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Heterogeneous recombination of atoms. Theory of the Smith-Linnett method

 

作者: Aleksander Jabłoński,  

 

期刊: Journal of the Chemical Society, Faraday Transactions 1: Physical Chemistry in Condensed Phases  (RSC Available online 1977)
卷期: Volume 73, issue 1  

页码: 111-120

 

ISSN:0300-9599

 

年代: 1977

 

DOI:10.1039/F19777300111

 

出版商: RSC

 

数据来源: RSC

 

摘要:

Heterogeneous Recombination of AtomsTheory of the Smith-Linnett MethodBY ALEKSANDER JABLO~SKIDepartment of Catalysis on Metals, Institute of Physical Chemistry,Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224 Warszawa, PolandReceived 2 1 st November, 1975A general differential equation has been derived which describes the concentration of atomsdiffusing and recombining in a cylindrical tube. It is shown that the previously published equationsare particular cases of that equation. The boundary conditions have also been generalized. In thecase of the Smith-Linnett method the simplifications of the general equation, leading to the one-dimensional linear equation, are the cause of a systematic error in calculation of the recombinationcoefficient. This error is considerable for very active and slightly active surfaces.In the case ofrecombination of atomic hydrogen, the error due to omission of the homogeneous reaction in theanalysis of experimental data is negligibly small when the pressure is lower than 0.1 Torr and theconcentration of atoms is lower than 10 %.The atomic hydrogen recombination coefficient y has been calculated for nickel films and forPyrex on the basis of the one-dimensional linear equation and on the basis of the model proposed inthe present work. It was found that the two methods of calculation differ by 3-6 % in the case ofsurfaces having the high activity (7 over and by 30-40 % in the case of surfaces having lowactivity (y below lo-"). Both methods give approximately the same fit to the experimental data.Heterogeneous recombination of atoms, one of the simplest catalytic reactions,has been reviewed in several pub1ications.l. Kinetic studies have shown that thisreaction is usually of the first order and that it takes place mainly according to theRideal me~hanism.~ A convenient measure of the rate of this reaction and of theactivity of the surface is the recombination coefficient, y, which is the ratio of thenumber of atoms reacting on the surface to the total number of collisions of atomswith the same surface.The kinetics of heterogeneous recombination is usuallyinvestigated by diffusion methods, an example of which is the Smith-Linnettmethod.4* This method has been used in studies on recombination of atomichydr~gen,~~ ' 9 9* 12* l 5 oxygen 5 * 6 * '* lo* 11* l4 and nitrogen.13 It consists ofmeasuring the decrease of concentration of atoms along the axis of a cylindricaItube (side arm) the wall of which is covered with the material to be examined. Theknowledge of this concentration profile makes possible the computation of therecombination coefficient for the material under examination.DISTRIBUTION OF ATOMS IN THE CYLINDRICAL TUBEIn this section we derive the differential equation describing the concentration ofatoms inside the cylindrical tube, and show that the previously published equationsare particular cases of this more general equation.The starting point is the equationof mass conservation in a multicomponent system :16h i p t = - div (nivi) +C VikUk,k11112 HETEROGENEOUS RECOMBINATION OF ATOMSwhere n, is the molar concentration of the ith component ; ol is the resultant velocityof the ith component ; V i k is the stoichiometric coefficient of the ith component in thekth reaction, and 24, is the rate of the kth reaction.Let the recombination reactionproceed in the stationary state and have the form : mA -+ A,,,. During this reactionthe number of particles decreases. The flow of atoms nu is accompanied by thedecrease of the number of moles of the atomic gas equal to no(m-- l)/m. 111 orderto maintain the constant pressure this decrease must be compensated for by con-vection of the gas filling the tube, i.e.,nu(nz- l)/in = N w, (2)where N is the total concentration of atoms and molecules, and w is the velocity ofconvection.Since the total flow of atoms, no, is the sum of convection flow. kiw,and diffusion flow, J, the following relationship results from eqn (2) :NmNm - n(nz - 1)Taking into account the first Fick law, J = -Dgrad?z, and assuming the inde-pendence of concentration of the diffusion coefficient D, eqn (l), considering atomsas the component “ i ”, is transformed into :lltr = J .r 1grad n = mu, J 1 NmD div 1 Nm - n(nz - 1)where u is the homogeneous recombination rate. Introducing cylindrical coordinatesx, r, we obtain finally the equation :U. (3)a2n 1 an a2n tn-1 -+--++,+ ax2 r ar 8r Nm-n(m-I)Let index “ s ” denote the component of a vector, perpendicular to the wall ofthe tube. For the stationary state at the wall the following equation must be obeyed :The number of collisions 2 of the flux of atoms with the unit surface area of the wallis equal to :yz = nu,.(4)I1 E4Z = - exp (- t2/2) + nv,Q(t),where t = ( v , , / G ) / C ; C is the mean velocity of atoms, and @(t) is the normaldistribution function. As a result eqn (4) becomes :Let t = t ( y ) be the root of eqn (5). Then on the boundary between the wall and thegas the following condition must be fulfilled :- -Usually it is assumed that the atomic gas is stationary with respect to the wall of thetube. Then 2 = nS;/4? and eqn (4) can be simplified to: t = y/J2n 2: g. As thesecond approximation l 7 only the first term in the series on the right side of eqn (5)A. J A B L O ~ S K I 113t.Hence the condition (6) is considered.can be rewritten a s :We obtain then : t = y[(l -y/2) Jz.)grad, n = - n[m - (m - l)n/N]/(SmR), (7)where 6 = 4D/(yFR) or 6 = 40(1 -y/2)/(yER) for the first or second approximationrespectively; R is the tube radius. Thus, the boundary conditions of eqn (3) forthe tube having the finite length, L, are as follows :an/& = 0 for r = 0 ;n = no = const. for x = 0 ;a??/ar = -n[m-(m- l)n/N]/(GmR) for Y = R ;[condition (7) for the wall of the tube]dn/dx = -n[m- (m- l)n/N]/(G’mR) for x = L.[condition (7) for the plate closing the tube]In the case of an infinitely long tube the first three conditions remain unchanged,whereas the fourth one becomes : n = 0 for x = co.The solution of eqn (3) is a very complex problem and, for this reason, in practicecertain simplifications are introduced.One of them is the assumption that an/dr = 0,which leads to a one-dimensional differential equation, has shown that insuch a case the catalytic activity of the wall of the tube must not be very high. Thesecond simplification consists in assuming that convection is absent. It follows fromeqn (2) that this is equivalent to the assumption that m = 1 . This simplification canbe inade when the concentration of atoms is low. The homogeneous recombinationis usually also omitted, i.e., it is accepted that u = 0 for a two-dimensional equation.In the case of a onedimensional equation it can be shown that u = yZn/2Rm+kn2(N-n), where k is the rate constant.Neglecting the homogeneous recombinationin this case leads to the relation : u = yEn/2Rrn.Table 1 shows the simplifications which should be made in eqn (3) in order toobtain the equations previously derived and solved. The most convenient methodSmithTABLE 1 .-LIST OF SIMPLIFICATIONS OF DIFFERENTIAL EQN (3) LEADINGTO THE PUBLISHED EQUATIONSlength ofauthor cylinder I?l 14 remarksSmith4Shuler, LaidlerTsu, BoudartMelin, Madix 2oGreaves, Linnett 21Wise, Ablow 22Dickens, Schofield,Walsh 23one-dimensional equationsinfinite 1 yFn/2Rfinite 1 yFn/2Rinfinite 2 yCn/4Rfinite 1 0 nonactive wall offinite 2 0 nonactive wall ofcylindercylindertwo-dimensional equationsWise, Ablow 2 2 finite 1 0Wise, Ablow 22 infinite 1 0Dickens, Schofield, finite 2 0Walsh 23Dickens, Schofield, infinite 2 0Walsh 2 114 HETEROGENEOUS RECOMBINATION OF ATOMSof analysis of experimental data is based on the linear one-dimensional equation foran infinitely long tube (first item in table 1).The solution of this equation can bepresented in the form :where b = yE/2RD. From the slope of the straight line obtained in the system ofcoordinates (In n, x> the value of y can be determined. This procedure is associatedwith the Smith-Linnett method.n/n0 = exp(- Jbx), (8)ANALYSIS OF APPLICABILITY OF THE ONE-DIMENSIONAL LINEAR EQUATIONThe published values of the recombination coefficient, determined by the Smith-Linnett method, range from to 10-l. This method cannot give correct resultsin the whole range of values of y since all the simplifying assumptions leading to theone-dimensional linear equation are not always valid. In order to evaluate quanti-tatively the systematic errors connected with the recombination coefficient determinedon the basis of eqn (8) we will use a model which is more similar to the actual structureof the side arm (fig.1). It is a cylinder consisting of two sections. The catalyticactivity of the first section is given by parameters y1 or 61 and its dimensions are :R,-radius and y-length. The second section corresponds to the examined surface.It is characterized by a similar set of parameters-?, 6, R and L respectively. Thecylinder is closed with the guide piston of the movable probe; the activity of thissurface is given by y' or 6'.The disturbing effect of the junction of the thermocoupleprobe is neglected, because in the case of the Smith-Linnett method the area of thisjunction is made as small as possible (-1 mm2). The analysis given below isaccomplished in the range of values of y between loA5 and 1, where the losses of atomson the wall are supposed to predominate over losses of atoms on the j ~ n c t i o n . ~ - ~This analysis is limited to recombination of atomic hydrogen only.s e c t i o n 2 s e c t i o n 1FIG. 1.-Model of the side arm.28Let us assume that the concentration of atoms is so low that the homogeneousreaction and the convection can be neglected. Then it is described by the equation[m = l , u = 01d2n 1 dn a2n -+--+- = o .dz2 r dr dr2 (9)This equation is subject to boundary conditions determined by the geometry of theintroduced model.n = no for z = 0,avt/dr = 0 for I' = 0,dn/dr = -n/R16,anfar = -n/R6dnpz = -n/R6' for z = L+y.for r = R1,O < z < y ,for r = R , y < z < L+yA.JABLONSKI 115It is also assumed that the function n and the derivatives an/ar and an/dz are con-tinuous. Let n(r, z, L) be the solution of the above boundary problem. Then thefunction n'(x) = n(0, y + x, d+ x ) describes the concentration of atoms at the thermo-couple junction as a function of its position, x. The function In [n'(x)/n'(O)] turnedout to be nearly linear,24 although having a different slope from the linear function (8),In [n(x)/n,] = - Jz x. This enabled the comparison of two methods of computationof recombination coefficient, i.e., on the basis of function (8), (ysL), and on the basisof the presented model (y).Fig. 2 shows the dependence of quantity (y-ySL)/y ontemperature and activity of the examined surface. It can be seen that in the case of f 0 --101 I ! 1 I I10-5 I 0-4 10-3 10-2 10-1 IYFIG. 2.-Relative difference between the atomic hydrogen recombination coefficient calculated on thebasis of proposed model and that resulting from eqn (8), (y-ySL)/y, as a function of surface activity.28It was assumed that y = d = 30 cm, R = 1.6 cm, R1 = 1.75 cm. The diffusion coefficient wascalculated from the Chapman-Enskog formula ;25 the H-H2 interaction was described by theLennard-Jones (6-12) potential with the parameters given by Khouw et ~ 1 .~ ~ Pressurep = 0.08 Torr.0 80 60 3 ns90 40 2010-5 10-4 10-3 10-2 10-1 0.5YFIG. 3.-Dependence of the relative concentration of atomic hydrogen, n'(0)/no, on the activity ofthe examined surface. The concentrations of hydrogen atoms measured at x = 0 in the case ofnickel film are fitted to the theoretical curves : (0) T = 298 K, (0) T = 245 K, (A) T = 215 K116 HETEROGENEOUS RECOMBINATION OF ATOMSvery active surfaces and in the case of surfaces having low activity the deviations areconsiderable. For very active surfaces the cause of deviation is the radial distributionof atomic hydrogen concentration, i.e., an/ar # 0. On the other hand, the assump-tion that the tube is infinitely long is valid since atoms quickly disappear along theactive surface.In the case of surfaces having low activity the value of derivativedn/ar is close to zero, but the assumption that the tube is infinitely long is no longervalid. This is the cause of deviation in the low activity region. It follows fromfig. 2 that the two methods of analysis of experimental data become equivalent whenthe activity of the examined surface is in the region 5 x(213 K < T < 453 K).Fig. 3 shows the dependence of the relative concentration n’(0)/no on the catalyticactivity of the examined surface. A decrease of activity at a constant temperatureshould be accompanied by an increase of the response of the probe in the positionx = 0, recording the concentration of atoms between sections of the cylinder.< y < 5 xTHE EFFECT OF HOMOGENEOUS RECOMBINATIONWe now evaluate the error arising from neglect of the homogeneous recombination.We assume that the relative concentration of atomic gas, a = n/N, does not exceed10 %.Let us introduce into eqn (3) the same simplifications as in the case of the one-dimensional linear equation (&/& = 0, zn = l), but let us also take into account thehomogeneous reaction, i.e., let u = ycn/2R+ kn2(N-n). The solution of theresulting equation for an infinitely long tube is given by the formula :1x = ---=In[; J b1 2 b + a y + 2[ b( b + a y - dy2)]*2b + a + 2[b(b + a - CI)]~]’where y = n/no, a = 2Nkno/3D, b = yZ/2RD, d = kn$/2D. As in the previoussection the calculations consist of the determination of the recombination coefficienton the basis of eqn (8), (ysL), and the comparison with the recombination coefficientdetermined on the basis of eqn (lo), (y).The rate constant of homogeneous recom-bination equal to 1.2 x 1OI6 cm6 mo1-2 s - ~ was used in these calculations. This valuehas been obtained by Larkin 27 for moist hydrogen which is usually used in theSmith-Linnett method. Fig. 4 shows plots of the dependence of deviation (ysL - y)/yon the total pressure of the gas in the side arm and the concentration of atomichydrogen at the inlet to the tube, ao. These plots make it possible to choose such apressure that this deviation becomes negligibly small. It can be seen that homo-geneous recombination may be neglected when the pressure is lower than 0.1 Torr,the recombination coefficient of the examined surface is greater than lo-’, and theconcentration of atomic hydrogen does not exceed 10%.Similar results could beexpected for other temperatures since the rate constant of homogeneous recombi-nation depends only slightly on temperature. According to Larkin 27 the activationenergy of this reaction in the temperature range 190-350 K is 1.1 3- 0.3 kcalfmol-l.The above calculations were carried out for 273 K.APPLICATION OF THE THEORY TO EXPERIMENTAL DATAA typical experimental apparatus was used for determining the activity of thinmetal films.24* 28 Hydrogen was prepared electrolytically, and was freed fromoxygen by passing the gas over heated palladinized asbestos.It was then saturatedwith water vapour up to -2%, to facilitate the dissociation of inolecules. Atomswere produced by pumping a steady stream of hydrogen at the pressure 0.08-0.09 Torrthrough an electrodeless radio-frequency discharge, 13.5 MHz. Surfaces having verA. JABLONSKI 117! I I :01 0 2 0 5 I 2 5 10p/Torrlo--\ cplTorr100x 10 ---.3a,=OI01 a, = 005a,=oo15(4-a0 2 0 5 I 2 5 10 O.O!o'Ip/TorrFIG. 4.-Relative difference between the atomic hydrogen recombination coefficient resulting fromeqn (8) and that resulting from eqn (lo), (~sL--"/)/Y, as a function of the pressure of the mixture ofatomic and molecular hydrogen. a. = tzo/N, R = 1.6 cm. (a)-"/ = ( b ) y = (c)y = lod33 4lo3 KITFIG.5.-Comparison of the temperature dependence of atomic hydrogen recombination coefficientfor nickel film on Pyrex, (0) ,with that for Pyrex, (A).*118 HETEROGENEOUS RECOMBINATION OF ATOMShigh and very low activity were chosen as examples of the application of the theorypresented above. It has been found that thin nickel films (thinner than loo&, attemperatures above room temperature, have a very high activity for the recombinationof atomic hydrogen.24* 28 Their activity is higher than that of thick films and foils.It has been also found that thin nickel films are poisoned at low temperatures (245 and215 K) as a result of the formation of the nickel hydride P - p h a ~ e . ~ ~ . 28 The recom-bination coefficient decreases then to the order ofA program was developed for calculating the recombination coefficient on thebasis of eqn (8) and on the basis of the proposed model.In this program the sumsof squares of deviations(fig. 5).S&) = [ln(n,/n*) + &I2 and S2(y) = (ln(ni/izO) - ln[n’(~,>/n’(O)]>~were minimized. Here the nf are the experimentally measured concentrations ofatoms, xi are the corresponding positions of the thermocouple probe, and no is theconcentration recorded for x = 0. The results of the calculations for nickel filmand for Pyrex are presented in tables 2 and 3. Those results, obtained on the basisof eqn (9), are also plotted in fig. 5 as a function of temperature. Fig. 6 shows some ofthe experimentally determined concentration profiles of atomic hydrogen covering thewhole range of observed values of y.This figure also indicates the correspondingdeviations between both methods of computation of the recombination coefficient.1 iTABLE COR VALUES OF ATOMIC HYDROGEN RECOMBINATION COEFFICIENT FOR NICKEL FILM. THESURFACE CONCENTRATION OF NICKEL = 2.6 pug cm-’no.1234567891011124534533 633 6329829824524521521 54534530.0900.0900.0900.0900.0900.0900.0900.8900.0900.0900.0900.090coefficient of 1calc. on the basis>f presented model3.68 x lo-‘3.01 x lo-’2 . 4 2 ~ lo-’2.02x lo-’1.26 x lo-’9.47 x 10-31.46 x 10-41 . 5 0 ~ 10-41.06 x 10-41.27 x 10-42.17 x2.09 x lo-’:econibination, ycalc. on the basisof eqn (8)3.46 x lo-’2.85 x lo-’2.28 x lo-’1.93 x1.21 x lo-’9 .1 4 ~ 10-31.07 x 10-41 . 1 0 ~ 10-47 . 5 5 ~ 10-59.36 x 10-52.08 x2.00x lo-’TABLE 3 .-VALUES OF ATOMIC HYDROGEN RECOMBINATION COEFFICIENT FOR PYREXp/Torr coefficient of recombination, ycalc. on the basisof presented modelcalc. on the basisof eqn (8)no. T / K12345678910111245345345336336329829829824524521521 50.0890.0910.0910.0910.0910.0920.0920.0920.0920.0920.0920.0929.46 x 10-51 . 1 4 ~ 10-41 . 0 6 ~ 10-41.62 x 10-41 . 1 9 ~ 10-49 . 4 9 ~ 10-51.11 x 10-41 . 1 2 ~ 10-48.88 x 10-59.11 x 10-59.18 x 10-58.88 x5 . 4 6 ~ 10-57.01 x 10-56 . 3 7 ~ 10-56.27 x 10-57.66 x 10-56.05 x 10-56 .2 4 ~ 10-56.46 x 10-56 . 2 2 ~ 10-51.13 x7.85 X lO-’7 . 5 8 A. JABLONS KI 119Those values are in good agreement with fig. 2. In the case of a surface having theactivity 1-4 x deviations within the range 3-6 % are observed. Larger deviationscan be seen in the case of surfaces having a low activity : about 30 % in the case ofpoisoned nickel film, and 30-40% with Pyrex. The poisoning of the nickel film at245 K resulted in an almost four-fold increase in the response of the movable probe,placed in the position x = 0. Such an increase in the concentration of atoms waspredicted earlier (fig. 3). This phenomenon also proves that the losses of atomson the probe are insignificant, because the nickel film was entirely behind the junctionof the thermocouple probe at the moment of poisoning.The validity of the theorydeveloped previously is confirmed more quantitatively by fitting the experimentallydetermined concentrations of hydrogen atoms at x = 0 and corresponding values ofy for nickel film to calculated curves on fig. 3. An excellent agreement can beobserved.Ni/Pyrex1460 2 4 6 8x/cmFIG. 6.-Concentration profiles of atomic hydrogen for nickel film.deviation = 6 % ; curve 4, y = 2.02 x lo-', deviation = 4.5 % ; curve 6, y = 9.47 x= 3.5 % ; curve 8, y = 1.50 xCurve 1, y = 3.68 xdeviationdeviation =28.8 %. The numbers of the curves are the same as in the fist columns of table 2 and 3.In order to determine which method gives a better fit to the experimental data theminimum sums of squares, S1(ysL) and S&), were computed and compared.Theywere nearly equal for most of the measurements. This stems from the fact that theshapes of the functions n(x)/n, = exp( -,/% x ) and n'(x)/n'(O) are almost identical ;as was stated earlier, in semilogarithmic coordinates the first function is linear, thesecond one is nearly linear. For this reason it cannot be decided experimentallywhich method of computation of y is more accurate. However, one can expect thatthe recombination coefficient calculated on the basis of the model presented in fig. 1is closer to its true value than that resulting from eqn (8), because this model is moresimilar to the real structure of the side arm than is the model of a catalytically uniformcylinder of infinite length.deviation = 26.7 % ; curve 9, y = 1.06 120 HETEROGENEOUS RECOMBINATION OF ATOMSCONCLUSIONSThe previously published differential equations describing the stationary stateconcentration of atoms diffusing and recombining inside a cylindrical tube areparticular cases of the more general equation which was derived in this work.Calculation of the recombination coefficient on the basis of the one-dimensionallinear equation gives correct results in the case of surfaces having moderate activity.In the case considered here, of heterogeneous recombination of atomic hydrogen, therecombination coefficient should be in the range 5 x When theactivity of the surface is outside this range a systematic error results from simplifica-tions made in the general equation.In the case of high activities this error is due tothe radial distribution of the atom concentration YE/& # 0) ; with low activities theerror is due to the fact that the length of the cylinder is finite.At pressures lower than 0.1 Ton and concentrations of atomic hydrogen lowerthan lo%, the error in the calculation of the recombination coefficient due to theneglected homogeneous recombination is negligibly small.< y < 5 xH. Wise and B. J. Wood, Adv. Atom. Mol. Phys., 1967, 3, 291. ’ V. A. Lavrenko, Recombination of Hydrogen Atoms on the Surfaces of Solids (Russ.) (Nauk.Dumka, Kiev, 1973).W. A. Hardy and J. W. Linnett, 11th Symp. Combust., Berkeley, 1966 (The CombustionInstitute, Pittsburgh, Pensylvania, 1967), p. 167.W.V. Smith, J. Chetn. Phps., 1943, 11, 110.J. W. Linnett and D. G. H. Marsden, Proc. Roy. SOC. A, 1956,254,489, 504.J. C. Greaves and J. W. Linnett, Trans. Faraday Soc., 1958, 54, 1323 ; 1959, 55, 1346, 1355.M. Green, K. R. Jennings, J. W. Linnett and D. Schofield, Trans. Faraday SOC., 1959,55,2152.P. G. Dickens and M. B. Sutcliffe, Trans. Faraday SOC., 1964, 60, 1272.P. G. Dickens, J. W. Linnett and W. Palczewska, J . Catalysis, 1965, 4, 140.lo P. G. Dickens and M. S. Whittingham, Trans. Faraday SOC., 1965, 61, 1226.l1 P. J. Crane, P. G. Dickens and R. E. Thomas, Trans. Faraday Soc., 1967, 63,693.l2 W. A. Hardy and J. W. Linnett, Trans. Faraday SOC., 1970,66,447.l3 M. L. Rahman and J. W. Linnett, Trans. Faraday Soc., 1971, 67, 170, 179, 183.l4 J. W. Linnett and M. M. Rahman, Trans. Furahy SOC., 1971, 67,191.l 5 W. Palczewska, Adu. Catalysis, 1975, 24, 245, and references contained therein.R. Haase, Thermodynamik der irreuersiblen Prozesse (Dietrich Steinkopf Verlag, Darinstadt,1963), p. 249; S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland, Amsterdam, London, 1969), p. 12.l7 H. Motz and H. Wise, J. Chem. Phys., 1960, 32, 1893.K. E. Shuler and K. J. Laidler, J. Chem. Phys., 1949, 17, 1212.l 9 K. Tsu and M. Boudart, Canad. J. Chem., 1961,39, 1239.2o G. A. Melin and R. J. Madix, Trans. Faraday SOC., 1971, 67, 198.21 J. C. Greaves and J. W. Linnett, Tram. Faraday SOC., 1959, 55, 1338.22 H. Wise and C. M. Ablow, J. Chem. Phys., 1958,29, 634.23 P. G. Dickens, D. Schofield and J. Walsh, Trans. Faraday Soc., 1960,56,225.24 A. JaModski, Thesis (Institute of Physical Chemistry, Polish Academy of Sciences, 1975).2 5 J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases andLiquids (J. Wiley,26 B. Khouw, J. E. Morgan and H. 1. Schiff, J. Chem. Phys., 1969,50, 66.27 F. S. Larkin, Canad. J. Chem., 1968,46, 1005.28 A, Jablodski and W. Palczewska, Bid. Acad. Polon. Sci., Ser. Sci. ?him., 1976, 24, 239.New York, 1954), p. 527.(PAPER 5/2277

 

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