BRUNO J . ZWOLINSK~ ANU HENRYEYKINL
2iU2
lines a t 51(% chromium. This suppurts the view that the cliroinium dispersion is three-dimensional and very great in this method of preparation. I t should be noted that three-dimensional dispersion does nof. ruean that the preparation is necessarily a iiiore effective catalyst. Acknowledgment.-It is a pleasure to acknowledge the support of the Sinclair Refining Company in connertion with this work. We a r e
also indebted to Ur. A. S. Russell of the Aluminum Research Laboratories, Aluminum Company of America, for catalytic measurements. Summary Magnetic susceptibilities have been used, together with X-ray and surface area studies, to elucidate the structure of supported oxides of chromium, molybdenum, tungsten and uranium. EVANSTON,TI.I.INOIS
_
~ C O N I K I U U T IFROM ON
THE
Vol. 69
RECEIVEDMAY19, 1947
._ ..
I
~
FRICKCIIEMICAL LABORATORY, PRINCETON UNIVERSITY, AND DEPARTMENT OH CHEMISTRY, UNIVERSITY OF UTAII]
The Non-Equilibrium Theory of Absolute Rates of Reaction’ BY
BRUNOJ. Z W O L I N Y KAND I~~
HENRY
EYRING
Introduction others.6 The success of the crude collision theory In every formulation of a theory of reaction and more so of the theory of absolute rates of rerates, two distinct problems are involved, the one action, which satisfactorily interprets chemical dealing with the forces acting between the par- reactions and many diversified physical phenomticles participating in the reaction, i.e., the acti- ena, is strong evidence for the correctness of the vation energy, and the other, with the calculation equilibrium hypothesis. All this, however, is a of the specific rate constants assuming the first is posteriori evidence that an adequate supply of known. Furthermore, to make the latter calcu- energetic or activated molecules is maintained lation possible, a basic assumption is made which during all stages of the reaction and more direct finds its origin in S. Arrhenius’slb classical pro- quantitative reasoning is desired. Our present posal that an equilibrium exists between normal knowledge of chemical dynamics is not sufficiently and activated molecules. This is equivalent to advanced for a detailed investigation of the indithe alternate assumption made in the quantum vidual collision processes which give rise to a very mechanical formulation of rate theory that an great variety of energetic molecules and to deterequilibrium exists between the various internal mine what fraction of these fortuitously propitious degrees of freedom of the initial configuration. collisions detefmine the concentration in the actiThe applicability of the equilibrium theory to vated state. An ingenious approach to this comchemical reactions or to any rate process, in which plex problem was made by H. A. Kramers.’ To rearrangement of matter involves surmounting a elucidate the applicability of the absolute rate potential barrier, requires further consideration theory for calculating the velocity of chemical rein view of a complete formulation of the theory of actions, he considered the effect of the Brownian motion of a medium on the probability of escape absolute rates of reaction.2 The simplest example of the inapplicability of of a particle (caught in a potential hole) over a pothe simple equilibrium theory is shown for the case tential barrier. Kramers’ results indicate that the of homogeneous gas reactions, where for SUE- theory of absolute rates of reaction gives results ciently low pressures molecular collisions are in- correct within 10% over a wide range of viscosity adequate for maintaining an equilibrium concen- values. Since he employed classical mechanical tration in the activated state. For most measur- diffusion theory in the study, i t is of interest to able reactions occurring under normal conditions take into account the quantized nature of moleof temperature and pressure, the assumption ap- cular levels. pears to be correct. Discussions of this. matter Method are to be found in papers by Marcelin,a WynneConsider reactants passing by a series of molecuJones and E y r i r ~ gGuggenheim ,~ and Weisss and lar collisions from a set of energy levels to a subse(1) Abstract from a dissertation submitted in May, 1947, t o the quent set of levels corresponding to the final states Graduate School of Princeton University in partial fulfillment of the of the products. It is assumed that the values for requirements for the degree of Doctor of Philosophy. Presented the specific rate of transition kij from level i to at the Atlantic City meeting of the American Chemical Society. level j are known, which, in principle a t least, are April, 1947. ( l a ) Allied Chemical and Dye Corporation Research FeUoa. calculable from the quantum mechanical theory (lb) S. Arrhenius. 2.physik. Chcm., 4, 226 (1889). of collisions.a Restricting ourselves to reactions (2) Also referred to as the “transition-state” method. See M. where we can neglect the concentration changes in Polanyi and M. G. Evans, Trans. Faraday SOL,S1,875 (1935). (3) A. Marcelin, Ann. Phys., 8, 168 (1915). (4) W. F. IC. Wynne-Jones and H. Eyring, J . Chsm. Phys., 8,493 (1935). (6) E. A. Guggenheim and J. Weiss, Trans. Faraday Soc., S4, 67
(leas,.
(6) Symposium on Kinetics, ibid., 3-81 (1938). (7) H. A. Kramers, Physica, 7, 284-304 (1940); see also S . Chandrasekhar, Rcr. Modcro Phys., 16, 1 (1948). ( 8 ) N. F. Mott and H. S. W. Maascr, “Theory of Atomic Collisions.” Oxford University Presa, 1933.
all species except A and designating the number in the ith level by Ai, we have the following set of n linear rate equaiions with constant coefficients
........................ = C ( k j i A j - kijAj) dl j+i ....... corresponding to the n possible energy levels of dAli
state of levels 3 and 4 with 1 and 4 designated as the levels of lowest energy. This model corresponds to the case of n = 4 in the general expressions given above. .The rate equations are
--
the reactants and products. Degenerate levels, like other levels, each carry a separate subscript. In each equation of the set (l),the summation extends over all n values except i. The solution is readily obtained for the set of linear differential equations of the first order with constant coefficients subject to the condition that n
E A ,= D where D is the total concentration. By assuming the solutions A i = B i e b ' where B i is a constant, b a parameter and t the time, and substituting in th.e set (l), a simultaneous set of homogeneous algebraic equations is obtained.
Proceeding according to the general method outlined above, the particular solutions take the form A, = Biebg (i = 1,
.... 4)
(6)
To make possible a solution of the characteristic 4th order determinant for the parameter b and subsequently, to obtain values for the constants Bi,....B4, certain assumptions have to be made with reference to the n(n - 1) = 12 specific rate .............................. constants and proper values chosen to represent their magnitudes. If we assume that a molecule in the activated ......................... state has the same probability for decomposition These equations are to be satisfied for all values along the reaction coordinate to any level of the of the Bi,...,Bn which do not all vanish. For this final state, then for the forward process kla = it is necessary and sufficient that the parameter b k u and kza = k2, and similarly for the reverse procbe a root of the auxiliary equation of the nth de- ess, it follows that k41 = k a and k82 = k31. In addigree, which is obtained from the set (2) by solving tion, if we limit ourselves to reacting systems of the equations for the Bi's by the method of deter- small heats of reaction, so that the energies of the minants. The auxiliary equationgis respective molecular levels in the initial and final kli + b ) / i l l . . . . . . . . . . . . . k,i . states are approximately the same, and combine iz1 the resulting relations between the kij's with those . . . ........................ = 0 (3) derived on the basis of the first assumption, the ........................... following set of relations is found to exist between kin 'bn - (Ckni b) the twelve specific reaction rate constants of the i#n By substituting the determined value of the k t h system Ria = kir = k41 p k4z root bk into equation (2), one is led to the sohkit = k n = kir = k31 (7) tions B j k = G k C j k , where for each j and k one obkia = k4s; kai = ksr tains a numerical value for c j k and G k is the Same arbitrary constant for all j's. Summing all the To obtain the last two relations where k12 = krs particular integrals, the general solution of set and k21 = k s , i t was further assumed that the transmission coefficients of the specific rate con(1) is stants of a pair of such similar transitions ( i e . , ............................... activation and deactivation, respectively) are apn n proximately equal. On the basis of the above Ai = 1;Bi,ebkt = Gk&?k' (4) k- 1 k-1 assumptions for the kij's, the characteristic 4th ................................ order determinant is readily diagonalized to yield In equation (4), the values of c i k and bk are known the following expressions for the parameter b in and by putting t. := 0 and the A i ' s equal to their terms of the four determining rate constants of the initial concentrations, the arbitrary constants are initial state kls, ka, k a and k2s'O readily calculated. Each A i is then a completely bi 0 determined function of time. br = - ( k n kti k13 kri) --a dd 48 Application to a Specific Model bs 2 The following simple case was chosen where the --a + 4 initial state consists of levels 1 and 2 and the final br =
I -(e 1
+
+ + +
(9) I t is readily shown that one of the bk roots of the seculardeterminant is zero and the remaining (n 1) roota are 111 negative as reauired from phymial connideretioni.
-
2
(10) The identity of hi is retained to make equation ( 8 ) appliable t o modifmation of the 4-level model in the following nection.
BRUNOJ. ZWOLINSKIAND HENRYEYRING
2704
where a
(klt
+
k2l
+ ksd +
B = [2kiz(ks:
in Table 11. The general solution for the case of equilibrium is simply
+ 3ki3 + + 2 k ~ ) k3l
2k11(2k21
+ + 3kd1
A i = 1y D 2.02 1 Ai - X lO-'D 2.02
ksi
Choosing the following plausible values for the kij's kiz
= 0.01
kza = 0.1 kir = 0.001
kzi = 1
the quantities bk and in turn Cij have been calculated and are given in Table I. With the particular integrals of equation (6) fuliy determined, it is only necessary to specify the supplementary conditions for evaluation of the arbitrary constants Glr...,G4, which will permit us to obtain the 'fAnLE 1
C,r
k
br
1 2 3 4
0 -1,306 -1.111 -6.737 x 10-3
Crk
C8k
1
0.01 1.437
1
-1 -6.959 x 10-3
0.01 -1.437 -1 6.959 x 10-3
-1 -1
CIt
1 1 1 1
explicit expressions as to how the population in the four levels varies with time of reaction. We will then be in a position to compare the rates of reaction under equilibrium and non-equilibrium conditions. Under conditions of equilibrium, the concentrations of the various species A , defined as tzll..,,tz4 are De-
ni =
4kT
(9)
e-ti/kT j-1
where 13 is the total concentration of the molecules A in the system and the statistical weights w for all levels are taken to be equal. From the principle of detailed balance, we have that k i j = kjie-fji/k'T where eji = ~j - ei, so that equation (9) is equally well written D 711 = -
,E$ 4
(10)
1-1
This represents the supplementary conditions (Le., wh.en t = 0) for the case of equilibrium. For the non-equilibrium case, it is assumed that the concentration of the species A in the lowest level of the initial state (level 1) is a t its equilibrium value and the concentrations in all the remaining levels are zero; thus, the supplementary conditions ,areA I = D/2.02 and A2 = A3 = A , = 0. The respective arbitrary constants for the two cases have been calculated and are summarized TABLE I1 Gt
Equilibrium
Noa-equilibrium
GI
D/2.02 0 0 0
0.2451 D -1.193 X 1O-'D 2.461 X 1O-'D
GI
G: G4
-0.2483 D
V o l . fi!l
(11)
1 A4=-D 2.02
where the concentrations in the various levels are constant. Similarly for the case of non-equilibrium, the general solution is A1 = D[0.2451eblt+ 1.103 X 10-3eb2t+ 2.451 x
+
lO-3ebac 0.24tiPeb1tJ As = D X 10-2[0.2451h1c 0.1714ebz2'0.2451ebrt 0.1714&11] (12) A , = D X 10-*[0.2451eb~t 0.1714ebzf 0.2451eb11 0.1714ebalI A4 = D[02451eblt- 1.103 X 10-3eb2f 2.451 X 10 - 3 e b l f - 0.24G3&1t]
-
+
+
+
-
where bl = 0, bz = -1.306, b3 = -1.111, b4 = -6.737 x To test the soundness of the equilibrium postulate of the activated complex theory of rate processes, it is required to show to what extent the equilibrium between the initial levels is disturbed, as molecules from the activated state rearrange or decompose into products of the final state. This is best demonstrated by formulating an expression for the ratio of the actual rate to the equilibrium rate for the process and calculating the variation of this ratio r with the amount of the substance reacted. Considering the forward rate to be determined by the sum of the velocities from levels 1 and 2 to the final state, the actual velocity is = (kir kidAi (ha kdAz (13) To calculate the rate of reaction by the theory of absolute rates of reactioh, it is assumed that the concentration of the species Ai in the initial state is given by the Maxwell-Boltzmann distribution function. Referring to this velocity as the "equilibrium" rate for the forward process, we have
+
~e
= (kl:
+ kid(Al + Az)
+
e- 4 k T
2
+
+
e-tr/kT
i-1
i-1
Hence, the ratio 'I of the actual rate to the equilibrium rate is
NON-EQUILIBRIUM THEORY OF' ABSOLUTE RATESOF REACTION
Nov., 1947
are the mole fractions in the initial state. Thermodynamically, the ratio of rate constants is
k4klr
+
= c(n/kT kir
(16)
2705
In practice, the conditions for rapid restoration of equilibrium concentrations in the activated state will be aided by the multitudinous number of energy levels in any reacting molecular system.
providing that the transmission coefficients of the various rate constants are the same. In addition, from the principle of detailed balance klr = k r l e - L p l / k T so the expression for the ratio 'I simplifies to
Modifications of the Four-level Model A more complete analysis of the equilibrium postulate of rate theory according to the general procedure outlined in this paper requires. consideration of the following additional points; (1) the effect of the transitions between non-adjacent levels, (2) the magnitudes of the specific rate conUsing the analytical expressions for the mole frac- stants k i j and finally, (3) the comparison of rates tions Nl and Nz as determined from the general of reaction under varied non-equilibrium consolution in equation (12) and the chosen values ditions for the initial state. The same four-level for klz = 0.01 and kzl = 1, the complete expression system was considered with the exception that for the ratio of the actual rate of reaction to the zero values were assigned to all specific rate constants between non-adjacent levels and the magniequilibrium rate is given by tudes of the rate constants between the top levels of the initial and final states [0.4902 - 0.1702e-1.a06f- 0.2426e-1.1i1f+ 0.4177e-6.7ai~10--'L '.'05 (Le., k ~ and s ka) were increased by a factor [0.2476 - 5.212 X 10-4e-LWBL + 0,2480e-8.737xlO-U1 ,' ',O',,J of ten. Similar assumptions were made in relating the rate constants between To test the validity of the equilibrium postulate adjacent levels as in the complete solution of the of rate theory on the basis of our simple four-level four-level case considered above. The relations model, we calculate how the velocity ratio devi- between the rate constants together with their ates from its theoretical equilibrium value of unity chosen t dues are : with the extent of reaction as determined in this k l r = ksi = k i d kdi = k2c = k a = 0 specific case by the concentration of substance A k n = ku kpl = k,c = 1 (19) in level 1 of the initial state. Employing equakir ku = 0.01 tion (18),the variation of the velocity ratio I' with Two cases of non-equilibrium are considered in the concentration A is plotted in Fig. 1. The reiation to this simplified model of the four-level calculations are summarized in Table 111. system. In Case I, it is assumed that the conTABLE I11 VARIATION OF THE RATIOOF ACTUALTO EQUILIBRIUM OF REACTION RATESWITH AMOUNT Amount of0
0.4908 0.87 0.7987 1.33 .8858 ,4885 ,4865 1.74 ,9134 ,4848 2.08 .9222 .4832 2.40 5 ,9251 .4754 3.98 10 ,9276 .3707 25.1 .9492 100 .2480 49.9 .9944 1000 W .9998 .2451 50.5 (A1)r-o = 0.4951 D where D is the total chosen concentration for the system. 1 2 3 4
The magnitude of the variation of the function
I? is of.extreme interest, even though the calculations are based on an extremely simple model. The deviation of approximately 20% in the velocity ratio a t the beginning of the reaction represents the over-all error to be expected in the specific rate constant as determined by the theory of absolute rates of reaction. Inasmuch as the calculations were made on an oversimplified model of a reacting system under drastic conditions of non-equilibrium, the results are most gratifying.
L
0
1 2 3 Amount of reaction, Fig. 1.
4
5
%.
centrations in the initial state have their equilibrium values and the concentrations in all the levels of the final state are zero. For Case 11, the concentration in the lowest level (level 1) of the normal state is a t its equilibrium value and the concentrations in all the other levels of the initial and final state are zero. Case I1 is similar to the non-equilibrium case investigated in the complete solution of the four-level model. The general solutions (similar to equation (12)) were obtained for the two casesof non-equilibrium and are shown plotted in Figs. 2 and 3 expressed as mole fractions of the respective speaes of the substance A where the mole fractions are defined as Na = &/(AI
+
HKUNU J . ZWULINSKIANIJ HENKYEYKING
27U6
r---
r.-3E:
.s8 1.000
2
1.0
3 0.998
'=
0.8
I
X
0
3 0.996
0.6
a
E
< X X 0.2 0.990;: qi0 H'
rl rl
10 20 30 40 50 Time, sec. concentration in level (1); __ , normal state; ------, final state.
0
Fig. 2.--
-.-.-,
0.992 >: 0.99og
x x 0.2
c i0
i
0
I
--.i-_-
10
20
Pi1
-.-..----L
30
40
Time, sec. Fig. 3.-- --.-.-, concentration in level (1); ---, mal state; ------, final state.
60 nor-
A2)! etc,. The general form of these curves in their dependence on time of reaction is characteristic of the general solutions obtained in various modifications of the n-level reacting system. Very rapid variation occurs with time for the various mole fractions and for both Case I and 11, equilibrium values are obtained in approximately 1000 seconds. These are given in Tables IV and V for the two respective cases of non-equilibrium. TABLE IV Case I, Initial Conditions: A , = 0/1.01; A2 = 0/1.01 X 10-2; AS Ai = 0 Nr X 10' N I X 10' N4 Ni 1 , sec. 0 1 10
100
LOO0 m a
0.9901 .9931 .9932 .9923 .9901 .9901
0.9901 .6798 .6732 .7676 ,9893 ,9901
(I
(I
59 9.91 1.679 0.9909 ,9901
0.42 .902 ,9834 ,9901 ,9901
Indeterminate.
TABLE V Case 11, Initial Conditions: A1 = D/l.Ol; Ad 0 L , sec. NI Nr X 10' NI X 10'
AI
Ap
=i
0 1 10 100
1000 m
1.0000 0.9951 ,9932 .9922 .9901 ,9900
Indeterminate.
0 0.4742 ,6722 .7667 .9779 .9900
a
73 10.9 1.687 1.002 0.9900
N4 (I
0.32 ,891 .9831 .9899 .9900
vol. t i 9
Of particular interest is the variation with the time of reaction of the mole fractions in level 2, since, in the absence of the transitions between non-adjacent levels, it determines the rate of the forward reaction. This is compared with the amount of reaction which is primarily determined by the coticentration in level 1. Tlie variation of the concentration A , expressed as a per cent. is also included in Figs. 2 and 3. In Case I, the mole fraction for level 2 drops to a minimum value of 67% of its equilibrium value in two seconds and the equilibrium value is attained in one thousand seconds. This represents the order of variation of 33% to be expected in the over-all specific rate constant for this modified version of the four-level system. A second modification of the original model was considered to study the effect of varying the magnitudes of the rate constants. It is similar to the model treated above with the one exception that the transition constants for the top levels of the initial and the final states (ie., kZ3and k32)were reduced by a factor of ten. Here, a general soh: tion was obtained for the non-equilibrium case, where the concentrations in the initial state have their equilibrium values and the concentrations in the levels of the final state are zero (which is similar to Case I of non-equilibrium used for our first modified model above). For this simplified treatment of the four-level system a maximum error of 8.2% in the over-all specific rate constant was found during the course of the reaction. A summary of the calculations based on modifications of the original four-level system of twelve rate constants is presented in Table VI. In columns 3 and 4, the maximum variation of the velocity ratio function I'is compared with the amount of reaction. This represents the error to be expected in the over-all reaction rate constant as calculated by the equilibrium theory of absolute rates of reaction. The small variation in the function I' of 8.2% from its equilibrium value of unity for the modified model ( 2 ) indicates that the values chosen for the rate constants between nonadjacent levels of our original model are slightly large. Comparison of the respective times for the attainment of a maximum variation in the function I' in this case indicates the important role played by the rate constants between nonadjacent levels in rapid restoration of equilibrium. By increasing the magnitude of the rate constants between the top levels of the initial and final states for the modified model (l),more rapid restoration of equilibrium concentrations is attained a t the expense of a larger error in the ratio I'. A comparison of the results for the two cases of nonequilibrium for the simplified model (1) indicates that for more drastic conditions of non-equilibrium as given by Case 11, the error in the over-all specific rate constant is increased by 2% together with a slightly longer time for restoration of the high equilibrium rate of reaction.
Nov., 1947
SPECTRA OF SUBSTITUTED VINYL AKOMATIC
MONOMERS AND
2707
YOLYMERb
TABLE VI Non-equilibriu m cases
-
Amount of reaction.
%
Time, sec.
0.80
0.87
1
Case I: A I = D/1.01 A2 (D/l.Ol) X lo-' Aa = A, = 0 Case 11: A1 = 0/1.01 A1 = A s = Ar = 0
.67
,87
2
65
.90
3
Case I : A1 = 0/1.01 A2 = (D/l.Ol) X IO-' Ai A, = 0
.918
.87
10
Case 11: A1 = D/2.02 A, = A, = 0 An
equilibrium rate
kli = kra = 0.01
Our modified model (1) of the four-level system especially for the Case I1 of non-equilibrium is in every respect analogous to the problem of simultaneous chemical reactions. The end species A1 and A , are the stable reactant and product, respectively, whereas A z and A 3 are the unstable concentrations. This case is an addition to similar ones treated by A. Skrabal" in his series of papers on the subject of simultaneous chemical reactions. This treatment applies directly to unimolecular reactions with or without an inert gas and to bimolecular or higher order reactions where one reactant is in much lower concentration than the others. Other cases which lead to non-linear equations provide special mathematical difficulties, but the results would be similar. (11) A. Skrabal. Mo-nafsh.. 84, 203 (1943). and earlier papers.
[(:ONTRIBUTION FROM THE
Summary The results obtained taking into consideration the quantized nature of molecular energy levels in testing the applicability of the equilibrium theory of absolute rates of reaction are essentially in agreement with the results found by H. A. Kramers. The general procedure employed complements the classical diffusion approach to this problem. Further application of this treatment to more complicated systems ( n > 4) is deemed necessary in order to indicate what refinements are to be made in the theories of reaction rates. The first author wishes to acknowledge his appreciation of the privileges extended to him by the University of Utah where the greater part of this research was carried out. PRINCETON, N.J. RECEIVED APRIL11. 1947
NOYES CHEMICAL LABORATORY, UNIVERSITY OF ILLINOIS]
Ultraviolet Absorption Spectra of Substituted Vinyl Aromatic Monomers and Polymers' BY H. A.
LAITINEN,
FOILA. MILLERAND T. D. P A R K S ~
In applying the ultraviolet absorption method of Meehan3 to the analysis of copolymers of butadiene with various substituted styrenes and other aromatic vinyl type monomers, i t was necessary to obtain the absorption spectra of the polymers of each of the monomers for calibration purposes. Since the absorption maxima most useful for analytical purposes lie in the wave length range 250300 mp, the data were restricted to this region. (1) This investigation was carried out under t h e sponsorship of the 0 5 c e of Rubber Reserve, Reconstruction Finance Corporation, in connection with the Government Synthetic Rubber Program. (2) Present address: Shell Development Company. Emeryville. California. (a) E. J. Mechan. J . Potymrr Sci., 1, 175 (1946).
Absorption data were also collected for some of the monomers in the same wave length range, inasmuch as it was necessary to prove the complete removal of mononlers from the polymers and copolymers. A small amount of residual monomer would lead to a large error because of its relatively intense absorption. Experimental The freshly distilled monomers, synthesized by the Organic Division,' were polymerized in bulk by irradiation with ultraviolet light. A 1 to 2 ml.sample of monomer in ( 4 ) C. S. Marvrl, cI al., THIS JOURNAL, 67, 2250 (1945); 68, 786 861, 1085, 1088 (19413): R. L. Frank. ef at., i b i d . . fi8, 1366. 1308 (1946).
