparatus was gas tight. It was filled with radon and then surrounded by asecond glass tube which was highly evacuated. It was shown that heliumaccumulated in the tube with the passage of time and had therefore been filledas a result of particles entering the tube.In 1908 Rutherford was awarded the Nobel Prize, not for physics but forchemistry. He entitled his lecture, ‘The Chemical Nature of the a-Particlesfrom Radioactive Substances’.Rutherford’s successful relationship with students and collaborators mustbe ascribed to h s enviable knack of keeping people happy. Certainly hislaboratory was a very cheerful place, as may be recognized from the followingextract :2I am sure that the laboratory tea-table, situated in the radioactive traininglaboratory, was far from being the least important bench in the laboratory.Rutherford provided tea and biscuits every day, and nearly always attendedhimself, sitting at the table, with the rest of us perched on stools and theneighbouring benches.It was a period of relaxation and general gossip, butthe meeting often resolved itself into an informal colloquium, with Ruther-ford taking rather more than a chairman’s part. It was here, too, and in thefrequent hospitality enjoyed by the research men in his own house, thatRutherford’s essential friendliness was most apparent. He had in a veryhigh degree that friendly and companionable spirit which is a notablecharacteristic of people who have spent much of their lives either in NewZealand or Canada. This undoubtedly had its effect in keeping the labora-13tory working as a more than commonly united family.Rutherford had,moreover, what is not typical either of England or of any Dominion, acurious insensitivity to differences of academic or social standing, withinfairly wide limits, and it probably was very good for the morale of a youngresearch student to see himself treated with as much-or as little-respect asan emeritus professor.During the tea break Rutherford loved chaffing his younger researchAs I was the only Scot amongst them I came in for a good deal of chaff onthe real or supposed foibles of Scotsmen.He thought them an over-scholarshipped and over-praised lot. ‘You young fellows come down herefrom across the border with such testimonials written by your Scotsprofessors that, why man alive! if Faraday or Clerk Maxwell were compet-ing against you they wouldn’t even get on to the short list’.The students at Manchester were kept abreast with current developments ofThe meetings, at which we were joined by large numbers of chemists andmathematicians who came to hear Rutherford, were preceded by anenormous tea-party, generally presided over by Lady Rutherford. Ruther-ford always addressed the first meeting of the session, giving with obviousenjoyment a summary of the main researches carried out in the laboratoryduring the preceding year.I am fairly certain that this inaugural address wasalways called the ‘The progress of physics, 1907-1908’ or whatever thesession might be, and that the choice of topics was always made in the sameway-or, as we should have said a little later, by the same selection principle.I am quite certain that in the atmosphere of sustained and justifiableenthusiasm which Rutherford created at these meetings, no young Man-Chester student could fail to feel that he was a member of a highly privilegedcommunity. I think we all felt that we were living very near the centre of thescientific universe, and maturer consideration has only strengthened myconviction that we were right. The sense of privilege naturally grew strongerin those who went on to do research, even in those of us who were doingrelatively humble tasks, and not, like Moseley, things that were going tomatter fundamentally in science for years to come.Although a relaxed and cheerful working atmosphere helps in research, thefinal consideration must be the viability of the research problem itself andagain Rutherford was unfailingly successful.He once said he had never given a student a dud problem! Napoleon isreported to have once said, ‘There are no bad soldiers, only bad generals’.Rutherford might have adapted this remark to some of his colleagues (andI think he certainly would have, if he had thought of it, for he had a sharptongue particularly in the dark room while counting scintillations), ‘Thereare no bad research students, only bad professors’.4It is no wonder that with the quality of his collaborators the work of theManchester group became known throughout the world.138 R.I. C. Re viewsstudents. Russell gives one illustration :3science in Friday afternoon physics colloquia.In 1908 Rutherford redetermined Avogadro’s number by counting thenumber of a-particles emitted by a known mass of radium in a given time.Although the value he obtained for Avogadro’s number was 40 per centdifferent from the known value at the time it was very close to the value whichis accepted today. Rutherford with Geiger developed a device to measurea-particles.The work was based on an observation Rutherford made whilst in Canada,namely, that during deflection experiments on a-particles the photographicrecords had sharply defined edges when the containing vessel was highlyevacuated but became broader, and the edges more diffuse, in the presence ofair.He asked Geiger and Marsden to determine whether a-particles werescattered through large angles.This research was to produce results whichgave an important insight into atomic structure. Rutherford records theoccasion :2I agreed with Geiger that young Marsden, whom he had been training inradioactive methds, ought to begin a research. Why not let him see if anya-particles can be scattered through a large angle? I did not believe thatthey would be.. . . It was quite a most incredible event that has everhappened to me in my life. It was almost as incredible as if you had fired a15 in shell at a piece of tissue-paper and it came back and hit you.Rutherford concluded that in order to deflect an a-particle through a largeangle the positive charge in the atom had to be concentrated in a small space.Thus, in a very simple way Rutherford had laid the foundations of thenuclear theory of the atom.Soon after, Niels Bohr, working with Rutherfordin Manchester, applied Plank’s quantum theory to Rutherford’s atom andshowed that, with certain assumptions, the spectrum of hydrogen could becalculated.Rutherford’s work in Manchester was much disrupted during the war yearsand it was not until 1919 that he succeeded in the transmutation of eIements-hitherto the dream of the alchemist. He discovered that some light atomicnuclei could be changed by the impact of energetic a-particles from radio-active substances.His findings were reported in the Philosophical magazine(1919, 37, 581).. . . From the results so far obtained it is difficult to avoid the conclusionthat the long-range atoms arising from collision of a-particles with nitrogenare not nitrogen atoms but probably atoms of hydrogen, or atoms of masstwo. If this be the case, we must conclude that the nitrogen atom is dis-integrated under the intense forces developed in a close collision with aswift a-particle, and that the hydrogen atom which is liberated formed aconstituent part of the nitrogen nucleus. . . . Considering the enormousintensity of the forces brought into play, it is not much a matter of surprisethat the nitrogen atom should suffer disintegration as that the a-particleitself escapes disruption into its constituents. The results as a whole supportthat, if a-particles-or similar projectiles-of still greater energy were avail-able for experiment, we might expect to breakdown the nucleus structure ofmany of the lighter atoms.cram 13Rutherford’s apparatus t o show the disintegration of nitrogen by a-particles.(Cavendish Laboratory)The results of his experiments can be summed up in the following equation :I$N + $H + ‘:O + :HIn April 1919 Rutherford succeeded Thomson as Cavendish Professor ofExperimental Physics at Cambridge.CAMBRIDGE 19 19-1 937Whilst at Cambridge, Rutherford again gathered around him a large numberof collaborators-people such as Blackett, Chadwick, Cockcroft, Ellis, Kapitzaand Oliphant.He continued his researches into the disintegration of lightelements and in his Bakerian lecture to the Royal Society (in 1920) announcedthat he had proved the long range particles to be hydrogen nuclei. In the sameyear he added the word ‘proton’ to our scientific vocabulary.In 1925 he became president of the Royal Society and was appointed to theOrder of Merit. In 1931 as a mark of his outstanding contributions to sciencehe was raised to the peerage, taking the title Baron Rutherford of Nelson.His maiden speech given in the House of Lords in May 1931 on the subject,‘Oil from Coal’, contained much argument that could still be of importancetoday.It is important to bear in mind that if our oil supplies are at any time sud-denly to be cut short, the greater part of our transport services, the ships etc.would soon become immobile.It is obvious, therefore, that it is of great1 40 R.I.C. ReviewThe Cavendish Laboratory.importance to consider carefully the question of the production of oil in thiscountry so as to be independent of other countries.It is in the national interest that researches into the utilization of coal forthe production of oil should be vigorously prosecuted.Rutherford, like any other person of distinction, found his time taken up inmany spheres of public life leaving him less than he would like for his researches.Nevertheless his students did not suffer. Although he showed little interest incvagg 14the laborious setting up of apparatus with which students began their researchcourse, he always found time to visit them once results arose and discussedtheir work with genial enthusiasm.He never gave a new student a technicallydifficult problem as he believed that however capable a person was he shouldhave the incentive of early success. To those students who had to read papersto the Royal Society he gave the following sound advice:Don’t show too many slides. When it is dark in the lecture room some of theaudience take the opportunity to leave!He was obviously greatly respected and admired by his students who intheir enthusiastic after-dinner gatherings of the Cavendish Society glorifiedhis achievements in songs:51 An alpha ray was I, contented with my lot;From Radium C I was set freeAnd outwards I was shot.My speed I quickly reckoned,As I flew through space,Ten thousand miles per secondIs not a trifling pace!For an alpha rayGoes a good long wayIn a short time t,As you easily see;Rutherford’s table at the Cavendish.(Cavendish Laboratory)R.I.C. Reviews 14Though I don’t know whyMy speed’s so high,Or why I bear a charge 2e.2 And in my wild career, as swiftly on I flew,A rarefied gas wouldn’t let me pass,But I pushed my way right through.I had some lively tusslesTo make it ionize,But I set the small corpusclesA-buzzing round like flies.For an alpha rayHasn’t time to stayWhile a trifling massOf expanded gas,That stands in awe,Of Maxwell’s law,Obstructs the road when I want to pass.3 An electroscope looked on, as I made that gas conduct;Beneath the field the gas did yieldAnd the leaf was greatly ‘bucked’.But in my exultationI lost my senses clean,And I made a scintillationAs I struck a zinc blende screenFor an alpha rayMakes a weird displayWith fluorescence greenOn a zinc blende screen,When the room’s quite dark,You see a sparkThat marks the spot where I have been.4 But now I’ve settled down, and move about quite slow:For I alas, am helium gasSince I got that dreadful blow.But though I’m feeling sicklyStill no one now denies,That I ran that race so quickly,I’ve won a Nobel Prize.For an alpha rayIs a thing to payAnd a Nobel Prize,One cannot despise,And RutherfordHas greatly scored,As all the world now recognize.Although during his time at the Cavendish Laboratory Rutherford was notdirectly involved with any major discoveries, it was in the Cavendish, in 1932,cram 143IA Friday evening discourse at the Royal Institution in 1934.(Royal Institution)that Cockcroft and Walton transformed lithium atoms into helium bybombarding them with protons accelerated through 600 000 volts.Throughout his life Rutherford wished only to solve the problem because itwas there and the attainment of the solution gave him complete satisfaction.The commercial exploitation of his successes was of no interest to him.As Kapitza relates :6In connection with Rutherford’s views on industry I remember a conversa-tion I had with him during a high table dinner at Trinity College.. . I wastelling my neighbour that every great scientist must be to some extent amadman. Rutherford overheard this conversation and asked me, ‘In youropinion, Kapitza, am I mad too?’‘Yes, Professor,’ I replied.144 R.I.C. Review‘How will you prove it?’ he asked.‘Very simply,’ I replied. ‘Maybe you remember a few days ago youmentioned to me that you had had a letter from the USA, from a bigAmerican company. In this letter they offered to build you a colossallaboratory in America and to pay you a fabulous salary. You only laughedat the offer and would not consider it seriously. I think you will agree withme that from the point of view of an ordinary man you acted like a mad-man!’ Rutherford laughed and said that in all probability I was right.Rutherford died at Cambridge on 19 October 1937, after a short illness, andwas buried in Westminster Abbey on 25 October. Sir James Jeans paid thefollowing tribute to him after his death:7Those of us who were honoured by his friendship know that his greatness asa scientist was matched by his greatness as a man. We remember, andalways shall remember, with affection his big energetic, exuberant person-ality, the simplicity, sincerity, and transparent honesty of his character, and,perhaps most of all, his genius for friendship and good comradeship.In his flair for the right line of approach to a problem, as well as in thesimple directness of his methods of attack, he often reminds us of Faraday,but he had two great advantages which Faraday did not possess-first,exuberant bodily health and energy, and second, the opportunity andcapacity to direct a band of enthusiastic coworkers. Great though Faraday’soutput of work was, it seems to me that to match Rutherford’s work inquantity, as well as in quality, we must go back to Newton. In somerespects he was more fortunate than Newton. Rutherford was ever thehappy warrior-happy in his work, happy in its outcome, and happy in itshuman contact.REFERENCES1 E. Rutherford, Dictionary of national biography (H. T. Tizard, ed.).2 H. R. Robinson, Rutherford: life and work to the year 1919, with personal reminiscences ofthe Manchester period, Rutherford Memorial Lecture 1942, Proc. phys. SOC., 1943, 55,161.3 A. S. Russell, Lord Rutherford: Manchester 1907-19, a partial portrait, RutherfordMemorial Lecture 1950, Proc. phys. Soc., 1951, MA, 217.4 P. M. S. Blackett, Memories of Rutherford, Rutherford Memorial Lecture 1954. (Ref. 7.)5 A. A. Robb, in Rutherford at Manchester, J. B. Birks (ed.). London: Heywood 1962.(Ref. 7.)6 P. L. Kapitza, Recollections of Lord Rutherford, Selected Lectures of the Royal Society,1967, 1, 131. London: Royal Society/Academic Press, 1967.7 Sir E. Marsden in Rutherford at Manchester (J. B. Birks ed.). London: Heywood, 1962.GENERAL REFERENCESThe collected papers of Lord Rutherford of Nelson, Sir James Chadwick (ed.), 3 vols.A. S. Eve, Rutherford. London : Cambridge University Press, 1939.N. Feather, Lord Rutherford. London & Glasgow: Blackie, 1940.I. B. N. Evans, Rutherford of Nelson. Harmondsworth: Penguin Books, 1943.J. G. Crowther in British scientists of the twentieth century, 43, 1952 and bibliographyLondon: Allen & Unwin, 1962.therein. London: Routledge & Kegan Paul, 1952.14

ISSN:0035-8940    

DOI:10.1039/RR9710400129

出版商:RSC    

年代:1971

数据来源: RSC

摘要:

RECENT ADVANCES IN THE CHEMISTRY OFNOBLE GAS ELEMENTS*N. K. JhaDepartment of Chemistry,l ndion Institute of Technology, New Delhi 29, lndiaSection 1 : New compounds . . , . . . . . .. . . 148Simple compounds, 148Complex compounds, 15 1New preparative methods, 157Structural and other physical studies, 159Formation of XeF2 and XeF4, 163Hydrolysis of XeF2, 164Other reactions of XeF2, 164Reactions of XeF4, 165Hydrolysis of XeF4, 166Reactions of XeO3, 166XeO3, 167Xe fluorides, 167Prolonging the lifetime of tungsten filaments, 167Separation of xenon and krypton, 167Section 2: Old compounds . . . . . . . . . . .. 157Section 3 : Kinetic studies and some other reactions . . . . .. 163Section 4: Applications . . . . .. .. . . . . .. 166Conclusion, .. . . . . . . . . . . . . . .. 167References . . . . .. .. .. . . . . .. .. 168Immediately after the synthesis of the first noble gas compound, XePtF6 byBartlettl in 1962, the chemistry of noble gases was explored very extensivelyand a remarkable amount of work was reported2 during the following year.Subsequent progress has not been as rapid as in the initial stages. Thoughthe ‘inert’ gases have been proved not to be chemically inert they are nobleand form compounds only with the electronegative elements oxygen andfluorine. Moreover, only xenon forms a large number of compounds;krypton is rather unreactive and the lighter members may still be regardedas inert. The chemistry of the radioactive element radon has only veryrecently begun to yield to experiment.Nevertheless, commendable progresshas been made in the structural and other physical studies of the noblegas compounds and their reactions with compounds of other elements. Manyreview papers and a few books have appeared since 1963, and a few recent onesare listed in the references.3--9The present review is concerned with the advances made in the chemistry of* A review of literature after December 1965.Jha 14noble gases since 1966. Section 1 is devoted to a discussion of new compounds,simple and complex. Section 2 includes a discussion of some of the simplecompounds reported during 1962-1 963 (designated ‘old’ here) ; only new pre-parative methods and advances made in their structural and physical studiesare considered.In Section 3 kinetic studies are reviewed and some newreactions of noble gas compounds are discussed. Section 4 deals with theapplications of these compounds.SECTION 1: NEW COMPOUNDSOf the new compounds reported the majority are complex. Some simplecompounds reported earlier without adequate evidence have now been charac-terized and are included here. The sub-section on complex compounds alsoincludes additional information relating to materials already reported.Simple compoundsMost of the progress made pertains to the xenon compounds only, but there islittle information about radon fluorides and predictions about the possibleformation and stability of argon and helium compounds have also been made.Xenon dichloride. It has been claimed that XeCl2 can be prepared by (i) inter-action of Xe and Clz at room temperature for a few days;l0 (ii) passing highfrequency discharge through a mixture of Xe, Fz and Sic14 (or cC14) at lowtemperatures;ll (iii) passing a mixture of Xe and Cl2 through a microwavedischarge;l2 and (iv) P-decay of 1291 incorporated into ICl2,13 129ICli +129XeClz.Of the four methods, the first is uncorroborated, while the fourth isgood only for providing evidence of the existence of XeC12.Xenon dichloride is said to be a white crystalline substance, which can bestored in a sealed glass tube for a long time and may be purified by sublimationat reduced pressure at room temperature.ll Its ir spectrum shows a structuredband at 319 cm-1.12 All the structures shown in the spectrum agree withcalculated values based upon a symmetrically linear XeC12.Xenon tetrachloride. This has been identified by Mossbauer spectroscopy fromthe /I-decay of 1291 incorporated into ICl;.l3 Like XeF4, the molecule is reportedto have a square planar configuration.Xenon dibromide.The reaction 1291Bri + 129XeBr2 has been observed in aMossbauer study.14 Based on the isomer shifts in the Mossbauer studies, theassumption of p a bonding in Xe di- and tetrahalides seems to be nearly valid.The charge per halogen atom calculated from the quadrupole splittings inthe Mossbauer spectra is: F, -0.72; CI, -0.52; and Br, -0.41.l3J4Xenon dzjluoride. More sophisticated experiments are being carried out toelucidate the structure of the xenon fluorides especially from the electronictransitions between various levels.One such experiment is the high resolutionHe1 and He11 photo electron spectra of XeF2.1g1 The spectra have been com-pared with the result of a more sophisticated calculation of the electron struc-ture of XeF2.192 Such spectra for XeF4 and XeFs are also being investigated.191148 R.I.C. ReviewXenon oxide dzjluoride. A colourless volatile material obtained by the reactionof Xe, 0 2 and F2 had been assumed to be XeOF2 and had not been fullycharacterized in 1963.l5 This reaction has been reinvestigated recently and ithas been shown that no stable oxide fluoride is produced and the materialobtained in the earlier work may have been the adduct XeFz.XeF4.16Xenon oxide difluoride had been deposited on to a CsI window at - 80 "Cby the interaction of XeF4 and H2O vapour sprays and was identified by an irstudy.17 Three bands at 747,520 and 490 cm-l were observed.The first band isattributed to Xe-0 and the latter two to Xe-F stretching by comparison of theXeOF2 spectrum with that of XeF2. The corresponding values for Xe-Fstretching in XeF2 are 567 and 496 cm-1.Xenon dioxide dzjluoride. Only mass spectroscopic evidence had been obtainedin 1963 for the existence of XeOzFz which was formed, along with XeOF4, inthe hydrolysis of XeF6.18 It has been found that XeOF4 (or XeF6) reactswith XeO3 to give a mixture of Xe02F2, XeFz and XeOF4.19 From thereaction mixture XeOzF2 may be purified by fractional distillation. Xenondioxide difluoride, XeOzF2, forms white crystals which melt at 30.8"C to acolourless liquid.It has a vapour pressure between that of XeO3 and XeOF4.Although XeO2F2 is more stable than XeO3, samples of XeOzF2 have beenfound to explode violently. With moist air rapid hydrolysis to Xe03 occurs.The first two ir and Raman bands at 537, 610, 851 and 882 cm-1 havebeen assigned to Xe-F and the last two to Xe-0 stretchings.lg The vibrationalspectra have been interpreted in terms of CzV molecular symmetry associatedwith a pseudo bipyramidal structure involving two axial F atoms, two 0 atomsand a lone pair of electrons in the equatorial position.20 It is interesting to notethat the structure is the same as that predicted by Gillespie21 on the basis ofvalence electron pair repulsion theory.Gillespie also predicted that Xe02F2may polymerize but this has not been found to occur.Xenon trioxide dzjluoride. Xenon trioxide difluoride, Xe03F2, has been gener-ated by the reaction of XeF6 with solid sodium perxenate at room temperatureand studied by mass spectroscopy.22 The volatility of Xe03F2 is similar tothat of Xe04 but is much greater than that of XeOzF2, indicating that Xe03F2is more symmetrical and probably non-polar. The shape of the molecule maybe trigonal bipyramidal with two axial F atoms and three equatorial 0 atoms,as predicted by Gillespie.21Xenon chloridejluoride. Recently, an attempt has been made to prepare xenonchloride fluorides by subjecting mixtures of Xe, Cl2 and F2 in ratios 1 : 1 : 1to 1 : 2 : 2 to high frequency excitation at 10 Torr or to uv irradiation atnormal pressure.The formation of xenon chloride fluorides was not observed,CIF3 and C1F5 were produced instead.23It has been proposed that XeCl2F2 might be stable at low temperatures andunder normal conditions it may decompose with splitting off of chlorine atomsto form XeF2 due to an intramolecular redox reaction, whereas a xenonchloride fluoride with hexavalent xenon must be thermodynamically unstableeven at low temperatures.24Jha 14Xenates and perxenates. A crystalline form of monocaesium xenate has beenprepared in appreciable amount by the action of aqueous XeO3 on aqueousCsOH in presence of fluoride i0ns.~5 It is stable in dry air and sparinglysoluble in ice-cold water.It shows the characteristic ir bands of the xenate ion.The earlier samples of monocaesium xenate and other mono-alkali metalxenates prepared were in the powder form.26A perxenate of americium has been reported.27 The reaction of solid XeF4with NaOH at 0-20 "C has been found to produce Na4XeO6.2H20, NaeXeO6.2HzO.yNaF ( y - 0.8) and Na4XeO6.2.5 NaF, according to the relativeproportions of the reactants. X-ray data indicate that Na4XeO6.2.5 NaF is achemical species and not a mixture of Na4XeO6 and NaF.28Helium compounds. Stable molecule ions containing He and F have beendetected29 by producing an ion beam of He,F$ in an electromagnetic separa-tor, retarding the ions to thermal velocities with the aid of a collector andidentifying the products by mass spectrometry and by their action on thecollector. The ion source was a 1 : 1 mixture of the reagents.The ions pro-duced from He and Fz were HeF+ and HeFi+; He and BF3 yielded HeF+ andHeF+ while He and RuF5 gave rise to HeFg+ ions only.Argon compounds. The feasibility of preparing argon compounds and theirpossible stability have been discussed in detail by Ferreira.30 Extrapolation ofX-F and X-0 bond energies for groups IVb, Vb, VIb and VIIb to 0 groupindicates that Ar-F bond should be as stable as Kr-F bond and that Ar-0bond should be more stable than Kr-0 bond. Ar-F and Ar-0 compoundsshould be thermodynamically as stable as Kr-F and Kr-0 compounds andtheir non-existence is to be related to kinetic considerations. Although activa-tion barriers should be about the same for Ar-Fz as for Kr-Fz reactions, thedecomposition mechanism may prove to be different.Theoretical investiga-tions have shown that ArFf may be sufficiently stable to allow the probableisolation of ArFPt+Fi.31Radon compounds. The possibility of preparing various radon compounds andtheir expected stabilities and formation enthalpies has been discussed byVasilescu.32 Considering the formation energies and differences in ionizationpotentials available for ClF3, BrF3 and XeF2, the value of AH* for RnF2 wasapproximated as -75 kcal mol-1. Similarly, -84 kcal mol-1 was approxi-mated for RnF4. By extrapolation, the existence of RnF6, RnOF4, RnO3,Na4RnO6 and Ag4RnO6 has been predicted. Although it is quite likely thatcompounds of Rn analogous to those of Xe may be even more stable than thexenon analogues, their preparation seems to be a very difficult task as radon isavailable only as a very highly radioactive isotope (222Rn, half life, 3.83 days).Apart from the health hazard the radioactive disintegration would cause atleast a 50 per cent loss of any radon compound in addition to its decomposi-tion by the radiations.Evidence from very small samples suggest that radonforms a chemical compound with fluorine, but the compound was notcharacterized.33 However, RnFz and RnF4 seem to have been prepared.34* The SI unit for H is J mol-1 (4.184 J = 1 cal).150 R.I.C. ReviewIt has been reported that radon can be oxidized in aqueous solution byH202, KMn04 or K2S20g35 but a reinvestigation of the reactions showed thatradon is not oxidized but is only mechanically trapped.36 Recently, it has beenclaimed that solutions of oxidized radon in BrF3, BrF5 and ClF3 have beenprepared.37 The halogen fluorides may be vacuum distilled from the solution atroom temperature without volatilization of radon, which indicates that theradon may be present as an ionic species in solution.Solids obtained onevaporation were not identified but they appeared to resemble the fluoridereported earlier.33 The solid liberates elemental radon on hydrolysis just asXeF2 liberates Xe on hydrolysis.Complex compoundsThe main progress in the chemistry of noble gases is characterized by thepreparation of complexes with Lewis acids and alkali metal fluorides.Some‘onium’ complexes have also been reported.Xenonium derivatives. Compounds containing xenonium cations such asPhXe+ and MeXe+ are reported to have been detected.38~39 Phenyl xenoniumcompounds, e.g. PhXeC104, are formed by the accumulation of Xe by /3-decayof 1311 or 1331 in compounds like PhI, Ph2IC104 and other diphenyliodoniumsalts labelled with 1311 or 1331. Accumulation studies, carried out under variousconditions, e.g. by using the iodine compounds in solid form, in aqueous ornon-aqueous solutions, and by inserting PhI into crystals of KC104, KBPh4,KBF4 etc., show that the yield of the ‘onium’ compounds depends on thenature of anion present. The stabilizing capacity of the anion for PhXe+decreases in the order: ClO,, BPh,, BF-, SO:-, NO, and Cl-.The ‘onium’compounds are stable in acid medium and not in alkaline or neutral medium.It is noteworthy that the product either from PhI or Ph2I+ salts containsPhXe+. The ‘onium’ nature of the products has been established by chromato-graphy and electrophoresis studies.Complexes formed by XeF6. Xenon hexafluoride resembles halogen fluorides inits ability to combine with Lewis acids and also in forming compounds withalkali fluorides. The complexes formed by XeF6 and reported during thisperiod are shown in Fig. 1. It should be noted that this is not a complete list ofall the complexes formed by XeF6.Most of these complexes have been prepared by direct contact between thereactants. Recently it has been shown that xenon hexafluoride reacts withuranium pentafluoride at room temperature to form the adduct UF5.XeF6.XeF6 also reacts with UF4 to produce an adduct approximating to the com-position UF5.1.75 XeF6 which, on prolonged pumping, produces the 1 : 1adduct. The 1 : 1 adduct is a pale yellow compound, soluble in anhydroushydrogen fluoride.lg3 The magnetic moment of the compound is 1.57 BM,which indicates the presence of pentavalent uranium.d-Spacings for x-raypowder diffraction pattern of this adduct have also been reported. Attempts toprepare XeFs-SiF4 adducts have failed.44 In some cases a solvent such asBrF5 has also been ~ s e d . ~ 6 Certain physical properties such as mp and thermalstability of some of these complexes have also been reported.These xenonJha 15(Ref. 40) M zXe F8MXe F, 2 NOF.Xe F6 (Ref.41)1:-RbCs) IN- NO,F.XeF, (Ref. 42:SnF, 4XeF,.SnF4 (Ref.43)XeF6 .ASF5 AsF - GeF. 4XeF,.GeF, -Fl (Ref. 46, 47) 2XeF6.AsF,(Ref.47) 2Xe F6.PF52XeF6.GeF, wf. 4))i l r F 5 XeF,.GeF,2Xe F6. Ir F5 2XeF6 .VF, (Ref. 45)(Ref.46) Xe F,. Ir F,Fig. I. Complexes formed by XeFs.hexafluoride complexes are unstable to heat, have good oxidizing andfluorinating power and are easily hydrolysed by water to give Xe(vr) in solution.Raman and ir spectra indicate that some may be formulated as having ionicstructures. The complex 2NOF.XeFs can be formulated as [(NO)2]2+[XeFs]2-,41and NOzF.XeF6 as N O ~ X ~ F ? . ~ ~ The 2 : 1 and 1 : 1 adducts of XeF6 and AsF5(or IrF5) may be formulated as [Xe2F11]+[MFs]- and [xeFg]+[MFg]- (M = Asor Ir), respectively.46 X-ray d-spacings have been reported in some cases butstructural analyses have not been done.44The structure of the XeFi cation has been determined from the crystalstructure analysis of a compound FllPtXe formed by the reaction of Xe, F2and ptF6.48 FllPtXe is formulated as [XeF5]+[PtFJ.The PtF, ion contains aregular octahedron of fluorine atoms whereas the fluorine atoms in XeF; arearranged in a slightly distorted square pyramid (Fig. 2a).The Xe atom is 0.34 A below the plane of the four fluorine atoms. It shouldbe noted that the structure is same as that predicted on the basis of five bondpairs and one lone pair of electrons in terms of the valence shell electron pairrepulsion theory.That the lone pair is sterically active is shown by the direc-tion of approach of the bridging F atoms of two PtF, groups (the dotted linesin Fig. 2a).Complexes formed by XeF4. Xenon tetrafluoride does not seem to form com-plexes with Lewis acids, e.g. it does not react with IrF5, AsF546 and VF5,45 norwith alkali fluorides. The results of the reaction between XeF4 and IF5 havebeen rather conflicting. One report is that XeF4 oxidizes IF5 to IF7.49 An-other reports that if specially purified XeF4 is used no reaction takes place atall.50 Yet another report claims that an adduct XeF4.IF5 is formed.51 Theadduct is stable up to 92 "C but is readily hydrolysed in air. The formula wasconfirmed by chemical analysis and by Raman and nmr spectra in CH3CN152 R.I.C.Review\\\\1.90 + a03 A I2-14 k.03 AFig. 2. (a) Structure of XeF;, (b) structure of XezF3f.solution. The complex XeF4.2SbF5 is reported but no conclusive evidence insupport of this formulation has been published. However, another complexXeF4.4SbF5 is claimed and its x-ray powder photograph has been reported.53The reactions of XeF4 with PF5, AsF5 and SbF5 in BrF3 have recently beeninvestigated by measuring the conductivities of the solutions. The breaks inconductivities correspond to solutions containing 1 mol XeF4 for 2 and 4 molAsF5; I mol XeF4 for 0.5 and 4 mol SbF5; and 1 rnol XeF4 for 1,4 and 6 rnolPF5.53 It is from these reactions that the above mentioned XeF4.4SbF5complex has been isolated.The interaction of XeF4 with XeFz has been reinvestigated54 to confirm theearlier results55 about the adduct (XeFz.XeF4) formed between the two.Complexes formed by XeFz.The XeF2 complexes reported recently are shownin Fig. 3 on p 154.These complexes are usually formed by direct interaction between thereactants. Solvents like CH3CN,62 BrF356 and BrF546358959 have also been used.XeFz.AsF5 is claimed to be produced by exposing a mixture of Xe, Fz andAsF5 to sunlight.57Some physical properties, e.g. mp and thermal stabilities of some of thesecomplexes have been reported. There seems to be some doubt regarding thenature of XeFz.AsF5. It has been described as a fluorine bridged molecularadduct by one group57 and another group reports that it is unstable andchanges rapidly to 2XeFz.AsF5, even while manipulating for ir or x-raysamples.59 XeFz.2IF5 does not have an ionic structure on the basis of nmr andRaman studies.64 On the evidence of Raman spectra the adduct XeF2.IF5 canbe supposed to be molecular in nature.50The adducts formed by XeF2 with noble metal pentafluorides are of threetypes: 2XeFz.MFg (M = Ru, Os, Ir, Pt); XeFz.MF5 (M = Ru, Os, Ir, Pt);and XeFz.2MF5 (M = Ru, Ir, Pt), the adducts of each type form iso-morphous series.Compounds of the first two types have also been formed byXeFz and AsF5. All these complexes are thermally stable at room temperature,except XeFz.OsF5 which decomposes spontaneously ca 20 "C: 3(XeF2.0sFs)-+ 2XeFz.OsFs + Xe + 2OsF6. On the basis of Raman and ir data these haveJha 15(Ref.4 6 , s ~ ) 2 XeF, .AsF,(Ref. 46,56,57,58,59) XeF,.AsF5Xe F, .2(2,2 ' b i p y r) (Ref. 62)A s f 5 1 1 2 . 2 ' bipyridine(Ref. so. 63, 176) XeF, .I F5 IF' F] pF' '=- XeF,.PF, (Ref. 56)(Ref. 6 3 , ~ ) XeFz .2 I F, XeF, .2PF5SbFS 1 ~ P l f 5 ( t l = P r , Ir, OS, Ru, E ' Rh)(Ref. 5 6 , ~ ) XeF, .S bF5(Ref. 52,60,61) XeF,.2SbF52 XeF,.MF5(Ref.46,s8,59.also 65 for Nb, Ta)(Ref. 60) XeF, . I .5S bF5(Ref. 60) XeF, .6SbF5XeF,.MF,XeF2 .2MF5(a~s0 Ref. 63 for Irand 65 for Nb, Ta & Ru)Fig. 3. Complexes formed by XeF2.been formulated as salts [Xe2F3If[MF6]-, [XeF]+[MF6]- and [XeF]+[M2F11]-,respectively.58~59A detailed x-ray analysis of XezF,f[AsF,]- shows that AsFG has nearly anoctahedral shape and XeZFi is a planar V-shaped cation.The cation Xe2Ficontains a bridging F atom and is symmetrical about that atom (Fig. 2b). Onthe basis of shorter (1.90 5 0.03 A) terminal Xe-F distances (cf. 2.0 A inXeF2) the cation is represented as F-XefF-Xe-FS.58959 XezFf salts arecharacterized by strong bands in Raman at - 575 and 591 cm-1 (stretchregion) and - 160 cm-1 (bend region).Raman spectra of the 1 : 1 complexes are similar to those of relatedestablished salts A+MF, (A = alkali metal and M = transition metal). Thesedepart slightly from the ideal salt spectra which suggest that MF, symmetry inthese cases is lowered by fluorine bonding of the anion with cation: F-Xef . . .F-MF-. Raman spectra of XeFf salts show an intense doublet in the region of600-612 cm-1.The doublet nature of the band is attributed to weak interact-ions between cations.59The crystal structure analysis of the complex XeF2.2SbFs has also beenreported.61 On the basis of shorter Xe-F bond length (1.84 3 0.04A) inXeF2. SbF5, this compound may be formulated as [XeF]+[Sb2F$, but it isalso found that a fluorine atom of Sb2Fll unit is very close to Xe (2.35 A, sumof the van der Waals radii of Xe and F = 3.5A), indicating considerableinteraction, and hence XeF2.2SbF5 may be regarded as essentially a covalentmolecule.61Conclusions regarding complex formation by XeF2, XeF4 and xeF6. XeF2,XeF4 and XeF6 seem to form most of their complexes by fluoride ion donation.It is apparent from the above list of complexes that the fluoride ion donor154 R.I.C.Reviewability is the least in XeF4. The fluoride ion donor abilities of these three fluor-ides has been nicely compared experimentally by Bartlett and Sladky.46Treatment of 1 : 1 : 1 mixture of XeF2, XeF4 and XeF6 with AsF5 in BrF5solution followed by the removal of BrF5 and excess of AsF5 yields a mixturecontaining [XeF5]+[AsF6]-, [Xe2F3]+[AsF6]- and XeF4. This shows that the F-donor ability decreases in the order XeF6 > XeF2 > XeF4. This reaction hasalso been suggested as a method for purification of XeF4 from XeF6 and XeF2contaminants .46One expects XeF2 to be the best fluoride ion donor as F- can be more easilyseparated from it because of low effective charge on Xe in XeF2. The order onthis basis should be XeFz > XeF4 > XeF6 but XeF6 is found to be a betterdonor than XeFz.This has been explained46 on the basis of the tendency ofnon-octahedral XeFs to change over to pseudo octahedral XeF5f.Complexes formed by XeOF4. Xenon oxide tetrafluoride forms a number ofcomplexes analogous to those formed by XeF6. These are shown in Fig. 4.XeOF4 forms complexes with K, Rb and Cs fluorides but not with NaF.Thermogravimetric studies indicate that a number of intermediates, e.g.3CsF.2XeOF4,6KF.XeOF4 etc., are formed before the final decomposition toalkali fluorides.66 It forms an adduct with AsF5 at -78 "C which is unstableat room temperature.66 XeOF4 also reacts with NO2F at about 100 OC.67Nearly all the complexes of XeOF4 dissociate completely in the vapour phase.The ir spectrum of solid NOF.XeOF4 shows bands due to NO+.41 Thus,XeOF4 may be acting as an F- acceptor in the complexes with NOF and alsowith alkali metal fluorides and as an F- donor with Lewis acids like VF5 andSbF5.CompZexes formed by XeO3.The first such complex reported was mono-caesium fluoroxenate, CsF.XeO3 which was formed by exposing CsF.XeFs40or CsF.XeOF466 to air. CsCI.XeO3 was prepared by mixing aqueous solutionsof CsCl and X e 0 ~ . ~ ~ A few more compounds, viz. MF.Xe03 (M = K, Rb, Cs)Fig. 4. Complexes formed by XeOF4.XeOF4.2SbF, (Ref.66) 1 SbF,RbF - 3RbF.2XeOF4 (Ref. 66)CrF : CsF.XeOF, (w. u)KF(Ref. 66) 3KF.XeOF4(Ref. 41) NOF.XeOF, 4 NOF2XeOF, .VF5 (Ref. 45)Jha 15and CsBr.XeO3 have recently been reported, prepared by mixing aqueousXeO3 and MF solutions followed by acidifying with aqueous HF, or neutral-izing aqueous XeO3 containing HF (obtained by hydrolysis of XeFs) withaqueous MOH and evaporating the solution in either case until crystals areformed.69 The stability of these compounds decreases with increase in atomicweight of the halogen: CsF.XeO3 > CsCl.XeO3 > CsBr.XeO3.It was realized in the beginning that such compounds are not equimolecularmixtures of MF and Xe03 because no lines due to MF or XeO3 showed inthe x-ray powder patterns of these compounds;40$66 they were assumed to bemolecular addition complexes of the type MF.Xe03 since no absorptionoccurred in the Xe-F band region though bands were present in Xe-0region.66These salts are much more stable than XeO3 and hence it is unlikely thatthey are simply loose molecular complexes of MF and XeO3.Recently anx-ray crystal and molecular structure analysis of one of these compounds,namely KF.Xe03, has been carried out70 which shows that the crystalstructure consists of infinite chains of XeO3 units linked by bridging F atoms,with K+ ions at non-bonding distances from 0 and F atoms. Thus, the formu-lation of this compound as molecular addition compound is incorrect, thecorrect formulation being nK+ (Xe03F-),. Other fluoroxenates should belikewise formulated.The x-ray analysis shows70 that the coordination around xenon is analogousto that found in XeFl and XeOF4, and may be considered to be a distortedoctahedron with three coordination sites occupied by oxygen atoms (nearlysame positions as in XeOs), two sites by fluorine atoms (Xe-F distances beinglarger than normal covalent Xe-F bonds but smaller than bridging Xe-Fdistances) and the sixth site occupied by the Xe lone pair of electrons.The absence of absorption in Xe-F band region is easily explained as theXe-F distances in these compounds are longer than the normal covalent Xe-Fdistance, hence the absorption due to Xe-F in these compounds would occurat lower frequencies. The stabilities of these compounds have been attributedto the special stability of octahedral coordination in Xe(v)r compounds.70A potassium salt containing Xe in (VI) and (VIII) oxidation states, namelyK4XeO6.2Xe03, which was reported earlier71 has been reinvestigated.72 It isprepared by ozonolysis of an aqueous solution of XeO3 and KOH.It is stableup to 201 "C but is sensitive to mechanical shock. It decomposes according tothe equation K4XeO6.2Xe03 = &XeOs + 2Xe + 302.Ir, x-ray diffraction and thermal study suggest that XeO3 is likely to be co-ordinated to the central perxenate moiety through an 0x0 bridge to give a salt,K4Xe3012.72Other complexes. All the compounds described so far are those in whichxenon is attached to F or (and) oxygen atoms. Some new compounds haverecently been reported in which xenon is attached to oxygen which is itselfattached to some other groups.The reactions of XeF2 with appropriate amounts of fluorosulphuric acid orperchloric acid at -78 "C or below produce compounds such as xenon(I1)fluoride fluorosulphate, FXeS03F ; Xenon@) bis(fluorosulphate), Xe(S03F)z ;156 R.I.C.Reviewand the corresponding perchlorates, FXeC104 and Xe(C10&.73 These seemto be formed by the replacement of fluorine atoms on XeFz by - OSOzF or- OC103 groups. All these compounds are colourless solids at room tempera-ture; their melting points have been reported. The fluorosulphates are kineti-cally more stable than the perchlorates but all are thermodynamically un-stable at room temperature. They decompose as shown:FXeS03F -+ X e + XeFz + S%OSF~Xe(S03F)z + Xe + SZOSFZThe perchlorates give Xe, 0 2 and C1207 with some ClOz.The x-ray structure of FXeS03F has been reported.73 The Xe is bicovalent,bonded on one side to F and on the other to one 0 of the S03F group.In thebis compounds Xe is bonded as 0-Xe-0.Similar compounds F X ~ O T ~ F S ~ ~ and Xe(OTeF&75 have been produced bythe reactions of XeFz with an equimolar amount of HOTeF5 and a five-foldexcess of HOTeF5, respectively. The tellurium compounds are much morestable than the above mentioned sulphur and chlorine derivatives. They maybe distilled under vacuum at room temperature without decomposition.FXeOTeF5 acts as a fluoride ion donor in the same manner as XeFz toproduce [F5TeOXe]+[AsFs]-.74 It is expected that the corresponding fluoro-sulphate and perchlorate will behave the same way.All these compounds of Xe react with water to produce the correspondingacid, xenon and oxygen.[Xe(OTeFs)z reacts slowly with HzO but reactsvigorously in strongly alkaline medium.741The reaction of XeFz with trifluoroacetic acid at -24 "C gives xenon@)fluoride trifluoroacetate, FXeOC(O)CF3 and xenon@) bis(trifluoroacetate),Xe[OC(O)CF3] 2 both of which are pale yellow solids and detonate on thermalor mechanical shock. Their ir spectra in acetonitrile solution have beenreported.76A new complex of approximate composition XeMnF5, a wine red solid, hasbeen reported to have been obtained during an investigation on the influenceof manganese trifluoride on xenon-fluorine reaction.181 The reaction betweenxenon and fluorine was carried out at 120°C in the presence of MnF3. Thevolatiles had been removed by prolonged pumping off at 50°C and theresidue left in the vessel had the composition XeMnF5.SECTION 2: OLD COMPOUNDSIn this section the 'old' compounds, i.e.the simple compounds reported priorto 1966 are discussed, and those compounds are included about which newpreparative methods and new structural and other physical data have beenobtained.New preparative methodsXeFz. A simple and elegant method of preparation of XeFz has been describedin which a mixture of xenon and fluorine (or FzO) in Pyrex glass is exposed tosunlight or diffused light at room temperature.77978 A slight modification ofthis method has been described in detail.79 Crystals of the difluoride can beseen within two hours in bright sunlight and in diffuse daylight within two days.Jha 15A static thermal method similar to that for the preparation of XeF4 andXeF6 has also been evolved to produce XeF2 under controlled conditions.8lThe advantage in this method is that the same vessel (Monel or nickel) can beused for the preparation of all the xenon fluorides; quartz or glass beingnecessary for preparing XeF2 by other methods.KrF2. A simple preparative method for KrFz has been reported by whichKr-F2 or Kr-OF2 mixtures are exposed to sunlight,@ but similar independentexperiments have failed to produce any krypton fluoride.83It is worth mentioning here that attempts to repeat the preparation ofkrypton tetrafluoride have only resulted in the isolation of K ~ F z .~ ~Manufacture of noble gas Jluorides. Various methods have been patented forthe manufacture of noble gas fluorides. The methods for production of xenonfluorides are by (i) contact of Xe with chlorofluorocarbons like CFzClz underhigh electric discharge86 and by treating Xe with (ii) NzF287 or (iii) 02F2.34A method similar to (i) has been patented for the manufacture of kryptonfluoride86 and one similar to (iii) for radon fluorides.34Gas mixtures of Xe or Kr with F2 have been found to react upon protonbombardment to yield the various fluorides of Xe or Kr.88The conditions for the synthesis of xenon fluorides have been studied againand it is reported that the main product from an Xe-F mixture in the moleratio 1 : 10 with the total pressure - 33 atm in a closed system is XeF2 at120 "C, XeF4 at 150 "C and XeF6 at 200 OC.8931 These conditions are rathermild compared to the others which require higher temperatures.Xe0F4.In earlier preparations it was assumed that the most satisfactory wayto produce XeOF4 was to treat XeF6 with H2O in the vapour p h a ~ e , ~ O ~ ~ ~ but ithas since been realized that the reaction is successful even when the reactantsare largely not in the vapour phase.92Xe-0 compounds. Radioactive xenates, XeO3 and Xe04 have been reportedto be formed by decay of l3lI and 1331 in some iodates, periodatesl3~93-95 andiodoxybenzene, CsHsIO2.96 The reactions are of the type 131105 -+ 131Xe03.Aqueous Xe03. A detailed method of preparation of pure aqueous Xesolution has been described by Appelman.97 In this, xenon hexafluoridehydrolysed and fluoride ion is precipitated by MgO.Mg2+ is removed wZr3(PO4)4 and the resulting H3P04, along with the last traces of fluoride,hydrous ZrO2.Perxenate. A method for the manufacture of perxenates has been patented.98Alkali metal perxenates are made by the alkaline hydrolysis of XeF6 or by thedisproportionation of alkaline solution of XeO3 and oxidation of such solu-tion by ozone. CU(III), Ag(rI), La(@, Zn(rr), Pb(Ir), UOZ(II) and Th(1v)perxenates are formed by adding the solutions of these ions to perxenatesolution. A laboratory method of preparation of sodium perxenate by the158 R.I.C. Reviewreaction of ozonized oxygen with aqueous XeO3 solution has been describedin detail.99Structural and other physical studiesxeF6. Among all the xenon compounds XeF6 has been the most controversialwith regard to structure. The other 15 known hexafluorides are all octahedraland so the same octahedral structure was expected for XeF6.At the very out-set it was realized that the valence shell electron pair repulsion model predicteda distorted octahedron whereas empirical MO calculations suggested a regularoctahedral structure.2 At that time electron diffraction studies indicated a dis-torted octahedral structure, whereas ir and Raman studies did not give anunambiguous result.It has now been established beyond any doubt that xenon hexafluoridepossesses a slightly deformed octahedral structure in the vapour phase. Thecause of this deformation is not well known yet. The various structural studiesare summarized below.Theoretical studies by Claxton and BensonlOO indicate that the structure ofXeF6 is determined by a net electron attraction term, causing distortion.Contrary to earlier conjectures it has been shown that MO calculations mayalso predict a distorted structure for XeF6.1019102Comparison of the measured equilibrium constant for the reactionXeF4 + F2 --f XeF6 with thermodynamic functions calculated for variousmodels of XeF6 suggests that the molecule has less than octahedral sym-metry.103Electron diffraction studies have again favoured a distorted octahedron forXeF685J04 but the distortion is much smaller than expected from Gillespie’smodel.21 In view of the small static distortion of the order of magnitude ofvibrational amplitude in XeF6 required to fit the electron diffraction data, analternative model of dynamic distortion symptomatic of Jahn-Teller inter-action was proposed and found to fit the diffraction data better.105 Realizingthat no single geometry is capable of accounting for the electron diffractionpatterns, Bartlett and BurbanklOG have offered an explanation for the reportedelectron diffraction data which involves intramolecular rearrangement betweenseveral molecular geometries ; there being little difference in energy betweenCzv and C3v configurations a transformation between the two along a path ofC8 configuration is assumed (Fig.5). By two successive transformations eitherC3v + Czv -+ CSv or Czv -+ CsV -+ Czv a molecular rearrangement may beaffected. A very good agreement between the synthetic radial distributioncurves obtained by this model and those reported has been claimed.A similarmodel has been used to explain the electron diffraction data of the relatedIF7.106 Another detailed analysis of the electron diffraction results183J84points to a similar model consisting of rapidly inverting non-Oh structures;however, a satisfactory model can be described as having a CsV structure, adistorted octahedron in which the xenon lone pair occupies the centre of one ofthe faces of the octahedron. The electron diffraction data gives a mean Xe-Edistance of 1.890 & 0.005 A.The Raman spectra of solid, liquid and gaseous XeF6 have been reported.107There are three bands in the Raman spectrum of monomeric XeF6 vapour asJha 1591CSlone pairc3 v@.fluorine atomsFig.5. Transformations of configurations in XeFG.expected for Oh symmetry, but one band is very broad compared with thespectra of other hexafluorides. Thus either the ground state vapour moleculespossess a symmetry lower than Oh or they have some very unusual electronicproperties that markedly influence the region of the spectrum usually con-sidered to be the vibrational-rotational. In the liquid and solid compoundmore bands are observed due to the lowering of symmetry by aggregation incondensed phases (see p 161). During an ir, far ir and microwave study108 morebands than expected for Oh symmetry were observed in the bond stretchingregion. Definite absorptions were not observed in the bond bending regionand no microwave absorptions occurred in the range 3.7 to 8.6 cm-1. Glassloghas analysed the reported Raman and ir data and has concluded that theground state vapour molecules of XeFs possess Oh symmetry but have unusualelectronic properties which influence the band widths.Magnetic susceptibility measurements show that XeF6 exhibits a tempera-ture-independent diamagnetism from 77 K to 325 K.llO*lll The deflection of amolecular beam of XeF6 in an inhomogeneous magnetic field indicates thatgaseous XeF6 at room temperature does not contain paramagnetic componentsgreater than - 1 per cent in abundance.l12 These results are not consistentwith Goodman's hypothesis113 that a low lying 3Tu state of XeF6 is appreciablypopulated at room temperature.The value of electric dipole moment of XeFG is found to be less than 0.03D114by deflection of a molecular beam of XeF6 in an inhomogeneous electric field.Such a small value of the dipole moment eliminates Gillespie's electron pairrepulsion model.The electrical conductivity of liquid XeF6 is (1.45 & 0.05) x 10-6 0-1at 50 "C and the dielectric constant is 4.10 & 0.05 at 55 "C.115 The low value ofthe dielectric constant accounts at least in part for the lack of ionization ofsalts, e.g.of CsF40 in liquid XeF6. A reported transition of XeF6 from acolourless crystalline solid to a pale yellow solid at 42 "C116 has been ascribedto premelting in impure sampIes.117A preliminary study of the heat capacity and other thermodynamic func-160 R.I . C. Reviewtions of solid XeFs has indicated that XeF6 is polymorphic; it exists in threemodifications.118 A similar detailed study has confirmed the above conclu-sion.llg In this study a more accurate value of the melting point of XeF6 hasbeen obtained, 322.63 5 0.10 K, and the enthalpy of fusion has been found tobe 5743 J mol-1. Two modifications, namely, monoclinic and cubic, have beenidentified crystallographically120 and they are assumed to correspond totwo of the three phases indicated by heat capacity data. A detailed x-raystructure analysis has been reported but crystallographic data was onlyobtained for the cubic form.120 Recently, during an x-ray structure analysis ofthe cubic form, it was also observed that the cubic form was stable in thetemperature range 301 K and 103 K.121J22 This observation is in conflict withthe heat capacity data according to which three modifications should havebeen observed in that temperature range.The single crystals of cubic XeF6 have been further investigated and it hasbeen found that cubic XeFs is stable from the melting point to 93 5 5 K;122thus, this phase is capable of existence at temperatures characteristic of all thephases inferred from the thermal measurements. In fact, the single crystals ofboth the monoclinic and cubic modifications were found to grow side by sidein the same capillary at 296 K.The stability of the cubic form of XeF6 hasbeen explained on the basis that a solid-solid transition would not be possibleat an observable rate between the two phases (cubic and monoclinic) becauseof the lack of a simple relationship between the two structures.122 The mono-clinic form apparently contains tetrahedra of xenon atoms linked by bridgingfluorine atoms120 whereas the cubic form contains ions of X e P and F-associated in tetrameric and hexameric rings of point group symmetries 4 and32.121It has been concluded122 that the cubic XeFs is a phase additional to thoseinferred from the heat capacity rnea~urements1~8J~ and the structuralmodifications do not involve the cubic phase; the transition observed duringthe thermal study most probably ensues from the monoclinic modification.Further work is evidently required to completely elucidate the nature ofpolymorphism in XeF6.However, in view of the stability of the cubic form ofXeF6 and its ability to coexist with other modifications, care must be taken toconsider the crystallographic phase present in addition to chemical purity,when making physical measurements on solid XeF6.XeFz, XeF4 (and XeF6). The linear structure of XeF2 and planar of XeF4 arewell established,2 hence very few new structural investigations have beenmade. Mossbauer study seems to have validated the assumption o f p bondingin XeFz and XeF4.13914 Recently, the v3 region of the spectrum of XeF2 hasbeen studied under high resolution. The Xe-F bond length obtained from thisstudy is 1.977 0.001 5 &l23 which compares well with the previous value. Anextensive study of the Xe-Fz system1o3 has yielded equilibrium constants forvarious fluorination reactions of Xe from which a few thermodynamic func-tions of these fluorides have been calculated.The values of the heats of forma-tion [AHf(g)] of XeF2, XeF4 and XeF6 have been found to be -25.9,- 51.5 and - 70.4 kcal mol-1, which lead to the Xe-F bond energy values of31.3,3 1.2 and 30.2 kcal mol-l in XeF2, XeF4 and XeF6, respectively. It may beJha 16noted here that the Xe-F bond energy in various fluorides of xenon remainsnearly constant whereas I-F bond energy seems to decrease with increase incoordination number of I in different iodine fluorides.85For XeF2 two more values of heat of formation have been reported. AHf(g)obtained by the measurement of the heat of reaction of XeF2 with ammonia isreported to be - 28.5 1 kcal mol-1.124 AHf(s) obtained by measuring theheat of combustion of XeF2 is found to be - 41.5 0.6 kcal rnol-1,125 whichyields the value of AHf (g) as - 28.4 kcal mol-l taking the heat of sublimationof XeF2 to be 13.1 kcal mol-1.126 These values are on the high side as com-pared to the data obtained by Weinstock et ~ 1 .~ 0 3Vapour pressure-temperature relationships have been studied for XeF2 andXeF4 by Schreiner et a1.,126 giving heats of sublimation of 13.1 and 14.4 kcalmol-l respectively (cf. earlier values of 12.3 and 15.3).127 Accurate triplepoints, obtained by the thermal arrest method, are 129 "C for XeF2 and 117 "Cfor XeF4. Earlier reported values were 140 "C and 114 "C, respectively.128 Avalue of 130 "C for XeF2 has also been obtained ~eparately.1~9XeOF4. The melting point of XeOF4 has been redetermined and found to be- 46.2 "C,66 the earlier reported values were - 41 "C90 and - 28 OC.91 Theelectrical conductivity at 24 "C is 1.03 x 10-5 Q--1 cm-1 which indicatessome self-ionization in the liquid and also accounts for the enhanced con-ductivity on addition of alkali metal fluorides.The dielectric constant at 24 "Cis 24.6 which lies between those of IF5 and B ~ F s . ~ ~ The dipole moment isreported to be 0.65 5 0.09D.lS5Normal coordinate analysis of XeOF4 and XeF4 has been carried out tocalculate various vibrational functions.130 A recent microwave study185confirms the C4v symmetry of this molecule and gives precise bond distances:Xe-F = 1.900 & 0.005 Xe-0 = 1.703 5 0.01 5 A, the 0-Xe-F angle =91.8 " 5 0.5 *.The Raman spectrum of XeOF4 in liquid HF is similar to thatof the pure substance.186KrF2. A Mossbauer study,131 an electron diffraction study85 and a rotationalfine structure analysis of 590 cm-l ir band of KrF2132 support the earlier con-clusion that the molecule is linear. A Kr-F bond length of 1.875 & 0.002 or1.867 & 0.002 A accords with the rotational structure satisfactorily; the valueobtained from electron diffraction measurements is 1.889A vapour pressure temperature study has given a value of 9.9 kcal mol-1 forthe heat of sublimation.133 The heat of formation of KrF2(g) has been deter-mined to be + 14.4 & 0.8 kcal mol-1. From this the Kr-F bond energy isestimated to be 11.7 kcal mol-1.134 KrF2 has a special importance: so far thisis the only fluorine compound known which is formed endothermically fromthe elements.Hence it follows that it is a stronger oxidizing agent thanelementary fluorine.0.01 A.Xe-0 compoundsXeO3. XeOs has a very low vapour pressure at room temperature but XeO3molecules have been observed in the vapour phase by mass spectrometry.187I62 R. I. C. ReviewElectrophoretic and chromatographic studies indicate that the hypotheticalH6Xe06 (aqueousThe enthalpy of formation of XeOs(aq) at 298.15 K has been estimated tobe 99.94 $ 0.24 kcal mol-l from the calorimetric measurements of enthalpiesof reactions of Xe03(aq) with HI(aq) and Iz(c) with HI(aq).137exists in solution either as XeO4+ or XeOgt.136Xe04.The molecular structure of xenon tetroxide has been investigated in thegas phase by electron diffra~tion.l7~ The data is compatible with the tetra-hedral structure proposed from analysis of the ir spectrum. The Xe-0distance is deduced to be 1.736 A as compared with 1.6 8, (approx.) from irresults.180 A detailed analysis of the vibrational spectrum has been reportedand vibration amplitudes have been calculated.lB8Xenate and perxenate. Structural data on xenate and perxenate ions in solidcompounds were available earlier but the aqueous solutions of these ions haveonly recently been investigated by Raman spectroscopy.l38 The investigationsindicate that the HXeO, ion is the predominant species present in aqueoussodium xenate solution and Xe0;- ions in the solutions of caesium perxenate.In perxenate solutions certain details of the spectra imply the presence of otherionic forms.A Mossbauer study has indicated that the measured isomer shift13is consistent with directed sp3d2 hybrid bonds for Xe04,-.SECTION 3 : KINETIC STUDIES AND SOME OTHER REACTIONSFormation of XeFz and XeF4Kinetic studies on the formation of XeFz and XeF4 from the gaseous elementsreveal that the reactions are heterogeneous and occur, for the most part, onthe fluorinated walls of the Monel reaction vessels or on the surface of addedmetal fluorides like COF3, NiFz and CaF2, which act as catalysts.139J40 Adetailed study of the catalytic formation of XeF2 has been carried out byBaker et al.141 They report that the formation of XeFz is catalysed by Pd andNi and other metals which form ionic fluorides, e.g.Co, Cu and Al, and not bymetals which form covalent fluorides, e.g. Ti, Zr, Mo, Ta, W, Re, Ir, Fe, Cr, V,Rh and Pt. The catalytically active metals become coated with an ionicfluoride layer. The Pd and Ni systems were investigated in detail; for bothmetals the reaction (Xe + Fz --f XeF2) was found to be zero order for xenonand fluorine with partial pressures > 50 Torr.That the rate of formation of XeFz is zero order in fluorine pressure has beenexplained on the basis that fluorine is known to be strongly adsorbed. That therate is zero order in xenon pressure is explicable only if the xenon is alsostrongly adsorbed at the surface, by chemical combination with fluorine.Baker et al.142 have demonstrated by radioactive techniques that xenon isadsorbed onto the metal surface in the presence of fluorine.The xenon isprobably chemically bound to fluorine in the adsorbed state and this mayconstitute the intermediate in the catalytic formation of XeF2. Attempts arebeing made to elucidate the mechanism of the photolytic preparation of XeFz.Flash photolysis of XeF2 has been carried out alone and in the presence of othergases to investigate the mechanism involved.143Jha 16A very interesting observation has been made during an investigation of thecatalytic role of magnesium fluoride and nickel fluoride of high surface area inthe reaction of xenon and fluorine (1 : 10 molar ratio) at low temperature,namely 120°C.In the presence of MgF2, xenon difluoride was produced asexpected89 but in the presence of nickel fluoride the only product was xenonhexafluoride.ls1 Xenon hexafluoride was produced even when the mole ratioof xenon to fluorine was only 1 : 5.Hydrolysis of XeF2The reaction representing the hydrolysis of xenon difluoride (XeF2 +H2O -+ Xe + 302 + 2HF) is a first order reaction. The rate constant in0.01 MHC104 solution is reported to be 2.58 x low2 min-1 at 25 OC; AH* andAS* are 19.6 kcal mol-land -8.1 cal OC-1, respectively. Mechanisms have beenproposed invoking XeO and Xe02.144 The rate constant in pure water isreported to bc (1.51 min-l at 25 OC.145 Another estimate ofthe activation energy for the hydrolysis is 18.4 & 2.1 kcal mol-1 which gives afrequency factor in the Arrhenius equation of 1.2 x 10l2 min-1.The frequencyfactor calculated from the number of active collisions of XeF2 and H2Omolecules is 0.9 x 1012 In this case the mechanism is supposed toinvolve the XeF. radical.Studies of the effects of pH and of different cations and anions on the rate ofhydrolysis show that the rate is minimal in the pH range 4-9; cations capableof forming stable fluorocomplexes, e.g. Th4+, A13+, Be2+ and La3+ and anionsNO,, HCO, and ClO, have an accelerating effect on hydrolysis.147 Cationsprobably form active (F-Xe-F-M)n+ complexes or else take up F- from XeFzto form MFn+ complexes, thus releasing an active XeF+ species.Anions alsopossibly form active intermediate fluoro complexes. Hydrogen fluoride alsoaccelerates the hydrolysis rate, perhaps by picking up fluoride ion to formHF;.l48It has been concluded that at least 97 per cent of XeF2 sample dis-solved in water is initially present in solution as molecular XeF2. Side reactionsproducing XeF(OH), HF, XeF+ and F- etc., are believed to occur only toan extent of about 3 per cent.1490.04) xOther reactions of XeF2Some of the reactions of XeF2 where definite products have been obtained areshown in Fig. 6. XeF2 acts as oxidative fluorinator in most of the cases.Xenon difluoride dissolves in acetonitrile with solvation but without furtherreaction at 25"C153 but at the boiling point of the acetonitrile fluorinationtakes place.49 Xenon difluoride also dissolves in dimethyl sulphoxide andpyridine giving some gaseous products; it does not dissolve in liquid NH3, butNz and F2 are slowly evolved with the formation of NH4F.49XeF2 dissolves in BrF5, HF, IF5, S02, CH3CN and WF6 without oxidationor reduction.But in the presence of a trace of a Lewis acid such as BF3, HF orSO2 it acts as an oxidative fluorinator,l51 e.g. 1 2 and XeF2 do not react inCH3CN but in presence of acid, iodine is oxidized; liquid SO2 and XeF2 do notreact but in the presence of a trace of BF3 the sulphur dioxide is converted tosulphuryl fluoride. The role of the Lewis acids may be explained by supposing164 R.I.C. ReviewSZOs F, (Ref. 151)ArF,(Ref. 49) AS F 5 -CCI,FCCIF,(Ref. 49) CCI F3 - - BrF, (Ref.150)(Ref. 152) KIO, IF 5 (Ref. 150)Fig. 6. Reactions of XeF2.that they facilitate XeF2 ionization, XeF2 + A --f XeFf + AF- or2XeF2 + A 4 Xe2F+ + AF-. These cations then act as oxidative fluorinators.Solutions of XeF2 in CH3N02, dioxane, cc14, THF, DMF, DMSO, pyridineetc. have been studied by ir spectroscopy.l54 It has been concluded that theXe-F bond is weakened with increasing donor strength of these solvents.The solutions of XeF2 in liquid NOF,3HF have been investigated bysolubility, conductivity and 9F nmr measurements.l55 XeF2 is highly soluble,rather unusually, in this solvent, e.g. 83.8 per cent at 80 "C and 73.2 per cent at16.8 "C. XeF2 remains as molecular species in this solvent.An interesting experiment on gas chromatographic analysis of XeF2 invapour phase using a KelF-10 oil Fluoropac 80 column has been described.156It is suggested that the retention properties of XeF2 should offer possibilitiesof preparative separation of XeF2, HF and F2.Reaction of molten XeF2 with TiF4 has been investigated by conductometricmeasurements which shows that XeF2 forms a 2 : 1 adduct with TiF4 whichhas been formulated as [FXe][TiF6][XeF].157Xenon difluoride reacts with anhydrous nitric acid at 20°C giving red-brown products which decompose rapidly and have not been characterized.Ithas been suggested that unstable FXeN03 and Xe(N03)~ are produced in thisreaction.76 Aqueous solutions of XeF2 oxidize iodate to periodatel52 andbromate to perbromate.l89XeF2 reacts with excess benzene in cc14 in the presence of small amounts ofHF to give fluorobenzene in 68 per cent yield.190Reactions of XeF4XeF4 reacts with NH3, MezSO, C5H5N, MeCN, CC~.LF-CCIF~ and AsF3 togive the same products as XeF2 but it is more oxidizing in nature as it appar-ently reacts with IF5 to produce IF7.49 The reaction of XeF4 with IF5 is doubt-ful as Bartlett et al.have observed5O that especially purified XeF4 does notreact with IF5. The solutions of XeF4 in liquid NOF,3HF have been investi-Jha 16gated by solubility, conductivity and 19F nmr techniques.155 The solubility is5.95 per cent at 18.4 "C and 21.22 per cent at 80.9 "C. These values are muchlower than those for XeF2.Hydrolysis of Xe F4During the hydrolysis reactions of XeF4 a transient yellow colour has beenobserved by many workers.The yellow colour was generally assumed to bedue to XeO and reactions involving XeO and Xe04 were proposed for thehydrolysis of XeF4.71 Recently, it has been proved that the transient yellowcolour may be due to XeOF217 and in the light of this finding the reactionschemes for the hydrolysis of XeF4 may have to be modified.Reactions of Xe03A thermal decomposition study showed that the decomposition of XeO3begins at 40 "C and is complete at 140 "C. The decomposition is smooth andtakes place without explosion,l58 contrary to earlier reports.The oxidation of PU(III) to PU(IV) by XeO3 in aqueous solution has beenstudied kinetically. The mechanism of reaction appears to involve either twosuccessive one-electron changes or one two-electron change to form a Pu(v)species, other than PuOg, which then reacts with PU(III) to form Pu(1v).l59 Asimilar study of the oxidation of Np(v) to N ~ ( v I ) indicates that a photo-chemical process is involved in the reaction.lG0 The decomposition of xenontrioxide in aqueous solution is increased by the addition of XeF2; the kineticdata have been explained by assuming the intermediate formation of H202.l61Xenon trioxide dissolved in tertiary butyl alcohol behaves like an acid and maybe titrated with standard K or Rb t-butoxide dissolved in t-butyl alcohol. Theacid may be represented as ButOXe03H.162Aqueous Xe03 in acidic or neutral medium oxidizes primary and secondaryaliphatic and aromatic alcohols to C02 and H2O.The reaction with tertiaryalcohols is slow; some tertiary alcohols apparently do not react at all.163J64Carboxylic acids, including hydroxy and dicarboxylic acids, are also oxidizedto C02 and H~O.165 Benzyl alcohol and benzaldehyde are oxidized to benzoicacid by aqueous Xe03.166Rough estimates for oxidation potentials have been obtained experi-mentally.71~167 In acid solution the Xe-Xe(v1) potential should be about 1.8 Vand that of Xe(vI)-Xe(vIIr) about 2.3 V. Recently, more accurate assessmentsof the electrode potentials of the Xe-XeO3 couple in acidic solution and the Xe-HXeO, couple in basic solution have been made; these values are 2.10 0.10and 1.24 & 0.01 V, respectively.l37SECTION 4 : APPLICATIONSAfter the synthesis of noble gas compounds potential applications were sug-gested.168 Since then, from time to time, this topic has been alluded to byvarious authors.6~44~169J70 A few reactions which are of potential practicalimportance have been carried out recently and are summarized below.166 R .I. C. Re viewXeO3The potential uses of xenon trioxide are based on its strong oxidizing powercoupled with the fact that the reduced product is xenon gas which does notcontaminate the oxidation products.Aqueous XeO3 oxidizes carboxylic acids and primary and secondaryaliphatic and aromatic alcohols to COz and HzO.l63-l65 Micro- and semi-micro-amounts of the acids and alcohols have been estimated by adding aknown excess of Xe03 and determining the excess of XeO3 iodometrically.Acids like acetic and succinic which are otherwise difficult to oxidize canalso be estimated.Other dicarboxylic, hydroxy and amino-acids have beenestimated in amounts as low as 100 pg165 whereas methanol, ethanol and pro-panol have been determined in amounts as low as 30 ~ 8 . ~ 6 3A direct spectrophotornetric titration between XeO3 and H202 has beenused to determine small amounts (- 50 pg) of HzO2. Even lower amounts ofH202 (- 0.9 pg) can be estimated by using a catalytic method which utilizesHzO2 to initiate the reaction between t-butyl alcohol and Xe03.171Xe fluoridesXenon difluoride acts as an oxidative fluorinator in the presence of a trace ofacid.151 In aqueous solution the difluoride can be used as an oxidant, e.g.oxidation of K103 to KIO4.152 XeF has been proposed for the detection anddetermination of I- and Cr3+.172 XeF2 oxidizes I- to 10, and Cr3+ to CrzO;-.Xenon fluorides have proved to be useful initiators in polymerization.87Advantage has also been taken of the fluorinating properties of the xenonfluorides.They may be used as fluorinating agents for inorganic and organiccompounds under certain conditions.Prolonging the lifetime of tungstep filamentsThe lifetime of tungsten filaments in incandescent lamps is prolonged by mixingxenon fluorides with the gas to fill the lamps.l73Separation of xenon and kryptonIt had been suggested earlier1G8 that Xe and Kr can be separated from eachother by fluorination reaction. The conversion of xenon to its fluorides isrelatively simple whereas krypton does not react under those conditions.Aproblem which may utilize such an application is the separation of radioactivexenon and krypton which are formed during the fission in nuclear reactors.Such a separation has been reported recently.182CONCLUSIONAs the advances have been made in the knowledge of the chemistry of noblegases the inherent danger in working with these compounds has also beenrealized, in view of the explosive nature of XeO3 and other compounds.A good account of safety precautions while working with noble gas com-pounds has been given by H o l l o ~ a y . ~ y ~ 7 ~ Recently, there has been anotherreport regarding the explosion hazard during work with fluorine containingJha 16xenon compounds.l75 Shock sensitivities of mixtures containing XeFz and XeF4were determined.XeFz and XeF4 are reported to be resistant to detonationbut partially hydrolysed XeF4 samples were found to be explosive. Explosionhazards are there when these fluorides come in contact with various materialslike paper, wool, sawdust, lubricants etc.There have been only a few recent general reports regarding the nature ofbonding in the noble gas compounds apart from the review literature quoted inthe reference section.177 A theoretical calculation based on configuration inter-action for XeF2 is reported to show that the valence bond structures whichincorporate the 5d,2 orbital at xenon contribute approximately 69 per cent ofthe total wave function and the resonance of F-Xe+F- and F-Xe+-F based onthe xenon 5pz orbital contributes only about 16 per cent of the total wavefunction.178 This brings out the importance of the valence bond structureswhich involve the 5d,2 orbital at Xe in covalent bonding.ACKNOWLEDGEMENTThe author expresses his gratitude to Professor R.D. 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Chem., 1970, 32, 1397.Jha 17

ISSN:0035-8940    

DOI:10.1039/RR9710400147

出版商:RSC    

年代:1971

数据来源: RSC

摘要:

GRAPH THEORY IN CHEMISTRYD. H. RouvrayChemistry Department, University of the Witwatersrand, Johannesburg, South AfricaIntroduction . . . . . . . . . . . . . . . . 173The representation of molecules . . . . . . . . . . 175Chemical nomenclature and documentation . . . . . . . . 176The enumeration of isomers . . . . . . . . . . * . 179The additivity principle . . . . . . . . . . . . . . 181Applications to bonding theory * . . . . . . . . . 183The representation of chemical systems . . * . * . . . 185The solid state . . . . * . .. . . . . . . . . 189Biochemical applications . . . . . . . . . . . . 192Conclusion. . . . . . . . .. . . . . . . . . 192References . . . . . . * . .. . . .. * . . . 193INTRODUCTIONA graph is an abstract mathematical concept probably most easily visualizedas a collection of points and lines, drawn so that pairs of the points areconnected together.Not all of the points are necessarily joined and so agraph need not be totally connected. Figure 1 shows two identically connectedgraphs. The type of line used to connect the points and the differing relativespatial orientation of the graphs is of no significance in graph theory. Whatis important is the way in which the points are joined together in the twographs. Labelling of the points makes it clear that these two graphs areidentically connected and therefore represent identical graphs.Graphs were first invented in the mid-1730s when Euler used idealizedFig. I. Two identically connected graphs. The type of line used, the length or angle of the lineor the relative spatial orientation of the graphs is of no importance here.Two such graphsare isomorphous.I 25 66 s, 5 I2 IRouvray 17networks to solve a celebrated problem of his time known as the Konigsbergbridge problem.1 They have since been employed in a variety of differentcontexts, such as Kirchoff's study of electrical circuits.2 More recently thetheoretical study of graphs has developed into an important branch ofmathernatics,3-8 with many applications to a large number of other disciplines.Thus today graph theory is making a significant impact in fields as diverse asecononiics,g psychology,lO nuclear and theoretical physics,llll2 linguistics,l3sociology14 and mathematical biology.15 Some of the major contributions tochemistry are discussed in this article.The use of graph-theoretical tech-niques in chemistry is becoming increasingly important, as graph theory hasbeen found to be sufficiently broad in scope to serve as a foundation for therepresentation and categorization of a very large number of chemical systems.It has also provided a valuable tool for obtaining useful chemical informationfrom such systems. In the future the great versatility of graph-theoreticalmethods will no doubt provide a framework for further significant develop-ments in all the major branches of chemistry.A few definitions basic to graph theory are given; more detailed definitionsare to be found elsewhere.16 A graph G may be defined as a set X of points,together with an operator f which generates the lines in the graph by mappingpoints of X into neighbouring points.A graph, which may be representedsymbolically as G = (X,F), thus gives all the neighbourhood relations for theset X. The function of r is to create from X a new set Y, consisting of un-ordered pairs of X as shown:i?{xlxa, xb, xc, . . .> --f { y / ( x k , xz), (Xr, Xa), (xb, xp), . . .>.The members of set X are the vertices of the graph; members of set Y,representing the lines in the graph, are edges. The number of members' ineach of these sets are related through the cyclomatic number p for thegraph, which is defined as:p = n e - n , + lwhere n, and n, represent the number of edges and vertices respectively. Agraph having ne = 0 will consist only of points with no lines and is termed anull graph; such graphs appear to be of little chemical interest at present.But if ne 2 1 the graph will contain at least one line and may then be used torepresent either a molecule or a chemical system.The cyclomatic number, which may take only integral values, is a con-venient parameter for categorizing different types of graph.Thus in Fig. 2four graphs are shown, each having a different cyclomatic number. Figure 2adepicts a (5,4) graph, so called since it contains five vertices and four edges.It has p = 0 which implies that the graph contains no closed loops or cyclesand is called a tree-graph or tree. Figure 2a could represent a very largenumber of molecules such as methane, methyl chloride or carbon tetra-chloride.In Fig. 2b p = 1 ; thus the graph contains one cycle and removal ofone edge would be necessary to reduce the graph to a tree. This particulargraph is called a multigraph as some of its vertices are connected by morethan one edge. In set notation this implies that the graph will possess repeated174 R.I.C. Review1 aCbdFig. 2. Four graphs used t o depict molecules. Graph (a) is a tree and could represent methane,( b ) contains one cycle and could represent ethylene, (c) contains seven independent cyclesand could represent coronene, (d) contains a large number of cycles and may be used t orepresent crosslinked polymer systems.members of set Y such as (xk, x,) and (xk, x,). This graph often depicts theethylene molecule.A more complex graph with p = 7, i.e. possessing a totalof seven independent cycles, is shown in Fig. 2c. This becomes a regulargraph when extended indefinitely since the same number of edges then meetat each vertex. The graph may be used to represent a coronene molecule or,when extended indefinitely in all directions, a single layer of graphite. Figure2d shows an irregular graph for which p is very large; such graphs are usefulfor representing crosslinked polymer systems.THE REPRESENTATION OF MOLECULESThe use of graphs to depict molecules was the first application of graphtheory to chemistry. Although largely taken for granted nowadays, it repre-sented a major advance and has even been claimed as possibly the mostfruitful scientific discovery ever made.17 The importance of this discoverycan hardly be over-emphasized since it has provided the necessary con-ceptual framework in which modern organic, and later inorganic, chemistrycould develop and flourish, and has thus been directly responsible for thefoundation and subsequent phenomenal growth of chemical industry over thelast 100 years.The originator of this mode of representation for molecules isdifficult to pinpoint as a number of workers had similar thoughts on thesubject at about the same time. The first use of the term ‘structural formula’Rorivray1217in its modern sense, however, is due to the Russian chemist Butlerov who in1861 wrote,l*. . . the chemical nature of a compound molecule depends on the natureand quantity of its elementary constituents and on its chemical structure .. .although the now famous description19 by KekulC of the discovery of thebenzene ring also shows his own interest in the problem at about the same time.The atoms were dancing before my eyes. . . . My mental eye. . . couldnow distinguish larger structures of manifold conformation ; long rows. . . alltwining and twisting in snake-like motion. But look! What was that? Oneof the snakes had seized hold of his own tail, and the form whirled mockinglybefore my eyes.This field of study is one rich in interest and controversy to the chemicalhistorian.It is now generally recognized that all covalent structures may be repre-sented by either a two-dimensional or a three-dimensional graph.The graphis basically a useful device for giving the neighbourhood relations in a mole-cule, i.e. it shows how the molecule is connected together. A graph thusprovides not only the chemical constitution but also the complete topology ofthe molecule it represents. The vertices give the time-averaged position ofthe atomic nuclei whilst the edges represent the valence bonds connecting thenuclei. An incidental advantage of this mode of representation is that it hasproved especially convenient when elucidating which of the physical orchemical characteristics of a molecule are essentially topology dependent.On the other hand a drawback has been the loss of all the stereochemicalfeatures of the molecule, such as chirality, although a number of attempts toovercome this difficulty have recently been made.20 A major advantage of therepresentation, however, lies in its exceptional versatility in both the depictionand categorization of molecular structure.Several paper~~l-23 and a numberof devoted to this aspect have appeared over the last few years.The representation has also been extended to the classification of inorganiccyclic compounds,*7 boranes,28 metallic compounds,29 aromatic hydro-carbons3O and molecules of biochemical interest.31CHEMICAL NOMENCLATURE AND DOCUMENTATIONGraph theory has been successfully employed in a number of problems whicharise from studies of chemical structure. Thus, in the field of chemicalnomenclature, a simple method of determining whether a given polycyclic,benzenoid molecule is cata- or peri-condensed, based on its characteristicgraph, has been proposed by Balaban and H a r a r ~ .~ ~ $ ~ 3 The characteristicgraph of the hydrocarbon is obtained by joining the centres of all the hexagonsin the molecule. If the characteristic graph is a tree the compound is cata-condensed whereas if it is not a tree the compound is peri-condensed. Thecharacteristic graphs for the triphenylene and perylene molecules, serving asexamples of the two cases, are shown in Fig. 3. This definition has led to anew systematic notation for benzenoid hydrocarbons.176 R.I.C. ReviewFig. 3. The characteristic graphs for (a) criphenylene and (b) perylene. Graph (a) i s a tree andche molecule cata-condensed whereas (b) is n o t a tree and the molecule is peri-condensed.(Double bonds omitted.)In recent years a large number of fairly extensive nomenclature systems,based on graph theory, have been devised both for the representation and forthe computer searching of structures.These developments have received aparticularly strong impetus in the last decade with the advent of high-speedcomputers with large storage capacity. Depending on the nature of theproblem to be tackled, these systems have represented molecular structuresby codes, matrices or polynomials. Coding was used by PennyZ4 who repre-sented the graphs of molecules by a system of numbers and typographicalsymbols. This representation has provided the basis for an efficient techniquefor searching files of chemical compounds stored in coded form.A similarmethod, developed by Meyer,35 has proved particularly well suited tosearching for molecules containing a given structural fragment within theirmolecular structure.A more ambitious project, initiated by Fugmann and others,36 hasattempted the representation of chemical concepts and relations in graph-theoretical language, depicted in a form suitable for storage by computer.The method is known as the Topological Representation of Synthetic andAnalytical Relations (TOSAR) of concepts. Any chemical paper to be com-puterized in this way is first subjected to expert analysis to display all theconcepts and relations contained in it in a suitable graphical form. Conceptsare depicted as points in this representation and relations form the edges ofthe graph.Figure 4 illustrates this method of reduction of a paper to graphicalformat showing the graph of a paper dealing with the fractional distillation ofAl(alky1)~ with the aid of added Al(alkyl)s-free propylene that has beenpreviously fractionally distilled and recovered from an oligomerizationprocess. The concepts and the relations are coded and introduced into thememory of the computer. The graphs may be retrieved at any time by relevantinquiry with the aid of suitable machine programmes. It is claimed that thismethod is capable of coding, storing in a computer and retrieving the contentsof any given chemical paper, however complicated it may be.Rouvray 17Propylene OligomerizationdistillationFractionaldistillationPropylene Al (a1 kyl),Fig.4. The graphical representation of a scientific paper. Vertices represent concepts andedges the relations between the concepts.Methods based on the representation of structures by matrices and poly-nomials have also proved their value in computerizing chemical information.A system due to S ~ i a l t e r ~ ~ is based on the so-called atom connectivity matrix(ACM) for a molecule. It is essentially a method of representing the graph ofthe molecule in matrix form as illustrated in Fig. 5a. Symbols for the atomsare placed along the principal diagonal of the matrix and the bonds placed inoff-diagonal positions corresponding to their location in the molecule. Thematrix may be expanded as a determinant to yield the characteristic poly-nomial of the m0lecule.~8 The notation has proved particularly well adaptedFig.5. Two convenient notation systems for the ethanol molecule. System (a) i s via t h eatom-connectivity matrix and system (b) i s the DENDRAL representation of the molecule.8 is used for t h e oxygen atom t o avoid possible confusion w i t h zero. The hydrogen atomshave not been explicitly considered in either system.I Ethanol178 R.I.C. Reviewto the storage and retrieval of structural information. Yet another system,developed by Lederberg and his co-workers,39 is known as the DendriticAlgorithm (DENDRAL) method. This uses a line representation for moleculesconsisting of letters, dots and other typographical symbols.Figure 5b showsan ethanol molecule in this system. The method is important as it has beensuccessfully extended to the interpretation of mass spectra by the matching ofstructures40 and also to the enumeration of structural isomers.41THE ENUMERATION OF ISOMERSIsomer enumeration is one of the oldest applications of graph theory tochemistry. In fact, the first attempts at enumeration came only a decade afterthe notion that graphs could represent molecules had gained general accept-ance. These early attempts were due to the noted English mathematician SirArthur Cayley, who in 1874 tried to find a general method for enumeratingisomers of the alkane homologous series.42 He represented all the isomers astree graphs, as shown in Fig.6, omitting the hydrogen atoms as these playedno part in his subsequent calculations. It soon became apparent, however,that no general formula explicitly relating the n in CnHzn+2 to the totalisomer count could be found. Cayley therefore directed his efforts to derivinga generating function which would give all the trees corresponding to any nvalue; this succeeded in yielding isomer counts for n values up to 13.43The method proved to be too cumbersome to get beyond n = 13 and, evenso, the last two enumerations proved to be in error. It was not until some 60years later that the American chemists Henze and Blair made the first majorbreakthrough in isomer enumeration by their introduction of an iterativetechnique.44145 Their method consisted essentially of evaluating the number ofways in which tree graphs of the hydrocarbons could be reconstituted to yieldnew configurations, assuming free rotation about all single C-C bonds.Although capable of giving results for all values of n, the method is laboriousin that each evaluation depends on the prior enumeration of the memberFig.6. Tree graphs representing the isomers for the first five members of the alkane series,CnHzn+2. (Hydrogen atoms omitted.)i n - - 3 - n = 4t Y Tn = 5R o u v r a y12*17Table I. Structural isomercounts of some members ofthe alkane series C n H 2 n + ~Member Formula Isomer countI23456783101520III23591835754347366 319containing one less carbon atom.Table 1 shows the isomer counts obtainedby these workers for small values of n.Shortly after the introduction of this iteration technique, which gave resultsfor the structural isomers in a number of homologous series, a theorem wasenunciated by P61ya46,47 which enabled the enumeration of many other kindsof series as well as other types of isomers such as geometrical isomers andstereoisomers. Because the theorem is rather sophisticated mathematicallyonly one small application in the enumeration of substituted benzene mole-cules will be outlined. The term cycle, as applied to a symmetry operationcarried out on the benzene molecule, is defined in terms of its order: a cycleof order one represents the coincidence of a given vertex with itself afterperformance of the symmetry operation, a cycle of order two represents theinterchange of two vertices after the operation, and so on. In benzene thereare cycles of order 1,2, 3 and 6.These cycles are collected together to form anexpression known as the cycle index, which for benzene has the form:1Z = 12 -. (f? + 4fz” + 3f:f; + 2f; + 2f6)where Z is the cycle index and f z represents n cycles of order m. The cycleindex may thus be regarded as a measure of how much symmetry a givenmolecule possesses. The configuration-counting series, which gives directlythe number of isomers, is obtained from Z by substituting for each f,: a termof the type (1 + xm) and then expanding as a power series in x thus:c = I + x + 3x2 + 3x3 + 3x4 $- xj + x6where C is known as the configuration-counting series and x is a variable.The coefficients in this series give the number of structural isomers of eachtype.For example, if two hydrogen atoms are substituted by some othergroup X, there will be a total of three possible isomers, since the coefficient ofx2 is three. These three correspond to the well-known isomers for the ortho,meta and para positions of benzene. The isomers for all the other cases areillustrated in Fig. 7.180 R.I.C. ReviewIX 3 x 2 3 x 3 3x4 x5 I ’ X ”Fig. 7. The possible substitution products for a benzene ring having one type of substituent,X, only. Determined by application of Polya’s Theorem.P6lya’s method has proved to be an extremely powerful tool in determiningisomer counts for series of all types.It has also been used to demonstrate theessential mathematical unity between structural, geometric and opticalisomerism.48 It is now being increasingly employed in the enumeration ofinorganic isomers and has recently been used for enumerating inorganiccomplexes49 and the boranes.50 Thus P6lya’s theorem is now regarded asone of the most fruitful graph-theoretical principles ever propounded. Theiterative method of Henze and Blair also continues to enjoy popularity as itis very easily adapted for computerization. This method too has been extendedto the enumeration of stereo isomer^^^ and, as mentioned ab0ve,~1 hasrecently been used by Lederberg and his co-workers in their DENDRAL methodof enumerating structural isomers. Much of isomer enumeration work hasbeen based on an implicit rather than an explicit use of graph theory.This isequally true for many of the applications described below.THE ADDITIVITY PRINCIPLEThe idea that the chemical components of molecules could display more orless fixed characteristics in series of molecules arose in the late nineteenthcentury. It soon led to attempts at calculating the properties of a moleculefrom those of its constituent parts. The concept on which this work was basedis known as the additivity principle and was first expressed in explicit form in1917.S2 It has since been used to determine a large range of physical andchemical properties of series of molecules such as heats of reaction, molecularvolumes or molecular refractivities. A number of expressions, all based ingraph theory, have been developed for deriving additive properties.ThusTatevskii and Papulov53 developed formulae for the heat of formation ofRoitvray 18compounds based on the assumption that the heat would be given by the sumof the energies of the individual bonds in the molecule. A slightly differentapproach was adopted by Bernsteins? who calculated heats of formation,assuming that this heat is given by the sum of all the possible interactionsbetween the atoms, both bonded and non-bonded, taken in pairs. The use ofgraph theory was made explicit by Smolenskiiss who employed the generalexpression :kL = 1f(G) = a0 + C aklXkjfor calculating the additive properties of the hydrocarbons.Here f(G) is theproperty to be calculated for each graph G of a set of graphs representing aseries of molecules, a0 and ak are physical constants and l x k l is a set ofsections of length k units for the molecule in question. Such formulae areusually found to give good results for the calculated properties of moleculeswithin homologous series.Another implicit use of graph theory was made by Dubois and his co-workers in their study of the relation between the structure and the chemicalproperties of a molecule.j6 They proposed the representation of molecules bylabelled, oriented graphs, such as that illustrated in Fig. 8a. The part of themolecule to be studied in detail is labelled as the focus and the rest of themolecule, known as the environment, is constructed around this focal pointin concentric spherical layers.The resultant three-dimensional graph may beconverted into matrix form as indicated in Fig. 86; such representation isconvenient for computerization of the inforniatioii contained within thegraph. The system on which this representation is based is known as theDescription and Automated Research of Correlation (DARC). It has beenused primarily in the search for correlations in the physical parameters ofseries of molecules. The approach has proved of value in studies of theFig. 8. Two representations of a molecule as ( 0 ) an oriented graph in concentric layers aboutsome focal point and (b) in matrix notation.182 R.I.C. Reviewinductive effect in non-conjugated systems57 and in studies of the carbonylstretching frequency in series of ketones.58.I1 20 3(1 4‘1 56 6APPLICATIONS TO BONDING THEORYThe manifold applications of graph theory, both in the valence bond and inthe molecular orbital approaches to the theory of bonding, cannot all bediscussed here. The uses of graph theory in this field are illustrated for a fewcases chosen at random. Thus Rumer’s rules59 for giving the allowed spectralterms for equivalent electrons with Russell-Saunders coupling, are based ongraphs representing the eigenfunctions corresponding to different distributionsof valence bonds (canonical structures) within a molecule. An implicit use ofgraph theory was also made in numerous early papers on valence bond theoryas in Pauling and Wheland’s study60 of the secular equations for the fivecanonical structures of benzene and the 42 such structures for naphthalene.An algorithm for determining the number of possible Kekulk forms for thehydrocarbons was reduced to a combinatorial problem by Gordon andDavison.61 A similar approach developed by Coulson62 used graph-theoreticalnotation for expanding secular determinants in terms of the structural com-ponents of the molecule under consideration.Some of the more recentapplications of graph theory to bonding have used graphs transposed intomatrix form.The most common method of transposing the graph of a molecule intomatrix notation is via the topological or structural matrix, better known ingraph-theoretical language as the adjacency matrix.The matrix is constructedFig. 9. Adjacency matrices for three different kinds of molecule showing how the topology ofthe molecule is reflected in the structure of the matrix.I 0 I 0 0 00 1 0 1 0 00 0 1 0 1 00 0 0 I 0 I2t 6x3 5 4 1;;;; i 1 1 0 0 0 0 01 0 0 0 0 01 0 0 0 0 0 a b0 4‘0 1 0 0 0 II 0 I 0 0 00 1 0 1 0 00 0 1 0 1 00 0 0 I 0 I1 0 0 0 I 0Rou way 18simply by inserting ones into all the (i, j) positions of the matrix correspondingto a bond between atoms i and j in the molecule and zeros for all the otherpositions. Figure 9 shows a number of adjacency matrices for differentmolecules. In Fig. 9a the matrix for a star-like molecule is shown, in Fig. 9bthat for a chain molecule and in Fig.9c that for a cyclic molecule. In each casethe structure of the molecule is clearly reflected in the matrix. Thus the non-zero entries for the star molecule appear only in the first row and column ofthe matrix whereas those for the chain molecule lie on either side of the princi-pal diagonal of the matrix. Such matrices have frequently been used in themolecular orbital approach to the theory of bonding.It was first demonstrated in 1956 by Giinthard and prima^^^ that the adja-cency matrix of a molecule may be used in place of the Hiickel matrix ofmolecular orbital theory. The proof given below is due to SchmidtkeG4 whostarted by representing a typical eigenvalue problem by the equation :/ H - ESI = 0where H is the Hamiltonian operator for the system in matrix form, S theoverlap matrix and E the energy, acting as a scalar multiplier.The matricesH and S are resolved into the following component parts thus:H=ctl+PMS = 1 f S Mwhere 1 is the unit matrix, CL and /3 are respectively the Coulomb and resonanceintegrals for the system (all as and all Ps are taken to be identical) and M isthe adjacency matrix for the molecule in question. Substitution of theseresolved expressions into the original secular equation now yields the result:where the expression in the inner bracket represenu the so-called Hiickelnumbers which give the energy eigenvalues for the molecule. The eigenvectorsof the system will also be the same as in the Hiickel treatment since thematrices H and M commute and must therefore have identical eigenvectors.This simple proof serves to illustrate the importance of the r81e which thetopology of the molecule, represented here through the adjacency matrix ofits graph, plays in determining the physical and chemical characteristics ofmolecules.The adjacency matrix has featured recently in a number of other contextsin bonding theory. The above approach is applicable to systems containingno n-electrons and may, therefore, be used in discussing inorganic as well asorganic molecules.G~ Heteroatomic systems may also be treated by resolvingthe graph of the molecule into a number of subgraphs, such that each containsvertices corresponding to one type of atom only.Thus the graph for thetetrahedrally symmetric system AB4 shown in Fig.IOa may be resolved intothe two subgraphs shown in Fig. lob. A full analysis of the species, repre-sented by the interaction of the subgraphs, may then be treated as a perturba-tion problem in quantum theory.64 The electronic structure of tetrahedral184 R.I.C. Re\,iewGaG2bFig. 10. Resolution of the tetrahedrally symmetric species AB4 shown in (a) produces the twosubgraphs shown in (b). The interaction of the subgraphs t o reconstitute the AB4 species isrepresented as GI @ GB.molecules, such as P4, has also been discussed by Kettle66 using a simpletopological equivalent orbital approach. The adjacency matrix was featuredprominently in a study by Kettle and Tomlinson67 of the electronic structureof the boron hydrides.In this latter case use was made of these matrices todemonstrate that the theory developed by these workers represented a topo-logically correct extension of Huckel theory to three dimensions.The use of appropriate operators applied to adjacency matrices can alsoyield important information about the system represented. Ruedenberg68 usedsuch a method to obtain bond orders for molecules and to derive a numberof basic theorems in molecular orbital theory including the well-knownCoulson-Rushbrooke theorem69 on charge orders in alternant molecules. Amore recent study by England and Ruedenberg70 utilized the adjacencymatrix in a calculation of the energy of localized rr-orbitals for a number ofaromatic hydrocarbons. As a result of their analysis they were able to relatelocalized molecular orbitals to Pauling bond orders and also give a newinterpretation of the resonance energy of these molecules.Other workers71have examined the appearance of excessively degenerate energy eigenvaluesin the Huckel approach by studying the irreducible representations of thegroup which map the graph of the molecule onto itself.THE REPRESENTATION OF CHEMICAL SYSTEMSThe use of graphs to represent complex chemical systems is comparativelyrecent. During the last five years graphs have been increasingly employed torepresent in a convenient and concise way a great deal of information aboutsystems which are often very complicated. Several representations whichhave been used to depict the interconversion of a penta-coordinate complex,ML5, into all of its stereoisomers via a pseudo-rotation process72 will be de-scribed.Such a process, first described by Be~ry,~3 leaves the molecule in arotated and permuted form of its original state. The literature contains manyexamples of this type of isomerization process, as described in a review byRamirez.74 Only the interconversions of trigonal bipyramidal moleculesRouvray131899 7 , i 5 sFig. I I. Graphical representations for the interconversions of a penta-coordinate ligand,MLs, via a Berry process. Figures denote possible isomers and figures with bars denote thecorresponding enantiomers.having five distinguishable coordinating ligands will be discussed, assumingthat each interconversion step depends at most on the result of the stepimmediately preceding interconversion and not on any earlier result.Under such conditions representation of the ML5 system is possible by useof the three-dimensional graph of the regular pentagonal dodecahedron.This suggestion was first made by Muetterties75 and later discussed in detailby Lauterbur and ram ire^.^^ In this representation, illustrated in Fig.Ila,the vertices of the dodecahedron are used to represent the 20 possibleisomeric forms of the MLs complex. Edge midpoints represent the 30 squarepyramidal transition states and some of the edges of the graph representinterconversion processes. It is not possible for all the edges to represent suchprocesses since the shortest closed cycle of isomerizations includes six isomersin all, which cannot be conveniently mapped around the pentagonal faces ofthe dodecahedron. Several improved representations have thus been proposedfor this system.One improved representation was described by Dunitz and Prelog76 and isillustrated in Fig.I l b . It is a bridgeless graph which is regular of degreethree, known as the Petersen graph. In this case the vertices of the graphrepresent either a given isomer or its optical antipode and are joined togetheronly if they contain no common isomeric forms. The edges thereby createdrepresent the possible interconversion steps and any sequence of inter-conversions is thus associated with a corresponding path between vertices ofthe graph. A cyclic path will lead back to the original enantiomer if thenumber of interconversion steps is even, and to the opposite enantiomer ifthe number of steps is odd.A second improved representation of this system,originally proposed by Dunitz and Prelog76 and later developed by Gielenand Nasielski,77 is illustrated in Fig. I l c . This makes use of a three-dimen-sional graph based on two adamantane skeletons. The insertion of two extravertices at the midpoints of two opposing faces of the graph provides a total of1 a6 R.I.C. Review20 vertices, so that each enantiomer may be represented by only one vertexIn addition to a vertical S6-axis the graph possesses a vertical Sz-axis whichpermits transformation of each isomer into its enantiomorph. The labellingsystem of the vertices is such that the shortest paths between pairs of enantio-mers encompass in all a total of six different isomers, as required for thisparticular system.The compact and concise nature of this representation of chemical systemshas made desirable its extension to other systems. Up to the present, worksimilar to that described has been carried out on hexa-coordinate systems fordiffering types of transition processes by Gielen and his co-workers.7*-80 Alsothe representation of the stereochemical displacement reactions occurring atthe phosphorus atom in phosphonium salts and cognate systems has beenstudied by De Bruin and others81 using a graph based on a hexasterane-typestructure.This work provided new insight into the accessible pathways andthe stereochemistry in this kind of displacement reaction. Muetterties82183has attempted to establish general principles for the topological representationof stereoisomeric processes which occur in molecular species via polytopalrearrangement, i.e.such that all stages of the transition may be representedby idealized polyhedra. From these studies useful information on the differ-ences in the stereoisomerizations for alternative mechanisms and the relative3.00.03,3 a3,3 CRouvray3,O3,3 bFig. 12. Representat 'n by graphical em-bedding of the reaction pathways in theal kylation of aliphatic ketones. Thegraphsdepict the cases where the reaction is(a) direction-specific, (b) partially direc-tion-specific and (c) non direction-specific.18energy relationships involved has emerged.In an investigation of the reactionpaths in the base-catalysed polyalkylation of aliphatic ketones, carried out byDubois and Panaye,84 use was made of a graphical embedding procedure in alattice network. In Fig. 12 three such alkylation graphs are shown. Figure 12adepicts a direction-specific alkylation process starting from an unsubstitutedacetone (denoted as O,O), through a number of intermediates to hexamethylacetone (denoted as 3,3). If no stage of the alkylation were direction-specificthe entire graph would be traced out as shown in Fig. I2b. A partial embed-ding lying between these two extreme cases is obtained for the polyethylationof neohexylmethyl ketone and is illustrated in Fig. 12c.The method isgenerally applicable in the study of multi-step syntheses and may be ofespecial importance in reactions where optimization of yields is a pre-requisite.Another important outcome of the representation of complex chemicalsystems by graphs is that, in some cases, the graphs have enabled the predic-tion of hitherto unsuspected reactions. Thus in a systematic survey of reactionsproceeding through six-membered transition states, carried out by Balaban,85it was shown that novel reactions were possible which could lead to newautomerizations, although the purpose of the investigation was only to givea complete graph-theoretical description of all the allowed thermal electro-cyclic reactions. In a similar way the study by Balaban and co-workers86 of themultiple 1,Zshifts in carbonium ions has enabled them to calculate not onlythe number of all possible carbonium ions which can be produced from suchFig.13. Graphical representation of the family of possible carbonium ions produced bymultiple I,2 shifts. Starting from the carbonium ion R1RzRsC-CR4R5, denoted here as1,2,3*4,5, the diagram closes itself at the optical antipode of this ion, 4,5*1,2,3.+I 1,2,3*4,52,3,4* I ,53,4* I ,2,5I 4,5* 1,2,3I ,2,5*3,4I ,5*2,3,4188 R.I.C. Reviewshifts in a given species but also the number of independent pathwayspermitting this kind of interconversion. The graph depicted in Fig. 13 showsall the possible products for the R ~ R z R ~ C - C R ~ R ~ carbonium ion, denotedhere by the shorthand notation 1,2,3*4,5, and the transitions between them.This scheme reveals that the branchng tree emerging from the starting point,indicated by 1,2,3*4,5, closes itself at the antipode of this member.Thesmallest circuit in this graph contains a total of six different transformations.This analysis has permitted these workers to devise complete sets of inter-convertible carbonium ions for a given species and also to compute theprobabilities for finding a given compound among the isomerization productsof any given ion.THE SOLID STATEThe Russians have made a number of the important contributions to theunderstanding of phase equilibria. This work, which extends back over40 years, has been concerned mainly with transitions in the solid state.Thefirst approaches to this study were made by K ~ r n a k o v , ~ ~ * ~ ~ who pointed outthe essential topological nature of phase diagrams, using several graph-theoretical concepts in his description. This analysis of phase equilibria waslater extended by Kurnakov*g and many of his co-workers, such asMlodzeev~kii.~~ They regarded the phase diagram basically as a closed topo-logical simplex, i.e. the n-dimensional analogue of a tetrahedron (n > 3).Multicomponent heterogeneous systems were similarly described by Palatnikand co-workers,91@ and D o m b r o v ~ k a y a ~ ~ ~ 9 ~ extended this work by analysingthe composition diagrams of multicomponent systems in which chemicalcompounds are formed. This was achieved by subdivision of the compositiondiagram into simpler forms by means of section, triangulation, tetrahedration,and so on.Furthermore, Levin95 and Klochko96 have indicated a similarityin form between a well-known graph-theoretical formula due to Euler relatingthe edges, faces and vertices of a polyhedral graph:V + F = E + 2,where V, F and E denote respectively the number of vertices, faces and edgesof the polyhedron, and the Phase Rule of Gibbs:P + F’ = C + 2,where P, F’ and C denote respectively the number of phases, degrees offreedom and components of a system in chemical equilibrium.A particularly useful way of representing phase diagrams in terms of graphtheory was devised by Seifer and Stein.97 Their method gives informationabout the general structure of any phase diagram and also any specificchemical compounds it may contain.The method has been used for storingand searching information pertaining to phase diagrams in the memory of acomputer.98 They started by pointing out the three main structural featuresof such diagrams, namely points, lines and fields. By representing the fieldsas vertices of a new graph it is then possible to construct a so-called dualgraph of the system by a process illustrated in Fig. 14. To each field is ascribedRouvray 18aNCN6IFig. 14. A n illustration of the procedurefor constructing the dual graph of a phasediagram. The null field i s denoted hereby N.a vertex (including the whole region which lies outside the diagram, known asthe null field) and the vertices are then joined as shown in Fig.14a. Thisgraph gives the relative positions of the fields and is used in the constructionof a second graph, shown in Fig. 14b, where the intersections of the dualgraph and the phase boundaries are now represented by vertices. The numberof intersection vertices is equal to the number of lines separating each fieldfrom the others. The last stage of the construction, illustrated in Fig. 14c,190 R.I.C. ReviewFig. 15. Three topologically equivalent fragments of a polymer network. The three casesdepict (a) no contact between opposite pairs of chains, (6) contact between the chains and(c) threading of the chains. Such graphs are helpful when investigating the properties ofpolymers.involves ‘colouring’ or marking the intersection vertices.Every such vertexoriginating from the same line in the phase diagram is assigned the samecolour, denoted here by the same value of 01. The vertices representing fieldsmay also be coloured using a different value of ,B for each as shown.Another area of study of the solid state where graph theory plays animportant r61e is in the field of macromolecules99 and polymers.100 In general,a polymer is produced when a p-functional chemical species reacts with aq-functional species, where both p and q have values greater than one. Inthe special case when p = q = 2 a polymer having a linear graph will beobtained; in all other cases branching and crosslinking is possible and thegraph of the polymer is likely to resemble that shown in Fig.2d. The differentkinds of crosslinking in polymeric structures have been discussed in terms oftheir graphs in some detail by Elias.101 Because studies on polymers frequentlyinvolve the way in which crosslinking occurs between molecules rather thanthe metrical properties of the molecules themselves, graphical representationof such systems has often proved very convenient, e.g. in providing a frame-work model for studies relating to the elasticity of polymer networks.102 In thislatter study the threading characteristics of polymers were considered and thethree graphs shown in Fig. 15 were used to depict segments of polymernetworks. All three graphs represented are topologically equivalent, thoughthey differ considerably in their chemical nature.In Fig. 15a no threading hastaken place, as is the case for the graph in Fig. 15b, although now there is aclose contact between opposite pairs of chains. In Fig. 15c threading of thechains has occurred. Such representation proved valuable in explaining anumber of properties of polymer networks such as their extent of swelling,i.e. their behaviour on dilution with inert solvents.l03The use of three-dimensional, regular lattice graphs to represent solids istoo well-known to merit comment here. However one model, based on alattice network, has a number of applications in different fields and deservesbrief mention. This is the Ising model of a crystallo4 and is the simplestpossible model of a crystalline system to exhibit a phase transition.In thismodel the lattice atoms may exist in either of two possible states; two inter-acting atoms have a negative interaction energy if they are in the same stateRouvray 19and a positive interaction energy if not. A second network can be constructedfor this model if all the interactions between the atoms are represented bylines joining them together. Where there is no interaction no connectionbetween them is made. Studies based on this latter network make possible thecalculation of a number of properties of the crystal. In this way the systemhas served as a model for ferromagnetic and anti-ferromagnetic crystals,l05binary alloys showing either phase separation or superlattice formation,l06lattice gasesl07 and for percolation and clustering problems.108J09 A widerange of physical and chemical problems associated with repeating networkscan thus be reduced to graph-theoretical format and solved by the use ofappropriate combinatorial techniques.BIOCHEMICAL APPLICATIONSGraph theory is also used in the field of biochemistry. A method for repre-senting any protein molecule as a tree graph was described by Hermans andFerro,llo who used their symbolism in computer programmes designed forthe calculation and subsequent modification of the atomic coordinates ofthese structures.Muncklll attempted the symbolic representation in termsof graphical networks of metabolic, endocrine and possibly other biochemicalsystems. He developed an algorithm for finding the structure of a networkconsistent with a prescribed set of displacements from steady-state values forany given components of the system.Graph-theoretical methods for theanalysis of multisubstrate enzyme reactions using kinetic data112 and for thegeneration of polymer sequences by use of fragmentation data113 have alsobeen proposed. In a number of cases use of graph theory has considerablyfacilitated the determination of certain physical or chemical parameters of asystem. Thus, in the derivation of the rate laws for enzyme-catalysed reactionsa graph-theoretical method due to Volkenstein and Goldstein114 greatlysimplified the more laborious method previously described by King andAltman.115CONCLUSIONI have attempted to give a concise account of some of the major applicationsof graph theory to chemistry.It is hoped that the examples chosen serve todemonstrate the great versatility and elegance of graph-theoretical methodsas applied in the various branches of chemistry. All of the applications ofgraph theory presented here, with a few notable exceptions, have beenconcerned primarily with the concise representation of existing chemicalknowledge. In exceptional cases entirely new information is obtainable fromgraph-theoretical techniques, e.g. in the field of isomer enumeration or inpredicting novel types of reaction. As the full potential of graph-theoreticalmethods is more widely appreciated this latter type of application will doubt-less become more prominent. Specific areas in which graph theory is likelyto have an important impact in future include the further study of problemsrelating to the Ising model, diffusion problems, the behaviour of polymers,the use of Feynman diagrams, and extensions of the theory to the field ofstereochemistry.192 R.I.C.ReviewA number of hitherto unsolved problems in graph theory could haveimmediate chemical consequences if solution proves possible. Thus the well-known cell-growth problem is clearly related to the properties of peri-condensed hydrocarbons and the full solution of the Ising problem wouldgive much information on systems described by this model. In fact, thewords of the mathematician Sylvester116 written in 1878 form an appropriateconclusion to this review :The more I study . . . the more I become impressed with the harmony .. .which exists between the chemical and algebraical theories. . . . There is anuntold treasure of hoarded algebraical wealth potentially contained in theresults achieved by the patient and long-continued labour of our uncon-scious and unsuspected chemical fellow workers.ACKNOWLEDGEMENTSThe hospitality of the Mathematical Institute, University of Oxford, wherethis project was completed during a sabbatical leave, was graciously extendedby Professor C. A. Coulson. Thanks are also due to Dr Robin J. Wilson for avery helpful critique of the manuscript.REFERENCES1 L. Euler, Comment. Acad. Petropolitanae, 1736, 8, 128.2 G. Kirchoff, Ann. Phys. und Chem., 1847, 72,497.3 D. Konig, Theorie der endlichen und unendlichen Graphen. Leipzig : AkademischerVerlag, 1936.Reprinted New York: Chelsea, 1950.4 0. 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Laue, Elemente der Graphentheorie und ihre Anwendung in den biologischen Wissen-and Breach, 1962.McGraw-Hill, 1967.Czech. Acad. Sci., 1964.1963.schafren. Leipzig : Akademischer Verlag, 1970.16 J. W. Essam and M. E. Fisher, Rev. mod. Phys., 1970,42,272.17 G. E. Hein. KekulP Centennial, Adv. Chem. Ser. 61 (R. F. Gould, ed.). Washington:Amer. chem. SOC., 1966.18 A. M. Butlerov, 2. Chemie, 1861, 549.19 A. Kekule, Annln Chem. Pharm., 1858, 106, 129; Chem.Ber., 1890,23, 1302.20 I. Ugi, et al., Angew. Chem., 1970, 82, 741 and references contained therein.21 W. J. Wiswesser, J. chem. Docum, 1968, 8, 146.22 H. Hiz, J. chem. Docum, 1964, 4, 173.23 J. Lederberg, Proc. natn. Acad. Sci. US, 1965, 53, 134.24 L. Pauling and R. Haywood, The architecture of molecules. San Francisco: Freeman,25 A. F. Wells, The third dimension in chemistry. Oxford: Clarendon Press, 1956.26 A. F. Wells, Models in structural inorganic chemistry. Oxford: Clarendon Press, 1970.Rouvray 193196427 I. Haiduc, Zh. strukt. Khim., 1961, 2, 374.28 W. N. Limcomb. Science. 1966.153. 373.29 G. B. Bokii, et ai., Zh. stiukt. Khim.; 1961, 2, 68.30 E. Clar, Chimia, 1964, 18, 375.31. F. Lynen, Naturw. Rdsch. Stuttg., 1970, 23, 263.32 A.T. Balaban and F. Harary, Tetrahedron, 1968, 24, 250533 A. T. Balaban, Tetrahedron, 1969, 25, 2949.34 R. H. Penny, J. chem. Docum, 1965,5, 113.35 E. Meyer, Angew. Chem., 1970, 82,605.36 R. Fugmann, et al., Angew. Chem., 1970,82,611.37 L. Spialter, J. chem. Docum, 1964, 4, 261.38 L. Spialter, J. chem. Docum, 1964,4,269.39 J. Lederberg, NASA Report CR-57029, 1964.40 A. M. Duffield, et al., J . Amer. chem. Soc., 1969,91, 2977.41 J. Lederberg, et al., J . Amer. chem. Soc., 1969, 91, 2973.42 A. Cayley, Phil. Mag., 1874, 67, 444.43 A. Cayley, Chem. Ber., 1875, 8, 1056.44 H. R. Henze and C. M. Blair, J . Amer. chem. Soc., 1931, 53, 3077.45 H. R. Henze and C . M. Blair, J. Amer. chem. Soc., 1931, 53, 3042.46 G. Polya, Acta math., 1937, 68, 145.47 G.Pblya, Z. Krist., 1936, 93, 415.48 T. L. Hill, J . chem. Phys., 1943, 11,294.49 I. V. Krivoshei, Zh. strukt. Khim., 1967, 8, 321.50 N. V. Emelyanova and I. V. Krivoshei, Zh. strukt. Khim., 1968, 9, 881.51 E. S. Allen and H. Diehl, Iowa St. Coll. J. Sci., 1942, 16, 161.52 H. R. Redgrove, Chem. News Lond., 1917, 116, 37.53 V. M. Tatevskii and Y . G. Papulov, Zh. $2. Khim., 1960, 34, 241.54 H. J. Bernstein, J. chem. Phys., 1952, 20, 263.55 E. A. Smolenskii, Zh.fir. Khim., 1964, 38, 1288.56 J. E. Dubois, Entropie, 1969, 27, 1.57 M. Chiang and T. Tai, Acta chim. sin., 1962, 28, 330.58 J. E. Dubois, et al., J . mol. Struct., 1969, 4, 403.59 G. Rumer, Nachr. Ges. Wiss., Gottingen, Math-Phys Klasse, 1932, 33760 L.Pauling and G. W. Wheland, J . chem. Phys., 1933, 1, 362.61 M. Gordon and W. H. T. Davison, J . chem. Phys., 1952, 20, 428.62 C. A. Coulson, Proc. Camb. phil. SOC. math. phys. Sci., 1950, 46, 202.63 H. H. Gunthard and H. Primas, Helv. chim. Acta, 1956, 39, 1645.64 H. H. Schmidtke, J. chem. Phys., 1966, 45, 3920.65 H. H. 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ISSN:0035-8940    

DOI:10.1039/RR9710400173

出版商:RSC    

年代:1971

数据来源: RSC

摘要:

Cumulative IndexVolume 1, 1968; Volume 2, 1969; Volume 3, 1970; Volume 4, 1971.Air pollution . . . . * . . . . . .. .. . . 3, 119Barrett, C. F., Air pollution . . . . .. . . . . . . 3, 119Bond, G. C., Catalysis in the context of chemistry . . . . . . 3, 1degradation . . . . .. . . .. .. .. . . 3, 45Betteridge, D., The teaching of chemistry in Victorian and Edward-ian times . . . . . . . . . . . . . . . . 3, 161Bourne, E. J . and P . Finch, Polysaccharides-enzymic synthesis andBriggs, G. G.-see Graham-Bryce, I. J.Cairns, A . C. H., Chemicals and the world economy . . . . 2, 41Chemical applications of ultrasonic absorption measurements inChemical education : problems of innovation .. . . . . 1,205Chemicals and the world economy . . . . * . . . . . 2, 41Chemistry and nutrition * .. . .. . . . . . . 2, 143Catalysis in the context of chemistry . . . . . . * . . . 3, 1the liquid state, Some * . . . . . . . . . . . 2, 59Chemistry and the consumer . . . . . . * . * . . . 1, 1Chemistry and the origin of life . . . . . . . . . . 2, 1Chemistry of tribology, The . . .. . . . . . . .. 1,135Computer simulation of some physicochemical problems . . 4, 97Chemistry and physics of enzyme catalysis, The . . . . . . 2, 117Chemistry of noble gas elements, Recent advances in the . . . . 4, 147Chemistry in Victorian and Edwardian times, The teaching of . . 3, 161Cragg, R. H., Lord Ernest Rutherford of Nelson (1871-1937) , , 4, 129Crossley, John, Dielectric relaxation and molecular structure inliquids .. . . . . * . . . .. . . . . . . 4, 69Currell, B. R. and M. J . Frazer, Inorganic polymers . . .. 2, 13Dielectric relaxation and molecular structure in liquids . . . . 4, 69Doonan, S., The chemistry and physics of enzyme catalysis . . 2, 117Economy, Chemicals and the world . . * . .. . . . . 2, 41Electrochemistry, Organic . . . . * . * . * . . . 2, 87Electron resonance in anisotropic solvents . . . . . . . . 3, 61Environment, Pollution of the . . . . . . . . . . . . 3, 85Enzyme catalysis, The chemistry and physics of . . . . . . 2, 117197Experiment, imagination and meaning . . . . . . . . 4, 4Farrar, W. V., Kathleen R. Farrar and E. L. Scott, The Henrys ofManchester . . . . . . . . . . .. . . . * 4, 35Fermentation-the last ten years and the next ten years .. . . 3, 135Finch, P.-see Bourne, E. J.Fish, H., Water pollution . . . . . . . . . . . . 3, 105Fleischmann, M. and D. Pletcher, Organic electrochemistry . . 2, 87Frazer, Alastair, Chemistry and nutrition . . . . . . . . 2, 143Frazer, M . J.-see Currell, B. R.Frost, B. R. T., Nuclear fuels . . . . . . . . . . . . 2, 163Fuels, Nuclear . . * . . . * . . . . . . . . . 2, 163Gowenlock B. G. and C. A . F. Johnson, Techniques of physicalmeasurement: vacuum technique . . . . . . . . . . 1, 107Graham-Bryce, I. J. and G. G. Briggs, Pollution of soils . . . . 3, 87Graph theory in chemistry . . . . . . . . . . .. 4, 173Hallam, H. E., Infrared and Raman spectra of inorganic compoundsHalliwell, H. F,, Chemical education: problems of innovationHogg, D.R . and R. B. Moyes, Practical aspects of programme writ-1, 39. . 1, 205Henrys of Manchester, The . . . . . . . . . . . . 4, 35ing . . . . . . . . .. . . . . * . . . 3, 27Infrared and Raman spectra of inorganic compounds . . . . 1, 39Inorganic polymers . . . . . . . . . . * . . . 2, 13Ives, D. J. G. and T. H. Lemon, Structure and properties of water 1, 62Jha, N . K., Recent advances in the chemistry of noble gas elementsJohnson, C. A . F.-see Gowenlock, B. G.4, 147Lal, M., Computer simulation of some physicochemical problems 4, 97Lemon, T. H.-see Ives, D. J. G.Lord Ernest Rutherford of Nelson (1871-1937) . . * . . . 4,129Luckhurst, G. R., Electron resonance in anisotropic solvents . , 3, 61Life, Chemistry and the origin of .. . . . . . . . . 2, 1Miall, L. M., Fermentation-the last ten years and the next ten yearsMoyes, R. B.-see Hogg, D. R.3, 135Nuclear fuels . . .. .. .. .. . . . . . . 2, 163Nutrition, Chemistry and . . . . . . . . . . . . 2, 143Oparin, A . I., Chemistry and the origin of life . . .. . . 2, 1Organic electrochemistry . . . . * . . . . . . . 2, 8719Petroleum chemicals today and tomorrow . . . . . .Physicochemical problems ; Computer simulation of some . .Pletcher, D.-see Fleischmann, M.Pollution, Air . . . . . * . . . . . . ..Pollution of soils . . . . . . . . . . . .Pollution of the environment . . . . . . . . . .Pollution, Water . . . . . . . . . . . .Polymers, Inorganic . . . . . . . . . . . .Polysaccharides-enzymic synthesis and degradation ..Practical aspects of programme writing . . . . ..Programme writing, Practical aspects of . . . . . .Recent advances in the chemistry of noble gas elements . .Roberts, Eirlys, Chemistry and the consumer . . ..Rouvray, D. H., Graph theory in chemistry . .Rowe, Geoffrey W., The chemistry of tribology . . . ,Rutherford of Nelson (1871-1937), Lord Ernest . . * ... . .. .. .. .. .. .. .. .. .. .* .. .. .. .. .Scott, E. L.-see Farrar, W. V.Smith, Ivor, Stereoviewing: visual aids for stereochemistry andmacromolecular structures . . . . . * . . . . . .Soils, Pollution of . . . . . . . . . . . . . .Stereoviewing : visual aids for stereochemistry and macromolecularstructures . . . . . . . . . . . . . . . .Structure and properties of water . . * . . . . . . .Teaching of chemistry in Victorian and Edwardian times, The . .Theobald, D. W., Experiment, imagination and meaning . . . .Transition metal ions in biological processes, Role of . . . .Ultrasonic absorption measurements in the liquid state, Somechemical applications of . . . . . . . . . . . .Vacuum technique . . * . . . . . . . . . . .Stereoviewing . . . . . . . . . . . . . . . .Visual aids for stereochemistry and macromolecular structures,Waddams, A . L., Petroleum chemicals today and tomorrow . ,Williams, R. J. P., Role of transition metal ions in biologicalprocesses .. .. * . .. a . .. ..Wyn-Jones, E., Some chemical applications of ultrasonic absorptionWater pollution . . . . . . . . . . . . . .measurements in the liquid state .. .. .. ..4, 14, 973, 1193, 873, 853, 1052, 133, 453, 273, 274, 1471, 14, 1731, 1354, 1294, 193, 874, 191, 623, 1614, 491, 132, 591, 1074, 194, 13, 1051, 132, 5919

ISSN:0035-8940    

DOI:10.1039/RR9710400197

出版商:RSC    

年代:1971

数据来源: RSC