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QUARTERLY REVIEWS ENERGY TRANSFER IN GASEOUS COLLISIONS By J. C. MCCOUBREY PH.D. D.PHIL. and W. D. MCGRATH PH.D. (IMPERIAL COLLEGE OF SCIENCE & TECHNOLOGY LONDON S.W.7) THEORIES of energy transfer have been given by Oldenberg and Frost 1 and by Massey.2 Certain special topics in this field have more recently been considered in a number of book^.^-^ The purpose of this Review is to present the unifying aspects and some of the recent work in this ex- panding field in particular such work as has been related to molecular structures. The scope has been limited to transfers involving vibration or rotation since these are of primary importance for chemical reactivity. Considerable attention has been paid to energy transfer involving low-lying energy levels. 1. General Introduction Low Energy Levels.-Many physicochemical processes are concerned with systems perturbed from.the equilibrium position by uptake of energy which then revert to equilibrium. A sudden input of energy may be transmitted to the system in some quite definite form e.g. by an increase in translational energy ; this input energy will ultimately become redistributed by collisions through all the available levels of the system a t a rate dependent upon the nature and number of these levels. A rapid change of an external parameter affecting the energy of the system followed by the subsequent shift of equilibrium which proceeds with a finite time lag forms the basis of a so-called relaxation process.3 Experimentally this finite relaxation of equilibrium or energy-transfer effect is studied in terms of the bulk properties of the gas.Examples of relaxation effects may be found in experiments on the change of velocity of sound with frequency at high frequencies or in the rapid expansion of gases out of jets or nozzles.* In these experiments 0. Oldenberg and A. A. Frost Chem. Reviews 1937 20 99. a H. S. W. Massey Repmts Prog. Phys. 1949 12 248. 3 H. S. W. Massey and E. H. S. Burhop '' Electronic and Ionic Impact Phenomena " * K. J. Laidler " The Chemical Kinetics of Excited Species " Clarendon Press 5 A. F. Trotman-Dickenson " Gas Kinetics " Butterworths London 1955. 6 I<. F. Herzfeld '' Thermodynamics and Physics of Matter " Ed. Rossini Princeton 7 I,. Bergman '' Der Ultrnschall ' I 5th edn. Herzel Ziirich 1949. 8 P. W. Huber and A. Kantrowitz J. Chem. Phys. 1947 15 275; W. Griffiths Clarendon Press Oxford 1952.Oxford 1955. 1955. J . Appl. Plrysics 1950 21 1319. U 87 88 QUARTERLY REVIEWS the steady-state condition set up depends to some extent upon the rate of redistribution of translational energy into internal molecular modes. In sound propagation or zerodynamic phenomena comparatively small changes in translational temperature are involved in the compressions or rarefaetions so that the internal modes into which energy must be equi- librated are not far from the ground state e.g. only one or two vibrational quanta above the zero-point half quantum. Higher Energy LeveZs.-The action of light upon molecules may in general bring about changes of both electronic and vibrational quantum numbers. Subsequent chemical reactivity of these molecules or their re-emission of light is conditioned by the way in which all or part of their excess energy may be transferred to other molecules.g * Under these conditions we may be concerned with vibrational energy levels ten to twenty above the xero-point energy.A number of thermal reactions such as occur in flames or oxidations involve the formation of transient species which contain sufficient energy to make them decompose again without taking further part in the main reaction. Removal of this excess energy serves to stabilise these inter- mediate The energy transferred in this case is vibrational in the region of the self-dissociation level. Magnitude of Energy Levels.-If the molecular potential-energy curve were a perfect parabola i . e . a perfectly harmonic oscillator all the vibra- tional energy levels of a molecule would be equally spaced.Real potential- energy curves refer to anharmonic oscillations ; the vibrational energy levels of such an anharmonic oscillator may be approximately expressed by two terms in a simple power series. The separation between levels is given (in units of cm.-l) l o by AG = V/C - x,(v/c)(~& + Z ) where v is the vibration frequency Q the quantum number x an anhar- monicity factor and c the velocity of light. It follows that the separation between vibrational levels AG decreases with increasing value of the quantum number. The magnitude of energy quanta will therefore vary considerably for the same molecule in relation to experiments on aerodynamic photochemical and reactivity phenomena. I n consequence the detailed mechanisms of energy transfer may vary considerably in these different cases.2. Transfer of Energy between Translation and Vibration or Rotation Quanta involving Low-lying Energy States of Molecules (a) A Study of the Dispersion and Absorption of Xi&-frequency Sound.- The most frequently used technique for studying transfer involving low- lying energy states has involved high-frequency sound in gases.ll9 12 When 9 G. Herzberg " Spectra of Diatomic Molecules " Van Nostrand New York 1950. lo A. G. Gaydon " Dissociation Energies " Dover New York 1950. 11 J. J. Markham R. T. Beyer and R. B. Lindsay Rev. Mod. Physics 1951 23 353. l2 W. T. Richards ibid. 1939 11 36. McCOUBREY AND MaGRATH GASEOUS COLLISIONS 89 a sound wave is propagated through a fluid the fluid is subjected to periodic cycles of compressions and rarefactions with accompanying periodic tem- perature changes.For a perfect gas the speed with which the adiabatic sound wave is propagated is given by Laplace’s classical equation where Cv is the total heat capacity a t constant volume and M is the mole- cular weight. In this classical treatment some absorption of sound occurs owing to energy losses from viscosity and heat conduction.ll When the frequency of the sound wave is low the time between successive compressions or rarefactions is long and all the energy states of the gas are excited by the sound wave at rates sufficient to maintain practically equilibrium distribution with reference to the fluctuating temperature pro- duced by the sound wave. At sufficiently high frequencies the time interval between the fluctuating temperature changes becomes markedly shorter than the time required for the redistribution of energy into the various internal modes.At this stage the effective specific heat of the fluid subjected to the sound wave becomes less than the maximum specific heat. In the limit a t very high frequencies sound propagation may involve none of the terms due to internal energy changes of the molecule. Physically this means that the velocity of sound in a gas will change with frequency between the low and high frequency limits if the energy- transfer process translational + internal modes is sufficiently slow. Such changes of velocity with f r s quency are usually called dispersions. In the high-frequency sound wave since the molecules are not able to redistribute their translational energp to internal modes before passing it on by collision there is a phase displace- ment of pressure relative to density and part of the sound energy is irrever- sibly absorbed by conversion into thermal energy.ll This leads to an absorption over and above that due to viscosity and heat conduction.At moderate frequencies (60-2000 kc./sec.) many experimentally observed changes of velocity with frequency or high non-classical absorptions are due to phase lags between translation and vibration ; dispersions have also been observed for phase lags between translation and rotation at still higher frequencies. Under given static conditions a t low frequencies the effective heat capacity may be written C and the velocity V,. At very high frequencies when none of the vibrational states is operative the corresponding parameters are C and V .We consider first the case of a gas with only one vibrational mode of frequency Y which becomes inoperative as the frequency is raised.ll7 l2 The energy E = hv is assumed to be such that only the first excited state is appreciably populated. At any instant there will be a number of molecules No in the ground state and a number N in the excited state ; No + N = N the total number of states occupied per mole. 25 = f l o exp (- hv/kT) V 2 = (RT/M)(l + RIG,) . - ( 1 ) At equilibrium Without assuming any mechanism for the interconversion of state we can write dN,/dt = EOlN - E1oN1 . (2) 90 QUARTERLY REVIEWS where k, and k, are the probabilities of exciting and de.excit,ing stlate 1 i.e. numbers of transitions per molecule per second.At equilibrium dN,,/dt == 0 and k,,/k, = nl/R0 = exp (- hv/kT) For displacement of the system from equilibrium by a small amount AN, within the limits of irreversible thermodynamics we may write l3 Here t measures the relaxation time necessary for the departure from equi- librium to be reduced to l/e of its initial value in the two-state case it can readily be shown l4 that l/z = k, + Elo ; since in our case hv/kT > 1 k, < k, and l/z N- El,.* At ideally low frequencies the sound wave increases the temperature by AT a t each peak compression and hence the population of the excited state by AR, since Ci AT = E Afll where Ci = Co - C is the total internal heat capacity. At ideally high frequencies the sound wave passes through the gas without substantial alteration in NJ.At intermediate frequencies the population of N increases a t peak compression by AN," an amount less than Anl and dependent upon w = 2nf where f is the cyclic frequency of the sound d(AN,)/dt = - ANl/z . * (3) AN," = ARl/(l + i ~ t ) ; Ciu = (C - Cm)/(l + ~ W Z ) . (4) The measured velocity V and absorption coefficient p vary with frequency according to the equations 11 12 15 Equations ( 5 ) and (6) are used to deduce z from experimental observations. ( b ) Analysis of Ultrasonic Measurements to obtain Energy-transfer Data. -In computing t from experiment by use of ( 5 ) and (6) the measured values must be corrected t o zero pressure by means of a suitable equation of state.16 Equation (2) implies no specific mechanism for the energy-transfer pro- cess. It is generally assumed that energy transfer is effected by two-body l3 S.R. de Groot " Thermodynamics of Irreversible Processes " North Holland Amsterdam 1952. l4 M. Eigen Discuss. Faraduy SOC. 1954 17 194. l6 A. Van Itterbeek and W. Van Doiminck Proc. Phys. Soc. 1946 58 615. * The general theory of relaxation for the two-state process A + B gives S. Petralia Nuwo cim. 1952 Suppl. to Vol. 9 1. kl* 4 1 7 = l / ( k l o + 7 ~ ~ ) . ~ ~ For the case of a harmonic oscillator the system will only approxi- mate to the two-state case in the condition that hv kT ; when this obtains 7 N l/klo. This condition is fulfilled in a number of experimental cases. For a multi-state relaxation process in a harmonic oscillator it has been shown that T = 1/(kl0 - kOl).l2 This relationIwillJprobably apply for vibrations where hv < kT.All of these expres- sions for 7 involve a certain amount of approximation. MWOUBREY AND MCQRATH GASEOUS COLLISIONS 91 collisions. If this hypothesis is correct then the probability factors El and k, will he proportional to the collision rate in the gas. The two-body collision rate is proportional to t'he pressure 11 which means that the relaxa- tion time z will be proportional to l / p . For birnolecular mechanisms the effect of reducing the pressure is exactly the same as increasing the frequency i.e. it prevents equilibrium of excited species from being established under the action of the sound wave. The usual experimental variable is the effective frequency a l p and the standard relaxation time is referred to one atmosphere pressure.l2? l5 Early experimental work in this field was mainly on inorganic molecule^.^^ More recent work has covered a wide range of organic molecules.ls When observations of relaxation time are made upon polyatomic mole- cules the supposition that the internal energy is contained in a single mode is no longer adequate since the number of normal vibrations of a molecule increases with the number of bonds. If each molecular vibration were excited separately from the translational energy a separate relaxation pro- cess would be expected for each mode. Such complexity is not generally observed. Usually the whole vibrational specific heat is governed by a single relaxation time ; for a few polyatomic molecules however two relaxation times are observed from ultrasonic meas~rements.1~ The experimental phenomenon of a single relaxation process for poly- atomic molecules may be interpreted in terms of two possible mechanisms.20 (1) Individual transfer processes for different modes of vibration 1 2 3 .. . are treated as independent ; if the separate relaxation times are all approximately equal the observed rela,xation time z [from ( 5 ) and (S)] is equal to z1 = z = z3' . . . Theoretical considerations suggest that this is rather unlikely (cf. 3.1). (2) Excitation is consecutive ; transfer is generally assumed to take place from translation to the lowest vibrational mode 1 with characteristic relaxation time tl from this mode to 2 and so on. The internal relaxation times zI2 t i 3 . . . are assumed to be much smaller than zl. I n this case 71 = zC,/(C + c + c + . . .) . (7) and the rate of transfer to all the internal modes on collision is controlled by a single value.I n equation (7) C is the heat capacity associated with mode 1 and (C + C + C + . . .) is the total internal vibrational heat capacity associated with the molecule due allowance being made for the degeneracy of the levels. This series excitation of vibrational energy must involve a considerable coupling of the anharmonic oscillators (simple harmonic oscillators vibrate independently). Such coupling enables the potential energy initially sup- plied to the molecule by collision to be apportioned to the various vibrators by rapid internal re-adjustments. It seems likely that not all molecular l7 R. A. Walker National Advisory Committee for Aeronautics Technical Note No. 2537 Washington 1951. 18 P. G.T. Fogg and J. D. Lambert Proc. Roy. SOC. 1955 A 232 537. 2o K. Schlifer Z . phys. Chem. 1940 B 46 212. D. Sette A. Busala and J. C. Hubbard J. Chem. Phys. 1955 23 787. 92 QUARTERLY REVIEWS vibrations will couple together readily. If coupling does not occur then internal redistribution of energy may be a much slower process than the direct excitation of separate vibrations by collision. This corresponds to the case of more than one relaxation time sometimes observed (cf. Table 2). Writing the single relaxation time l/t N kl, we define a transition probability per collision averaged over the whole system where 2 is 4 n ~ r ~ ( n R T / M ) ~ / ~ the number of collisions made by one molecule in one second,21 n is the molecule density a t one atmosphere pressure o the Pio = ki,/Z = 1/zg .* (8) TABLE 1. Observed relaxation times z relaxation times for lowest mode zl and numbers of collisions Z; required to de-excite lowest quantum hv for a selection of molecules obtained from ultrasonic data Molecule 0 . . . C12 . . . co,. . . * - . . S F . . . CH,. . . CF4 . . . cc1 . . CH3F . . CH3C1 . . CH,Br . . CHJ . . CC12F2 . . CBr2F2 . . cis-C2H2C12 . trans-C2H,CI2 C2H4 . - C2F4 . - c~cZO-C~H . C6H6 . . ?%-C,H14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 18 20 20 20 36 109 100 100 100 100 100 100 27 27 15 100 100 100 100 96 100 z x 1 0 7 (sec.) - 42 116 9.3 16.7 5.89 8-4 6.6 0.83 1.8 1.0 0.45 0.78 0.29 2.5 0.25 0.082 0-14 2.7 1.5 32 - r x lo7 (see.) - 42 108 8.7 13.5 1.76 6.0 2.4 0-23 8.9 0.62 0.36 0.16 0-16 0.05 0.71 0-049 0.016 0.020 0.62 0.28 - 500,000 32,000 108,000 7 000 14,000 1400 6000 1300 130 6000 190 76 165 47 7 00 31 10 11 600 200 20 400 1580 560 667 589 527 363 1306 435 218 1048 732 61 1 533 2 60 165 810 173 227 190 740 405 < 100 Ref.22 3 24 23 25 26 27 26 28 28 28 28 28 28 29 29 30 18 18 18 31 31 31 2 1 J. R. Partington " Advanced Treatise on Physical Chemistry " Vol. 1 Longmans 22 H. 0. Kneser Ann. Phys. 1935 21 682. 2 3 A. Van Itterbeek P. De Bruyn and P. Mariens Physica 1939 6 511. 24A. Eucken and R. Becker 2. phys. Chem. 1934 B 27 235. 25 A. Eucken and H. Jaacks ibid. 1935 By 30 85 ; A. Eucken and E. Niimann 26 A. Eucken and S. Aybar ibid. 1940 B 46 195. 2 7 C. L. O'Connor J . Acoust. SOC. Amer. 1954 26 361. 2 S P . G. T. Fogg P. A. Hanks and J. D. Lambert Proc. Roy. SOC.1953 A 219 29 T. D. Rossing and S. Legvold J . Chem. Phys. 1955 23 1118. 30 J. C. McCoubrey J. B. Parke and A. R. Ubbelohde PTOC. Roy. SOC. 1954 A 3 1 J. D. Lambert and J. S. Rowlinson ibid. 1950 A 204 424. Green and Co. London. ibid. 1937 By 36 163. 490. 223 155. McCOUBREY AND McORATH GASEOUS COLLISIONS 93 collision diameter and M the molecular weight. The quantity Zio is the number of collisions required to de-excite one quantum of the lowest vibration hv,. Alternative collision efficiencies sometimes defined are written Here P, and k, refer to average values for the whole molecule obtained from the relation z N l/k,,. For diatomic molecules Ze is equal $0 Zi,,; in general zio = ZeffC1/(CI + Q + C + * - *) Tables 1 and 2 give a selection of relaxation times and collision efficiencies Also included are values of the lowest A value of V / C of 100 cm.-l for translation-vibration transfers.spectroscopic vibration " frequencies " v,/c. corresponds to an energy change 0 -j 1 of 286 cal./mole. TABLE 2 . Observed relaxation times for molecules in which double dispersion is indicated by experiment at 30" c Molecule I I I CH2C12 . . . . . I 9.46 ' 0.195 19 1 l-C,H,Cl . . . 36 I 0.535 ~ 1 1 2-C,H,CI2 . . . 1 0.425 i 6.6 Prom Table 1 it appears that there is a wide spectrum of values of Z; ranging from about lo6 for oxygen to about 10 collisions for some organic molecules. For polyatomic molecules exhibiting a single relaxation the energy- transfer probability depends broadly upon the magnitude of the lowest vibration frequency. Molecules with low vibration frequencies are more readily de-excited.There is however no strict parallel between Zio and vl. When molecules are divided into classes of common spectroscopic type then the correlation in any class between Zi and v1 becomes more nearly quantitative. Thus the molecules CH,F CH3C1 CH3Br and CHJ in which the lowest frequency is a carbon-halogen stretching show an excellent linear relationship between log Zio and v1.I8 Another property of the lowest vibration frequency which affects the value of Zi is whether or not the oscillation is infrared-active. l 8 Molecules for which the lowest vibration is infrared-active have been differentiated from those in which the lowest vibration is active only in the Raman spec- trum. This may be due to differences in the electrical behaviour of the vibrations which are perturbed by the dipole moment of the approaching molecule in a collision (see Fig.1 ) . 32 T. S. Rao and J. C. Hubbard J . Acoust. SOC. Amer. 1955 2'4 321. 94 QUARTERLY REVIEWS 0 200 400 600 800 I000 7200 Lowest vibration frequency (crn.-’l FIG. 1 (Substantially copied from P. G . T. F o g and J. D. Lnmbert Prcc. Roy. SOC. 1955 A 232 537.) Collision lifetime for lowest vibrational state in polyatomic molecules (Fogg and Lambert’s data). Infrared inactive vibrators (1) CHCl:CCl ; (2) CF,:CF ; (3) cis-CHC1:CHCl; (4) CHCl,; ( 5 ) CCl,; (6) CH,Cl (cf. Table 2) ; ( 7 ) CP,. Infrared active (8) CH,:CNI ; (9) CH,:CHBr ; (10) CH,:CHCl ; ( 1 1) CH,:CF ; (12) CH,:CHF ; (13) CHJ ; (14) CH,F ; (15) CH,Br ; (16) CH,C1 ; (17) CH,:CH ; (18) CH,F. ( c ) Translation-Vibration Transfer in Two-component Systems.-Values of the overall relaxation time z may be obtained for mixtures from the equations (12) and (13).This value of z corresponds to effects due to collisions between all possible species AA AB and BB. If a molecule A itself exhibits a dispersion addition of a molecule B may cause some shift of the dispersion region in terms of frequency. I f collisions AB are more efficient in transferring internal energy to A than collisions AA the dis- persion shifts to higher frequencies. If the mixture is sufficiently dilute effects due to BB interactions are negligible. It is possible to segregate effects due to interactions AA and AB. If only binary collisions are effective then their effect is given 1 5 9 l 7 by an equation of the form where xA and xB correspond to mole-fractions.The value of TAB obtained from this equation refers to a hypothetical mixture of A and B in which only AB collisions are effective and in which the partial pressure of B is one atmosphere. By analogy with the heatment given above for pure gases rAB should be multiplied by a factor [C,/(C + C2 + C + . . .)IA t o give where l / x = xA/xAA xB/xAB MPCOURREY AND McQRATH GASEOUS COLLISTONS 95 represents the relaxation time associated with a deactivation of the From this quantity ABtl we may define a collision efficiency *l3.P; for lowest vibrat'ional mode of A by a inoleciile of B. de-excitation of a vibrational quantum of A by B where ZAB is the collision number for AB encounters in the above hypo- thetical gas. Numbers of collisions required to de-excite vibrational quanta are given for a range of molecules in Table 3.""Pi0 = ABk',o/ZA 1 /ABZio ; "nkio 2 IjABT * (10) TABLE 3.* Energy-transfer data expressed in number of collisions to de-excite lowest vibrational quantum of parent gas by additive ABZ',, obtained jrom ultrasonic data -_______ -.- . ~ ~ ~ _ _ ______ Temp. (OK) . . . . Parent . . . . . I_ __- Additive Self . . . . H . . . . H e . . . . D,. . . . . N,. . . . . CH . . . . H,O . . .. . D,O . . . . C,H5*CH3 . . CH,*OH . . . n-C,H1 . . . NH . . . . n-C5H1 . . . Ref. . . . . 293 I 293 ~ 293 ~ 0 2 . . - 500,000 20,000 150,000 103,000 - - 400 - - - - - 390 22 ~ cos ~ 14,000 160 1000 3000 6 - - - __ 30 - - - 26 33 203 CO 108,000 500 1500 1500 2100 130 270 23 35 - - - 23 24 33 34 293 N,O -~ ~ 7000 600 1500 450 8 00 70 190 90 - - - - 450 25 33 288 700 70 550 1400 24 - - - - - 11 8 - 30 35 * Where the data for one compound are drawn from different sources the figures may not be self-consistent t o better than a factor of two.Table 3 shows the wide range of effects found with additives in ultra- sonics. Light molecules such as H or He have a high efficiency in trans- ferring to their translational modes the vibrational energy of other molecules. Molecules which possess permanent dipoles such as NM, CH,.OH or H,O are in general found to be efficient. Molecules of quite high mole- cular weight such as toluene n-pentane or n-hexane may also show a very high efficiency in energy transfer. In several cases such as C0,-H,O and C1,-CO incipient chemical reactivity produces a high transfer efficiency.33 V. 0. Knudsen and E. F. Frirke J . Acozrst. SOC. Amer. 1938 10 89 ; 1940 12 244. 34 D. Sette and J. C. Hubbard ibid. 1953 25 994. 35W. T. Richards and J. A. Reid J . Chem. Phys. 1934 2 206. 96 QUARTERLY REVIEWS In general binary collisions are responsible for energy transfer though evidence for the importance of three-body collisions in mixtures a t high concentration of additives has been obtained in a few cases.38 ( d ) Ultrasonic Study of the Transfer of Energy between Translation and Rotation.-A treatment similar to that given above for translation-vibration may be applied to transfers occurring between translation and rotation. Since the magnitude of rotational energy quanta is small (often <50 cal./mole) redistribution of energy to and from rotational states is a much more efficient process than for vibration.Experimentally it is necessary to go to much higher .frequencies t o observe dispersion or absorption due to this cause. 3 7 9 38 A relaxation time for the translation-rotation process may be evaluated from equations similar to (5) and (6) and the product of this time with the number of collisions per molecule per second 2 gives an approximate number of collisions Z, required to effect transfer between translation and rotation. Experimental values are given for a few diatomic molecules in Table 4. TABLE 4.-Relaxation times z and numbers of collisions Z required to effect translational-rotational energ?] transfer in diatomic molecules calculated from ultrasonic data Molecule N . . . . 0 . . . . Air . . . . H . . . . H . . . . D . . .. . Temp. (" R) 302 303 305 298 273 273 ltelaxation time t x 10s ( E N . ) 0.12 0.524 0.304 0.229 1-9 1.7 2.2 2.0 No. of collisions zeff 6 30 21 16 320 285 390 250 Ref. ~ 37 38 39 40 41 41 ( e ) Other Experimental Methods of Study for Transfer at Low Quantum Levels.-Heat capacity lag in gas dynamics. In the flow of gases about obstacles such as aerofoils compressions and rarefactions occur with enthalpy changes due to loss or gain of mass motion. If gas initially contained in a chamber is expanded through a faired nozzle isentropically it undergoes an enthalpy decrease. If the gas is now brought to rest rapidly a t the nose of an impact tube its enthalpy is increased rapidly and part of the heat capacity (the internal modes) may fail to reach equilibrium immediately on the compression.I n the impact tube the final state of the gas then involves an irreversible return to equilibrium and the gas suffers an increase in entropy X and a 36 R. A. Walker J . Chem. Phys. 1951 19 494. 37 A. J. Zmuda J . Acoust. SOC. Amer. 1951 23 472. 38 W. J. Thaler ibid. 1952 24 15. 30 C. Ener A. F. Gabrysh and J. C. Hubbard ibid. p. 474. 40 E. S. Stewart Phys. Review 1946 69 632 ; J. E. Rhodes ibid. 1946 70 932. 41 E. S. Stewart and 5. L. Stewart J . Acoust. SOC. Amer. 1952 24 194. McCOUBREY AND McGRATH GASEOUS COLLISIONS 97 decrease in pressure p - p from the initial values X and po. The observed pressure defect between chamber and impact tube and hence the entropy increase A 8 = R logpo/p2 will vary with the ratio of the compression time to the relaxation time. The maximum entropy increase occurs for an instantaneous compression.Hydrodynamic equations relating A 8 to t have been obtained for different types of impact tubes. This technique has been used experimentally to determine translation- vibration relaxation times for gases such as N, CO, CF,Cl, H,O and C3H8,8 covering values ranging from lo- sec. in nitrogen to (6 x sec. in propane. Data obtained in this way refer to the same process as ultra- sonic measurements and are in reasonable agreement with them. A shock wave is a positive pressure wave which moves through the gas with a velocity greater than the normal velocity of sound in the unshocked gas. Shock velocities are therefore equal to or greater than the mean molecular velocity. The region of abrupt temperature pressure and density rise in the front of a gaseous shock wave is known as the shock front.The thickness of this shock front region is of the order of a few mean free paths. As the shock passes through the gas the molecules in the shock front have their translational rotational and vibrational energy increased in a period of the order of a few collision times. If there is a lag in equilibrium the heat capacity effective in the shock front is initially that of a monatomic gas (translation only) and behind the shock front it rises to the equilibrium value. The effect of such a lag is to distort the shock front; the initial increase in the translational temperature is greater and the increase in density less than for the same gas in equilibrium. If the rotational or vibrational energy is not appreciably excited instantaneously with the shock the shock front will be thicker than for the equilibrium value.An optical method has been reported for measuring the thickness of the shock front and distortions in it from a study of the total density change across the front.42 Results have been used to determine the number of collisions Z, required to obtain rotational equilibrium in shock fronts in diatomic gases. Values of ZeE- 10 for 0 and N are closely similar but differ slightly from those found in ultrasonics ; s 7 7 38 there is also an indication of two or more distinct relaxation times for these gases. Lines in the microwave spectrum correspond to transitions between closely spaced internal energy levels e.g. rotation-rotation or inversion transitions (e.g. AE = 1-2 cal./mole). The breadth of a microwave line A v ~ .~ . due to pressure broadening depends upon the time interval over which the radiating processes are unaffected by collisions. A v ~ . ~ . = 1/2nt where the collision time t = 1/45 nncf2p.B.v. Here v is the velocity of the molecule cp,B. the collision diameter and n the number of molecules per C.C. Collisions of importance for microwave pressure broadening involve energy exchange between the colliding molecules leading to a change in Study of moving shock waves. Pressure broadening of microwave lines. *2 E. F. Greene and D. F. Hornig J. Chem. Phys. 1953 21 617. 98 QUARTERLY REVIEWS the internal state of the radiating molecule. Such collisions can have attributed to them a characteristic diameter Measurement of A v ~ . ~ . as a function of pressure gives op.A.which is it relative measure of the efficiency of molecules in transferring energy between their translational modes and the appropriate internal states of the radiating molecule. Pressure broadening of the NH inversion (3 3) line shows that the effective collision diameters of molecules such as N, H, or He for this process are close to those deduced from gas kinetic properties such as vis- cosity indicating a ready transfer of energy. Polar molecules such as NH, HCCN and CClN give op.B. > G gas kinetic. This is probably due to long- range electrostatic interaction with energies comparable to the transition energy. High values of for self-broadening in pure rotation spectra are also observed for a number of polar gases such as OCS CH,F and which determines t . 4 3 H,0.43 3.Theoretical Treatments of Energy Transfer in Low-lying States (a) Translation-Vibration Transfers.-Early quaptitative notions of energy transfer arise from the concept of an adiabatic collision. If the collision process is sufficiently slow readjustments of translational energy in a single collision do not occur quickly enough to affect the fast internal movements of the molecule but act only upon the centres of gravity of the molec~les.~~ I n order for the translational energy changes to influence the internal modes during a collision collisions must be non-adiabatic. The time taken for the translation energy change to occur must be of the order of the time taken for the internal vibration. For vibrations of frequency Y this time is 1/2nv. The time taken for the molecule to change its kinetic energy is x/v where z is the distance travelled in the repulsion potential field which converts the kinetic energy into potential energy and v is the average velocity of the molecule.Thus when 2nvxjw is of the order of unity or less appreciable .energy transfer may occur. This simple relation shows that the probability of energy transfer depends upon the frequency of vibration upon the repulsion potential -which deter- mines how quickly the kinetic energy is dissipated and upon the molecular velocity of approach i.e. molecular mass and temperature ; faster molecules should be more efficient. Landau and Teller 44 have extended this concept to obtain the prob- ability for 1 -+ 0 transitions by collision in a diatomic molecule. The transition probability for the collision of two molecules of reduced mass ,u along a line of centres distance r apart in a repulsion field is found to be proportional to exp (- 4n2v/orv).Values of the velocity v giving appreciable transition probabilities are much greater than gas kinetic 43 W. Gordy W. V. Smith and R. Trambarulo “ Microwave Spectroscopy ” Wiley New York 1953. 4 4 L. Landau and E. Teller Physikal. Z. So7~ljeftcnion 1936 10 34 ; C. Zener PTOC. Camb. Phil. SOC. 1933 29 136. McCOUBREY AND McGRATR GASEOUS COLLISIONS 99 average velocities. The probability of such velocities’ occurring is given by the usual Boltzmann term exp (- ,uv2/8kT). The ma,jority of transitions will occur for inolecular collisions having a velocity which mutually satisfies both of thcsc conditions. This velocity is obtained by miiiiinising the total exponent term 4n2v/av + pv2/2kT.The value of v which satisfies this is o = (4n2kTv/o(p)l/3. Substitution of this value of the velocity in the total exponential gives P, proportional to exp - 3(2n4pv2/a2kT)1/3. A formal quantum-mechanical treatment of the collision problem of translation-vibration transfer was first given by Jackson and M ~ t t ~ ~ using the method of distorted waves. Recent develop- ments of this have been made without substantial change in the physical picture.46 4 7 9 ** For the two-molecule interaction the transition prob- ability calculated for approach of free rotationless particles in a potential field based on U(r) = U exp - ar to produce a change in internal quantum number from 1 -+ 0 is given by an equation similar in essentials to that of Landau and Teller.The effective repulsion potential U(r) is strictly the potential interaction between the nearest atoms of the two molecules. This may be approximated by the potential between molecular centres in the region of repulsion if the molecules behave as spherically symmetrical and if the internally effective force is proportional to the intermolecular force. By using a suitable molecular velocity distribution function the tran- sition probability per collision statistically averaged over all molecules is found to be Here the quantity pint is the effective reduced mass of the oscillator being de-excited. On the assumption that the main dependence of P, upon mass fre- quency and temperature comes from the exponential term the form of (11) may be used to interpret some of the main features of experimental results.(1) P, and hence Z, depend exponentially upon the magnitude of the frequency. The form of (11) a,pplies strictly only to diatomic molecules with a single vibrational state. For polyatomic molecules we may use the transition probabilities Pio referring to the lowest mode of vibration and apply (11) in the same way to show the qualitative dependence of Zio upon the frequency. Equation (1 1) requires a quantitative dependence of log Zio upon ,u1/3v2/3. Experimental work provides little confirmation of this relation and the extensive work of Lambert l8 and of Rossing and Legvold 29 suggests a closer dependence upon v (cf. Fig. 1). 45.J. M. Jackson and N. F. Mott Proc. Roy. Soc. 1932 A 137 703. 46 K. F. Herzfeld R. N. Schwartz and Z.I. Slawslry J. Chem. Phys. 1952 20 4 7 T. L. Cottrell and N. Ream Trans. Paraday Soc. 1955 51 159 1453. 48 K. Takayanagi and T. Kishimoto Prog. Theor. Physics Japan 1952 8 497. 1591 ; K. F. Herzfeld and R. N. Schwartz ibid. 1954 22 767. 100 QUARTERLY REVIEWS (2) PI in (11) increases markedly with a decrease in the reduced mass of the colliding pair of molecules. This gives a qualitative explanation of the efficiency of light molecules such as H or Be (cf. Table 3). (3) Making certain assumptions about inter- and intra-molecular forces we can calculate transition probabilities from (1 1) to give reasonable agree- ment with experiment at one reference temperature. 47 ( b ) Quantitative Comparison of Theory with Experiment.-Temperature coeficients. Equation (11) may be used to obtain values of a directly from experimental data for P, at different temperatures by plotting log P, or log Plo as a function of l/T1/3 and deriving a from the slope ; a so obta,ined measures the exponent of the repulsion potential effective in energy-transfer collisions.It is also possible to calculate a from the average repulsion potential constants obtained from equilibrium and transport properties of gases by comparing for example the Lennard-Jones 6 12 potential which has been fitted to these properties with the exponential repulsion potential. The values of a obtained in the two different ways do not agree ; 47 e.g. a (energy transfer) for methane and benzene is a t least three times greater than a ( v i s ~ o s i t y ) . ~ ~ Thus although it may be possible by adjustment of parameters to obtain agreement between PI as calculated by equation (11) and as found experimentally at one particular temperature it will not in general be possible using these parameters to obtain agreement a t other temperatures.A priori calculations of P, using the average repulsion potentials usually assumed between molecular centres are not adequate. There is evidence for an additional highly specific temperature-independent factor p required as a multiplier in (1 1) to bring about agreement between theoretical and experimental values of Introduction of such a “ steric ” factor suggests a reason for the high degree of specificity found in energy-transfer phenomena. Transition probabilities may not depend in any simple way upon vibration frequencies and may be complicated by specific steric factors.More complex interactions. The theoretical treatment given above makes no attempt to explain several significant phenomena found in energy transfer. (1) The effects of dipole moment. More elaborate calculations indicate that these depend mainly upon the additional attractive forces between dipoles which give rise to increased velocities of approach. 5O Faster relative velocities give more instantaneous collisions which are more efficient. It is well known that water has a remarkable efficiency for transferring translational energy to vibrations of carbon dioxide. Widom and Bauer 51 have recently shown that when two substances have incipient chemical reactivity the shape of the intermolecular potential curve is altered in such a way that much more efficient energy transfer results.(3) The effects of additives of high molecular weight containing a number (2) The effect of additives with incipient chemical reactivity. 49 J. W. Arnold J. C. McCoubrey and A. R. Ubbelohde in course of publication. 6o Z . I. Slawsky and F. W. De Wette Physica 1954 20 1169. 51 B. Widom and S. H. Bauer J . Chem,. Phys. 1953 21 1670. McCOUBREY AND Mc3RATH GASEOUS COLLISIONS 101 of low-lying internal vibrations. Herzfeld Schwartz and Slawsky 46 have shown that in such cases transfers of energy may occur which involve the internal states of the heavy molecule in such a way as to produce an easier alternative route for the energy transfer e . g . a vibrational resonance mech- anism. Such an explanation may well account for the high efficiency of molecules such as toluene or n-pentane in de-exciting carbon dioxide or ethylene (cf.Table 3). ( c ) Transfer of Energy between Translation and Rotation.-For molecules other than the hydrogen isotopes and diatomic hydrides rotational quanta are small; this means that appreciable numbers of energy levels will be populated even at low temperatures. The rate of populating and de- populating all the rotational levels by exchange with translational energy has been treated by a single relaxation process. A quantum collision method similar to that used in deriving ( l l ) but with an angle-dependent repulsion potential has been employed to calculate probabilities for rotation-translation transfers. For diatomic molecules of molecular weight >20 a considerable simplification is obtained.52 Effects of mass and also of temperature cancel in these rotational transfers (since both translational and rotational energies depend upon these quantities).Since the collision time is short relative to times of rotation and since therefore these transfers are insensitive to repulsion potential we may write an approximate equation where do is the internuclear distance and cr the collision diameter. Prob- abilities calculated from this equation give reasonable agreement with experi- ment for N and 0,. This equation will not however hold for H and D where the rotational quanta are large and where ortho- and para-states may be differentiated. More detailed quantum collision calculations made for H and D 53 54 give calculated transition probabilities in good agreement with experiment.Separate relaxations are attributed to ortho- and para- states and to different rotational transitions within t'he para-state. This corresponds with the experimental observation that for H the relaxation region is broader than that for a single p r o ~ e s s . ~ ~ ~ 41 4. Energy Transfer in Higher Energy States ( a ) Photochemical Phenomena.-Flash photolysis. Norrish and his co workers 55 56 have shown that when C10 or NO is flash-photolysed mole- cular oxygen in its electronic ground state is formed with a considerable population in the fifth sixth and seventh vibrational levels. The efficiencies of additives in deactivating these vibrationally excited species from the sixth to the fifth quantum have been examined. Free radicals and 5 2 R. Brout J . Chem. Phys. 1964 22 1189. 53 J.C. Beckerle ibid. 1953 21 2034. 54 K. Takayanagi and T. Kishimoto Progr. Theor. Physics Japan 1953 9 578. 55 R. G. W. Norrish and B. A. Thrush Quart. Rev. 1956 10 149. 6 6 F. J. Lipscomb R. G. W. Norrish and B. A. Thrush Proc. Roy. SOC. 1966 A 233 429. 1 02 QUARTERLY REVIEWS molecules with vibrational levels close to those of the excited oxygen are most efficient. Collision numbers range from 10' for N to <500 for NO,. Fluorescence in small molecule.s. Molecules excited by light may pass into states of higher electronic energy. Owing t o the accompanying cha'nge in shape of the electronic potential -energy curves such transitions normally involve changes in the vibrational quantum number as follows from the Francl-Condon prin~iple.~3 For example when molecular iodine is irradiated with mercury light of wavelength 5431 A the excited electronic state is produced in the twenty-sixth vibrational Examination of the fluorescence spectrum of iodine suggests that de-excitation of these high vibrational levels by collision with inert gases occurs much more readily than de-excitation of lower levels in similar collisions.Similar observations have been made on ~ulphur.~8 If a complex molecule is excited to a higher electronic state with an increase in vibrational quantum number it may in t'he absence of collisions leave this state by alternative processes (1) It may revert to the ground state with emission of energy as fluorescence. (2) It may pass over into a third state by accumulating vibrational energy from the rest of the molecule in a critical degree of freedom.From this state it returns to the ground state by a radiationless transition. Collisions with the excited molecule may remove electronic energy directly by quench- ing thereby decreasing the fluorescence yield. This direct quenching is not of immediate interest for the present Review. Alternatively collisions may remove vibrational energy thereby stabilising the excited molecule and increasing the fluorescence yield. This process can be discussed on the same lines as other vibrational traii~fers.5~ If a molecule with translational temperature T possesses vibrational energy AE in excess of the equilibrium value for T, its effective vibrational temperature TILvib.) is defined as T + AE/Cvib.. On collision with another molecule X it may lose vibrational energy attaining a vibrational tempera- ture T2(vib.) = T + AE'lC,,,.and increasing the temperature of X from T to T = T + AE - AE'/Cx. The efficiency of this process which increases the fluorescence yield may be written as When AE - AE' = 0 y = 0. When T = T2(vib.) equilibrium is established and y = 1. Even when y = 1 the amount of energy transferred AE - AE' cannot equal AE ; the magnitude of AE - AE' depends upon the heat capacity of X and for T = T2(vib.) the ratio ,h' = (AE - AE')/(AE) could only be unity if the heat capacity of X were infinite. Values of the quantities AE and Tl/vib.) must first be obtained experi- mentally from the decrease in fluorescence yield in the presence of a quencher a t different temperatures and wavelengths of exciting light. Values of 1'2(vib) and hence AE - 4E' and y can then be obtained from experiments Fluorescence in complex molecules.Y = (T2 - Tl)/(T2(vib.) - G 7 F. Rossler Z. Physik 1935 96 251. 58 E. Durand J . Chem. Phys. 1940 8 46. 59 H. G. Curme and G. K. Rollefson J . Amer. Chein. Soc. 1952 '74 28 ; R. S. Neporent Zhur. Pis. Khim. 1950 24 1219. McCOUBREY AND McGRATH GASEOUS COLLISIONS 103 on foreign gases which increase the fluorescence yield of P-naphthylamine excited by 2652 A light. Typical values of y are H ( O - l ) C5Hl (0-2) CHCl (0-5) SF6 (0.5) NH (O-9).60 Out of an excess energy AE of 8400 cm.-l in 0-naphthylaniine only 70 cm.-l is taken up by helium in an efficient collision whereas N removes 190 sIE'6 570 CHC1 1000 and C,H, 1280 cm.-l. Clearly energy transfer involving vibrations in this region of the potential-energy curve (probably Q > 5 for some modes) is not a com- pletely efficient process.(b) Atom Recombination Reactions.-When atoms or radicals recombine the molecule resulting from the primary step of recombination will contain sufficient energy to cause redissociation. Only when some of this excess energy is removed in collisions will the molecule become stable. In the case of atom recombinations the excited '( molecule '' having only one bond will have a very short lifetime; the excess energy cannot be shared into other bonds but stays in the single vibrational mode so that in the absence of collisions the (' niolecule " will redissociate before completion of a single vibration i.e. in a time of about l O - l 3 sec.61 Only if a collision with a third body which can remove energy occurs within this period 'will a quan- tised stable molecule result'.6 2 In discussing the energy transfers involved here we are considering the removal of energy from a molecular species in which the vibrational energy is not quantised ; this is different from most of the other cases considered in this Review and the results may not be directly comparable with those of Sections 4(a) or 4(d). Two recent determinations of the rate of recombination of iodine atoms in the presence of a number of different molecules acting as third bodies have been made by using ;t flash of light of high intensity; 63 recom- bination following the dissociation of iodine in a shock wave has also been studied. 64 Recombination reactions have been analysed in terms of third-order rate constants for the reactlion X -t X + M -+ X $- M.The rate con- stants show that although simple atoms and molecules can facilitate tjhe reaction complex molecules are considerably more efficient and remove energy easily. It is suggested that this increase of ability to transfer energy is due to the high van der Waals attraction of the polyatomic molecdes and this is substantiated by a plot of the velocity constant for recom- bination of iodine atoms against the boiling point of the added molecular gases (taken as a measure of the van der Waals forces).63 Fig. 2 shows this plot for the case of iodine-atom recombinations. The value of t'he temperature coefficient of the reaction I + I -t M -+ I + M in terms of an Arrhenius plot yields an activation energy of the order of - 2 kcal./mole i.e. the three-body complex is more stable than M.Boudart arid J. T. Dubois J . Chew. Phys. 1955 23 223. E. Rabinowitch and W. C. Wood Tmns. Faradmy SOC. 1936 32 907. 6 1 G. E. Kimball J . Chem. Phys. 1937 5 310 ; E. P. Wigner ibid. 1939. 7 646. c 3 K. E. Russell and J. Sirnons Proc. Roy. Soc. 1953 A 217 271 ; M. I. Christie R. G. W. Norrish and G. Porter ibid. 1953 A 216 152. 6 4 D. Britton N. Davidson and G. Schott Discuss. Famclay SOC. 1954 17 58. H 104 QUARTERLY REVIEWS the reactants. The value of this activation energy is fairly coiistant for a number of different molecules acting as third 64 Such constancy for large and small molecules may indicate that some of the marked stabilis- ing effects of complex molecules are to be found not in energetic but in specific stei-ic factors or alternatively in the removal of the excess energy of I into the numerous vibrational modes of the complex molecule.100 200 300 400 Boiling po/nts o f additives (OK) FIG. 2 (Substantially copied from Ti. E. Ilussell and J. Simons Z'roc. Boy. SOC. 1963 A 217 271.) Horizontal axis Boiling points (" K) of additives. Vertical axis Log k x Third-order mte constant k (mole-2 cm. sec.-l) for recombinatz'on of iodine atoms in presence of third body 1M as a function of the van de7. Wads forces of M (boiling points). (I) Neon ; (2) argon ; ( 3 ) nitrogen ; (4) oxygen ; ( 5 ) methane ; (6) ethylene ; (7) propane ; (8) cyclopropane ; (9) ethyl chloride ; (10) n-pentane ; ( 1 1 ) ethyl bromide ; (12) benzene ; (13) p-xylene ; (14) mesitylene. (c) Vibrationally Excited Reaction Intermediates.-Another problem similar to tliat of the stability of newly formed diatomic molecules is con- cerned with the lifetime of small radicals containing a total vibrational energy in the molecule in excess of the dissociation energy of a bond.These species are formed in the course of certain reactions and may act as chain carriers. In the absence of contact with other molecules such species have only a transitory existence and will dissociate into atoms or radicals with excess translational energy. For example in the hydrogen-oxygen reaction the encounter H + 0 gives rise to the radical HO,* containing energy in excess of the dissociation limit ; 65 this energy may be shared between the two bonds but will rapidly revert into a single bond causing dissociation ; i . e . the radical will have 6 5 C.J. Danby and C. N. Hinshelwood J . 1940 464. McCOUBREY AND McGRATH GASEOUS COLLISIONS 105 a very short lifetime. If another collision involving HO,* can occur suffi- ciently rapidly after the first encounter the excess energy may be removed and a stable HO radical may be formed. Patrick and Robb 6 6 have shown experimentally that about 1 in lo3 collisions between H and 0 results in the formation of stable HO,. Hinshelwood and his co-workers have shown experimentally that CO and H,O are considerably more efficient than N or H in removing energy from HO,*. Walsh 67 has suggested an explanation for the differences in efficiency of various molecules in terms of their vibrational structure. It seems likely that a form of resonance transfer involving removal of energy from HO,* direct io vibrational energy of the additive is involved.This would be possible if a quantised energy jump - AE in HO,* is close to the available energy uptake of a bond in the additive molecule a factor which would tend to favour H,O and CO rather than N or H,. In the oxidation of n-hexane it has been shown experimentally that the use of hydrogen as a diluent gives rise to an inhibitory effect which can be explained on the basis of energy transfer from the energy-rich hydro- peroxide radical R*O,** to hydrogen. 68 Such energy transfer phenomena may well underlie the remarkable effects of structure on hydrocarbon oxidation .69 In a number of addition or radical recombination reactions vibrationally hot complexes may be formed which will if their lifetime is sufficiently long undergo de-energising collisions to yield stable products.Experi- mental a,nd theoretical work '09 71 has shown that the lifetime of vibrationally excited species increases markedly with the number of degrees of freedom ; i.e. the energy wanders away from the dissociating bond into other bonds and will require a time long with respect to the time between colli-' bions to return in the form available for dissociation. During this lifetime a number of collisions may occur sufficient to remove the excess energy. The recombination of large radicals to give stable products seems to occur on practically every collision. 72 (d) Effects of Energy Transfer on the Rate of Unimolecular Reactions.- Certain gaseous reactions which are unimolecular at sufficiently high pres- sures become biniolecular a t low pressures.Unimolecular reaction is possible when collisions maintain a stationary Boltzmann state with a fraction of molecules independent of pressure possessing the critical energy for reaction. A small proportion only of this fraction of activated molecules is chemically transformed the majority 6 6 C . R. Patrick and J. C. Robb Discuss. Faraday SOC. 1964 17 98. 6 7 A. D. Walsh Puel 1954 33 247. 68 N. J. H. Small and A. R. Ubbelohde J. 1952 4619. 6D A. R. Ubbelohde Rev. Inst. franp. du Pe'trole 1949 4 315 ; A. D. Walsh Discuss. 70 G. B. Kistialiowsky ibicl. 1954 17 94 ; D. Garvin and G . €3. Kistiakowslry 7 1 It. A. Marcus ibid. p. 355. 7 2 R. Gorner and G. B. Kistiakowsky J . Chem. Phys. 1961 19 55 ; K. J. Ivin and Faraday SOC. 1951 10 320. .J. Chem. Phyls. 1952 20 105.E. W. R. Steacie Proc. Roy. SOC. 1951 A 208 25. 106 QUARTERLY REVIEWS being deactivated by collision before undergoing chemical decomposition. As the pressure falls the number of collisions decreases and ca>n become insufficient t o maintain the Boltzmann concentration of activated molecules owing to their concurrent removal by reaction. At sufficiently low collision rates this results in a bimolecular-type '* Addition of a chemi- cally inert gas a t this stage may be used to restore unimolecular kinetics. The ratio of the pressure of the reacting molecule itself required to restore the rate constant to the high-pressure limit relative to the pressure of additive which brings about this same change in rate constant measures the relative efficiencies of the two molecules in producing the steady-state concentration of activated molecules by collisions.If collision diameters are known we may express the efficiencies of the additives relative to the reacting molecules themselves on a collision for collision basis. Ref. . . . . 75 TABLE 5 . Selected values of relative eficiencies of gases in energy transfers with reacting molecules in unimolecuhr reactions 76 77 78 Reactant additive . . Temp. Self . He . . H . . Ne . . A r . . N . . CH . H,O . co . C,H,Me SF6 . C6H3Me3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VcloPropane 492" 1 0.05 0.12 0.07 0.07 0.24 0.74 - - - 1.10 oms9 448" 1 0.07 0.10 0.12 0.2 1 0.2 1 0.38 0.44 - - 1-12 1-23 50.5" 1 0.07 0.09 0.15 0.23 - - - 0.39 0.44 - - .4zomethane 310" 1 0.07 - - - 0.2 1 0-20 0-46 0.25 - - - N20 663" 1 0.66 0.47 0.20 0.24 1.5 1.3 - I - - - 79 Table 5 shows the high relative efficiencies of polyatomic molecules and the comparatively small effect of small molecules even including H and He which by contrast are found to be efficient in low-energy transfers.The numerical spread of efficiencies is however much smaller than in the case of low-energy transfers. Different theoretical treatments of unimole- 7 3 L. S. Kassel " The Kinetics of Homogeneous Gas Reactions " Chemical Catalog v 4 N. B. Slater Phil. Tra?is. 1953 A 246 57 ; Proc. Roy. Soc. 1953 A 218 224. 7 5 H. 0. Pritchard R. G. Sowden and A. F. Trotma,n-Dickenson ibid. 1953 A 76 Idem ibid. 1953 A 218 116. 7 7 H. S. Joliiison J. Amer. Chmz. SOC. 1953. 75 15G7 5763. 78 D. V. Sickinan and 0. I<.Rice J. Chem. Phys. 1936 4 608. 79 M. Voliner arid H. Froehlich 2. phys. C'lteni. 1932 By 19 89 ; M. Volmer and Co. New York 1932. 21'7 563. M. Bogdan ibid. 1933 B 21 257. McCOUSREY AND MvGRATH GASEOUS COLLISIONS 107 cular reaction rates have been given by Kassel 7 3 and by Slater 74 resulting in the same overall expressions for the rate constants. The treatments differ in the detailed mechanism assumed for the activation process. It does not appear possible a t present to discriminate between these two theoretical models from experiment ; more extensive measurements are required. Both of these treatments make the assumption that all gas- kinetic collisions are effective. Neither is immediately suited to a dis- cussion of the relative effects of foreign molecules in bringing about energy transfer.(e) Comparison of the Results obtained from Low-energy and High-energy Transfers.-Whereas experimental results on low-energy transfer can be used to give direct numerical values of the transition probability of a single fairly well-defined molecular process the complexity of the phenomena involved in higher-energy transfers makes it difficult a t present t o adduce a numerical value for the transition probability of a particular molecular step; most of the results are of a comparative nature and are referred to a convenient norm. Transfers of energy involving large vibrational quanta low in the potential-energy curve may require between say 10 and lo6 collisions in different pure gases depending upon the size of the quantum and upon highly specific niolecular interactions.Large variations in the effect of additives have been observed; both light and heavy molecules can be efficient in transfers with the same molecule. For transfers involving higher-energy quanta which are still well below the dissociation level efficiencies may again range from loG collisions required to de-excite the large sixth quantum in oxygen to one or two collisions required to de-excite the small 26th quantum in iodine. In the neighbourhood of t)he dissociation energy level the quanta are small and energy transfer between translation and vibration should occur readily in all cases since the non-adiabatic condition is automatically fulfilled [cf. Section 3(a)]. There is however quite a considerable variation in the efficiencies of additives in transfer with the same excited molecule large complex molecules being more efficient than small ones.It seems clear [cf. Sections 4(a) ( b ) (c)] that large molecules can effect transfer with the excited species in larger amounts of energy and by more numerous routes than can small molecules (e.g. via their internal modes as well as with translation) and this will affect the relative rates of the observed processes. From the observed relative efficiencies we can say that if small molecules transferred the appropriate energy on every collision as calculated from kinetic theory then large molecules would have collision dianeters for the same transfer greatly in excess of the simple kinetic theory diameters. Alternatively the complex molecules may be considered to transfer energy on every collision as calculated from kinetic theory and the light molecules then appear to require several collisions for each effective transfer.At present no decision between these alternatives is possible from experiment t,liough the results of Section 4(d) suggest a limiting unit efficiency for polyatomic molecules in transfers with reacting polyatomic species. 108 QUARTERLY REVIEWS Rapid redistribution of energy within a molecule by vibration coupling is usually observed in low-energy transfers. In the case of higher-energy states preliminary evidence suggests a rapid exchange of energy between vibrations Finally we ought to say something about the present links between theory and experiment. In general some measure of prediction is possible from theoretical treatments of low-energy transfers from equations such as (1 1).I n the case of higher-energy states although theoretical work gives information about the relation between the stability of excited species and their internal structure 7 1 9 74 s1 little has been done to examine the effects of intermolecular forces on energy transfer or to study the detailed potential- energy surfaces involved in collisional transfers. The importance of these effects in a wide variety of molecular and radical reactions together with a growing body of experimental results of more theoretical interest makes it probable that the quantitative aspects of this important branch of reaction kinetics will develop considerably in the near future. but this important point requires much further study.81 R. H. Lindcluist and G. K. Itollefson J . Chem. Phys. 1956 24 725. B. F. Gray and H. 0. Pritchard J . 1956 1002 ; R. M. Barrer Trans. Parnday SOC. 1948 44 399.

ISSN:0009-2681    

DOI:10.1039/QR9571100087

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

GRIGN'ARD AND ORGANOLITHIUM REAGENTS DERNED FROM DIHALOGEN COMPOUNDS By I. T. MILLAR PH.D. F.R.I.C. and H. HEANEY B.A. (DEPARTMENT OF CHEMISTRY UNIVERSITY COLLEGE OF NORTH STAFFORDSKIRE) IN this Review an attempt is made to outline the methods available for the preparation of Grignard and organolithium reagents from dihalogen compounds and the uses which such compounds have found in synthesis. Many of these reagents are of organodimetallic type. Related compounds can often be prepared by adding lithium to a multiple linkage or by replacing hydrogen linked to carbon by lithium or by MgX but the scope of such methods is clearly restricted particularly in respect of control over the point of attachment of the metal. First the two halogen atoms may be brought into reaction with metal selectively.For instance in the synthesis of the tertiary phosphine (I) from o-bromo- benzyl bromide,l the first step is effected by magnesium in ether followed The reagents to be discussed are of value mainly in two ways. by chloromethyl methyl ether and the second by magnesium and ethyl- magnesium bromide in ether followed by chlorodiethylphosphine PEt,Cl. Of course this type of double synthesis is impossible if the functional group introduced in the first stage interferes with subsequent formation of a Grignard reagent; but this limitation may be evaded in some cases by using an interconversion reaction with an alkyl-lithium in place of magnesium in the second stage for formation of an organolithium reagent in this way is not inhibited by the presence of hydroxyl carboxyl amino- or thiol groups; and this method is often effective in replacing halogen which fails to react with magnesium or lithium.Secondly some dihalogen compounds give organodimetallic derivatives of magnesium or lithium in a single step. Such derivatives when treated 8- Br 0 Br Bun Li - . Et10 C m Li Li Ph& CPh OH OH Q-p Ph 1 Mann a,nd Millar J . 1951 2205. 109 110 QUARTERLY REVIEWS with a monofunctional co-reactant permit two functional groups to be introduced in one laboratory operation or when treated with a difunctionel co-reactant may give a cyclic product e.g. (II).2 Procedures of the latter type have been used parbicularly in the syn- thesis of a number of heterocyclic derivatives of Group IV and V elements which are not easily made by other methods. Elegant syntheses of a number of novel hydrocarbons (e.g.methylenecyclopropane di- and tetra- phenylene and some macrocyclic types) utilise Grignard or lithium reagents derived from dihalogen compounds. Choice of Reagent and Preparative Methods.-The types of reagent behave similarly in carbonyl addition or in elimination of metallic halide on treatment with typical co-reactants although there are qualitative differences in reactivity (e.g. towards olefins and azomethines) so that the two types are in part complementary. Whether one prepares a Grignard or an organolithium reagent from a given dihalogen compound therefore normally depends on the reactivity of the halogen compound on yield and convenience and on the need for bringing one or both of the halogen atoms into reaction. The order of reactivity I > Br > C1> F holds for reaction with metal and interconversion with organolithiums and €or Wurtz-type coupling.Dibromides are generally chosen in both the aliphatic and the aromatic series. Treatment in ethereal solution with magnesium is convenient and satisfactory for most aliphatic dihalides and the more reactive aromatic dihalides. Progressively more forcing reagents are magnesium alloyed with copper ; magnesium or an alloy previously heated with iodine ; and magnesium used with a reactive halide such as ethyl bromide as an “ en- training ” agent. More forcing still is the use of tetrahydrofuran as solvent. Nevertheless ease of reaction does not necessarily lead to a better final yield for a given reagent ; particularly with aromatic dihalides the reaction is best performed under nitrogen.Extensive surveys of the preparation and use of Grignard reagents are a~ailable.~ A number of compounds which fail to react with magnesium can be brought into reaction with lithium. General methods for the formation of organolithium compounds and the reactions of these compounds have been reviewed by Braude and by Wittig.5 As an alternative to the use of metallic lithium an exchange reaction between a preformed alkyl- or aryl-lithium and the dihalogen compound may be used RLi + R’X + R’Li 4- RX. The position of equilibrium 2 Gilnian and Gorsich J . Arne?.. Chem. Soc. 1955 77 6380. Kharasch and Reinmuth “ Grignard Reactions of Nonmetallic Substances ” Prentice-Hall Inc. New York 1954 ; Yoffe and Nesmeyanov “ A Handbook of Mag- nesium-Organic Compounds ” Pergamon Press London 1956.Braude in “ Progress in Organic Cheniistry ” edited by J. W. Coolr Butterworths Scientiftc Publications London 1955 Vol. 111 p. 172. 5 Wittig in “ Newer Methods of Preparative Organic Cheniistry ” Interscience Publications New York 194s ; see also “ Annotated Bibliography on the IJse of Organolithiuin Compounds in Organic Synthesis ” Lithium Corporation of America Inc. Minneapolis 1949 (and supplements). MILLAR AND HEANEY GRIGNARD AND ORGANOLITHIUM COBIPOUNDS 111 in this reaction depends on the relative electronegativities of the groups R and R’ favouring the formation of the organolithium compound in which the metal is linked to the more electronegative of the organic radi- c a l ~ . ~ ~ The preferred compounds for use in this reaction are phenyl-lithium in ether and n-butyl-lithium in ether or light petroleum.The properties of these compounds have been investigated especially by Wittig and by Gilman and their co-workers. Phenyl-lithium is in general less reactive than 12-butyl-lithium and is more prone to side reactions; but it has proved valuable in bringing about intramolecular cyclisation with di( bromo- methyl) compounds (see below). n-Butyl-lithium in ether effects replace- ment of halogen in some dihalogen compounds which are unaffected by magnesium or lithium and acting in homogeneous solution is more suitable than the metals for use in replacing one halogen atom only. Since ethers are slowly cleaved by alkyl-lithium compounds low temperatures and brief reaction times are desirable. Petroleum solutions do not suffer from this limitation but interconversion is slower in this solvent.The halogen- metal interconversion with organolithium compounds has been reviewed.6 Mechanisms of Formation and Reaction.-Relatively little is known about the mechanisin by which Grignard and organolithium reagents of any type are formed. Direct reaction of a halide with metal presumably involves a nucleophilic attack on halogen by the metal ; a “ bimolecular ” reaction at the metal surface has been postulated for organic ha81ides and lithium and the greater reactivity of this metal has been related to its smaller lattice energy and interatomic ~pacing.~ Kharasch and Reinmuth 3 have put forward persuasive arguments for a scheme in which reaction with magnesium is initiated by radical-type unsaturated centres at the metal surface.The halogen-metal interconversion with organolithiums is thought to involve nucleophilic attack by tJhe anion of the organolithium on posi- tively polarised halogen ; in reactions at low temperature configuration can be retained in compounds which have halogen linked to asymmetric carbon atoms or unsaturated centre^.^ The reactions of lithium and magnesium compounds derived from dihalogen compounds are generally normal except when the reactive groups are vicinal (as in the reagents from o-dihalogeiiobenzenes) or are separated by a conjugated carbon chain (as in o- and p-xylylene dihalides) some specific cases of these structural effects are considered below. “ Free ” radical paths may well contribute to a greater extent in reactions of double Grignard reagents ( L e . molecules containing two MgX groups) than in those of simple type.Orgaiiolithium compounds are more reactive than the corresponding Grignard reagents in additions a t multiple bonds and in reactions involving elimination of metal halide probably because the C-Li bond is more polar than the C-MgX bond and becausc the smaller lithium atom has less steric effect. Studies on mechanisms of reaction of Grignard reactions 5a Rosenberg J . Axe?. Clieiti. Soc. 1954 76 4389. 6 Jones and Gilman in ‘‘ Organic Reactions ” John Wiley and Sons Inc. New York 1951 Vol. V I p. 339. 7 Sunthaiikar and Gilman J. Org. Chem. 1951 16 8. 112 QUARTERLY REVIEWS (which are complicated by the equilibrium 2R.MgX + R,Mg + MgX,) have been reviewed by Kharasch and Rei~imuth,~ and organolithium com- pounds by B r a ~ d e .~ Limitations in the Use of Dihalogen Compounds.-The major difficulty is due to dehalogenation which may be either intramolecular leading to an unsaturated or cyclised product or intermolecular yielding in the limit polymeric material. However if the reaction conditions are suitably chosen yields of the organometallic reagent are satisfactory in many cases. Some aromatic double Grignard reagents are only slightly soluble in ether but their solubility is increased by addition of benzene and they then usually react nicely with typical co-reactants such as carbon dioxide. In the aromatic series the reactivity of dihalogen compounds with magnesium is lower than that of corresponding monohalogen compounds. The difficulty of obtaining extensive reaction at a second halogen atom in one ring which is also shown in interconversions presumably arises because a carbanion or anionic complex resists attack by a nucleophile.However some older claims of complete unreactivity to magnesium or lithium may require revision since the early workers could not fractionate halogen compounds efficiently and did not have oxygen-free nitrogen high-purity metals or solvents such as tetrahydrofuran. Characterisation of Reagents.-Early workers characterised their double Grignard reagents by hydrolysis. Isolation of the resulting hydrocarbon or halogenated hydrocarbon is inconvenient ; and the yield of reagent indicated in this way or by the amount of metal consumed or of ionic halide present after hydrolysis is rarely approachable in terms of final product yield in typical syntheses.More useful is treatment with excess of carbon dioxide best as a slurry of the solid and ether which is con- venient and gives yields of carboxylic acid more like those attainable in other syntheses. This method or treatment with benzophenone to give a tertiary alcohol or diol has also been commonly used with organolithium compounds. The side-reaction with carbon dioxide giving a ketone although troublesome with lithium compounds is suppressed with Grignard reagents a t low temperatures which may however cause difficulty with double Grignard reagents owing to insolubility. Another difficulty with such compounds is illustrated by the behaviour of o-phenylenedi(magnesium halides) which on carboxylation give benzoic acid but no phthalic acid probably as a result of chelation in the intermediate complex.Compounds of this type have been characterised by treatment with iododimethylarsine.8 Treatment with a mercuric halide to give an organomercurial has also been used for the estimation of organodimetallic reagent^.^^ l o Types of Compound The classification adopted below is based on the dihalogen compounds Interest in the latter resides largely from which t,he reagents are derived. 8 lleaney Maim and Millar J. /1956 1 4692 and unpublished results. 9 Hilpert and Gruttner Ber. 1914 47 177. lo Wittig and Herwig Chem. Ber. 1954 87 1511. MILLAR AND HEANEY BRIGNARD AND ORGANOLITHIUM COMPOUNDS 113 in their value in synthesis so yields are given in a number of typical cases. Aliphatic Dihalides.-The reactions with magnesium of all the members of the series Br*[CH,];Br from n = 1 to n = 14 have been investigated.Reactivity to magnesium (and to lithium) decreases with increasing chain length and with n = 16 the compound does not react even with activated magnesium.ll With excess of magnesium the major primary product when n > 4 is the double Grignard reagent BrMg*[CH,];MgBr ; these reagents are more soluble in ether the longer the chain length and a t n = 10 complete miscibility is attained. l2 The principal side-reaction is the inter- molecular condensation summarised ,as Br*[CH2lnWa*Br + (m - l)MgBr where the compounds €or which m = 2 3 . . . etc. are formed in progres- sively diminishing yields. In the presence of excess of metal these higher dibromides also form double Grignard reagents l3 the higher members probably as a result of the entraining action of the simpler compounds.The compounds having n = 1-3 give derivatives of no value in synthesis. Methylene dibromide gives a reagent which fails to react with typical Grignard co-reactants other than hydrolysing agents possibly because it is too in- soluble ; l4 di- and tri-methylene dibromide undergo both intra- and inter- molecular elimination of magnesium bromide the former giving ethylene and the latter after carboxylation cyclopropane propylene and suberic acid.3 With higher compounds intramolecular condensation diminishes not being observable with pentamethylene dibromide and the reagents derived from the compounds having n = 4-10 have all been used in syn- thesis. The most satisfactory techniques for reducing " coupling " have been the use of ( a ) activated magnesium and high dilution,lj and (b) ordi- nary magnesium and wet (O-lyo aqueous) ether.g l6 I n this way yields of final products have been raised from 30-50% to 50-70% though the advantage of using " wet ) ' ether in the preparation of BrMg-[CH,],*MgBr has been contested.By reaction with suitable chloro-compounds tetra- and penta-methylene- di(magnesium bromide) have been used to prepare five- and six-membered heterocyclic derivatives of arsenic l7 phosph~rus,~ antimony bismuth l7 lead,l' and silicon ; 3 l 8 organotin halides give distannanes.19 With cadmium chloride an organocadmium compound (probably a linear polymer) is formed which reacts with half-ester acid chlorides to give good yields of diketo-diesters.12 These reagents have also been used in syntheses involv- ing other purely organic co-reactants.3 9 l6 The inaccessibility until recently mBr*[CH,],*Br -+ (m - 1)Mg + I1 Chuit Helv. Chim. Acta 1926 9 264. l2 Kreuchunas J. Amer. Chem. Soc. 1953 75 3339. I3 von Braun and Sobecki Ber. 1911 44 1918. l4 Fidler Jones Clark and Stange J. Amer. Clzem. SOC. 1955 77 6634. l5 Luke; and BIBha Chem. Listy 1952 46 683. I G Brown and Jones J. 1946 781. l7 Gruttner and Krause Eer. 1916 49 437 2666. l9 Ziminer and Mosl6 Chem. Ber. 1954 87 1255. West J. Amer. Chem. SOC. 1964 76 6012. 114 QUARTERLY REVIEWS of the higher dibroniides (n = 6-14) limited their use,ll> 16 l8 although yields of the double Grignard reagents improve with increasing chain length at least LIP to the decamethylene compound.12 l5 Very restricted use has been made of the single Grignard reagents the probable inter- mediates in the coupling reaction.An excess of hexamethylene dibromide when heated with magnesium gives dodecamethylene dibromide by the route BrfCH,],*MgBr + BrfCH,],-Br + Br*[CH,],,*Br (30%) and heptamethylene dibromide similarly yields tetradecamethyleiie di- bromide. 2o Dilithium compounds prepared from aliphatic dihalides have been investigated only in recent years. The restriction in the series Br*[CH,],*Br that n > 4 for the satisfactory preparation of a dimetallic compound again holds and yields of the higher compounds are similar to those obtainable from double Grignard reagents. 21 Methylenedilithium is obtainable only in very poor yield by the action of the metal on methyleiie dibromide a t c2 low temperature ; 21 it is of interest that this compound may be prepared by the pyrolysis of methyl-lithium.22 Ethylenedilitlhium could not be prepared from ethylene dibromide either by the action of the metal 21 or by the use of phenyl-lithi~m.~~ The higher compounds have been used in high-yield syntheses of disilanes 21 spirosilanes such as [CH2]4)Si([CH2]4,18 and cyclic derivatives of Unsaturated dihalides tend to undergo dehalogenatioii with magnesium ; thus 1 4-dibromobut-2-ene gives butadiene in nearly quantitative yield. 24 The highly strained hydrocarbon methylenecycbpropane has been pre- pared 25 in this way from 3-chloro-2-chloromethylpropene by using tetra- hydrofuran as solvent (Cl*CH,),C:CH -+ LCH,12 > C:CH2 (17%). Diha1ogenobemenes.-Although a number of dihalogenobenzenes (e.g. o- m- and p-di-iodo- and m- and p-dibronio-benzene) have been shown to react with substantially more than one equivalent of magnesium the double Grignard reagents can be obtained only in poor yield and have been littlle used.Such reagents from p-dibromo- 26 and p-dibromodeutero- benzene 27 have been employed but more valuable than the former is p-phenylenedilithium obtained by means of n-butyl-lithi~m.~~~ 28 It has recently been shown that 3-bromo-4-iodotoluene with an excess of mag- nesium yields a double Grignard reagent.29 p-Bromo- Much wider use has been made of single Grignard reagents. 2" Mdler and Schutz Brr. 1938 71 689. a l West and Rochow J . Urg. Chin. 1063 18 173'3. 2 2 Ziegler Yagel and Patheiger Z. anoiy. Chert&. 1955 282 345. 3 3 Wittig and Harborth B e y . 1944 77 306. 23a Torssell Acta Chern.Scand. 1954 8 1779. 3 4 Khitrik J . Gen. Cliem. (U.,S.S.K.) 1940 10 2098. z s Gragson. Greenlee Dci-fer arid Boord J . Arney. C'llenz. Soc. 1953 75 3344. 2 6 voii Brauii Irmiscli and Nelles Re?,. 1933 66 1471. !!7 Ucst and Wilson J . 1946 239. z8 Gilman Langhain and Moore J . A r t i t ) / . C'liem. Soc. 1940 62 2327. 29 Hart and Maim Chem. and Id. 1956 574. MILLAR AND HEANEY GRIGNARD AND ORGANOLITHIUM COMPOUNDS 115 phenylmagiiesium bromide with a variety of co-reactants gives the expected products generally in yields of 40-50 yo. All the chlorophenylmagnesium bromides and iodides have also been used as have the m- and p-fluoro- phenylmagnesium bromide. n-Butyl-lithium in ether appears to be particularly effective for replace- ment of a single halogen atom by lithium in dihalogenobenzenes.p-Di- bromobenzene gives p - bromophenyl-lithium in 50-70y0 yield 2% 3O and m- and p-chlorophenyl-lithium can similarly be prepared from the brorno- chlorobenzenes. Fluorine is not susceptible t o interconversion and m-di- fluorobenzene is inetallated a t a C-H bond by methyl- or phenyl-lithium.6 A study of the effect of hydroxyl and methoxyl substituents in dibromo- benzenes and -naphthalenes has shown that such substituents activate halogen in interconversions in the order o > p > m.' Studies on 0- halogenophenyl-lithiums are of particular interest having contributed to the recognition of the " benzyne " (cyclohexadienyne) inter- mediate. Treatment of o-bromofluorobenzene in furan with lithium amal- gam gives 1 4-epoxy-1 4-dihydronaphthalene (IV) in good yield and it was suggested that this arose.by elimination of lithium fluoride from o-fluorophenyl-lithium followed by Diels-Alder addition to furan of the resulting reactive " benzyne " intermediate [represented by (III)] 31 the strincture of which presumably involves an electronically excited acetyleiiic state together possibly with dipolar canonical forms. o-Fluoro- o-chloro- and o-bromo-phenyl-lithium have been prepared 32 by treating the corresponding o-bromohalogenobenzenes with ethereal n-butyl-lithium a t temperatures below - 50". All are highly labile (0-Br-C,H,Li > o-Cl*C,H,Li > 0-P*C,H,Li) and their coupling reactions and addition to furan have been interpreted as involving the intermediate (III). 3 2 9 33 The o-halogenophenylmagnesium bromides can also yield this intermediate.33 The above o-halogenophenylmetallic compounds may be prepared in good yield except the o-bromo-compounds (which are available in 20-30y0 yield by treatment of o-dibromobenzene with n-butyl-lithium 32 or with magnesium in ether or tetrahydrofuran 8 33) and o-fluorophenyl- magnesium bromide ; all react in " normal " fashion with typical co- reactants under suitable conditions. The formation and reactions of the " benzyne " intermediate have recently been reviewed by Wittig. 33a G i h a n and Melvin J. Amer. C'hena. Soc. 1950 '72 996. 3 1 Wittig and Pohmer Angew. (!hem. 1955 67 348. 32 Gilman and Gorsich J. Aniet.. Chenz. Soc. 1956 78 22.27. 33 Wittig and Pohmer Chem. Bet.. 1956 89 1334. 33a Wittig Suornen Kem. 1956 29 A 283. 116 QUARTERLY REVIEWS Dihalogen Derivatives of Diary1 Ethers Diphenylamine and Dibenzy1.- o-Bromophenyl 2-bromo-4-methylphenyl ether reacts slowly with mag- nesium to give ultimately a high yield o€ the double Grignard reagent which has been treated with various aryldihalogenostibines thus affording a series of 10-aryl-2-methylphenoxstibines.A compound of this type 10-p-carboxyphenyl-2-methylphenoxstibine (V ; X = Sb R = Me R’ = p - H02C*C,H,) has been resolved ; it probably owes its asymmetry to folding of the tricyclic system about the O-Sb axis.34 Di-o-bromophenyl ether gives a dilithium compound with n-butyl- lithium which with phenyldichlorophosphine gives 10-phenylphenoxphos- phine (V; X = P R = H R’ = Ph).35 One or both halogen atoms in di-p-bromophenyl ether may be replaced by metal if ethereal n-butyl- lithium is used in appropriate amount,28 and the same would certainly be true for all these compounds in which interconversion is assisted by the o- or p-ether linkage.00’-Dibromodiphenylamine similarly gives a dilithium compound which on carboxylation gives the 2 2’-dicarboxylic acid (84 yo). 36 2 2’-Dibromodibenzyl with n-hutyl-lithium gives 2 2’-dilithiodibenzyl which with dichlorophenyl-arsine or -phosphine gives in 20-25% yield the cyclic arsine or phosphine (VI ; X = As or P R = Ph) these being isodimorphous. 37 This organodilithium compound has also been used to prepare the first cyclic derivatives of boron having carbon-boron bonds within the ring by reaction with n-butyl borate.38 Diha1ogenodiphenyls.-As in the preceding series the members which have proved the most useful are the 2 2’-compounds yielding organodi- metallic derivatives of value in the synthesis of several elusive cyclic structures.With magnesium 2 2‘-dibromodiphenyl is slowly and partially con- verted into a slightly soluble double Grignard reagent ; when heated with cupric chloride in ether (Krizewski-Turner reaction) this yields diphenylene (4%) and tetraphenylene (s-tetrabenzocyclooctatetraene ; 16yo).39 The dibromo-compound reacts smoothly with n-butyl-lithiuni in appropriate amount to give either the mono- or the di-lithium compound ; the latter with dichlorodiphenylsilane gives the cyclic silane (11).2 32 With phenyl- lithium the dibromo-compound gives triphenylene in low yield (apparently via 2-bromoterphenyl) and diphenyl but no diphenylene.40 An alternative route to 2 2’-dilithiodiphenyl is afforded by direct reaction between 2 2’- 3 4 Campbell J .1947 4. 35 Mann and Millar J. 1953 3746. 3G Jones and Mann J. 1956 786. 3 7 Mann Millar and Smith J. 1953 1130. 38 Letsinger and Skoog J. Amer. Clbem. SOC. 1955 77 5176. 39 Rapson Shuttleworth and van Niekerk J. 1943 326. 40 Barton and McOmie J. 1966 796. MILLAR AND HEANEY GRICNARD AND ORGANOLITHIUM COMPOUNDS 117 di-iododiphenyl and lithium. This procedure gives an excellent yield and has been employed both in the synthesis of P-phenyl-9-phosphafluorene (by reaction of the product with phenyldichlorophosphine) 41 and in a more recent synthesis of diphenylene. The dilithium compound with mercuric chloride gives a nearly quantitative yield of diphenylmercury which on being heated with silver powder gives diphenylene (54%).10 3 3’-Dibromodiphenyl reacts with magnesium only with the aid of an entraining agent giving the double Grignard reagent.However 3 3’- dilithiodiphenyl is readily obtained by the use of n-butyl-lithium ; the metal fails to react.42 4 4’-Dibromo- and 4 4‘-di-iodo-diphenyl yield double Grignard re- agents with magnesium,43 44 but these are less useful in synthesis than 4 4’-dilithiodiphenyl obtainable by the action of n-butyl-lithium on the dibromo-compound. 19 28 I n the same way partial replacement of bromine may be effected in satisfactory yield.28 The reactions with magnesium of four of the six homonuclear dibromo- diphenyls have been very fully investigated by Case.45 The most satis- factory results were obtained with a magnesium-copper alloy and the extent of reaction is given as 2 4 > 2 5 > 3 5 > 3 4.The last com- pound failed to react. Coupling with the remaining compounds was not extensive ; double Grignard reagents were formed in poor yields but failed to be carboxylated normally. It is clear that these reagents have little value in synthesis and as with dihalogenobenzenes it is hard to cause a second homonuclear halogen atom to react with metal. Dihdogenonaphtha1enes.-1 2-Dibromonaphthalene reacts with mag- nesium only in the presence of an entraining agent and yields mainly a double Grignard reagent as adduced by the formation of naphthalene on hydrolysis. However this reagent gives only a trace of dicarboxylic acid on carboxylation. Reaction of 1 4-dibromonaphthalene with magnesium does not require an entraining agent and with one equivalent of metal a single Grignard reagent is formed which is carboxylated normally in good yield; excess of metal leads to partial formation of a double Grignard reagent which fails to react with carbon dioxide.I 5-Dibromonaphthalene gives a double Grignard reagent with all proportions of magnesium and after reaction with an excess of metal affords on carboxylation an almost quantitative yield of the 1 5-dicaJrboxylic acid. Similarly both halogens are reactive in 1 6- and 1 7-dibromonaph- thalene the former compound also giving 5- bromo-2-naphthyl magnesium bromide in satisfactory yield which reacts normally with carbon dioxide and with cyanides.46 With “ mixed ” dihalogenonaphthalenes there is no difficulty in bringing one halogen only into reaction with magnesium.41 Wittig and Geissler Annalen. 1953 580 44. 42 Snyder Weaver and Marshall J. Arner. Chem. SOC. 1949 71 289. 4 3 Malinovskii and Pokrovskii Trticly Go~’kov. Gosudarst. Pedagog. Inst. 1940 No. 5 4 4 Kern Gehm and Seibel Malcromol. Chern. 1965 15 170. 45 Case J . Amer. Chem. SOL 1936 58 1246. 46 Zalkind Ber. 1934 67 1031. 51 ; Chein. A h . 1943 37 3077. 118 QUARTERLY REVIEWS The reactions of the above dibromonaphthalenes with lithium and n-butyl-lithium have not been recorded although the exchange reaction with some dibromonaphthols has been studied.' Vicinally substituted naphthalenes would be expected to yield a " naphthyne " intermediate the existence of which has been postulated. Dihdogen Derivatives of Anthracene and Naphthacene.-g 10-Dibromo- anthracene undergoes replacement of one or both halogens on treatment with lithium or n-butyl- or phenyl-lithium depending on the ratio of reac- tants used.The products react satisfactorily with carbon dioxide alkyl halides and ethylene oxide although yields from the dilithium compound are poor. The 2-methyl compound behaves similarly bromine a t the 10- position being replaced preferentially. The bromine of 9-bromo-10-chloro- anthracene is also replaceable best by phenyl-lithi~rn.~' 9 10-di- bromoanthracene in its behaviour on halogen-metal interconversion. 4* 9 10-Di-p-bromophenylanthracene and 5 11 -di-p-bromophenyl-6 12-di- phenylnaphthacene both yield double Grignard reagents by the entrain- ment procedure ; the reagent from the second compound (a substitut>ed rubrene) has the distinction of giving a black solution in ether.49 Halogen-substituted Benzyl ]Halides.-The high re,activity of benzyl halides leads to the predominant formation of a coupled product on treat- ment of such halides with lithium or its simple alkyl and aryl derivatives ; thus 2-bromobenzyl bromide with phenyl-lithium gives 2 2'-dibromodi- benzyl providing the most satisfactory route to this compound.38 A similar reaction has been observed with 1 - bromo-2-bromornethylnaphtha- lene. 50 However reaction with magnesium enables halogenobenzyliiiag- nesium halides to be prepared in good yields.lY Di(hdogenomethy1)aryl Compounds.-The xylylene dihalides all fail to give derivatives of value in synthesis. Mann and Stewart have shown 51 that o-xylylene dichloride with magnesium in ether gives a poly-o-xylylene probably via the highly reactive o-quinodimethane (VII) 6 ll-Dibromo-1 2 3 4-tetrahydronaphthacene resembles t WB) z Similar mechanisms have been offered for a number of related cases in which an MgX residue is separated from a group capable of forming a stable anion by a suitable conjugated system.51 p-Xylylene dibromide also gives a polymer with magnesium ; 52 this reaction presumably pro- 47 Mikhailov and Bronovitskaya Zhur.obschchei Khim. 1952 22 157 ; 1953 23 48 MikhaZlov and Chinaeva ib<cZ. 1952 22 1887. *9 Dufraisse and Morgoulis-Molho BUZZ. SOC. chinz. Prunce 1940 7 930. 51 Mann and Stewart J. 1054 2826. 5 2 Cmothers Chenz. Rev. 1931 8 373. 130. Hall and Turner J. 1955 1242. MILLAR AND HEANEY GRIGNARD AND ORCANOLITHIUM COMPOUNDS 119 ceeds by an analogous mechanism involving p-quinodimethane.It is of interest that this dihalide with sodium in ether gives tri-p-xylylene and m-xylylene dibromide similarly yields di-m-xylylene which could not be obtained by the action of lithium or phenyl-lithi~m.~~ The above mech- anism of polymer formation is not of course available for m-xylylene dihalides. Hall Lesslie a'nd Turner 54 observed that 2 2'-di( broniomethy1)- diphenyl (VIII) reacts with magnesium or phenyl-lithium in ether to give 9 10-dihydrophenanthrene (IX) presumably via an intermediate organo- monometallic compound (VIII) BrH2C CH2Br L/ (1x1 This procedure phenanthrenes affords a general method for the synthesis of 9 10-dihydro- from which if desired phenanthrenes may be obtained by dehydrogenation.The preferred reagent is ethereal phenyl-lithium the necessary di( bromomethyl) compounds being prepared either by bromina- tion of the appropriate dimethyl compound with N-bromosuccinimide or from dicarboxylic esters by reduction to diols with lithium aluminium hydride followed by treatment with hydrogen bromide.55 A poor yield is obtained in the synthesis of 9 10-dihydro-4 5-dimethoxyphenanthrene probably as a result of steric interference between the methoxyl groups inhibiting the close approach of the CH,Br and CH,Li groups in the diphenyl intermediate. 56 A particularly interesting case is the conversion of 2 2'-di(bromo- methyl)-1 1'-dinaphthyl (X) into 9 10-dihydro-3 4-5 6-dibenzophen- anthrene (XI) ; use of an optically active dibromide results in the formation of an active product of opposite rotation activity in both compounds being dependent on the non-coplanarity of the two naphthyl ~ystems.~O 1 8- Di( bromomethy1)naphthalenes undergo a similar reaction yielding ace- naphthenes.57 Where intramolecular cyclisation is impossible or difficult a similar reaction leads to interniolecular coupling giving rise in suitable cases to G3 Httker. McOniie and Norman J. 1951 1114. 5 4 J . 1950 711. 5 5 Beaven Hall Lesslie Turner and Eird J. 1954 131 ; Bcrginann and Pelchowicz 56 Hall and Turner J. 1951 3072. 8 7 Bergmann and Szmuszkovicz J. L4?~iei.. C'hem. Soc. 1953 75 2760. J. Oyy. Chem. 1954 19 1387 ; Badger Jefferies and Kirnber J. 1957 1837. I 120 QUARTERLY REVIEWS some interesting large-ring compounds. Thus 2 7-di( bromomethy1)riaph- thalene yields di(naphtha1ene-2 7-dimethylene) (XII) ; 58 and 2 2'-di- (bromomethy1)dibenzyl gives 1 2-5 6-9 10-13 14-tetrabenzocyclohexa- deca-1 5 9 13-tetraene (XIII).50 These strainless non-planar structures are obtainable in yields of 20-60y0.(XI I) (XI I I) Although not strictly within the purview of the present section it is of interest that cis-1 2 3 6-tetrahydro-o-xylylene di-iodide with phenyl- lithium gives bicycZo[4 2 Oloct-3-ene (XIV) (75%) from which cycloocta- 1 3 5-triene may be prepared.6o Dihalogen Derivatives of Heterocyclic Compounds.-Few results have been reported with di halogenofurans. Single Grignard reagents have been prepared from 2 5-di-iodofuran In the thiophen series halogen in the 2-position is notably reactive and all the 2 3-dihalogenothiophens yield single Grignard reagents ; that from the dibromo-compound when subjected to the Krizewski-Turner reaction yields 3 3'-dibromo-2 2'-dithienyl (36%).The 3 4-di-iodo- compound also gives a single Grignard reagent .62 2 5-Di-iodothiophen gives a double Grignard reagent and affords 2 5-dilithiothiophen in good yield with ethereal phenyl-lithium.6 3 5-Dibromopyridine which fails to react with magnesium gives 3- bromo-5-lithiopyridine with n-butyl-lithium but the second halogen resists replacement ,63 2 6-Dibromopyridine behaves similarly with n-butyl- lithium,63 but surprisingly gives a double Grignard reagent ,64 Several symmetrical dibromodibenzofurans have been converted into dilithium derivatives by the action of n- butyl-lithium ; such derivatives obtainable in good yields are of potential value in synthesis.6 2 8- Dihalogeno-9-ethylcarbazoles behave similarly.6 Treatment with an equi- molecular quantity of n-butyl-lithium would almost certainly give the monolithium compounds in these cases.and 3 4-di-iodo-2 5-dimethylfuran.61 68 Baker McOmie and Warburton J. 1952 2991. 58 Bergmann and Pelchowicz J . Amer. Chem. SOC. 1963 75 4281. 6o Alder and Dortmann Chem. Ber. 1954 87 1492. 61 Hurd and Wilkinson J . Arner. Chem. SOC. 1948 70 730. 8 2 Steinkopf and Kohler Annalen 1937 532 250. 6 3 Gilman and Spatz J. Org. Chem. 1951 16 1485. 64 Proost and Wibaut Rec. Traw. chirn. 1940 59 971.

ISSN:0009-2681    

DOI:10.1039/QR9571100109

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

STRUCTURES OF ELECTRON-DEFICIENT MOLECULES By H. C. LONGUET-HIGGINS M.A. D.PHIL. (DEPARTMENT OF THEORETICAL CHEMISTRY UNIVERSITY OF CAMBRIDGE) The Structure of the Boron Hydrides.-In the last three decades or so a new branch of inorganic chemistry has come into being-the chemistry of electron-deficient compounds. This development is most clearly appreci- ated if one takes as a starting point Sidgwick's monumental book " The Electronic Theory of Valency ".l The concepts and generalisations put forward by Sidgwick in 1927 are still adequate for describing a large portion of structural chemistry namely that of compounds in which every atom or ion has a completed valency shell of two eight or more electrons. Sidg- wick's classification of chemical bonds into those which are essentially ionic and those which are essentially covalent has been extended and deepened of course by the work of Pauling,2 Mulliken, and others but the general validity of this classification has remained unchallenged a t least for the types of compound with which Sidgwick was mainly concerned.Neverthe- less even in 1927 it was recognised that there were some relatively simple substances which defy classification within the Lewis-Langmuir-Sidgwick scheme ; one substance in particular was an outstanding anomaly namely the simplest hydride of boron diborane. and his colleagues had established beyond doubt that this hydride has not as might have been expected the formula BH, but is dimeric B,H6. In order to appreciate the structural problem raised by the existence of B,H, it is necessary to consider the postulates on which classical valency theory was founded.It was implicitly assumed that any chemical bond whether ionic covalent or intermediate is a bond between two atoms only; also that if a molecule contains one or more essentially ionic bonds then the substance will be non-volatile high-melting and usually soluble in liquids of high dielectric constant. Diborane is a substance of high volatility resembling other non-metal hydrides such as ethane rather than ionic hydrides such as calcium hydride in its physical properties. On the basis of the above postulates the bonds in the molecule should be essentially covalent. This conclusion however leads to an immediate difficulty a molecule composed of eight atoms requires a t least seven " two-centre bonds " to hold it together ; seven covalent bonds demand fourteen valency electrons and in diborane the number of valency electrons is only twelve.In this sense therefore B,H is an " electron-deficient " molecule. The early attempts to fit diborane into the classical scheme were to The classical work of Alfred Stock Sidgwick " The Electronic Theory of Valency " Oxford Univ. Press 1927. Pauling " The Nature of the Chemical Bond " Cornell Univ. Press 2nd edn. Mulliken J. Ghern. Phys. 1934 2 782; J. Clzim. phys. 1949 46 497. Stock " The Hydrides of Boron and Silicon " Cornell Univ. Press 1933. 1940. 121 122 QUARTERLY REVIEWS say the least unsatisfactory. Pauling 2 suggested that diborane is a reson- ance hybrid between structures involving two-electron one-electron and no-electron bonds and he supported this with the observation that such bonds would be most easily formed between two atoms of similar electro- negativity the electronegativities of boron and hydrogen being nearly equal.It was difficult to understand however why in these circumstances diborane should be diamagnetic and also colourless since a molecule with many resonance structures is usually coloured. It soon became clear that the electronic structure of diborane would not be understood until definite evidence became available as to the geometrical disposition of the atoms in the molecule. (The situation might have been compared with that facing a theoretical chemist invited to account for the properties of a material whose empirical formula is compatible with several isomeric structural possibilities.) It had been assumed on the basis of slender physical evidence that diborane is geometrically similar to ethane ; early X-ray studies were consistent with this hypothesis and electron diffraction appeared to favour it.7 There were however indications that the ethane-like structure was not entirely satisfactory.Researches on the chemistry of the alkyl diboranes showed that it is possible to replace only four of the hydrogen atoms in B,H by methyl groups and it was well known that BMe was monomeric. The infrared* and the Raman9 spectrum of diborane also presented acute difficulties of interpretation in terms of an ethane-like model ; there were more strong absorption bands than could be accounted for without postulating low-lying electronic excited states a hypothesis for which there was no independent evidence.These unsatisfactory features of the ethane-like structure led to the re-examination of an alternative geometrical structure for B,H which had been suggested in the early days by Dilthey.lo It was shown 11* 12* l3 that many of the above anomalies disappear if diborane is supposed to possess a bridged structure in which four of the hydrogen atoms are situated a t the ends of the molecule being attached to the two boron atoms by normal covalent bonds the other two hydrogen atoms being situated in the middle of the molecule. This left open of course the question how the remaining four valency electrons could hold together the central bridge but such a linkage could be conceived in terms of equivalent resonance structures covalent l1 or ionic l2 or both.13 It was furthermore possible to generalise this structural hypothesis to digallane and the volatile metal borohydrides and the same hypothesis led to an explanation of the non-volatility of aluminium hydride and to the prediction l4 that beryllium should also form Schlesinger and Burg Chem.Rev. 1942 31 1. Mark and Pohland 2. Krist. 1925 62 133. Bauer Chem. Rev. 1942 31 43. Stitt J . Chem. Phys. 1941 9 ‘780. Anderson and Burg ibid. 1938 6 586. lo Dilthey Z. nngew. Chem. 1921 34 696. l1 Nekrassov J . Gen. Chem. Russuh 1940 10 1021 1056. l 2 Syrkin and Dyatkina Acta Physicochim. U.R.S.S. 1941 14 517. l3 Longuet-Higgins and Bell J. 1943 250. l4 Longuet-Higgins J. 1946 142. LONGUET-HIGGINS ELECTRON-DEFICIENT MOLECULES 123 a non-volatile hydride a prediction which was subsequently verified.15 A quantitative theoretical study of the vibrational spectrum of B2H6 lent very strong support to the bridged structure.l6 Before we discuss the electronic nature of the bridge it will be well to refer to more recent evidence which confirms the bridged structure since upon it depends the validity of the current electronic theory. Price l7 examined the rotational lines in certain infrared absorption bands of B2H and found that they resembled very closely the corresponding bands in the spectrum of ethylene. I n particular the rotational lines show a two-fold alternation of intensity which is immediately intelligible on the basis of a two-fold symmetry axis through the boron atoms but would be unin- telligible in terms of a three-fold axis which would lead to an altered inten- sity in every third rotational line.Price’s infrared work has been extended and confirmed by other workers.l* 19 2o The most recent and perhaps most conclusive evidence for the bridged structure comes however from a study of the nuclear magnetic resonance spectrum. 21 The isotopic species shows three main regions of resonance absorption due respectively to the llB atoms the terminal hydrogen atoms and the bridge hydrogen atoms (the last two regions of absorption are close together). The fine structure of these absorption bands is particularly interesting. The absorp- tion due to the terminal protons is split into four equally spaced lines by the magnetic field of the neighbouring boron atom ; this is because the 1lB nucleus has a spin p which can be oriented in four different directions bridge protons is split into seven components by the pair of llB nuclei whose joint spin is 3 and can be oriented in seven different ways (3 2 1 0 -1 -2 -3).The absorption due to each llB nucleus is split into three main lines by the two terminal protons attached to it (the relevant component of their joint spin can be 1 0 or -1) ; each of these main lines is further split into three closely spaced lines by the bridge protons whose joint spin is also unity and can be oriented in three ways. The agreement between the observed and expected spectrum is perfect and leaves no doubt whatever of the correctness of the bridged structure. Having reviewed the evidence which led to the bridged structure for diborane we can now consider the electron distribution in this molecule. If we allow that the terminal hydrogen atoms are attached by ordinary two-electron covalent bonds the structural problem is reduced to explain- ing how the remaining four valency electrons can hold the bridge together (that the terminal bonds are “normal” is strongly suggested by their force constants as calculated l6 by Bell and Longuet-Higgins).If one (3 2 1 2 -1 2 -3) in the external magnetic field. The absorption due to the l5 Finholt Bond and Schlesinger J . Arner. Chem SOC. 1947 69 1199. l6 Bell and Longuet-Higgins Proc. Roy. SOC. 1945 A 183 357. 1 7 Price J . Chem. Phys. 1947 15 614; l8 Webb Neu and Pitzer ibid. 1949 17 1007. l9 Anderson and Barker ibid. 1950 18 698. 2o Lord and Nielsen ibid. 1951 19 1. 21 Shoolery Discuss. Farnday SOC. 1955 19 215. 1948 16 894. 124 QUARTERLY REVIEWS accepts the existence of equivalent alternative resonance structures as a sufficient condition for stability then one can invoke resonance between forms such as (1-VIII) to account for the diamagnetism and stability of H H H H H H H H H H H H \B/' B/' \ B \ B / '.\+ B \B/ \B/ &/ / / \ / \ \ / / \ / \ \ H H H H H H H H H H H H (1) (11) (111) (IV) (V) (VI) (VII) (VIII) the system.However the close proximity of the two boron atoms (1.78 A) shows that there must be substantial overlap between the atomic orbitals of these atoms and that resonance forms such as (IX) and (X) should not H H+ H H H+ H -/ B- B '- B- B -/ \ \ H+ H / H H / \ H / H (IX) 4x1 be excluded from consideration. The resulting picture is unsatisfactorily complex and gives one little insight into the real nature of the bridge linkage.Recognising this Pitzer 2 2 suggested a description of the bridge in which he compared it with the double bond in ethylene. In the ethylene molecule the bond between the carbon atoms involves four valency electrons two of which move in a " molecular orbital " of type CT which is symmetrical about the C-C axis and the other two of which move in a molecular orbital of type n formed by the sideways overlap of two 2p orbitals one on each carbon atom. Pitzer proposed that one should regard the diborane bridge as a " protonated double bond " in which the function of the four electrons is to bind the boron atoms together the two hydrogen nuclei being situated at the points of greatest electron density in this bond. This description has an attractive simplicity but suffers from the drawback of suggesting that the bridge protons should be acidic whereas in their reactions diborane and similar molecules behave l5 as though they contained the hydride anion H-.The next step towards a description of the bridge was to consider ex- plicitly the 1s atomic orbitals of the central hydrogen atoms.23 A general principle of molecular-orbital theory 24 is that two orbitals or groups of orbitals of the same symmetry will interact to give two new orbitals which are respectively more and less strongly bonding than the original orbitals. This idea can be applied directly to the present situation. To make this clear we will number the atoms of the bridge as shown in Fig. 1. Pitzer's cr orbital is then ( G ~ + a,)/1/2 and hisn orbital (pl + p2)/1/2 (the factor l/v'2 is inserted for rough normalisation).The 1s orbitals of 2 2 Pitzer J . Amer.. Chew,. SOC. 1945 67 1126. 23 Longuet-Higgins J . Chirn. phys. 1949 46 276. 2 4 Coulson Quart. Rev. 1947 1 144. LONGUET-HIGGINS ELECTRON-DEFICIENT MOLECULES 125 the bridge hydrogen atoms must be considered together ; the combination (s’ + s ” ) / 4 2 has the same symmetry as the CT orbital and the combination (s’ - s ” ) / 4 2 has the antisymmetry characteristic of a n orbital. By com- bining these with the orbitals of the double bonds we obtain the new com- binations * y1 = (01 + (72 + 8’ + s”)/2 y2 = (231 + 232 + 8’ - s”)/2 y3 = (01 + g 2 - 8‘ - 0 / 2 ? y4 = (PI + P - s’ + s”>/2 the first two of which are illustrated diagrammatically in Fig. 1. Now there is another general principle of molecular-orbital theory namely that if one assigns four electrons in pairs to each of two molecular orbitals yl and y2 then a precisely equivalent description 25 28 is to assign them instead to the orbital combinations (yl + y 2 ) / 4 2 and (yl - y 2 ) / 4 2 .Applying this principle to the orbitals depicted in Fig. 1 we obtain two new localised molecular orbitals of the form x1 = (01 + 231 + u2 + P2 + 2S’)/242? x 2 = bl - P1+ a2 - P2 + 2s”)/242 These two orbitals are concentrated on either side of the central bridge and by assigning an electron pair to each of them we obtain two “ three-centre ” bonds 23 each involving the two boron atoms and one of the hydrogen atoms. The idea of interpreting electron-deficient structures in terms of many- centre bonds has been taken up and extensively applied to all the known boron hydrides by Eberhardt Crawford and Lipscomb.27 To Lipscomb and his colleagues is due the credit for having established firmly and in considerable detail the structures of the higher boron hydrides B4H10 2sa B6H8,2eb and B5Hl ; 28c the structure of BloH14 was established crystallo- es Lennard-Jones Proc. Roy. SOC. 1949 A 198 14. 2* Pople Quart. Rev. in press. 27 Eberhardt Crawford and Lipscomb J. Chem. Phys. 1054 22 989. ““(a) Nordman and Lipscomb J. Amer. Chem. Xoc. 1953 75 4116 ; ( b ) Dulmage and Lipscomb ibid. 1951 73,3539 ; (c) Lavine and Lipscomb J. Chem. Plzys. 1954 22 614. * The qualitative justification for taking equal weights in these combinations is that the electronegativities of boron and hydrogen are close together. 126 QUARTERLY REVIEWS graphically by Kasper Lucht and Harker.29 The structures of these molecules are themselves interesting but perhaps of even greater interest are the underlying structural principles.The logical foundation of Eber- hardt Crawford and Lipscomb’s theory is the description of diborane out- lined in the last paragraph. The two localised molecular orbitals x1 and x2 can be regarded as formed from hybrid atomic orbitals (al & p l ) / @ and (02 & p 2 ) / z / 2 on the two boron atoms and the 1s orbitals of the hydrogen atoms thus (51 - Pl I 0 2 - P2 + +.s”)/2 (see Fig. 2) x2 = {- 4 2 d2 One might therefore have taken as a starting point the four hybrid orbitals on the boron atoms and considered how they overlap with one another and with the two hydrogen orbitals. It is clear that these six atomic orbitals will overlap in two sets of three and that it would therefore be possible to consider each set of three mutually overlapping orbitals independently of the other set.Eberhardt Crawford and Lipscomb pointed out that when three atomic orbitals all overlap one obtains three molecular orbitals of which only one is strongly bonding ; this was the fundamental principle of their structural theory. I n discussing a particular molecule one is a t liberty to hybridise the atomic orbitals of a boron atom in any appropriate manner provided that one subsequently takes into account the overlap of every hybrid with the orbitals of all neighbouring atoms. Let us then see how these ideas apply to the higher boron hydrides. Other Boron Compounds.-The next simplest hydride after B2H is tetraborane B4H,,.The geometrical structure of this molecule indicated in Fig. 3 is such that each boron atom is surrounded more or less tetra- hedrally by the four atoms indicated. I n setting up the molecular orbitals it is therefore appropriate to start with tetrahedral hybrids on the boron atoms and then to consider the mutual overlap between these and the 1s orbitals of the hydrogen atoms. The resulting situation is one in which every full line in the diagram represents a pair of mutually overlapping 29 Kasper Lucht and Harker Acta Cryst. 1950 8 436. LONGUET-HIGGINS ELECTRON-DEFICIENT MOLECULES 137 atomic orbitals ; these give rise to 7 two-centre bonds involving fourteen valency electrons. The remaining eight electrons can then be assigned to three-centre bonds of the same type as in diborane; these bonds sjerve to bind together the atoms joined by broken lines.All the electrons being assigned to bonding orbitals we obtain a closed-shell structure and the molecule is diamagnetic. FIG. 3 FIG. 4 B,H1 presents a more complicated situation. In Fig. 4 the lines radiating from each atom make roughly tetrahedral angles with each other. The external B-H bonds use up sixteen electrons and ten are left to be assigned to the rest of the system. Of these six are required to form three-centre bonds involving the bridge hydrogen atoms lcaving four to hold the boron frame- work together. This is achieved by the formation of two three-centre bonds of a different type involving the mutual overlap of two groups of three tetrahedral hybrid boron orbitals.The geometry of this molecule is that of a square pyramid. The four neighbours of each basal boron atom are again disposed in a nearly regular tetrahedron. The external B-H bonds consume ten electrons and the B-H-B bridges between them another eight leaving six to bind the apical boron atom to the base of the pyramid. To see how this is done it is simplest to consider separately the 28 2px and 2p9 valency orbitals of the central boron atom the 2217 orbital being already involved in a bond to a hydrogen atom. The 28 2px and 2py orbitals have the same symmetries as the combinations (see Fig. 5 ) Again we obtain a closed shell. The molecule B,H can be described 3O in similar terms. (tl + t + t 3 4- t4)/2 ( - t + t + t - t d / 2 (4 + t - t 3 - t4)/2 c1 respectively I CL FIG.6 with these to give six molecular orbitals of 30 Longuet-Higgins J . Roy. I'nst. Cherra. 1953 77 179. 128 QUARTERLY REVIEWS which three will be bonding and three antibonding [the seventh molecular orbital arising from the seven atomic orbitals 28 2px Zpy t, t, t, t4 namely (tl - t + t - t4)/2 does not involve the central boron atom and is in fact antibonding between the basal atoms]. There is therefore room for three pairs of bonding electrons and this is the number which remains after the attachment of the hydrogen atoms to the framework. The structure of B,,H, has been interpreted in detail by Eberhardt Crawford and Lipscomb but the underlying principles are the same and there is no need to describe this molecule in detail here. Lipscomb and his colleagues have also investigated a B hydride 31 and a B hydride 32 and have suggested structures for them.It will be seen from these examples that the structural principles apply- ing to the electron-deficient molecules are rather different from those appro- priate to saturated molecules. The reason for this is essentially as follows. I n saturated molecules composed of the atoms hydrogen carbon nitrogen oxygen and other atoms possessing a t least half the number of electrons required for a stable valency shell the valency shell of each atom can be completed by the formation of two-centre bonds alone. Further if the total number of electrons in a molecule exceeds the total number of valency atomic orbitals then there will be unshared pairs-for example in water there are six valency orbitals altogether and eight electrons so that we obtain 2 two- centre bonds and two unshared pairs.The filling of the valency shell of each atom then imposes geometrical restrictions on the disposition of its neighbours. These restrictions arising essentially from Pauli’s exclusion principle require that different electron pairs shall avoid one another as far as possible. In a molecule such as diborane however the usual stereo- chemical restrictions become relaxed ; 33 each boron atom is in proximity to five other atoms although the four electron pairs contributing to its valency shell are as already explained disposed in what can be regarded as a tetra- hedral manner. An interesting molecule whose geometrical structure has been established by Atoji and L i p ~ c o m b ~ ~ but not hitherto accounted for is boron mono- chloride B,C14.In this molecule the boron atoms are linked together in a regular tetrahedron and to each is attached an outlying chlorine atom. In this molecule there are clearly insufficient valency electrons to allow each full line in Fig. 5 to be interpreted as a two-centre electron-pair bond. The explanation of this structure is probably as follows. To start with it is natural to regard each boron atom as forming a two-centre bond with the 3pa orbital of the neighbouring chlorine atom. This uses up eight valency electrons. Disregarding for the moment the 3pn electrons of the chlorine atoms we are then left with eight valency electrons to hold together the boron tetrahedron. These electrons can be assigned t o molecular orbitals compounded from the six equivalent bonding orbitals xii between the boron 31 Eriks Lipscomb and Schaeffer J.Chem. Phys. 1954 22 754. 32 Dickerson Wheatley Howell Lipscomb and Schaeffer ibid. 1956 25 606. 33 Platt ibid. 1945 22 1033. 34 Atoji and Lipscomb ibid. 1953 21 172. LONGUET-HIGGINS ELECTRON-DEFICIENT MOLECULES 129 Now the permissible combinations of these equivalent orbitals are atoms. fully determined by symmetry. They are respectively (x12 + x 1 3 + Xl4 + x 2 3 + X24 + x34)/'d6 (XlZ + x34)/\/2 T2 ( X I 3 - x24)/+ { k 1 4 - x23)/'d2 (XI2 - x23 + X34 - x4l)/' (2x13 + 2x24 - XI2 - x 2 3 - x34 - X14)/2/12. { The symbols A, E and T are those conventionally used to distinguish the different symmetry classes. The first molecular orbital designated as A, is strongly bonding as it involves all the equivalent orbitals overlapping with the same sign.The next three orbitals denoted as T, are also bond- ing ; for example (x12 - xS4)/42 binds atom 1 to atom 2 and atom 3 to atom 4. The next pair of orbitals labelled E presents an ambiguity. They have the same energy for reasons of symmetry but if one expresses them in terms of the boron s and p orbitals one finds that for example SO that each of the E orbitals of the tetrahedron consists purely of 2p boron orbitals. Now it is not difficult to show that there are two combinations of the hitherto unconsidered 3pn orbitals of the chlorine atoms which also have the symmetry E. As a consequence there will be " back-co-ordination" from the chlorine atoms into the B4 tetrahedron. This back-co-ordination will partly fill the E orbitals of the tetrahedron and render them no longer available for other electrons.We conclude that the eight electrons avail- able for the boron tetrahedron will occupy the orbitals labelled A and T but not those labelled E . In summary the reason why only eight and not twelve valency electrons are used in B-B bonds is that two of the mole- cular orbitals which would otherwise be bonding within the tetrahedron are instead involved in accepting 3pn lone pairs from the chlorine atoms. The Solid Borides.-An equally interesting problem this time in solid structure is presented by the hexaborides of which a typical member is CaB,. Crystallographic studies by Pauling and Weinbaum 35 established that the structure of the crystal is analogous to that of sodium chloride the sodium ions being replaced by calcium atoms and the chloride ions by regular octahedra of boron atoms each octahedron being linked to six neighbouring octahedra at a distance of 1.72 A which is also the interatomic distance within each octahedron.Longuet-Higgins and Roberts 36 inter- preted this structure as follows. One may hybridise each boron atom in a digonal manner to give two hybrids one pointing into the octahedron and the other outwards leaving two 2p orbitals with axes perpendicular to the axes of the hybrids. The outward-pointing hybrids of neighbouring octa- hedra are then regarded as giving rise to two-centre electron-pair bonds between the octahedra and this leaves fourteen valency electrons per ( x l 2 - x 2 3 + 2 3 4 - x41)/2 = {- (2$)x>l + (2?)y)2 + (2?)x)3 - (2$)y)P}/2 35 Pauling and Weinbaum 2.Krist. 1924 87 181. 36 Longue -Higgins and Roberts Proc. Roy. SOC. 1954 A 224 336. 130 QUARTERLY REVIEWS CaB unit. Now we consider the molecular orbitals available for binding together the atoms of an octahedron Here again symmetry plays a simpli- fying r61e and one finds that from the remaining three orbitals on each atom of the octahedron there arise eighteen molecular orbitals of which seven are bonding and the remainder antibonding. The forms of these orbitals are roughly as shown in Fig. 7 where each arrow represents a directed atomic orbital and they suffice to accommodate fourteen valency I I I 4 7 T~(20the1-s also) 7iU(2others also) (a) m Cb) (6) FIG. 7 ( a ) A boron octahedron i n CaB,. ( b ) The seven bonding molecular orbitals. electrons. The final picture therefore is one in which all the valency electrons of the crystal are associated with the giant lattice of B octahedra the calcium atoms being required to contribute their valency electrons to this system.We thus obtain a closed-shell structure for the boron lattice with calcium ions located in its interstices. It is interesting to note that the corresponding borides of lanthanum and cerium which have three and four ionisable electrons respectively are not insulators like CaB but conduct electricity. 37 An interesting corollary to these results is that the hypothetical hydride B,H, whose existence has been discussed would in all likelihood not have a closed-shell structure; two more electrons would be required to fill the bonding orbitals.25 36 Another interesting electron-deficient system which has been discussed in terms of the molecular-orbital theory is the icosahedron of boron atoms in which every boron atom is linked to five other atoms within the icosa- hedron and to one atom outside.This structure occurs in boron carbide BqC,38 and also probably in elementary boron.39 Longuet-Higgins and Roberts 40 investigated these structures and were led to the conclusion anticipated by Eberhardt Crawford and Lipscomb 27 that an icosahedral hydride B1,H, would require two extra electrons to complete a closed shell. The reader is referred for details to their original paper. Electron-deficient Carbon Compounds.-Electron deficiency is not con- fined to boron compounds. It occurs whenever the number of valency electrons in a molecule is less than 2(n - 1) where n is the number of atoms 37 van Stackelberg and Neurnann Z.phys. Chem. 1932 19 23 314. 38 Zhdanov and Sevastyanov Doklady Akud. Nuuk S.S.S.R. 1941 32 432. 39 Hoard Geller and Hughes J . Amev. Chenz. SOC.. 1951 13 1892. 40 Longuet-Higgins and Roberts Proc. Roy. SOC. 1955 A 280 110. LONGUET-IIIGGINS ELECTRON-DEFICIENT MOLECULES 131 in the molecule. Examples of electron-deficient substances not containing boron are the polymerised alkyls of metals of Groups I 11 and 111 ; for example Al,Me is ele~tron-deficient,~~ and so is the polymeric BeMe,. In these substances physical studies have established that pairs of neighbouring metal atoms are linked through bridges involving the methyl It is not yet known whether the three-centre bonds which occur in these substances involve the fourth orbital of the methyl group or one or more of its hydrogen atoms ; l4 the former possibility seems more likely but experi- mental work is needed to settle the question.For example a study of dimeric trimethylaluminium by nuclear magnetic resonance would be of great interest. If the central bridge involves hydrogen atoms in three- centre bonds one would expect three regions of proton resonance absorption one due to the terminal methyl groups one to the bridge hydrogen atoms and one due to the remaining protons of the bridged methyl groups. The absence of three regions of absorption would indicate either that the bridge does not involve hydrogen atoms or that bridged and unbridged protons are too rapidly exchanged to give separate signals in the spectrum. Tetra- meric tetramethylplatinum presents a similar problem which has been only partially resolved by X-ray analysis.43 It is also possible to regard as electron-deficient certain unstable car- bonium ions some of which are intermediates in chemical reactions 44 and others of which have been identified in the mass spectrograph.Of the latter an intriguing example 45 is the species CH5+ which contains six atoms and only eight valency electrons. It is possible of course to represent this molecule as a resonance hybrid of five structures each containing an unbound proton or to formulate it in terms of molecular-orbital theory by assuming for example a bipyramidal configuration. At present however it is not possible to be sure of the molecular configuration which in any case is probably very easily deformed ; hence the most satisfactory point of view is to regard the electrons as constituting a neon-like closed shell in which the five protons take up a configuration of minimum potential energy presumably distorting the electron cloud to some extent.Examples of electron-deficient intermediates are becoming more com- mon. There is evidence 46 that the propyl cation is not a three-carbon chain but a cyclopropane molecule with an additional proton attached to the ring. One manner in which such a system could be formed is illus- trated in Fig. 8 ; if a mechanism of this sort is realised it provides a ready explanation for the great ease with which the hydrogen and carbon atoms in such carbonium ions are " shuffled "-an effect which has been known for many years. Another such system is the carbonium ion shown by Roberts Mazur and Chambers to be an intermediate in the solvolysis of cyclobutyl 41 Pitzer and Gutowsky J .Amer. Chew. SOC. 1946 68 2204. 4 2 Snow and Rundle Acta Cryst. 1951 4 348. 43 Rundle and Sturdivant J . Amer. Chem. SOC. 1947 69 1561. 4 4 Dewar Bull. SOC. chim. France 1951 18 C 79. * Hamill personal communication. 4 6 Rylander and Meyerson J . Amer. Chem. Soc. 1956 78 5799. 132 QUARTERLY REVIEWS chloride or cyclopropylmethyl chloride 47p 48 (see Fig. 9). Winstein and his collaborators 49 50 have postulated the occurrence of analogous inter- mediates in the reactions of certain norbornyl derivatives. It is also usual to regard as electron-deficient certain inorganic complexes containing a metal atom linked to an unsaturated hydrocarbon radical or molecule.Of these well-known examples are the ethylene complexes of bivalent platinum 44 9 5 and the transition-metal cydopentadienyls of which ferrocene was the first to be dis~overed.~~ In these compounds however the bonding is rather different in type from that occurring indiborane. For H I H I FIG. 8 FIG. 9 example the linkage of an ethylene molecule to a platinous ion involves the p orbitals of both partners and the d orbitals of the Then electrons of the olefin form a dative bond to the metal which in turn donates a pair of d electrons into the antibonding n orbital of the carbon-carbon double bond. The resulting situation is alternatively described by saying that each carbon atom forms a covalent bond to the platinum atom and if this point of view is adopted the electron deficiency is formally removed.The structure of ferrocene is also describable in terms of d-p bonding between the metal and the two hydrocarbon rings; the details of this structure have been fully discussed in the literature; 53 54 again it is a matter of preference whether one classifies such molecules as electron- deficient or not. Summary.-We can set down the following generalisations about the structures of electron-deficient molecules ( 1 ) The most restrictive and perhaps the most clear-cut definition of electron deficiency is that the number of valency electrons is less than 2(n - 1) where n is the number of atoms in the molecule. This definition covers the boron hydrides the volatile metallic borohydrides the non-ionic metal hydrides the polymerised metal alkyls and certain unstable car- bonium ions.(2) In such molecules the idea of an electron-pair bond can be retained if we allow that some bonds may involve atomic orbitals from more than 4 7 Roberts and Mazur J . Amer. Chem. Soc. 1951 '73 2509 3542. 48 Roberts and Chambers ibid. p. 5034. 49 Winstein and Trifan ibid. 1949 71 2953. Winstein Walborsky and Sehreiber ibid. 1950 72 5795. 51 Chatt and Duncanson J. 1953 2939. 52 Kealy and Pauson Nature 1951 168 1039. 63 Moffitt J . Amer. Chem. SOC. 1954 '76 3386. 54 Dunitz and Orgel J. Chem. Phys. 1955 23 954. LONGUET-HIGGINS ELECTRON-DEFICIENT MOLECULES 133 two atoms in the molecule. Thus in diborane the central bridge comprises two three-centre bonds each of which involves both the boron atoms and one of the bridge hydrogen atoms. (3) Although the polymerised alkyls presumably contain many- centre bonds it is not well established whether the bridged alkyl groups are bound by carbon orbitals or hydrogen orbitals though the former hypothesis seems more likely on theoretical grounds.(4) Certain complexes formed by metals with unsaturated hydrocarbons are superficially electron-deficient but in many cases it is possible to assign to them conventional covalent- bond structures it being understood that in such cases the bonding involves the d orbitals of the metal and the p orbitals of both partners. (5) As regards their chemical properties electron-deficient molecules normally behave as Lewis acids-all of them react with water and other Lewis bases. This is because 2 two-centre electron-pair bonds represent a more stable situation than one unshared electron pair and one three- centre electron-pair bond.

ISSN:0009-2681    

DOI:10.1039/QR9571100121

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

ULTRASONIC ANALYSIS OF MOLECULAR RELAXATION PROCESSES IN LIQUIDS By R. 0. DAVIES D.PHIL. (DEPARTMENT OF PHYSICS QUEEN MARY COLLEGE LONDON) and J. LAMB PH.D. (ELECTRICAL ENGINEERING DEPARTMENT IMPERIAL COLLEGE LONDON) Introduction.-Propagation of a sound wave through a liquid takes place adiabatically except at very high frequencies which are outside the range a t present available experimentally. Thus as the pressure in the liquid alternates about the static or ambient pressure a t a frequency f (= w/2n.) there will be a corresponding variation in the temperature a t any point. If c is the phase velocity a t a frequency f then the variations in excess of pressure in a plane progressive sound wave can be expressedby p = p exp (- ccx) exp [ico(t - x/c)] where 12 is the distance measured in the direction of propagation p is the amplitude of the pressure variations at the source (x = 0) and a is the amplitude absorption coefficient of the liquid through which the wave is propagated.This Review is concerned with the interpretation of absorption data for pure liquids and their binary mixtures. In the first place there is a con- tribution to the total absorption due to the shearing motion which occurs in the propagation of a plane wave this is generally termed the " classical " absorption * and is expressed in terms of the shear viscosity 7 by 8n 2 ~ j (sec.2 cm.-l) classical 3pc3 The quantity (a/f2)classjcal is independent of frequency for a particular liquid a t a given temperature except in the case of highly viscous liquids in which the shear viscosity is frequency-dependent.Other contributions to t,he absorption of sound waves in liquids arise from time-dependent molecular processes. Thus whenever the molecules can reside in two or more equilibrium states which difler in energy (although they may have the same volume) it is possible for this equilibrium to be perturbed by the sound wave. Consider for example the case of a first- order reaction A + A for which kl, k, are the appropriate rate constants. The important parameter as far as the sound wave is concerned is the relaxation time z = (k12 + I C ~ ) - ~ . At sufficiently low frequencies (CUT <Q 1 ) Litovitz et al. J . Acoust. SOC. Amer. 1951 23 $5 ; 1954 26 566 577. * There is an additional contribution arising from the energy lost from the sound wave owing to thermal conduction but this is negligible except in mercury and liquid helium.134 DAVIES AND LAMB ULTRASONIC ANALYSIS OF MOLECULAR RELAXATION 135 tlhe molecular distribution between states A and A follows the variations in the sound wave. On the other hand at very high frequencies (cot> 1) the internal equilibrium is sensibly unaffected by the sound wave. As the frequency increases from a low value (wz < 1) there is an increasing phase retardation between the population of a given energy state and the excess of pressure of the sound wave. This is accompanied by a decrease in the magnitude of the changes in population density with the result that the absorption per unit of wavelength p (= KC,”) passes through a maximum near a “ characteristic frequency ” fc (4 + E2,)/2n. There is an accom- panying decrease in the quantity (oc/f2) from its low-frequency value to a limiting high-frequency value when f > fc (see Fig.2 ) . Such a behaviour is termed ultrasonic relaxation and with certain approximations can be represented by an equation of the form Here the term B represents the combined effects of absorption due to shear viscosity and any further relaxation processes having a characteristic fre- quency much greater than the value of ,fc for the particular mechanism considered. Measurements of the absorption of sound waves in liquids have revealed the existence of relaxation processes which can be given a molecular inter- pretation. Moreover the detailed study of the behaviour a t different tem- peratures and over a frequency range sufficient to delineate a major part of the relaxation enables the parameters of the reaction in question to be evaluated for processes which in many cases are not readily observable by other methods.Prom the chemical point of view the field of study can be termed the investigation of fast reactions. A particular feature of the ultrasonic method of observing such reactions is its sensitivity to very low population densities (of the order of 1 per cent. and less) in the higher- energy state. Experimental results on which an analysis can be based have in the main originated in the past few years. The measuring techniques which have been developed are now used to an increasing extent by physical chemists thus providing a further analytical method for the study of mole- cular behaviour which is complementary to the established techniques of infrared spectroscopy dielectric absorption etc.Simultaneously with ex- perimental developments there has been corresponding progress in the theoretical understanding of relaxation processes based on the methods of irreversible thermodynamics. Familiarity with the theory is essential to the chemist who desires to correlate experimental results with molecular behaviour and the purpose of this Review is to present a summary both of the fundamental theory and of tjhe outstanding experimental results with their inolecular interpretation. Andreae and Lamb Proc. Phys. Soc. 1956 69 B 814. 3 Meixner Ann. Physik 1943 43 470 ; Acoustica 1952 2 101 ; Davies and Jones Phil. Mag. 1953 2 (Suppl.) 370; Davies Proc. Roy. Soc. 1954 A 226 24. K 136 QUARTERLY REVIEWS The technique which provides the greatest accuracy in determining the absorption coefficient is the pulse method this is generally employed a t frequencies above 5 to 10 Mc./sec.and accuracies of &2% in ( a / j z ) have been achieved up t o frequencies as high as 200 M~./sec.~ I n the frequency range 1 to 10 Mc./sec. the method of acoustic streaming which has been recently developed 6 has considerable advantage over other measuring techniques particularly for low-absorbing liquids absorption measurements can be obtained to within &5y0 accuracy. A reverberation technique is employed a t frequencies in the range 100 lsc./sec. to 1 M~./sec.~ but corrections which must be applied to the recorded results are mainly responsible for a relatively poor accuracy of measurement-ca. & 15% in (cc/f2).In connection with the experimental observation of relaxation processes it should be mentioned that an increase in the phase velocity takes place with increasing frequency over the relaxation region. It can readily be seen why in general it is more profitable to measure the absorption rather than the velocity for if acetic acid a t 25" c is taken as an example the total change in velocity over the relaxation region is only 1.7% whilst the absorption falls from (cc/j2) = 132,000 x 1O-l' sec2 cm.-l at low frequencies t o 140 x A further point of importance is the extent of the frequency range required to delineate a relaxation. The relaxing part of (a/f2) is expressed by A/[1 + (f/j'c)2] and this falls from 0.9A to 0.1A in one decade of fre- quency. In practice a somewhat wider range of frequency is to be desired and this must of course be roughly centred about fc in order to permit the results to be analysed satisfactorily.sec2 cm.-l a t high frequencies-a factor of 1000. The thermodynamic theory of relaxation and the propagation of sound Static Thermodynamic Relations.-It is convenient to collect here some essential thermodynamic reasoning about a simple fluid system containing a single chemical reaction mechanism or more generally a single ordering process. It is clear that to define the state of such a system we require three independent variables. In the first instance we shall choose the temperature (T) the pressure ( p ) and an ordering parameter (x) which in the case of a chemical reaction specified by the stoicheiometric formula Ci Y ~ M ~ = 0 can be taken as the degree of reaction.I n this case therefore the change in mole number of the i-th constituent due to the reaction is given by dN = vidx. The dependent thermodynamic variables are the entropy (X) and the volume ( V ) conjugate respectively.to T and p and the affinity ( A ) conjugate to x. I n the case of a chemical reaction we have A = - C ~ V ~ L . L ~ . * ( 1 ) Pinkerton Proc. Phys. SOC. 1949 62 B 286. Andreae Hensell and Lamb ibid. 1956 69 B 625. Piercy a,nd Lamb Proc. Roy. SOC. 1954 A 226 43 ; Piercy J . Phys. Radium Karpovich J . Acoust. SOC. Amer. 1954 26 819. 1956 17 405. DAVIES AND LAMB ULTRASONIC ANALYSIS OF MOLECULAR RELAXATION 137 m d for all cases we assume the existence of a Gibbs function G(T,p,x) such that for any change of state in the system Thus - A = (aG/ax),, is the change in Gibbs function due to the ‘‘ re- action ” per unit change in x a t constant T and p ; it is sometimes denoted by AG.‘The first-order derivatives of the dependent variables X V and A are of three types (i) The thermodynamic coeficients Cp = T(aX/aT),,, a = (l/V)@V/i3T),,, and K = (- l/V)(ilV/ap)T,z. It should be noticed that they are the heat capacity expansivity and compressibility of the system taken at $xed z. They are sometimes called the “ frozen ” “ glassy ” or “ instantaneous ” coefficients. (ii) The reaction coeficients AX = (aS/az),, and AV == (aV/ax),,,. For the chemical case we have AX = xi visi and AV = Ci vpi where si and vi are partial molar quantities. AS and AV are the entropy and volume changes due to the reaction per unit change of x.(iii) The ordering coeficient /3 = (aA/ax),,T. This gives a measure of the curvature in the plot of G against x a t constant T and p . For the case of a chemical reaction we have from equation (1) that - dG = SdT - Vdp -+- Adz . ’ (2) p = - xtxj ViVj(a,LAi/aNj)p,T. * (3) - AG = A(p,T,x) = 0 . (4) The condition for thermodynamic equilibrium is that G should be a minimum for variations a t constant p and T. This condition is therefore that which relation fixes the equilibrium value of x as a function of T and p . Variations in the dependent va’riables are given by the equations ’ ( 5 ) In the spegial case for which x is fixed we have of course the familiar I- * dX = (C,/T)dT - Vxdp + AXdx dV = VxdT - VKdp + AVdx dA = AXdT - AVdp + Bdx relations > - * (6) } (7) (dS) = (C,/T)dT - Vadp (dV) = VadT - VKdp Consider now the case in which changes are made so slowly that x is allowed to attain its equilibrium value a t each instant that is changes subject to A = 0 or dA = 0.Solving the last member of eqn. (5) for dx and substituting the result in the first two members we then have (dS) = [(C - TAS2//S)/T]dT - [ V ( M - AXAV/DV)]dp (dVIA = [V(a - ASAV/pV)]dT - [V(K - AV2/pV)]dp We now introduce a notation which distinguishes the (‘ equilibrium ” thermodynamic coefficients by means of a bar and denotes the difference between (‘ equilibrium ” and “ frozen ” coefficients by 6 dC E C, - C T(aS/aT),, - T(aX/aT),, 138 QUARTERLY REVIEWS Equation (7) can therefore be written in a form analogous to eqn. (6) . (7’) (dS) = (C,/T)dT - VZdp (dV)A = VEdT - V<dp and the ‘’ excess ” or “ difference ” properties are given by 6% == - ASAV/OV .- (8) I 6C = - T AS2/P i 3 ~ =’ - AV2/PV This is an important result because it shows how the ‘‘ excess ” thernio- dynamic coefficients can be related to the reaction coefficients A S and hl; together with the ordering coefficient ,8. dC, da and 6~ are not independent but from equation (8) we find the identity dK.dC = TV(d%)’ . . (9) This can be used to reduce any thermodynamic expression involving the “ excess ” properties to one involving only two of them. We shall find it convenient to take da and dC as the independent excess variables. Then from the elementary expression for the adiabatic compressibility we have By use of eqn. (9) this can be thrown into the form dKs/Zs = (7 - 1)(8%/2 - 6cp/c,)2/[(dc,/c,)(1 - dC,/C,)] .(16) where 7 = C,/Cv is the ratio of the principal specific heats. The relation (10) will be used later. The arguments just outlined could of course be carried through in terms of independent thermodynamic variables other than p and T. Indeed it is an economy of expression (which will be found valuable later) not to be tied down to any special choice of variables. Let the independent thermodynamic variables (like p and T) be denoted by x and their con- jugate dependent variables by X (note that x for example is a contracted way of referring to more than one quantity) ; we then have a free energy P such that - d F Xdx 4- Adz . * (11) . (12) 1 - * dX = cdx + Adz also d d = A’& + /3dx where c stands for the theriiiodynaniic coefficieiits (aX,/ax), A( = A’) for tlic reaction coefficients (aX/az) and /3 for the ordering coefficient (aL4 / a x ) .Eliminat,ing dx from the two members of (12) we have d S - (c - A p 1A’)d.t. -$- a/j-u,4 . . ( 1 3 ) The equilibrium coeficientw E . z ( a X / a ~ ) ~ appear as the coefficient of d.c in eqn. (13) so that - (14) & z== 6 - c = - $-1l‘ , DAVIES AND LAMB ULTRASONIC ANALYSIS O F MOLECULAR RELAXATION 139 Equations (1 1)-( 14) summarise the static thermodyiianiics of a relaxing or reacting system. Rate Equations.-It is now necessary to consider the time-dependent behaviour of a relaxing system. It therefore seems reasonable to regard A as a kind of internal driving force acting on the system and whenever A is non-zero trying to change x so as to bring it back to equilibrium. The simplest rate assumption consistent with this idea is that the system satisfies the linear relation We know that in equilibrium A = 0.dx/dt = LA . with L a constant. It inay be helpful to note that this is precisely the assumption required by the Prigogine-de Groot theory of irreversible thermodynamics,8 since the rate of entropy production in the system can easily be shown to be (dx/dt)(A/T). Moreover in the case of chemical reactions equation (15) although formally very different from the usual assumptions about forward and backward reaction rates can in fact be shown to have equivalent macroscopic consequences. Instead of using L it is convenient to introduce a new rate parameter z having the dimensions of time and defined by z = - 8-lL-l. Then eqn. (15) can be written as - pzdx/dt = A .. (16) To obtain the “ thermodynamic equations of motion ” which give the time-dependent law connecting x and X we must eliminate A and x from equations (12) and (16). This yields the form dX = [C - A ( 1 + ~d/dt)-lD-lA’)]dx . which with eqn. (14) becomes or These important relations give the dynarnical behaviour of a relaxing I n order to see their Then system in a form convenient to use and remember. significance let X and x be referred to equilibrium values of zero. we can write eqn. (18) in the form X + z(dX/dt) = EX + zc(dx/dt) . * (19) This shows that if a fixed “ force ” of xo be applied to the system (initially in equilibrium) the “ response ” X(t) is an instantaneous jump of cxo followed by an exponential relaxation to the final response Exo with a time-constant of z.Having olhined tJhe general formula ( 1 8 ) in a somewhat contracted This is illustrated in Fig. 1 . 8 Prigogine niid Defug. “ Thcl.inoclS.n~ti~iique Chimiyue ’’? Desoer LiBge 1950 ; de Groot “ Thermodynarnies of Irreversible Processes ” North Publ. Go. Amsterdam 1951. 9 Davies and Lamb Proc. Phys. SOC. 1956 69 B 293. 140 QUARTERLY REVIEWS notation we may recover the appropriate dynamical equations with say p and T as independent variables. In the neighbourhood of an equilibrium t = O t FIG. 1 Response of a relaxing system to a “step ”. -4 curve of X / x = c + ( E - c)[l - exp ( - t / ~ ) ] state with co-ordinates Vo To etc. they may be written as - { “6 4- 1 + BK z(d/dt) )}(P -Po) Before using equations such as these to describe the behaviour of an actual system they must be supplemented by information giving the mech- anical and thermal exchanges between the system and its environment.(For example we might be required to describe the adiabatic approach to equilibrium in a constant-volume container. The left-hand sides of equa- tions (20) then vanish and we are left with the integrable system of two equations for the two unknowns p and T.) The derivation of eqn. (18) shows that we are not tied to any special independent variables. However it is important to notice that although the “ dynamic ” coefficients have always the same form the value of z depends on the choice of inde- pendent variables.9 Formally this arises because p = (&4/8~),,~ is involved in the definition of z so that a change in the choice of x generally implies a change in the value of z.Thus if we write z,,T for the value of z sppro- c* EZ c + &/[l + z(d/dt)] . (21) DAVIES AND LAMB ULTRASONIC ANALYSIS OF MOLECULAR RELAXATION 141 priate when p and T are independent variables etc. we have the dynamic expressions * (22) The relations between various possible z's have been worked out. It can be shown for example that z,,~ = (C,/(?D)~D,T. Sound Propagation in a Relaxing Fluid.-Now that the dynamical relax- ation behaviour of a simple '< reacting " system has been established we can use the result to find the effect of the relaxation on the propagation of a sound wave. Let u be the instantaneous velocity of a particle of the fluid. The net force per. unit area acting on an element of the fluid of thickness dx is - (ap/ax)dx ; the (mass x acceleration) of the element is po(au/at)6x where po is the density.1 . * K* = K + 6 K / P + z,,,(d/dt)] Ks* = KS + d K x / [ 1 + T,,~(d/dt)] Consider the propagation of a plane sound wave in the x-direction. Hence by Newton's law of motion poau/at = - ap/ax . * (23) Let d be the dilatation of the fluid so that the mass included in unit area of thickness 6x is po( 1 - d)dx. This changes a t a rate - po6x. ad/& which must be equal to the net rate of flow of mass into the element viz. - po(i3u/2c)dx. Equating these two quantities we have the continuity equation aupx = aapt . (24) azp/ax2 = - poa2d/at2 . * (25) From equations (23) and (24) we find that and it remains to consider the equation of state of the fluid it is a t this point that the work of the previous sections is relevant.It can be shown that except a t frequencies so high as to be out of range of present experi- ment the transmission of sound through a fluid is " adiabatic ', This means that virtually no heat is transmitted down the temperature gradients which exist a t any instant in the direction of the sound wave. Hence the relation between the pressure and the dilatation is the time-dependent adiabatic relationship implied by the second member of eqn. (22). It can be written in a form similar to that of eqn. (19) Here the " z " is actually zp,s of eqn. (22). (25) and (26) we have the wave equation in the form Eliminating d between equations As usual we seek solutions of eqn. (27) having sinusoidal time variation of frequency w/2n by substituting in it a solution proportional to 9 e exp 142 QUARTERLY REVIEWS [i(cot - Ex)] and then solving the resultant equation for k.that k must satisfjT the characteristic equation It is found k2/co2 = pLs( 1 + iuncS/kS)/( 1 + icor) = P.Y exp (- ie) say . (28) co2 Thus the propagation number k is complex and the solution sought is pro- portional to Be exp {i[cot - I k I (cos 0 / 2 ) x ] - I k I (sin 0 / 2 ) x } . Consideration of the form of the solution shows that the phase velocity (c) or wavelength (A) and the amplitude absorption coefficient per unit length (a) are given by * (29) 1. c = w / ( l E I cos e/s) 3L = 2n/(l E I cos 0/2) or and The most convenient theoretical measure of the absorption is not 01 but the dimensionless absorption coefficient per wavelength (p) which is given from eqn. (29) as Finally tan 0/2 can be found from eqn.(28) so that where x = ur = c o ~ ~ ~ and r = 6Ks/Es-sometimes called the '' relaxation strength ". The dependence of ,u and of c on em is shown in Fig. 2 on a a = I Ic I sin r3/2 ,U = MA = 2n tan 0/2 . * (30) iu = (zn/xr){[1 + 24(1 - r)2 + 2x2(1 - r ) + 2 2 ~ 1 4 - 1 - x2(1 - r ) > (31) 0*0540*04 / / \ 0 - - A / 0.05 0.7 0.2 0.5 7 2 5 70 20 X FIG. 2 Absorption and velocity associated with a single relaxation (r = GKs/Ks = 0-1). Curves of A (a/f2)[co.fc d(1 - r ) ] ; B c/co and p as functions of x = WT. logarithmic scale for a moderate value of &cs/Ks (= 0.1). spicuous and important feature is the hump on the curve of ,u. mum occurs a t a frequency cornax given by The most con- The maxi- CL),,,'C~,,~ = (1 - ~ K s / < s ) - * . ( 3 2 ) DAVIES AND LAMB ULTRASONIC ANALYSIS OF MOLECULAR RELAXATION 143 and the value of the maximum is The expression for pmax in eqn.(33) can be expanded in powers of In practice it generally happens that ~ K ~ / K ~ < 0.4 ; then with an accuracy better than 1% we have (34) It is clear that equations (32) and (33) can be used to find the thermo- dynamic parameter ~ K ~ / K ~ and the rate parameter zD,? from reasonably accurate measurements of sound absorption as a function of frequency. The Introduction of Chemical Reaction Parameters Formulae for Reactions in Ideal Solution.-The main interest in obtaining the parameter ~ K ~ / ' T is that the work of p. 136 shows that it can be related to the reaction coefficients of chemical interest AV and Ax. Indeed by ecp. (10) it may be expressed in terms of 6a and 6C together with certain measurable equi- librium properties of the fluid ; 6a and 6C are in turn related to AT' and A# by the formulze of eqn.(8). The only other unknown quantity appearing in this chain is the ordering coefficient 0. It is the object of this section to show how /I can be found for the special case of reactions in ideal solution. The definition of /3 and its expression in eqn. (3) show that ,8 is a solution property of the system the evaluation of which depends on knowing the variations in chemica,l potential with composition. The simplest and most useful case to consider is that of the ideal solution specified by pi = pio + RT In xi where xi is the mole fraction of the i-th constituent. becomes where Y = Xi v and N = X i N i . The mole numbers Ni will be determined by the initial number of moles and the degree x to which the reaction has proceeded in equilibrium Finally x will be fixed by the equilibrium condition Civ,pi = 0 which can be written as (Ape is often called AGO = - RT In K the Gibbs free-energy change of the pure constituents at the same pressure and temperature.) The expression for ,8 comes from eliminating N N and x from equations (35) (36) and (37).The two most important ones are (i) The unimolecular reaction - M + M = 0 with v1 = - 1 v2 = 1. Equation (3) then /3 == - R l ' ( C i ~ i ~ / N i - v 2 / N ) . (35) Ni = Nio -+- V ~ X * (36) (37) rIi(NiO -+ Y i X ) Y ( = (No + V X ) ~ exp (- Apo/RT) . Each case must be dealt) with on its merits. 144 QUARTERLY REVIEWS Here we find that . (38) (N," + N2")' exp (AGOIRT) (ii) The dimerisation reaction - M + 2M2 = 0 with Y = - 1 v2 = 2.RT [I + exp (AG0/RT)l2 - p = In this case* * (39) RT [ l + 4 exp (AGo/RT)]3/2 - p z _.___ ~ _ _ _ _ _ _ _ _ _ ~ _ _ (2NI0 + N p ) ' exp (AGOIRT) Application and Summary of the Theory.-In order to make clear the practical use of the material of the last three sections we shall now consider how one could start from absorption data taken over reasonably wide bands of frequency and temperature and make inferences concerning the reaction mechanism responsible for the absorption. We shall assume that appro- priate corrections have already been made for the " classical " absorption due to viscosity and thermal conductivity and that we are left with experi- mental values for ,u(co,T) due to relaxation only. We must first check that the frequency dependence has the correct form [eqn.(31 I] for a single relaxation mechanism. If so we may consistently make ASSUMPTION 1 The relaxation is due to a single mechanism. Without any further assumption we then obtain the two macroscopic parameters associated with the mechanism AV and AS. Take the maximum value of ,u at a given temperature [pu,,(T)] and the corresponding frequency [~In,x(T)l* (i) Let us write m = 2,umax/n. Then from eqn. (33) the thermostatic parameter r(T) = &cS/ks is given by This is the only piece of thermostatic (as opposed to kinetic) information obtainable directly from experiment. (ii) A kinetic parameter z( T)-the adiabatic-isobaric relaxation time- characteristic of the mechanism is obtained from eqn. (32) which with the help of (40) can be written as r(T) = 8 ~ ~ / t ? ~ = m((1 + &rn2)i - i m ] .* (40) 1 / 2 2 = comax2{1 - m(1 + i r n 2 ) t + +m2] . - (41) Later we shall mention the Eyring-like discussions which are usually given in connection with observed values of ~ ( 5 " ) . Here we wish to con- sider how ~ K ~ / K ~ (a function of T) can be used to obtain some thermostatic information. To do this we must first introduce a definite assumption about the stoicheiometric nature of the mechanism. Let us take the simplest and most important case. * In the ca.se of dimerisation in solution with N,O moles of solvent it follows from ( 3 5 ) and (37) that Y = - (2Nlo + N O + N,O) /3 is the positive root of the quadratic equat,ion where ( = (2N10 + N,0)/(2N10 + N20 + N,O) and K = exp (- AGo/RT). Thk; equation gives the dependence of p on the strength of solution (5) and the equilibrium constant K .Y y ( 1 - &()2/(K -1 8f - 452) = Y ( l - 5) + ( K + 4)2/4K DAVIES AND LAMB ULTRASONIC ANALYSIS OF MOLECULAR RELAXATION 145 ASSUMPTION 2 The reaction involved is the unimolecular reaction - N + M = 0 and it takes place in ideal solution. This means that we may take over formula (38) for p which involves the unknown Gibbs energy change for the "pure constituents" AGO depending in general on T. If this expression for 48 is substituted in eqn. (8) (remembering that AS = AH"/T) we obtain 6C, da and 6~ in terms of the three reaction parameters AVO AH" and AGO. Thus eqn. (10) eventually expresses the measurable quantity 8 ~ ~ / k ~ from (40) in terms of these three unknown temperature-dependent reaction parameters.The only relevant relation between them * is the Gibbs-Helmholz equation a(AG"/T)/aT = - AH"/Tz * (42) It is clear that further assumptions must be made before information can be obtained about the reaction coefficients. The simplest and commonest assumption is ASSUMPTION 3 The volume change of the reaction (AV") is zero. For isomeric reactions this is often a very reasonable assumption because the kinds of molecular mechanism which have been studied do not usually involve large changes of size. It has immediate implication that da = 6~ = 0. Hence 6~~ can be expressed entirely in terms of 6C and we find in fact from equations (40) and (10) that The right-hand side of eqn. (43) is an experimentally accessible function of temperature only. Returning now to equations (38) and (8) we have the final formula (44) m exp(AGo/RT) R [l+exp (AG"/RT)I2 where @(T) is an experimentally accessible function of T and the right- hand side contains the unknown functions AGO and AH" connected by eqn.(42). The system (42) and (44) can be integrated to give AG'"(T) = 4RT tanh-1 tan ${C + T o ~ T ~ [ ' I ( T ) ] d 5 " / T } - (45) 1 AHO(T) = ~RT-~[CD(T)] sec {(I + y o J T d [ ~ ( ~ \ i d ~ / ~ } where C is an arbitrary constant of integration. are therefore not sufficient to fix the reaction coefficients. used to estimate them by making a further assumption. AH") is independent of temperature. The ultrasonic data alone It is nevertheless ASSUMPTION 4 The entropy change of the reaction (AS") (and therefore The Gibbs energy change of the reaction is thus taken to be a linear * If data at various pressures were available the relation aAGo/pa = AVO would become relevant.We do not consider this case any further. 146 QUARTERLY REV1 EWS function of T AG" = AHo - TAXo and it becomes a question of fitting formula (44) which now involves two constants AHo and ASo to the experi- mental results. It should be emphasized that even if the experimental data fit eqn. (44) exactly for an appropriate choice of AH" and As" this does not prove the constancy of AH" and AS" but merely shows the con- sistency of assuming that they are constant the integration constant of eqn. (45) cannot be removed by taking thought. The simplest version of eqn. (44) occurs under the further specialisa.tioi1 of ASSUMPTION 5 The entropy change of the reaction (As") is zero.This is often quite a reasonable assumption. In any case the resulting forin of eqn. (44) usually gives a good qualitative picture of the variation of 0 (which is nearly proportional to p,,,) with temperature. The form of eqn. (46) is shown in Fig. 3 where 6C is the product of R and the right-hand side of eqn. (46). It is sometimes known as Schottky's function because it was FIG. 3 The yelaxing specific heat for u. two-state epuilibi.izwn with AS0 = 0. introduced by him as the contribution to the specific heat due to a single excited energy state above a non-degenerate ground state. If the variation of ,urnax with temperature is known one can tell immediately whether AH" is larger or smaller than about 2.6RT ( i . e . 1.5 kcal./mole at 300" K). To sum up under conditions and assumptions which we have attempted to display exactly it is possible to use ultrasonic .absorption measurements in order to obtain (i) an exact measurement of a relaxation time (z) associated with simple molecular mechanisms and (ii) reasonable estimates of the thermodynamic reaction coefficients (AH" and As") of these mechanisms, DAVIES AND LAMB ULTRASONIC ANALYSIS OF MOLECULAR RELAXATION 147 Experimental investigations on ultrasonic absorption in liquids Relaxation Due to Rotational Isomerism.-Whenever the molecules of a liquid can exist in two or more isomeric forms which differ in energy then it is possible for the equilibrium to be perturbed by a sound wave in a manner discussed in the last section.Thus in the neighbourhood of the characteristic frequency fc the absorption per unit wavelength (p) passes through a maximum value which for the two-state process with A V = 0 is related to the energy difference between the two forms by eqn.(44). This equation shows that an essential requirement for the existence of an ultrasonic absorption due to the isomerism is that t'he different configura- tional forms of the molecule shall d#er in energy. Although evidence for the existence of rotational isomers can be found from other measurements for example from infrared studies it appears that the strength of the ultra- sonic relaxation is particularly sensitive to the number of molecules in the state of higher energy ( N 2 ) . Prominent relaxations have been observed in cases where less than 1% of the molecules reside in t,he upper state cor- responding to AH" values of 3 kcal./mole or more.Further as we have shown it is possible to make reasonable estimates of AH" and 48" from experimental measurements of absorption over a range of temperature. This is not always possible by other methods. The connection between observed ultrasonic absorption and the exist- ence of different energy states due to rotation isomers has been established only recently so that ninny of the results ava'ilable at present are of a preliminary nature. In many cases work has not been carried out over a sufficiently wide range of temperature to permit estimates of AHo and AX" although the mechanisms responsible for the observed absorption can be described with reasonable certainty. However the following cases will serve to illustrate the value of the ultrasonic method of investigating rota- tional isomerism.Once the mechanism responsible for the absorption h& been established it is merely a questiqn of extending the measurements over a sufficient range of temperature and frequency in order to obtain the reaction coefficients (p. 144). Acraldehyde (CH,:CH*CHO) is a typical example of an unsaturated aldehyde in which a relaxation has been observed.10 This is attributed to a perturbation by the sound wave of the equilibrium (47) Unsaturated aldehydes and ketones. H H H H- C H I I H -C I c \ / C I c . (47) These planar configurations of the molecule are stabilisecl by conjugation lo de Groot and Lamb Trans. Paraduy Xoc. 1955 51 1676 ; Nature 1956 177 1231. 148 QUARTERLY REVIEWS of the bond between the central carbon atoms which can be regarded in terms of the resonating structures H H I+ \ / C I1 C H-C H I \ / I H-C H c t - - c In this case the characteristic frequency of the relaxation at 25" is in the region of 250 Mc./sec.as is evident from Fig. 4 which includes details of other similar liquids. It has been suggested lo that the smaller the resonance I 2 5 70 20 50 700 200 f (Mc./sec.) FIG. 4 Relaxation in unsaturated aldehydes. A Cinnamaldehyde (25") ; B crotonaldehyde (25") ; C crotonaldehyde (50") ; acraldehyde (25" c ) . [By courtesy of de Groot and Lamb Nulztre 1966 177,1231.1 energy (i.e. the weaker the conjugation) the smaller is the energy barrier which has to be overcome in bringing a molecule from a favourable position through a non-stabilised intermediate form to a second favourable position and hence the higher is the characteristic frequency.The replacement of the hydrogen in the trans-position on the top carbon atom by a methyl group to form crotonaldehyde (CH,*CH:CH*CHO) strengthens the con- jugation and lowers the characteristic frequency from somewhat above 200 Mc./sec. for acraldehyde to 30 Mc./sec. for crotonaldehyde a t the same DAVIES AND LAMB ULTRASONIC ANALYSIS OF MOLECULAR RELAXATION 149 temperature. This effect of stronger conjugation can be offset by an increase in steric hindrance if for example the hydrogen of the aldehyde group is itself replaced by a methyl group to form a ketone. An interesting case arises in mesityl oxide [(CH,),C:CHCO-CH,] where no relaxation is observed over the entire frequency range 250 kc./sec. to 200 Mc./sec.the correspond- ing value of (cc/f2) being constant a t 34 x 10-l' set. cm.-l for a temperature of 25" C. However construction of a scaled molecular model for mesityl oxide shows that the higher-energy state to the right of the corresponding equilibrium (47) is untenable owing to the physical size of the interacting methyl groups. The absence of any relaxation in this case substantiates t,he molecular interpretation described above. A comprehensive study has been made of the ultrasonic absorption in these and other related liquids and will shortly be published. A summary of the results at present available is given in Table 1. TABLE 1. Sound absorption parameters for certain aldehydes 1017 B 1 1 1 io-6c' (sec.z cm-l) (Mc./sec.) (cm. set.-') ~ Croton- ~ ~ o ~ 674 aldehyde 355 loppmax Cinnam- 1 aldehyde 1 25 1 722 j (Mc./sec.) 0.35 0.40 6.8 11.8 lo*//,ax 3.1 4-8 0.072 0.020 29 30.3 1-268 1-29 34 ~ 70-0 ~ 1.165 ~ 1.45 Methyl formate .. . . Ethyl formate . . . . . Methyl acetate . . . . . Ethyl acetate . . . . . 78 ~ 15.7 ~ 1.561 ~ 0.885 25" 25 20 20 Esters. Ultrasonic relaxation has been observed in a number of osters,lll 12 a summary of the relaxation parameters for four such liquids being given in Table 2. TABLE 2. Relaxation parameters for four esters I Liquid I Temp. The explanation of these relaxations is that the sound wave perturbs These planar positions of the molecule 11 Pinkerton Ultrasonics Conference Brussels 1951 p. 1 17 (Med. VZuumche Akad. k. Wet.) ; Huddart Thesis London 1950; Biquard Ann. Physique 1936 6 195; Liebermann Phys. Rev.1949 75 1415; 1949 76 440. an equilibrium of the form (49). 18 Karpovich J. C h . Phys. 1964 m 1767. 150 QUARTERLY REVIEWS are stabilised by conjugation of the carbon-oxygen bond which is possible owing to the lone-pair electrons of the oxygen atoms. T 7 - 2 I . ('19) 0 l o Measurements over an extended temperature range are available oiily for ethyl acetate for which AH" M 3 kcal./mole whilst the activation energy in the backward direction (see p. 158) is AH,$ = 5-7 kcal./mole. The value of AH" is consistent with the fact that pmax increases with increasing tem- perature and the value of 6C is therefore on the right of the hump in the specific-heat curve for a 2-state equilibrium (Fig. 3). An increase in the size of the groups R1 and R2 diminishes ,urn and therefore increases Arlo.A marked increase in characteristic frequency is found in passing from a formate (It1 = H) to a corresponding acetate (R1 = CH,) owing presum- ably to increased steric interaction. On the ot,her hand only a slight increase infc is found in passing from the methyl ester (It2 = CH,) to the corresponding ethyl ester (R2 = C,H,) in this case increased conjugation of the C-0 bond partially offsets the effect of increased steric interaction 011 the variation of f c . cycloHexane derivutiues. Pronounced relaxation occurs in certain deriv- atives of cyclohexane with characteristic frequencies in the neighbourhood of 100 to 200 kc./sec.l2* l3 Owing to the limited accuracy of the rever- beration technique which is employed to measure the absorption in liquids a t these relatively low frequencies i t has not been thought worth while to make measurements a t different temperatures.The molecular mechanism responsible for the observed relaxations a t room temperatures is however clearly defined and it is to be hoped that further efforts to improve the experimental accuracy will be successful thus permitting direct evaluation of the energy parameters. Both " boat " and " chair " forms of cyclohexane are possible although the " boat " form does not appear to be present in appreciable proportions under ordinary conditions. 14 l5 No relaxation has been observed in cyclo- hexane for frequencies up to 100 Mc./sec.16 from which it can be inferred that the population in the higher-energy " boat " form is very small indeed ; the conversion from " boat " into " chair ? ' form is extremely rapid.There is however a suggestion of incipient relaxation in the region of 200 Mc./sec.l6 Directing attention to the chair form of the niolecule we must dis- tinguish between one set of axial C-H bonds which are parallel to the 1 3 Lanib and Sherwoocl Trans. Paraday Soc. 1955 51 l G i - 4 . ? 4 Pitzer et nl. J . Amer. C'hem. Soc. 1947. 69 957 2488 ; Hassel IZP,search 1!150 l 5 McCoubrey and Ubbelohde Quwt. Rcu. 19.51 5 364. l6 Heasell and Lamb Proc. Phys. SOC. 1956 69 B 869. 3 504. DAVIES tlND LAMI3 ULTRASONIC ANALYSIS OF MOLECULAR RELAXATION 1.51 principal axis of the carbon ring and the other set of six C-H bonds which make an angle of 109” 28’ with this principal axis the latter being termed equatorial bonds. If now a methyl group is substituted for a hydrogen atom then t8he two alternative “ chair ” forms Fig.5 are of different energy. The calculated energy difference for methylcyclohexane is 1 *6 lical./mole. F I G . 5 Alternative ‘ chair ” forms for metkylc3’cloiLe.z.ane. It is this equilibrium which is perturbed by the sound wave giving rise to the observed relaxations. Since in cyclohexane and 1 l-dimethylcyclo- hexane the alternative chair forms are of equal energy there is no observed relaxation. The corresponding absorption values a t 25 O are ( x / j 2 ) = 192 x 10-l‘ cm.-l for cyclohexane and 127 x cm.-l for 1 1-dimethylcyclohexane. I n the case of methylcyclohexane at 16” the (ct/’2) value falls from 200,000 x 1O-l‘ sec2 cm.-l a t very low frequencies (below 100 lsc./sec.) to 118 x Measurements have been made on a complete series of dimethylcyclo- hexanes.No relaxation was observed in cis-1 2- trans-1 3- and cis-1 4-dimethylcyclohexane since in each of these cases rotation of one methyl group from an axial to an equatorial position is accompanied by rotation of the second methyl group in the reverse direction (equatorial to axial). I n the case of cis-1 3-dimethylcyclohexane the methyl groups in the higher energy (axial) positions encounter consider- able steric interaction since both are on the same side of the plane of the carbon ring. An energy differcnce of order 5 kcal./mole has been assigned to the isomers of this molecule so that the expected relaxation is presumed t o occur a t very high frequencies outside the present range (as for example with the “ boat ” and “ chair ” process for cyclohexane).It is noteworthy that a relaxation has been observed in cyclohexene 1s which has been attributed to perturbation of the equilibrium between the boat and p~eudo-chair forins. The double carbon-carbon bond in cyclo- hevene causes the two hydrogen atoms pointing together from the (‘ fore ” and “ aft ” positions in the boat form to be further apart than in cyclohexane. The consequent reduction in steric interaction may explain why this type of relaxation is found in cyclohexene but not in cyclohexaiie where it pre- sumzbly occurs a t frequencies above 200 Mc./sec. I n connection with these relaxation processes mention should be made sec2 cm.-l a t 100 Mc./sec. An analysis of the results is given in Table 3. L 152 QUARTERLY REVIEWS of measurements recently reported for the shear viscosity of ethylcyclo- hexane as a function of frequency.17 The shear viscosity of the liquid has been found to decrease with iiicreasiiig frequency over the same range in which the ultrasonic relasatioii occurs.It is very likely that the change in shear viscosity arises from the saiiie inolecular rnechanisin as is responsible TABLE 3. Sound absorption iiL cycloAe,rane and its deyit:atlves c No r lasation 4 Umubstituted . . ~ 500,000 500,000 Relaxation above 1 Mc./sec. 0.017 0.019 - Methyl- . . . . Etl1J.l- . . . . 1 1 3-Trimethyl- . ~- "000 cis-1 2-Diinethyl- . Ko re1atsa.t ion ohserved ~- < 300 truns- 1 2-Dimethyl- 0.004 150 ~ 55,000 cis-1 3-Dimethyl- . ~ 102 XO relaxation observed trans- 1 3-Diiiiethyl- No relasat,ion observed cis-1 4-Dimethyl- .~ 87-G KO relaxation observed trans-1 4-Dimethyl- 0.0023 112 for the ultrasonic behaviour described nbovc. Physically it is possible to visualise that in shearing motion preference would be given to that chair forin in which the C-CH bond is in the plane of the carbon ring. These preliminary results on the variation of shear viscosity with frequency are particularly interesting and it is hoped that this aspect will receive further attention in future work. Measurements of sound absorption in triethylamine over the frequency range 23 to 192 Mc./sec. mid a t teniperatures from 35" c1 to 70" c have demonstrated the existence of a relaxation process which is attributed to the presence of rot,utional isomers.l* Curves of absorption a t selected temperatures are shown in Fig. 6. Aiialysing the results in the Triethylnmine.17 Clausnizer and Kneser International Coinmission on Acoustics 2nd Congress 18 Heasell and Lamb Proc. Roy. SOC. 1956 A 237 233. 1956 Paper I.B.2. DAVIES AND LAMB ULTRASONIC ANALYSIS OF MOLECULAR RELAXATION 153 manner discussed earlier we find that AH" = 3.4 kcal./mole and AXo = 4.7 cal./(mole deg.). Accordingly the experimental region is to the right of the hump in the specific-heat curve (Fig. 3 ) plnax increases with increasing Y 0.02 0 01 20 50 I00 200 500 f(Mc./sec.) FIG. G Absorption in triethylumine. The curves are tlieoretical and the points experimental. A 25" B 33" C 45" c. [By courtrry of Heasell and LZLIII~ Proc. Roy. SOC~. 1056 A 237 233.1 temperature. The activation energy associated with the backward reaction is AH,$ = 6.8 kcal./mole (see p.168). A study of t,he molecular model of triethylainine reveals the existence of three possible configurations shown in Fig. 7. It is considered that the equilibrium between two or possibly FIG. 7 Rotatioiaal isomeys of triethylamine. [By courtesy of Heasell and I ~ i i i b Proc. Roy. Soc. 1956 A 237 233.1 three of the configurational states is responsible for the relaxation. It is noteworthy that no evidence was found for relaxation in diethylamine where the two ethyl groups can be accommodated in the basal plane this is riot possible in triethylamine as can be readily verified from a molecular model. Additional support for this explaiiatioii is given in ref. 18. 154 QUARTERLY REVIEWS iVethyZbutanes. The siniplest type of rotational isomerism is due to alternative orientations of groups around the carbon-carbon bond in ethane and substituted ethanes and has been described in a previous Review.15 I n these types of molecule there are in general three equilibrium positions which may or may not differ in energy depending upon the symnietry arrangernent,s.Where energy differences exist between the alternative states i t is to be expected that this will give rise to ultrasonic relaxation as has been demonstrated.lg It appears that in order t o observe the relaxation peak a t convenient frequencies of the order of 10 Mc./sec. it is necessary to work at temperatures near 200" K. Relaxation was observed for 2-methylbutane 2 3-dirnethylbutane and also for 2-methylpentane but was not found for 2 2-diinethylbutane where the three equilibrium states are of equal energy nor for the straight-cha.in compound n-pentane.Typical results for 2-methylbutane are shown in Fig. 8 froill which it is evident c-q-c-c i FIG. 8 Sound absorption in 2-niethylbutane. A 180" B 190" C gooo I) 210" E 220" K. per wavelength. [By courtesy of Young ttiitl Petrauskas J . ( ' J N ) ~ . Phys 1958 25 943.1 p in this Fig. is in decibels that as the temperature increases from 1130" K the value of plllaX increases initially and then falls with subsequent temperature increase above 190 O K. This behaviour is characteristic of the specific-heat curve for a two-state process and indicates that the maximum in the specific heat (6G,) occurs around 185" K. Since the maximum corresponds approximately t o the condition AH"/RT = 2.5 it follows that AN" is approximately 0.9 kcal./mole.Activation energies for the relaxations in 2-methylbutane and 3-methyl- peiitaiie were found to be about 4.7 lical./mole a value of 3-3 kcal./mole was found for 2 3-dimethylbutane. I n a mixture of 3-methylbutane and n-pentane the curve of absorption against temperature exhibits a peak at the same temperature as that in pure 3-methylpentane but reduced in proportion to the reduction in the number of molecules of 3-methylpentane l9 Young and Petmuskas J . Ghem Phys. 1956 25 913 ; Chen and Petrauskas ibid. in the press. DAVIES AND LAMR U14TRASONIC ANALYSIS OF nlOLECULSR RELAXATION 155 which are present in t,he mixture. This simple behaviour in solution confirins that the reaction is unimolecular. It is significant that in these liquids as in the other cases of rotational isomers previously discussed the observed relaxations can be described by a single relaxation time in each case.This fact alone could perhaps be used as evidence for an intra- as distinct from an inter-molecular equilibrium as the mechanism responsible for the relaxations. Vibrational Relaxation.-The majority of ultrasonic investigations in gases have been concerned with relaxation due to the time-delay in establish- ing equilibrium of the internal vibratory motions of the molecules. I n practically all cases it has been found that t,he observed relaxatlion involves the whole of the vibrational specific heat and is characterised by a single relaxation time. The explanation which is accepted a t present is that the energy is coupled into the inolecule via the mode of lowest vibrational frequency and then spreads rapidly to the other vibrational modes.Physi- cally the tinie-delay in establishing equilibrium is due to the fact that many collisions are required for the molecule to lose one quantum of vibratJional energy. Effects of a similar nature are to be expected in liquids and t,he relatively high values of (cc/'2) for non-polar liquids are attributed to this mechanism. I n liquids however the vibrat,ional relaxation time is markedly affected by dipole interactions so that in general it is only possible to observe t!liis type of relaxation in non-polar liquids ; methylene chloride (results for which are discussed later) is an exception. A complete analysis is available for carbon disulphide which has a characteristic frequency of 78 Mc./sec.at 26" c . ~ The relaxation at both 25" and - 63" is characterised by a single relaxation time and involves the whole of the vibrational specific heat,. The corresponding curves of absorption per unit mavelength are given in Fig. 9. There arc Carbon disulphide. 0.25 0.20 Q I5 Y 0. ro 0.05 - * f- i 5 IQ 2b sb IiO W O SbO f(Mc./sec.) FIG. 9 Relaxation of the vibrational specijc heat of carbon disulphide at (A) 25" am! ( B ) 63" c . The curves are theoretical and the points experimental. [From results of Andreae Hensell and Lamb Proc. f'hys. SOC. 1949 69 R 625.1 , 156 QUARTERLY REVIEWS three vibrationa'l modes of this molecule and the total vibrational specific heat (6C) is obtained by summing the Einstein specific-heat contributions of each mode 6C = C nRx2e-"/(l - e-2)2 where z = hv/kT and the degeneracy factor n is unity for the two stretching vibrations and 2 for the bending vibration.Table 4 shows the excellent agreement found be- tween the calculated values of ,urnax obtained on this basis and those derived from experiments by fitting the best single relaxation-time curve to the measured absorption values after allowing for the velocity dispersion. TABLE 4. Relazatio,n parameters for carbon disulphide 2 5 O 7 8 0.262 0.260 ~ - 63 ~ 31 ~ 0.134 ~ 0.133 ~ 3.933 2.683 * This is the contribution of the relaxing process ; it forms the major part of the total absorption per unit wavelength. The " classical " absorption due to shear viscosity has been subtracted from the measured absorption before making the above comparison with theory. Benzene.Incipient relaxation has been observed in benzene l6 as shown The results a t present available do not cover a sufficient portion in Fig. 10. \ \ 5 70 W 50 100 2eO f (Mc./sec ) F I G . 10 T'ibrational relaxation i?i ( A ) ?nethylene dichloride and ( B ) "enzene at 25' C. [Vrom results of Heawll and Lanib P r o ( . I'/I?/s Soc. 19.56 69 B. 809 and A411dleap ibid. 1957 70 B 71 ] of the relaxation region to permit an analysis to be made but it is presumed that the effect is due to relaxation of the vibrational specific heat. This questlion can be resolved only by measurements at higher frequencies. Methylenc! d ichloride. The relaxation observed in liquid methylene dichloride 2O is of interest since this is the only compound in which a multiple dispersion region hzs been est,ablished in the gas phase.I n methylene dichloride vapour there are two dispersion regions 21 and it is concluded that the lower-frequency relaxation involves all the vibrational modes 2o Andreae Proc. Phys. SOC. 1937 70 B 71. 2 1 Xette Busala and Hubbard J . C'hem. Phys. 195.5 23 585. DAVIES AW LAMB ULTRASONIC ANALYSIS OF MOLECULAR RELAXATION 157 except the mode of lowest infrared frequcncy which is itself responsible for the higher-frequency relaxation. It appears that a similar pattern of hehaviour occurs in the liquid. The observed relaxation (Fig. 10) (of which only part has been covered experimentally) can be accounted for by all the vibrational modes except that of the lowest infrared frequency. Further experiments at different temperatures should help to clarify this point.Carboxylic Acids.-Historically acetic and propionic acids \$.ere the first liquids in which ultrasonic relaxation was observed and for which complete series of experimental results are availahle covering a substantial part) of the relaxation ( p ) curve at) different temperatures.22 The values of f c and pmns are given in Table 5 for an approximate temperature a t 20' c. The Table also includes reaction parameters associated with the relaxation assuming (see p. 144) that we have a uniniolecular reaction in ideal solution with constant values of AH" and ASo. TABLE 5. Somcl crbsoiption in cn~bo~rylic cxcids near 20" c - 3.9 4.4 It is known that acetic acid forms double molecules united by two hydrogen bonds in both the vapour and the liquid phase although the heat of reaction is only known reliably for the vapour phase and for dilute solutions in non-polar solvents.It is natural therefore to consider the possibility that the relaxation is due to perturbation of the monomer-diiner reaction namely A + A + (A1) ; 2CH,*CO,H + (CH,*CO,H),. Using this equation which is of the first order to the left and second order to the right Freedman 2 3 has found approximate agreement between the AH" value which is obtained from analysis of tjhe expcrimental results and that estimated for the pure acid using an extrapolation of the Moelwyn-Hughes formula AHo,ol,,t,on = AHov,ipo,lr x (e + 2 ) / 3 r where I is the dielectric constant of itbe solvent. Freedman then uses the value I = 7.1 for acetic acid in this formula. Although the lnt ter procedure is somewhat dubious the agreement between the two values of' AH" is good and until recently the conclusion was accepted that the relaxation in acetic acid and possibly also in the other carbosylic acids was due to pertIurbation of the rnonomer- dimer reaction.However in order to clarify this issue nieasureinents have recently been ninde of the nhrorption in solutions of acetic acid.24 It was found that a second relaxation region occurs predominantly a t low con- centrations of acstic azid aiid with a ch-tracterjstic frequency an order of O 2 Lamb and Pinliorton Proc. Roy. Soc. 1949 A 199 114 ; Lamb a i d Huddart, 2 3 Freedmiin J . Chein. Phys. 1933 21 1784. 2 1 Piercy and Lamb T'ram. Fccrctday Xoc. 1956 52 930. Trans. Paraday Sac. 1950 46 540. 158 QUARTERLY REVIEWS magnitude higher than that for the pure acid.This second relaxation has been associated with the monomer-dimer reaction which leaves the explena- +,ion for the relaxation in pure acetic acid still open to question. One possi- bility is that in the pure acid the process responsible is the breaking of a single hydrogen bond which is supported by evidence that hydrogen links are formed in crystalline formic acid as distinct from double molecules. 25 _____ . . . . . . . . . . . . . . . . . . . . . . . . Rate parameters frequency factor and activation energy I n cases where the relaxation strength is small corresponding to a velocity dispersion of say less than I% it is unnecessary to distinguish between the various relaxation times z,,~ z,,~ etc. and for a unimolecular relaxation with AV = 0 a single mean relaxation time z = (k, + k2J1 can be used without further qualification.Then the characteristic frequency is fc = (k12 + k2,)/2n. Let the free energies of activation in the forward and reverse directions be respectively AGl$ = AH,$ - TAX,$ AGJ = AH2$ - TAS2$ Then on it rate-theory basis the transition rates would be written AH2% (kcal./mole) - 8-5 7.5 5-5 5-6 5.7 6-8 8.5 4-7 exp (AX,$/R - A H I / R T ) exp (AX,I/R - AH,$/RT) The equilibrium constant K = kl,/k2 = exp (- AGo/RT). Provided that AG"/RT < 3 say k12 can be neglected in comparison with k, to within 5% and then fc w k2J2n = (1/2n)(kT/h) exp (ASJ/R - AH$/RT) = (F2/2n)(kT/h) exp (AH$/RT) The heat of activation AH2$ in the reverse direction is thus given by where F = exp (AS21/R). TABLE 6. Activation energies activation entropies an.d frequency factors for various liquids near 20" c Acetic acid .. Propionic acid . Crotonaldehyde* . Cinnamaldehyde* . Ethyl ac . . Triethylamine . . Toluenet . . . 2-Methylbutane . 1.2 0.8 0-3 0.2 0.2 9.4 0-31 1.6 AS,* [cnl./(mole deg.)] + 0-37 - 0.45 - 2.1 - 3.0 - 3.2 + 4.5 - 2.3 -+ 0.9 * Calculations from recent results of Mr. M. 8. de Groot. t Beyer J . Acowt. SOC. Amer. 1955 27 1. - 2 5 Holtzberg Post; and Fankuchen A c t ~ C y s t . 1953 6 127. DAVIES AND LAMB ULTRASONIC ANALYSIS OF MOLECULA4R RELAXSTION 159 the slope of the line representing a plot of log (f,/T) against 1/T. A sum- mary of values is given in Table 6 from which it is evident that the factor F is within an order of magnitude of unity in all these cases. Carbon disulphide cannot be considered in this light since the vibrational relaxation is not a 2-state process.Nevertheless i t is perhaps worth noting that if the experimental results are treated in this fashion one arrives at the values AH,$ = 1 lical./mole F = 0.3 x The large discrepancy between this value for F and those in Table 6 shows that if the cause of a relaxation is sought and the estimated F is of the order of unity it is very unlikely that the relaxation is one of vibrational energy transfer. This fact seems to illustrate that the cause of the relaxation in toluene must be found in an explanation other than that of vibrational energy. Experimental investigations on solutions A significant feature of the sound absorption in binary mixtures of liquids is the shape of the curve of a/' against concentration. If the experimental results refer to frequencies which are well removed from a relaxation region then at any particular concentration a/' will be inde- pendent of frequency.I n these circumstances four main patterns of behaviour can be distinguished. Sagging curves of ( x / f 2 ) versus concentration of t8he type shown in Fig. 11 are obtained for mixtures of liquids which are not associated Type A . 20 44 60 80 I00 CC14 (moles%) FIG. 11 Absorptioli ~ P L solutions of Type A ; ( a ) benzene nrtd ( b ) clilorofown in cni*bora tetrachloride at 25" c. [By courtesy of Sette -3'ttoz.o cin2. (Supp!.) 1050 7 318.1 2ooo L either in the pure state or in the mixture e.g. for a mixture of two non- polar liquids.26 If x / f 2 for liquid X is much greater than m / f 2 for liquid Y deeply sagging curves of type (a) are found whilst liquids of almost equal absorpt,ion generally exhibit rather flat curves of type (b).It is a'ssumed that the rna8jor loss in each liquid is due to relaxation of the vibrational specific heat and the observed results are in reasonable agreement with theory . p8 Sette Nuoeo cim. (Suppl.) 1950 7 318. 160 QUARTERLY REVIEWS Type R. If the mixture consists of molecules X a.nd Y which assockte with each other then the curve of (o(/f2) against concentration may exhibit a large hump a t some intermediate concentration. It has been suggesbed that this effect is related to an interaction between the molecules of the 20 40 60 80 700 A/coho/(mo/es %I F I G . 12 Absorption in sol7ction.s of T y p e B ; ethyl rclcohol in water. A 22.5 B 37.5 C 52.5 Mc./sec. [I%y coiirtry of Storty Pror.Phys. Soc. 1052 65 B !)43.] two species as for example in the case of ethyl alcohol-water mixtures 3 7 (Fig. 12). On the other hand the view has also been put forward that this type of behaviour may he associated with the phenomenon of crit,ical mixing 28 (see Type D). Type C. Consider a mixture of two liquids X arid Y with the following properties the molecules of X do not associate either mutually or with molecules of type Y whilst the molecules of Y associate with each otlher 0 5 10 75 20 Phenol (mo/es %) FIG. 13 Absorption in soluiioiis of Type C ; phenol in carbon tetrachloride. [By courtesy of Ahier aiid Mez Z. Satiirforsch l’J52. 7a 300 j ~ ~~~ 2 7 Storey PTOC. Phys. ~Soc. 19.52 65 B 9-13. 28 Piercy International Commission on Acoustics 2nd Congress 1956 paper I.B.8.DAVIES AND LAMB ULTESSONIC SNALYSIS OF MOLECULAR RELAXATION 161 but not with those of type X. In siich a mixt,ure the ( ~ / f ~ ) versus con- centration curve may cxhibit a small peak a t low concentrations of Y as for example in a mixture of phenol and carbon tetrachloride 29 (Fig. 13). This effect is attributed to the association of molecules of type Y as for instance in a monomer-dimer equilibrium. I n a number of cases a critical mixing phenomenon occurs so that the liquids X and Y are only partially soluble in each other. For temperatures above the solubility curve thc liquids are completely miscible whilst below this curve they separate into layers of each other. In these circumstances a pronounced maximum in the curve of (a/f2) against con- centration occurs at temperatures near the consolute temperature and this is attribut,ed to the formation of molecular clusters.A typical example is shown in Fig. 14 for nitrobenzene in n - l ~ e s a i i e . ~ ~ A surprising feature Type D. I I I I 1-5 0 0.2 04 0.6 08 7.0 01 Nitrobenzene (mole fraction) FIG. 14. Absorption in solutions of T y p e D ; nitrobenzene in n-hexnne. A absorption at 25' ; B solubility. [By courtesy of Sette ,V'rto~o cijj'r. 1955 1 800.1 of the ultrasonic absorption in such mixtures is the persistence of the maxi- mum absorp t,ioii associated with the critical mixing phenomenon into the range of temperature far removed from the consolute temperature. The sound absorpt'ion is therefore sensitive to the existence of critical mixing even a t temperatures more than 30" c removed froin the critical conditions. An important conclusion which can be drawn from the work on mixtures is that a completely satisfactory explanation of the behaviour can be found only if future experimental observations are extended to cover not only variations in temperature and concentration but also a frequency range sicflcient to determine a substantial part of the relaxation. The authors great{ly appreciate the kindness of Dr. J. H. Andreae Dr. A. A. Petrauskas and Mr. M. S. de Groot for personal communications of their work which at the time of writing is awa,iting publication. 39 Maier and Mez 2. Natwjorsch. 1952 Ya 300 ; 1965 10a 997 ; Eppler ibid. p. 744. 30 Xette Nitoco citn. 1955 1 500.

ISSN:0009-2681    

DOI:10.1039/QR9571100134

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

NUCLEAR QUADRUPOLE COUPLING AND CHEMICAL BONDING By W. J. ORVILLE-THOMAS PH.D. (THE EDWARD DAVIES CHEMICAL LABORATORIES UNIVERSITY COLLEGE OF WALES ABERYSTWYTH) RECENTLY the adequacy of modern valence theory has been severely tested by the results provided by the newer spectroscopic techniques including microwave 1-4 and nuclear quadrupole resonance spectroscopy. I n some instances such as the determination of bond lengths and dipole moments of gaseous molecules a much higher accuracy has been achieved ; in the case of nuclear quadrupole coupling constants however the information obtained is of a radically new kind. Nowadays chemical bonds are classified as covalent or ionic ; moreover covalent bonds can have bond-orders of one two or three. It is believed that in certain environments a bond can possess a hybrid character i.e.a bond can embody covalent and ionic character and in addition have a non-integral bond-order. Modern spectroscopy has provided results of such high accuracy that it is now clear that the properties of a chemical bond such as a carbon- chlorine bond vary with its molecular environment. No two carbon- chlorine bonds in differing molecular environments are ever exactly the same ; they differ albeit minutely in properties such as length or ionic character. The determination of the electronic distribution in chemical bonds is one of the most important aims of chemical physicists. I n general this is difficult since most bond parameters such as length and moment are connected with the properties of the molecule as a whole. If it were possible to insert it charged probe at different points within a molecule and to measure the forces exerted upon it or its potential energy then the charge dis- tribution could be studied.Dailey6 has pointed out that quadrupolar nuclei act as " built-in " probes but since their relative position within the molecule is fixed they give information about the electronic distribution a t one point only. The purpose of this Review is to examine the tlype of information about W. Gordy W. V. Smith and R . Tramburulo " Microwave Spectroscopy " John M. W. P. Strandberg " Microwave Spectroscopy " Methuen and Co. Ltd. C. H. Townes and A. L. Schawlow " Microwave Spectroscopy " McGraw-Hill D. J. E. Ingram " Spectroscopy at Radio and Microwave Frequencies " Butter- H. G. Dehmelt Amer. J. Phys. 1954 22 110. B.P. Dailey J. Phys. Chem. 1953 57 490. Wiley and Sons Inc. New York 1953. London 1954. Boolr Co. Inc. New York 1955. worths London 1955. 162 OLCVILLE-THOMAS NUCLEAR QUADRUPOLE COUPLING 163 the chemical bond which can be obtained from quadrupole coupling constants. The significance of nuclear quadrupole coupling constants in relation to the theory of valence has been emphasised by Townes and Dailey 7 and by Gordy.8 Nuclear Quadrupole Coupling.-FVhen the spin I of an atomic nucleus is greater than one- half its distribution of positive charge is non-spherical in shape. With respect to the spin axis the ellipsoid of positive charge can be either prolate or oblate (Fig. 1). In the case of the prolate distribution of charge it can be irnagiried that some of the positive charge has migra.ted t Spin axes Nucleus.quadrupole moments. leading to an excess of positive charge at the North and South poles (repre- sented by positive signs) balanced by a defect of positive charge a t the equator (represented by negative signs). The opposite state of affairs occurs with the oblate distribution of charge. The nuclei then possess electric quadrupole moments Q which measure the deviation of the nuclear charge from spherical symmetry. In terms of the nuclear dimensions Q = p r y 3 cosz o! - 1) d t J where p is the nuclear charge density r is the distance from the centre of gravity of the charge to the element of volume dt and 0 is the angle between r and the spin axis. A positive value for Q indicates that the nucleus is elongated along the spin axis (Fig. la) ; a negative one indicates that it is flattened along this axis (Fig.lb). In atoms and molecules nuclei are embedded in an electronic cloud. When the electrons and other charges outside a particular quadrupolar nucleus have cz non-spherical charge distribution there is an interaction between the nuclear field and the external field. A nucleus possessing a quadrupole moment situated in an inhomogeneous electric field possesses a potential energy which depends upon the orientation of the quadrupole C. H. Townes and B. P. Dailoy J . C'hem. Phys. 1919 17 782. W. Gordy Discuss. Fnraduy SOC. 1955 19 14. 164 QUARTERLY REVIEWS nioinent with respect to the external field. The possible orientations of the spin axis of a nucleus in a molecule relative to the axis of rotation of the molecule as a whole are quantised.When the quadrupole moment & irit,eracts with an inhomogeneous electric field of the molecule each of the allowed orieiitations of the nucleus may possess a different potential energy. There will then be a number of different energy states which are responsible for the hyperfine structure of rotational lines. This hyperfine structure is resolvable in the microwave region and was first observed by Good in the case of ammonia. FIG. 2 Q t ~ i c l r ~ p ~ l a r nucleus in a n inhomogeneous Jield. The quadrupole coupling energy of a single nucleus is directly propor- tional to the quantity e&q where e is the charge on the proton Q is the nuclear quadrupole inoinent and q the field gradient is a 2 V / k 2 where V is the potential a t the nucleus arising from all charges outside the nucleus and z refers to a fixed axis.When the nucleus is situated in an electric charge distribution which is spherically symmetrical the field gradient q is zero. The nuclear quadrupole coupling constant eQq can be calculated from the hyperfine structure of rotational lines or from nuclear quadrupole resoiiaiice spectra when both Q and q are finite. Effectively q is a measure of the departure from spherical symmetry of the charge distribution at the nucleus due to the electrons and other nuclei present in the same molecule. I n those cases l - where values for & have been obtained by such methods as atomic beam techniques the determination of the nuclear quadrupole coupling constant e&q enables an accurate value of q to be obtained. This quantity depends on the environment of the quadrupolar nucleus in the molecule and is therefore intimately connected with the type of valence bonding surrounding the nucleus.In the region near the nucleus wave- functions which otherwise have proved satisfactory have not previously been tested. Townes and Dailey argue that since s electrons and filled iiiner shells have spherical symmetry and since d and f electrons do not penetrate W. E. Good Piiys. Rev. 1946 70 213. ORVILLE-THOMAS NUCLEAR QUADRUPOLE C’OUPLING 165 sufficiently near the nucleus the quadrupole coupling coiistani is largely due to the p electrons present in the valence shell. On this basis an insight into the character of chemical bonds can be obtained from the values of quadrupole coupling constants. Determination of Nuclear Quadrupole Coupling Constants.-The majority of nuclear quadrupole coupling constants eQq have been obtained from studies of the rotational spectra of polar molecules in the microwave region or by pure quadrupole resonance studies on solids in the radiofrequency region.Coupling constants can also he obtained by molecular beam methods. Cavity oscillators known as klystrons generate energy in the microwave region. A siniple form of microwave spectroscope (Fig. 3) consists of a klystron as radiation source aii absorption iVicrozcave spectroscopy of gases. Khjstron ’ power WPP/Y Vacuum fpequency measurement 7 Sweep ampli fie P windows I -7- FIG. 3 Block diagram of nzicrowave spectroscope. cell terminated by mica wi~~c~ows a radiation detector an amplifier of the detected energy and an indicator such as a cathode-ray oscilloscope.A comparison of the apparatus for microwave spectroscopy with that for infrared spectroscopy reveals that the klystron corresponds to the infra- red source a length of rectangular wave-guide to the collimating mirrors and cell and crystal detector to the thermocouple. It has a narrow frequency range coniparable with that of the absorption bands in the micro- wave region. Hence during the search for spectral lines the klystron frequency is varied manually. To examine a line in detail use is made of the fact thatl the resonant frequency of a klystron tube can be shifted slightly 1)y changing the negative potential on the repeller electrode. By applying the voltage from a saw-tooth generator to this electrode the klystron frequency can thus be swept over a short range.By applying the saw-tooth voltage also to the X plates of the oscilloscope its horizontal sweep becomes proportional to the frequency of the klystron. When the Radiation from a klystron is almost monochroniatic. 166 QUARTERLY REVIEWS amplified voltage from the detector is applied to the Y plates of the oscil- loscope a continuous and rapid record of the absorption line being studied is reproduced on the oscilloscope screen. When the rotational spectra of molecules containing a quadrupolar nucleus in an inhomogeneous field are examined with the high resolution afforded by a microwave spectroscope the rotational levels are found to be split. This hyperfine structure is caused by the coupling of the nuclear spin axis by means of the quadrupole moment to the molecular axis of rotation through the electric field gradient Q effective at the nucleus.Some twenty years ago Casimir lo developed the theory of the inter- action of nuclear quacirupole moments with surrounding electrons. The energy of interaction EQ depends on the product of e&q and known functiocs of the quantum numbers I J and K,ll where J is the rotational quantuin number and K the quantum number associated with the component of angular momentum along the symmetry axis x . The frequencies in Mc./sec. of the individual lines making up the hyperfine pattern are easily obtained by subtracting the value of the inter- action energies E, of the lower rotational state J from those EQ’ of the upper rotational state J + 1 with the appropriate selection rules and dividing the result by Planck’s constant h.For example for a symnietric- top molecule the effect of centrifugal distortion being omitted the frequencies of the hyperfine components would be given by where B = h/(8n21b) I b being the moment of inertia about the B axis. The positive identification of two lines in the hyperfine structure of a rotational transition enables the nuclear quadimpole coupling constant t o be calculated. The theoretical and observed hyperfine structure of part of the J = 7-8 rotational transition for the methyl iodide is shown in Fig. 4.12 Hyperfine structure in microwave spectra has been observed and resolved for a number of atoms including nitrogen chlorine bromine iodine arsenic and sulphur. If the quadrupole splittings are not small compared with the rotational frequencies l3 it is necessary to use second-order perturbation theory in order to get the correct energies.14 I n the microwave region the energies associated with the nuclear orientations are observed indirectly as a perturbation of the rotational spectra of molecules in the gas phase.I n solids direct transition between levels corresponding to different nuclear orientations can be observed by means of pure nuclear quadrupole spectro- lo H. B. G. Casimir “ On the interaction between atomic nuclei and electrons ” Teyler’s Tweede Genootschap E. F. Bohn Haarlem 1936. l1 A. Nordsieck Phys. Rev. 1940 58 310 ; B. T. Feld and W. E. Lamb ibid. 1945 67 15 ; D. K. Coles and W. E. Good ibid. 1946 70 979 ; J. H. Van Vleck ibid. 1947 71 4 6 5 ~ . l2 W. J. Orville-Thomas J. T. Cox and W.Gordy J. Chem. Phys. 1954 22 1718. 13 0. R. Gilliam H. D. Edwards and W. Gordy Phys. Rev. 1918 73 635; C. H. Townes F. R. Merritt and B. D. Wright ibid. p. 1334. l4 J. Bardeen and C. H. Townes ibid. pp. 627 1204. y = 2B(J + 1) + (EQ’ - EQ)/h Nuclear quadrupole resonance spectroscopy. ORVILLE-THOMAS NUCLEAR QTTADRUPOLE COUPLIKG 167 scopy the frequencies being observed directly in the radiofreqnency region. The first observations of nuclear quadrupole resonance spectra were made in 1949 by Dehmelt and KrUger,l5 a,nd numcroiis investigations have since been made.1 4 5 16 17 Frequency - FIG. 4 Theoretical and observed hyperjine lines of the J = 7-M transition of C13,.12i'I at 2.5 nvn. wcivetength. The spectrometers used for detecting pure nuclear quadrupole resonance are very similar to those used for nuclear magnetic resonance except that no magnet is necessary the external magnetic field being replaced by the internal crystal field of the specimen.Generally a radiofreqiiency hidgc - - FIG. 5 Block diagram of piwe mclear quad upole resonance specti~oscope. or a type of regenerative oscillator is employed. l* Radiofrequency power from a signal generator is fed into two almost identical resonating circuits which are balanced against each other in a bridge circuit (Fig. 5)) the sample l6 H. G. Dehmelt and H. Kruger Naturwiss. 1950 37 111. l6 R. Livingston J . Chem. Phys. 1952 20 496. l7 M. Davies and W. J. Orville-Thomas Ann. Reports 1954 51 7. R. Livingston Ann. New York Acctd. Sci. 1052,55 800 ; Phys. Rev. 1951 82 289. M 168 QUARTBRTLY REVIEWS being placed in a coil forming the inductive component of the specimen circuit.The resultant output of the two tuned circuits is fed into an amplifier and the detected signal displayed on an oscilloscope or fed into a recorder. As the frequency of the input power is varied a signal is passed on to the detecting system when the specimen absorbs radiofrequency energy and so causes the bridge to become unbalanced. These resonant frequencies can then be measured very accurately. Nuclear quadrupole resonance spectra are obtained for solids as a result of the interaction between the nuclear quadrupole moment and the stutic crystalline field. I n liquids and in gases on account of the rapid movements of the particles the gradientl a t the nucleus varies rapidly and its average value is zero ; hence no pure quadrupole spectra are obtained for them.Nuclear quadrupole resonance spectra have been obtained for a number of molecules containing the following nuclei log llB 14N 33S 37Cl 63Cu 'j5Cu 69Ga 71Ga 7 5 A ~ 79Br PIBr 121Sb 123Sn 12'1 1291 *!)lHg and 209Ri. The absorption frequencies are given by the product of eQq and a function of I and M I the component of the nuclear spin I along the axis of the crystalline field. For a nucleus such as chlorine with I = 3/22 there are two energy levels corresponding to the values of M I f- 3/2 and 5 3. There is only one transition frequency for each chlorine isotope The frequencies of these transitions in chlorine compounds have been deter- mined by Livingston.l* The ratio of the frequencies for the isotopic species 35Cl and 37Cl gives the very accurate value of 1.26878 &- 0.00015 for tjhe ratio &(C135) &(C13').Nuclear Quadrupole Coupling in Atoms.-Before discussing how coupling constants can be interpreted to give information about molecular structure we shall see what factors determine the field gmdient a t a nucleus. The simpler atomic case will be dealt with first. Most electrons in atoms are arranged in groups of closed shells. The charge distribution of such shells is spherically symmetric and produces zero average field at the nucleus. (In effect q the field gradient at the nucleus due to the closed shells is zero.) Atoms may possess in addition to these closed shells a number of valence electrons. Let us consider an atom containing one valence electron of charge e. ORVILLE-THOMAS NUCLEAR QUADRUPOLE COUPLING 169 The potential V at the nucleus due to the electron is V = e/r = e / d ( x 2 + y2 + x 2 ) where r is the distance from the nucleus to the electron whose co-ordinates are x y x.a2V/az2 = e(3z2 - r2)/r5 = e(3 C O S ~ 8 - l)/+ since x = r cos 8 and 0 is the angle between the fixed x axis and the radius vector r. When this quantity is averaged over the orbital occupied by the electron we obtain the contribution of the valence electron to q the potential gradient a t the nucleus zlix. We have then where 4 is the wave-function for the electron. If the electron is in a central field 4 is separable into a radial factor and a function depending upon the angles. where l is the orbital angular momentum of the electron and (l/r3)av. is the average value of the inverse third power of the distance between the electron and the nucleus.I n the case of atoms the field gradient q can often be calculated from experimental data because measurable quantities such as the nuclear mag- netic hyperfine structure l9 and the fine-structure splitting of atomic spectra 20 21 depend upon J 4(i/r3)+* dt. Por atoms of known quadrupole moment & the field gradient a t the iiucleus can be determined from the value of the atomic coupling constant eQq (atom).22 These experimental values differ by some loC)'' from the calculated q values. In deriving the expressions connecting q with fine- structure splittings or with magnetic hyperfine structure it has been assumed that tlhe closed shells of electrons are spherically symmetrical. The inner shells are however polarised by the valence electron and as a result the electrons in the inner shells tend to keep as far as possible from the position of the valence electron.This polarisation effect produces at the nucleus a contribution to q which is opposite in sign to that attributable to the va,lence electron. Thus inner shells to some extent shield the nucleus from fields due to the valence electrons. When there is more than one valence electron the problem is more difficult and only a less exact evaluation of q is possible. lo In many cases no fine-structure splitting or nuclear magnetic hyperfine structure data are available. However approximate values of the field IoR. G. Barnes and W. V. Smith Phys. Rev. 1954 93 95. 2o R. Bacher and S. Goudsmit " Atomic Energy States " McGraw-Hill Book Co.New York 1932 ; C. E. Moore Atomic Energy Levels National Bureau of Standards Circular 467 U.S. Government Printing Office Washington D.C. 1949. 21 H. A. Bethe and R. F. Bacher Rev. Mod. Phys. 1936 8 226. 2 2 L. Davis B. T. Feld C. W. Zabel and J. R. Zachariss. Phy8. RCPI. 1948 73 On integration eqn. (1) yields q = - we/(2ze + 3 ) 1 w 3 ~ ~ . - ( 2 ) 526. 170 QUARTERLY REVIEWS gradient q can be obtained by substituting hydrogen-like electronic wave functions in eqn. (1). computed the relative value of q for various atomic states n I m = 0 (where n I and m are the usual electronic quantum numbers). Their results show that with increasing n there is a steady decrease in the value of q and a similar effect occurs when the value of l increases. The effect of changing I how- ever is much more marked owing to a correspondingly larger change in screening.Even in the lighter elements fluorine and sodium where screen- ing effects are less important the field gradient at the nucleus decreases rapidly with increasing n or I. Atomic-beam experiments lead to values for the nuclear quadrupole coupling constants e&q in atoms. These values vary widely for 35Cl 79Br and 1 2 7 1 eQq is - 109.74 + 769.62 and - 2292439 Mc./sec. respectively. Nuclear Quadrupole Coupling in Molecules.-The calculation of the field gradient a t a particular nucleus within a molecule is a complex task. An explicit solution of the problem would entail an intimate knowledge of the distribution of charges in the molecule. The contribution from each nucleus and electron would have to be summed to give the correct resultant field gradient a t a specific nucleus.Essentially this entails a determination of wave-functions for the electrons since the effect of adjacent nuclei on the field gradient a t a particular nucleus is small. An explicit solution to this problem with use of an accurate wave-function has been achieved only for the hydrogen molecule. 23 White 24 nnd Bassoinpierre 25 have recently attempted t o calculate q a t thc deuterium atom in DCN and a t the nitrogen atom in HCN respectively and calculations of this type will undoubtedly become more accurate and be of great value in the future. Theoretical estimates of various molecular parameters such as bond-order and dipole moment have been obtained by the use of approximate wave- functions. I n c2 similar fashion useful inforniat ion about the field gradient can be obtained since q depends to a lmge extent on a small number of parameters appertaining to bond structure.If we consider a molecule containing several electrons the contribution to Q of the i-th electron is By this method Townes and Dailey where yi is the wave-function for the orbital of this electron in thc molecule. Approximate solutions for the niolecular case can be obtained by assuming that the molecular orbitals y t can be replaced by atomic orbitals $i for the atonis in molecules. This approximation was introduced by Townes 2G and devcloped in a classic paper by Townes and Dailey ; the theory has l)een used with reasonable succcss in ;L iiuniher of instance^.^ 2 7 9 28 z3 A. h’ordsieck Phys. Rw. 1940 58 310. z 4 R.L. White J . Chern. I’hys. 1955 23 253. z 5 A. I3assornpierre7 Disc7cf~cc.. Fn?*ctdcty Roc. 1955 19 260. 2 6 C. H. TOWIICS Ph!/c. Rev. 194i 71 $109. 2 7 W. Gordy H. King and A. LZ. Burg ibid. 1960 78 512. 28 C. C. Looinis and $1. 7V 1’. Strandberg ibid. 1931 81 798. ORVILLE-TIIOMAS NUCLEAR QUADRUPOLE COUPLING 171 I n their analysis Townes and Dailey showed that the magnitude of the field gradient a t a nucleus depends very largely on how the p-type orbitals of lowest energy are occupied by the valence electrons associated with the nucleus. The main contributions to p = azV/az2 a t a particular nucleus A in a molecule arise from ( a ) valence electrons associated with the chemical bonding in the molecule arid having a high probability of being near nucleus A ; (b) lone-pair electrons associated with nucleus A ; (c) the electrons and nuclei present in the rest of the molecule (these are essentially a t distances of more than an atomic radius from nucleus A) ; and ( d ) polarisation of the inner closed shells of electrons which surround nucleus A.At first sight it seems that contributions of type (c) are the only extra feature in the molecular as compared with the atomic case but since the wave-functions of the valence electrons of an atom in a molecule are very much changed by the formation of chemical bonds in effect contributions of types (a) and (b) are modified to some extent from the relatively simple atomic case. Townes and Dailey estimated the contributions of valence electrons to the field gradient at a nucleus by expressing the wave-function of a valence electron forming part of a covalent bond as an expansion in terms of atomic wave-functions Usually the coefficients of the lowest-energy atomic orbitals are expected to have the greatest values in the molecular wave-function of the valence electron i.e.ari will be largest for the lowest allowed values of n and 1. Substitution of eqn. (4) in eqn. (3) gives an expression which represents the contribution of the valence electron to q. An estimate of the magnitude of each term in this expression has been obtained by Townes and Dailey who showed that the dominating term will be the state of lowest allowed total quantum number n ( a ) and (b). Ccntributions of electrons in the valence shell. y = C . * (4) i.e. q (Valence electron) m eai2 1 +(- 3 73-)+* cos2e - 1 dt = I ai 12pi where qi represents the field gradient arising from an electron whose atomic wave-function is +i.This is reasonable since energy considerations will ensure that the fractional importance I ai l 2 of this atomic state will be large and in addition the value of the field gradient for this state of lowest allowed energy is considerably larger than those for the higher-energy atomic states. The approximate effects of other charges present in the molecule on the mag- nitude of the field gradient a t a nucleus can be calculated in a simple fashion. If for example we assume that a neighbouring ion at a distance of 2 A from the nucleus has an average charge of el2 then it produces a value of q = 6 x 1013 e.s.u. a t the nucleus ; a charge of e placed 1 A from the nucleus contributes 9.6 x 10l4 e.s.u.to p. These contributions ase srnall compared with the contribution of a valence electron in a low energy y state and so may be neglected. ( c ) . Contributions due to the other charges present in the molecule. M* 172 QUARTERLY REVIEWS (d). Polavisation eflects on the innw shells. I n the last section it was seen that a neighbouring ion makes only a small contribution to the field gradient a t a nucleus. In addition to this direct effect a neighbouring ion will distort the electronic dist,ribution about a nucleus. This polarisation of the inner shells will contribute to (I. This problem has been analysed by Towiies and Dailey who showed that the contributions to q from the polarisation of a particular shell amounted to less than 1% of the value due to a single electron of this shell.The indications are that such dis- tortions could increase the contribution to q of a neighbouring ion by a factor of ten. If as in the preceding section we haye an ion of average charge e / 2 at a distance of 2 from the quadrupolar nucleus then its contribution to q could be increased by polarisation effects to 6 x 1014 e.s.u. This is still a rather small contribution amounting to less than 27; of the value of the field gradient due to a single valence p-electron in iodine. Hence contributions to the field gradient at a nucleus due to neighbowing ions or to polarisation effects can frequently be neglected. This analysis of the importance of the various contributions indicates that if the lowest-energy atomic state is a p state the field gradient a t a nucleus due to a single valence electron is (to a first approximation) q = I a I 2qp where I u 12 is the importance of the p wave-function in the molecular wave-function and qp represents the value of the field gradient for a p atomic state.Overlap and Nuclear Quadrupole Coupling.-If y represents the moleculizr wave-function of a bonding electron its contribution to the field gradient a t a nucleus A is given by eqn. (3). If A is joined to B by a chemical bond the electron will be in a molecular orbital whose wave-function can be expressed as y = a+(A) + h/+(B) Qualitatively we can divide the electronic density of the bond into three parts. The electronic density near the nucleus A which depends on the importance of the atomic orbital +(A) in thc molecular orbital y will provide the dominant contribution to the field gradient at A.A second contribution to q arises from the charge cloud in the overlap region. This cloud is approximately the distance of one covalent radius away from the nucleus A and because of the inverse variation with r makes only a small contribution to q. The electronic density of the bonding electrons near nucleus B will affect the field gradient at A even less. The effect of overlap on the field gradient a t a nucleus can be calcu- lated 29-31 by conventional ineans (by use of t,he overlap integral normally used in theoretical chemistry) ; this procedure has been shown 32 to under- estimate the field gradient due to the bonding electrons. This application of molecular-orbital theory is incorrect since it implies that when the covalent bond is formed an appreciable amount of charge density is removed from regions close to the nucleus but as this would need considerable energy 28 P.Shatz J . Chem. P h p . 1964 22 60.5. 31 I d e m ibid. p. 1974. 30 Idem ibid. p. 755. 32 W. Gordy ibid. p. 1470. ORT'ILLE-THOMAS NUCLEAR QUADRUPOLE COUPLING 173 it seems irnprobab1e.l Gordy has pointed out that the charge cloud con- centrated in the overlap region can be formed with the expenditure of less energy by a redistribution of the charge cloud in the outer parts of the atomic orbitals. These outer distortions would be expected to have only a small influence on the field gradient a t the nucleus because of the inverse- cube variation of q with r. These qualitative considerations and some experimental data on quadrupole coupling in halogens seeni to indicate that near the nucleus the electronic charge of a homopolar bond can be divided equally between the two atoms so that on the average there is one electron in each of the atomic orbihls $(A) and &B).This is equivalent to putting the overlap integral equal to zero when we are concerned with the calculation of quadrupole coupling. It seems then that the molecular wave-functions hitherto found satisfactory for calculating the properties of valence bonds are not appropriate for the calculation of field gradients near the nuclei. Ionic Character and Hybridisation.-Two other factors that affect the electronic distribution near a nucleus in a niolecule in particular its p - character (as compared with an atom) are (a) the degree of ionic character of the bond and (6) the degree of hybridisation of the bond-forming orbital.( a ) Ionic character. The effect of ionic character on q can be seen as follows by considering the simple case of a diatomic molecule such as tliitllous chloride TlC1. The electronic configurations of the ground states of the two atoms are T1 (. . . 6s26p) C1 (. . . 3s23p5). It might be assumed that TIC1 possesses a o bond formed by using a p orbital of each atom. If we consider the bond to be homopolar there is on the average one electron each in the orbitals $(T1:6p) and $(C1:3p). I n these circumstances the thallium nucleus is surrounded by filled electronic shells plus a single extra electron in the Tk6p orbital similarly the chlorine nucleus is sur- rounded by filled shells with the exception of the valence shell which has a defect of one C;1:3p electron.Filled inner shells have spherical symmetry and do not contribute to q ; hence the field gradient at the thalliuni nucleus arises from a single electron in the 6p orbital i.e. qRlol. = qDT1. For the chlorine nucleus qMol. = - qpcl the negative sign indicating a defect of one 3p electron from cz closed-shell configuration. On the other hand if the atoms are held by an ionic bond Tl+Cl- the field gradients a t t3he nuclei will be zero since apart from polarisation effects the charge distribution is spherical. If the ionic structure Tl+Cl- has a fractional importance x the importance of the covalent structure T1-C1 will be (1 - x). Each structure will contribute to q an amount given by the product of its fractional importance and the value ofq for the bond structure concerned.T1 + rqTlt Hence for thallium qM01. = (l - x)q~ where eQq is the nuclear quadrupole coupling constant arising from one valence p electron. This quantity is related to the coupling constant measured for atoms by atomic-beam techniques. 174 QUARTERLY REVIEWS Similarly for the chlorine nucleus e&qclMO1. = - (1 - x)eQqDC1 . * (6) where eQqTIMol. and eQqC1,,,. are tlie experimental values for the nuclear quadrupole coupling constants. By convention the quantity multiplying - eQq,” or - eQqpc* in eqns. (5) and (6) is known as the amount of unbalanced p elcctrons U, oriented along tlie bond. The quadrupole coupling constant is then - U p multiplied by the coupling per p electron eQq, and hence I n TlCl U p = - (1 - x) for T1 arid U = (1 - x) for C1.ion Cl- there are a number of p electrons whose effects cancel out. this case U = 0. differing molecular environments are given in Table 1. up = - e&q,o,./eQqp * * (7) I n the chloride In The quadrupole coupling constants of 35Cl and 79Br in a number of The small values TABLE 1. Quadrupole coupling constants of 35Cl and i9Br in molecules I Molecule ~ eQq (Mc./$ec.) I Molecule I eQq (Mc./sec.) C1 (Atomic) . . BrCl . . . . ICI . . . . CH,*Cl . . . SiF,Cl . . . TlCl . . . . KCl . . . . - 109.74 ( - e&qp) - 103.6 - 82.5 - 74.8 - 43.0 - 15.8 0.041 79Br (Atomic) . BrCl . . . . SiF,Br. . . LiBr . . . . KBr . . . . 768.8 ( - eQy,) 876.8 440.0 37.2 10.24 obtained for the coupling constants of the halogen nuclei in potassiuni chloride and potassium bromide indicate very clearly that these molecules must be essentially ionic.Thallous chloride and litliiuni bromide must also be essentially ionic. The other molecules listed in Table 1 have bonds to halogen which have a mixed nature. They are primarily covalent in nature but have some ionic character. Ionic character is not the only factor which tends to lower the value of eQq at a nucleus in a molecule below the value found for the same nucleus in an atom. The experimental data available reveal that the quadrupole coupling constants for chlorine atoms predominantly covaleiit,ly bound lie near - 80 Mc./sec. and are considerably lower t)han those to be expected if the chlorine atoms used pure p orbitals for bonding. For such pure p bonding the value obtained for eQqnfol should approximate to that obtained €or atomic chlorine vix.- 109-74Mc./sec. A goodexample is iodine monochloride ICl where the small difference of 0-4 unit between the electronegativities 33 of the two atoms leads one to expect only a small amount of ionic character in the bond. Notwithstanding this the coupling in iodine monochloride is appreciably less than the value obtained for atomic 33 W. Gordy and W. J. Orville-Thomas J. Chem. Phys. 1956 24 439. (b) Hybridisation. ORVILLE-THOMAS NUCLEAR QUAURUPOLE COUPLING 175 chlorine. chlorine atomic orbital used for bonding occurs. In these cases it has been assumed that hybridisation of the The molecular orbital for a bond A-R has the wave-equation y' = qm) + b q m ) +(A) = d s 4 s + d P 4 P + 2 W d If the atomic orbital +(A) is hybridised its wave-function can be written in the form which implies that the orbital has s character of an amount s and d character of amount d.When a bond possesses ionic character a + b and if we define the ionic character i of the bond by a well-known convention as i = a2 - b2 it can be shown that for a covalently bonded halogen atom If a quantity p be defined as eqn. (8) can be arranged to where s and d give a measure of the s and d character of the bonding orbital +(A) of atom A. When u2 > b2 the coupling atom A carries a negative charge. If the extent of hybridisation ( s and d ) and the ionic character i are known then eqn. (9) can be used t o calculate p. If the quadrupole inonleiit of the nucleus A is known then the value for eQqMOl. gives a direct estimate of Q the field gradient a t the nucIeus A.In practice the amount and type of hybridisation and i are unknown and the procedure is reversed. The experimentally determined values for the nuclear quadrupole coup1 ing con- stants in molecules are used in an attempt to obtain an insight into the character of the bond i.e. hybridisatioll and ionic character. Unfortu- nately relation (9) contains three unknown quantities s d and i and only one measurable quantity p. By use of quadrupole data alone then it is not possible to separate effects due to hybridisation from those due to ionic character. The leading workers in the field agree that the effects of d hybridisation are probably small and can be neglected i.e. d = 0 in eqn. (9). This leaves us with two unknowns and one experimental datum. I n order to overcome this final difficulty attempts have been made to derive a relation between the ionic character of a bond A-13 and the electro- negativity difference between the atoms A and B.Curves connecting ionic character and electronegativity difference (XA4 - XB) have been put forward for bonds to halogen atoms by Gordy * 3 2 9 34 and by Dailey and T0wnes.~5 These curves are used to give an estimate of the ionic character possessed by a bond which value when substituted in eqn. (9) enables a value for the hybridisation factor to be obtained. The curves of Gordy and of Dailey and Townes disagree most markedly e&qMol. = [l - s + d - i(l - s - d)]eQqD . * (8) P == I e&qm leg$% I i(1- s - d ) +- s - d = 1 - p . - (9) 3 4 VC'. Gordy J . Chem. Phys. 1951 19 792. 35 €3. P. Dailey and C . H. Townes ibid.1955 23 118. 176 QUARTERLY REVIEWS in the region of small ionic character. In this region the quadrupole coupling constants are extremely sensitive to the degree of hybridisation assumed. Gordy points out that for the homopolar molecules C1 and Brz i = 0 and the observed p = 1. Substitution of those values in eqn. (9) shows that either s = d or s = d = 0. Similarly for crystalline iodine if a correction is made for cr~ss-bonding,~~ s % d to a good approximation. It should be pointed out that these results are obtained from solid-state data and the situation in the vapour phase need not necessarily be the same. Molecules such as KCl and KBr are generally regarded to be completely ionic i.e. i = 1. The experimental values obtained for the quadrupole coupling con- stants give p M 0.Insertion of these values in eqn. (9) shows that d = 0. On this basis Gordy argues that the experimental evidence indicates that hybridisation does not exist either in the pure covalent or in the pure ionic state unless s w d. Consequently Gordy reduces eqn. (9) to i = l - p . * (10) These equations are taken to give a measure of the ionic character of the type that puts the negative pole on the coupling halogen atom. tJsing relation (lo) Gordy has obtained values for the ionic character possessed by bonds to halogen atoms in a number of diatomic and polyatomic niole- cules. These values (Table 2 ) are plotted against the corresponding electro- negativitly differences in Fig. 6. The curve obtained by Gordy levels off Electronegativity difference FIG. 6 Plot of ionic character against electroncyaticity diffee,*ence.A Gordy.s B Dailey and T o ~ n e s . ~ ~ a t the top for molecules such as those of the alkali-metal halides which are considered to be completely ionic. This curve has led Gordy t o pro- pound the approximate rules i = +(XA - X,) for I X - X I < 2 and i = 1.00 for 1 XA - X I > 2. Regardless of whether (1 - p ) represents 3 6 H. Robinson H. G. Dehmelt and ITr. Gordy J . Chem. Phys. 1954 22 511. ORVILLE-THOMAS NUCLEAR QUADRUPOLE COUPLING 177 the true ionic character of a bond Fig. 6 shows a systematic variation of the coupling ratio I e&qM, /eQq I with electronegativity difference. Gordy derived relation (10) on the assumption that the atoms use pure p atomic orbitals in bond formation. The possibility of small amounts of hybridisation up to - 3% of s or d character or even larger amounts if s and d hybridisation are approximately equal cannot however be ruled out.This view of pure p bonding by the halogens is not shared by Townes and Dailey 3’ who made the first attempt to interpret qusdrupole couplings. The quadrupole coupling constants for covalently bonded chlorine are some 250/ less than the value €or atomic chlorine. Townes and Dailey regard this as good evidence that a C1 sp-hybridised atomic orbital is used in bond- ing. They do not believe that this diminution in e&q is due entirely to the ionic character of the bond. I n their original paper Townes and Dailey obtained an expression for the nuclear quadrupole coupling constant in a diatomic molecule containing chlorine. Fcr 13jCl eQq = - 82.5 Mc./sec.This expression is equivalent to From Pauling’s curve of ionic char- acter agailrst electronegativity difference 38 Townes and Dailey obtained a value of i = 0.08. Substitution of these values in eqn. (ll) together with the value e&q = -t 109.74 IIc./sec. gives ( s - d ) = 0.18. Hence it is quite impossible to distinguish the effccts due to s hybridisation from those due to d hybridisation. It has ususlly heen assumed that very little d hybridisa- tion occurs and d is put equal to zero in equations (9) and (11). On this basis Towiies and Dailey claim that the chlorine bonding orbital in iodine monochloride and other chlorine compounds is an <~-p hybrid containing some 18% of s character. These eytimates have been revised to include bromine and iodine. The latest interpretation is condensed in the following rule 35 “ The halogen bonds are liybridised with 150/ of s Character when- ever the halogen is more electronegative by 0.25 unit than the atom to which it is bonded.eQq,, rn (- 1 + s - d ) ( l - i)eQq . (11) Otherwise there is no hybridisation.” If d hybridisation is ignored relations (9) and (11) reduce to Townes and Dailey have used this equation in conjunction with their “ hybridisstion rule ” to obtain the ionic character values given in Table 2 and Fig. 6. This curve was obtained from quadrupole coupling constants obtained for diatomic molecules in the gaseous state. Townes and Dailey point out that small deviations from the curve of Fig. 6 may occur owing to changes in the effective electronegativity of an atom situated in different molecular environments (cf.ref. 33). The amount of ionic character in a bond is also affected by the internuclear distance and by hybridisation. The relation between ionic character and elect-o- negativity difference is therefore not expected t o be precise. Gordy’s relation and the S-shaped curve of Dailey and Towiies do not differ greatly 37 B. P. Dailey Discuss. Fnraday SOC. 1055 10 255. 38 L. Pauling “ Tho Nature of tho Chemical Bond ” Cornell Univ. Press Ithaca eQq = (- 1 -t s)(l - i)e&q (12) 1940. 178 QUARTERLY REVIEWS except in that part of the curve where [ Ax I < 1. I n this region there is a paucity of data ; the only points available are provided by FC1 and FBr. The interpretation of the nuclear quadrupole couplings in these molecules is complicated since the chlorine and bromine atoms are positively charged.I€ the coupling halogen atom carries a positive charge the bond is posi- tively ionic. When the bond is completely ionised to AfB- two electrons are missing from the valence shell of the halogen atom A. The iiuclear quadrupole coupling constant arises from a defect of two p electrons in an otherwise spherical shell and eQq = - 2eQqp. Since the interaction between s-Character Ionic assunicd (yo) character TABLE 2. Ionic character of diatomic halides Dailey and Townes Nolecule eQq (Mc./sec.) (ohs.) 35CC1 ( - e&qp) BrCl . . . IC1 . . . TIC1 . . . KC1 . . . RbCl . . CSC'I . . . FCl . . HCl . . . ''Br ( - e Q ~ p ) BrCl . . . LiBr . . . NaBr . . K B r . . . DBr . . . FBr . . . "'1 ( - e Q ~ p ) DI . . . LiI . . . NaI . . . KI .. . 1 - P Gordy . . . . . . . . . I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - 109.74 - 103.6 - 82.5 - 15.8 0.04 0.774 3 - 146.0 - 789.76 876-8 37.2 58 10.244 533 1089 - 1823 - 2292.84 - 198.15 - 259.87 - 60 ~ 0.0 15.0 15.0 15.0 15.0 15.0 0.0 15.0 0.0 15.0 15.0 15.0 15.0 0.0 15.0 15.0 15.0 15.0 0-06 0.12 0.83 1.00 0.99 0.97 0.26 0.29 0.1 10 0.944 0.91 1 0.985 0.186 0.329 0.065 0.900 0.867 0.970 0.06 0.25 0.99 1.00 0.99 0.97 0.40 - - 0.932 0.925 0.987 0.308 - 0.205 0.914 0.887 0.974 cz p valence electron and the nucleus is larger when the atom is positively charged Dailey and Townes 35 introduce a correction factor c such that eQq = - 2(1 + c)eQq When a bond possesses some positive ionic character eqn.(12) is modified to eQq = [(- 1 + s - d ) ( l - i) - 2(1 -t- c)i]e&q For the halogens c M 0.15. This equation was used to obtain the values o€ i for FC1 and FBr given in Table 2 and plotted in Fig. 6. Dailey and Townes attempt to justify their postulation of a compara- tively large amount of hybridisation in covalently bonded halogen atonis (when Ax 2 0.25) by pointing out that (i) bond energies of molecules con- taining elements of the first two rows of the Periodic Table have shown the presence of s hybridisation ; (ii) quadrupole coupling data for compounds of the Group V elements indicate quite strongly the presence of s hybridisa- ORVILLE-THOMAS NUCLEAR QUADRUPOLE COUPLING 179 tion in the bonding orbitals ; (iii) quadrupole coupling data demonstrate unequivocally that certain sulphur bonds are hybridised ; (iv) polyatomic molecules for which coupling constants are known appear to fit the S-shaped curve reasonably well provided multiple-bonding and bond-interaction effects are allowed for.To some extent Gordy’s belief in the absence of hybridisation in halogen compounds is based on the values obtained for the quadrupole coupling constants of Cl, Br, and I in the solid state. Gordy makes the assumption that since these are homopolar molecules the coupling constants for the isolated gas molecules will not be far different from the solid-state values. This seems a very reasonable assumption but it is open to criticism.35 In the interpretation of the coupling constants obtained for the alkali metal halides the values obtained for the ionic character being almost unity are insensitive to the assumption or otherwise of s hybridisation.For these molecules Dailey and Townes in accordance with their “ rule ” assume 15% of s hybridisation. On simple chemical grounds it seems un- necessary to invoke hybridisation to explain the properties of these molecules which are as in the case of potassium chloride completely ionic. Gordy in his treatment interprets the coupling constants of these molecules as indicating the absence of hybridisation. The molecules BrCl and IC1 are extremely important to the problem under discussion since the quadrupole coupling constants have been meas- ured for both nuclei in these molecules. Quite clearly the amount of ionic character in the bond should be independent of which coupling con- stant is used for its estimation.No s hybridisation is assumed to be present in BrCl by Gordy or by Dailey and Townes. I n this molecule the coupling in the bromine atom indicates 10% of ionic character and that in the chlorine atom indicates only 6%. The discrepancy here is small and could be caused by a small amount of d hybridisation. The ionic character of iodine chloride is found by Gordy to be 23% when the coupling in the chlorine atom is used and 24% with the coupling in the iodine atom. Dailey and Townes assuming 15yo of s hybridisation of the chlorine bonding orbital obtain vajlues of 12% and 24%. The postulation of s hybridisation in this case makes the agreement worse. The Electronic Structure of Polyatomic Molecules.-In terms of valence- bond language the mesomeric state of a molecule can be expressed as a combination of unperturbed structures.Estimates of the fractional im- portance of the contributing structures can be obtained from bond-length and dipole-moment data. These estimates can be refined by the use of nuclear quadrupole coupling constants. In the following sections it will be shown how the coupling constants to be expected for various combina- tions of resonating structures can be calculated. By comparing these “ theoretical ” values with the coupling constant found experimentally for the molecule concerned the combination of structures most in accord with all the available data can be chosen. We have seen in previous sections that the main contribution to the field gradient q comes from the “ unbalanced ” p Symmetric moZecules.180 QUARTERLY REVIEWS electrons in the valence shell of an atom. In practice this is the condition encountered most frequently. Let us consider chemical elements which have in their valence shells s and p electrons only. If the bond A-B is aligned along the x axis it can be shown that where N, N, and N represent the effective electron populations of the px p, and p orbitals of atom A. e&qnmA = c- ( N + N ) P + ,Y,le&q,* ' * (13) A comparison of eqns. (7) and (13) shows that u p = (Nx + Nv)/2 - N = eQqmi./(- e&q,) I n certain cases the atomic couplings have been accurately determined and values for e&q are known (Table 4). A large number of determinations of quadrupole couplings in molecules have also been obtained. Hence for certain compounds such as the halides U is known experimentally.When the bond A-B possesses some ionic character and the bonding orbital of A is hybridised the number of unbalanced p electrons along the bond is given by N = (1 i)(l + s - d ) The positive sign is used for i when the bond has polarity in the sense A-B+ and the minus sign when the polarity has the reverse sense. If the change in nuclear screening caused by the formal charges on A and B is taken int'o account for an atom like C1 bonded to a less electronegative element the relation becomes N = (1 + i)(l + ~ ) ( l + s - d ) where c is the correction factor for the change in screening caused by the charges on A- and B+. A table of values for c has been given by Townes and Schawlow 3 who state that each stage of ionisation modifies the field gradient q by a factor (1 + c ) ; c is 0.15 for the halogens arsenic and antimony 0.20 for oxygen and sulphur a,nd 0.30 for nitrogen.Positive ionisation A+B- increases q by pulling all the electrons closer to the nucleus and negative ionisation A-B+ decreases q. The field gradient for A in the structure A+B- is then q( 1 + c ) and in the structure A-B+ it is q / ( 1 + c ) where q refers to the neutral atom. The number of unbalanced p electrons U, for various types of bond are given in Table 5 . When the structure of a bond is intermediate be- tween two or more of the types listed the resultant 77 (hybrid bond) is obtained by summing the product of U multiplied by the fractional importance for each type contributing to the mixed character of the bond. For example consider a chemical bond which can be described as a resonance hybrid of the two structures A:B and A-B+.If the bond A-B has an ionic character i the fractional importances of the covalent and ionic structures are (1 - i) and i respectively. Hence the net unbalanced p electrons is 77 (Effective) = (1 - i)(l - s + cl) + i(0) = (1 - i)(l - s + d ) Structure A:B . A:B . A-B+ A+B- A-I-B- A=B . AgB . TABLE 3. U for various types of bond. (After Townes and Schawlow 3 Electron configuration of a s2p5 (like C1) s"5 szp6 (like C1-) szp4 (like C1+) s2p4 (like 0) s2p3 (like N) Bonding orbital used by A Pure p spd Hybrid - sp Hybrid 0 Bond ; spd hybrid T Bond ; Pure p u Bond ; spa hybrid r Bond ; Pure p Type of bond Single covalent Single covalent Single ionic Single ionic Single ionic Double Covalent Triple Covalent 'yy 1 l + s - a 182 QUARTERLY REVIEWS Similarly if a bond possesses ionic character in the opposite sense i.e.the contributing structures are A*B and A+B- where A uses sp hybrid orbitals then U (Effective) = (1 - i)(l - 5) + i[2(l - s)(l + c ) ] = (1 - s)[l + i(1 + ZC)] Townes and Dailey 7 and Townes and Schawlow have estimated the percent,age importance of the various valence-bond structures contribut'ing to a niolecule from bond-length and bond-angle data modified by con- sideration of molecular dipole moments and quadrupole couplings. By this means a combination of resonating structures is chosen which gives a value of U which is near the observed value obtained by using eqn. ( 7 ) . I n these interpretations it is assumed that the contributions to the field gradient from p-type wave-functions will be so predominant that the d-orbital con- tributions can be neglected.The chosen combinat>ion is not of course unique. An attempt is made t o choose a combination which is reasonably in accord with all the experimental data. This procedure can only be applied when the quadrupole coupling constant produced by an excess of one p electron along the axis of the bond i.e. eQq, is known. Table 4 gives values for e&q for various isotopes. Some of these have been obtained from measurements of atomic spectra whilst others have been estimated from observed coupling constants in a variety of molecules. I n Table 5 some examples are listed of the bond structures and the expected values of Up derived for certain molecules. The observed values for U are included for comparison.Asymmetric molecules. When the hyperfine structure of the rotational spectra of linear and symmetric-top molecules are analysed only one coupling constant eQq is obtained. For an asymmetric rotor or for the quad- rupole spectra of solids two independent coupling constants are needed in the analysis of the spectra. These can be expressed as eQq, and The quadrupole coupling constants are obtained experimentally with refer- ence to the principal axes of the molecule a b and c . These coupling constants can then be resolved along il new set of reference axes 2 y x with for example the z-axis along a chemical bond by a rotation of axes.39 By comparison with eqn. (13) we can define the nuclear coupling with reference to the new axes x y and x c = (qxx - qyy)/qzz where qxx = a2V/ax2 qYv = a2V/ay2 and qzz = a2V/ax2 e&qaz = - [(Ny + N z ) / 2 - NzleQqp eQqZz = - [ ( N + fly)/2 - NzleQqD eQqyy = - [(Nx 4- Nz)/2 - NyIeQqp Using these relations we obtain the asymmetry parameter The asymmetry parameter E can be obtained from the measured quantities a2V,@a2 a2V/ab2 a2V/ac2 where a b and c are the principal axes of the 39 3.K. Bragg Phys. Rev. 1948 74 533. ORVILLE-THOMAS NUCLEA4R QUADRUPOLE COUPLING 183 molecule. This value of E can be compared with the quantity on the right- hand side of eqn. (14) with assumed values for N, Nv and N,. I n this manner supplementary information about the way the valence orbitals are filled is obtained. Goldstein and Bragg 40 have used the asymmetry parameter to show that the C-C1 bond in CH,:CFCl has only about 5% of double-bond character and not 1576 as had previously been s~rmised.~l This method of obtaining an estimate of the double-bond character of conjugated C-C1 bonds has been developed by B e r ~ o h n .~ ~ The quadrupole coupling of the 33S nucleus in hydrogen sulphide has a large asymmetry parameter 43 of E r= - 0.60. For a long time it has been supposed that because the inter-bond angle in hydrogen sulphide is approximately 90" the sulphur bond-forming orbitals are nearly pure p in character. The large asymmetry parameter obtained for this molecule demonstrates unequivocally that this view is wrong. Gordy 44 has discussed TABLE 4. Quadrupole coupling constants for various nuclei due to one valence p electron. (After Townes and Schawlow 3 Nucleus I eQq (Mc./sec.) I I Nucleus loB .. . llB . . . 14N. . . 1 7 0 . . . 2 7 ~ 1 . . 33s . . . 35s . . . - 10.9 5.3 - 10to - 2 3.3 - 37.5 55 - 39 35c1 . . 3 ~ 1 . . 37ci . . 75As . . 7 9 S ~ . . 79Br . . *lBr . . eQq (Mc./sec.) 109.7 - 1400 - 769.8 - 643.1 Nucleus I eQqp (Mc./sec.) 1131n . . 1 1 5 1 ~ . . 1271 . . 1291 . . I2lSb . . 123Sb . . zolHg . . - 886.2 - 899.1 2000 2500 2292.8 1688 - 1000 " multiple bond " mole- the bonding in hydrogen sulphide in terms of the cular-orbital model and has reconciled the observed bond angle with an s character for the two bonds of 7%. This is almost twice the value expected from the inter-bond angle of 92". Similarly in arsine the observed coupling indicates that the arsenic bonding orbitals must have about 9% of s char- acter as compared with a value of 4% obtained from a consideration of the inter-bond angle.44 Quadrupole Coupling in the Solid State.-Pure quadrupole coupling data in solids can be obtained for much more comdex molecules than can be L investigated in the gaseous state.The interpretation of the coupling data obtained from direct quadrupole spectra of crystalline substances is however complicated by the presence of solid-state effects. These effects are added to the uncertainties of not knowing the exact contributions to chemical 40 J. H. Goldstein and J. K. Bragg Phys. Rev. 1950 78 347. 41 L. 0. Brockmay J. Y. Beach and L. Pauling J. Amel.. Chem. SOC. 1935 57 42 R. Bersohn J . Chem. Phys. 1954 22 2078. 43 C. A. Burrus and W. Gordy Phys. Rev. 1953 92 274. 4 4 W. Gordy in " Chemical Applications of Spectroscopy " Interscience Publishors 2693; 1937 59 2181.New York Ch. 2. Nucleus TABLE 5. Structure of molecules and values of unbalanced p valence electrons Ref. Afolecule FCl IC1 IC1 TlCl TlCl SiH,Cl SiH,Cl ASH AsCl Contributing structures F:Cl F-Cl+ 1:Cl II-Cl- 1:Cl I+Cl- T1:Cl TI+Cl- T1:Cl Tl+Cl- H,Si-Cl H,Si+Cl- H,Si-= C1+ H,Si-CI H Si+Cl- AS-H /H As-Cl /cl \H C1- As+-Cl \Cl \Cl Hybridisation None 15% s None 15% s None 15% 15y0 u bond None 9% 8 10% Up for each structure 1.00 2-30 0.85 0 1.00 0 0.85 0 1.00 0 0.85 0 0.40 1.00 0 - 0.30 - 0.25 - Oq.30 Importance (%) 75 25 85 15 75 23 18 82 20 80 30 40 30 40 60 100 50 50 Xet Up 1.37 0.72 0.75 0.15 0.14 0.38 0.40 - 0.30 - 0.28 Up (obs.) 1-33 0.753 0.753 0.144 0.144 0.36 0.36 - 0.28 - 0.29 ORVILLE-THOMAS NUCLEAR QUADRUPOLE COUPLING 185 bonding of ionic character and hybridisation.The probleni is simplified when crystal-structure data are available. The large asymmetry parameter E = 0.15 found in solid iodine by Dehmelt 45 reveals the presence of weak cross-bonding between the I molecules in the crysta’lline sheets. Townes and Dailey 46 believe that the covalent bond from a given iodine atom resonates between its partner in the molecule and the two nearest neighbouring atoms belonging to other I molecules in the crystal. These auxiliary cross-bonds are each calculated to have an importance of 9% ; the main molecular I bond thus having an importance of 82%. An alternative explanation has been put forward by Robinson Dehmelt and Gordy 36 who point out that the sp hybrid bonds suggested by Townes and Dailey do not explain the formation of planar sheets of I molecules in the crystal.Robinson Dehmelt and Gordy sug- gest that the cross-bonding arises as a result of the employment of the 5d orbitals to increase the valence of iodine. If it is assumed that the iodine atoms use hybrid spd orbitals for bonding the planar arrangement of the atoms in the crystal is more readily explained. Where the crystal structure of a substance is unknown a complete analysis is impossible. Useful information has been obtained by studying a series of similar c0mpounds.~7 In interpreting these results it is generally assumed that the solid-state effects can be neglected since they are not large for slightly polar molecules and do not vary much from one member of the series to the next. Pure quadrupole spectra of aliphatic chlorine compounds have been obtained by Living~ton.~* He was led to the conclusion that the replace- ment of hydrogen by a more electronegative atom such as a halogen reduced the ionic character of the C-Cl bond.The ionic character of a C-C1 bond in an aliphatic chlorine compound was increased by the replacement of a hydrogen atom with a more electropositive group such as CR,. A great deal of information of value to organic chemists has already been obtained. Of particular interest is the variation in the C1 coupling in substituted chlorobeiizeiies with the nature and position of the sub- stit~ent.4~ A close correlation was found with Hammett’s substituent parameter a.50 A relation between the resonance frequency and cs is to be expected since both parameters depend on the electron density of the bonding electrons.Meal 5 l has shown that the correlation is not as good when the substituent group is a complicated polar group; presumably these groups lead to large interactions in the solid state. Bray and Barnes,52 in a similar fashion have compared the resonance frequencies 4 5 H. G. Dehmelt Naturwiss. 1960 17 398. 4 6 C. H. Townes and B. P. Dailey J. Chem. Phys. 1962 20 35. 4 7 A. L. Schawlow ibid. 1954 22 1211. 48 R. Livingston ibid. 1951 19 1434 1613 ; 1952 20 1170. 48 E. Bright Wilson Ann. New York Acnd. Sci. 1952 55 943. 50 L. P. Hammett “ Physical Organic Chemistry ” McGraw-Hill Book Co. Inc. 5 1 H. C. Meal J. Amer. Chem. SOC. 1952 74 6121. 52 P. J. Bray and R. G. Barnes J. Chem. Phys, 1964 22 2023. New York 1940 Ch. 7. 186 QUARTERLY REVIEWS of the *IBr isotope in bromobenzene derivatives with the electron density in the vicinity of the carbon atom to which the bromine is bonded.Studies have been carried out on the Group I V tetra halide^.^^ 47 The possibility of the links to the halogen atoms having some double-bond character complicates the analysis of these results. There are clear indica- tions however that there is an increase in ionic character in going from carbon to tin with a corresponding decrease in double-bond character from silicon to t,in. Duchesne and Monfils 53 have shown that the average values of the quadrupole coupling constant for the chlorine atoms in varying relative positions as one goes from C,H,Cl to C,CI are linearly related to the number of chlorine atoms. They suggest that the increase in the coupling constant in going from m-dichlorobenzene to o-dichlorobenzene is due to the twisting of the C-Cl bonds in the non-planar ortho-compound.B e r ~ o h n ~ ~ however suggests that an inductive effect is primarily responsible for the change in the coupling constant. Other valuable solid-state studies are listed by Dehmelt. 54 Applications.-The use of coupling constants in obtaining information on the electronic structure of molecules has been emphasised in this Review. In certain cases coupling constants give direct information on the valence states of atoms in molecules. When nitrogen is triply bonded as in am- monia and the X*CN series the 14N coupling constant is about - 4 Mc./sec. (Table 6). The molecules CH,*NC and NNO possess quadrivalent nitrogen atoms whose coupling constants are found to be very small since the sur- rounding valence-shell electrons are nearly spherically distributed.In this manner coupling constants afford a new and powerful source of information about the type of bonding associated with atoms in molecules. TABLE 6. Quadrupole coupling constants of 14N and 33S Molecule H*CN . . CH,*CN . H*CZEC.CN eQq (Mc./sec.) Molecule ~~ NH . . I1 I - 29.1 (S) 1 1 I As values for groups of substances containing structural linkages in common accumulate it will be possible to relate certain ranges of coupling constant with specific structural units. The value of coupling constants in this respect is illustrated by the widely differing coupling constants obtained for 14N for the -C=N and -NN-C links (Table 6). The great similarity in electronic structure of the C-S bonds in H*NCS and OCS is emphasised by the approximately equal values obtained for the 33S coupling constants.The value obtained for the I4N coupling constant proves conclusively 55 that the molecule is H*NCS and not H*SCN. 53 J. Duchesne and A. Monfils J. Chem. Phys. 1954 22 562. 54 H. G. Dehmelt Discuss. Faraday Soc. 1955 19 263. 5 5 G. C. Dousmanis T. M. Sanders and H. J. Zeiger J . Chenz. Phys. 1953,21 1416 ; C. H. Tomes and S. Geschwind Phys. Rev. 1948 74 626. ORVILLE-THOMAS NUCLEAR QUADRUPOLE COUPLING 187 A number of substances have been studied in the gas phase and in the solid phase. No great variation in coupling constant is found (Table 7). TABLE 7. Nuclear quadrupole coupling constants in the solid and the gaseous state Molecule CH,*I . . . . . .I C N . . . . . . ICI . . . . . . CH,*Br . . . . . CH2C12 . . . . . CH3*C1 . . . . . CF3.Cl . . . . . . Nucleus 1271 1271 1271 '$Br 3 5c1 35c1 3 5c1 eQq (Mc./sec.) Solid 1753 2549 3037 528.9 71-98 68.40 77.58 Gas 1931.5 2420 2944 577.0 78 75.13 78-05 There is fair agreement between the values measured in the solid and in the gas phase. Such comparisons can be used to study the changes in valence bonding associated with a change in phase. Valuable information on the crystal structure of substances can be obtained by single-crystal studies. For example the number and symmetry of non-equivalent lattice sites in a crystal occupied by the same atomic species can be obtained since the field gradients a t the nuclei are not iden- tical.56 For many substances only a single Cl resonance line is observed but in certain crystals a large number of closely spaced lines are observed.*8 In addition the Zeeman effect of nuclear quadrupole resonance lines can be used to determine the direction of covalent bonds with respect to the crystal lattice.This information is of great value in simplifying the X-ray analysis of crystal structures. Additional solid-state information can be obtained by studying the shapes widths und the temperature-dependence of nuclear quadrupole resonance lines. Dehmelt and Kriiger 57 attribute the temperature depen- dence of the C1 quadrupole spectra in trans-dichloroethylene to a variation in the amplitude of the torsional oscillations of the molecule which thereby changes the average field gradient effective a t the chlorinc nucleus. It is obvious that a great deal of new information about the solid state will be obtained from nuclear quadrupole resonance studies.Conclusion.-It is clear that the measurements of nuclear quadrupole coupling constants are of great importance for the elucidation of problems concerned with the nature of valence bonds. This new type of experimental evidence will supplement data such as bond energies dipole moments and polarisabilities which have hitherto served to test electronic wave-functions Moreover it is important to point out that coupling constants are sensitive to the electronic distribution near thc nucleus and not to the electronic 5 6 H. G. Dehmelt Z . Physik 1951 130 385. 5 7 H. G. Dehmelt and H. Kruger ibid. 1951 129 401. 188 QUARTERLY REVIEWS distribution in the overlap region of the valence bond. In this region near the nucleus the adequacy or otherwise of wave-functions has not previously been tested. It is clear that a difficulty arises if wave-functions are normalised in the usual manner. For bonds possessing a mixed char- acter the values obtained for the amounts of ionic character and sp hybridisa- tion are not unique since it is impossible to obtain two unknown quantities from a single quadrupole coupling measurement. The author thanks Dr. Delia M. Agar Dr. J. W. Linnett and Dr. N. Sheppard for their helpful advice.

ISSN:0009-2681    

DOI:10.1039/QR9571100162

出版商:RSC    

年代:1957

数据来源: RSC