112 E. R. Andrew -...

112 E. R. Andrew

B. The point of entry into the intermediate stateConsider first the thread model. Substitution for and from table 1 in (42)

gives after rearrangement:B_He

r 2 > A' L(a/2-1 ) L

(A3)

This value of B is substituted in d(j>/dB, and both are then put in (13). After expanding h(as a power series in /?, and substituting for B from (A3) to whatever degree of approximation is required, the ratio pT = hJHc is found as a power series in ( ) * , of which (14) shows the important terms. A similar procedure for the laminar model gives the corresponding ratio pL as a power series in (A'/£)* the initial terms being given in (15). Numerical solution of (13) for particular values of shows that (14) and (15) are accurate to about 0*1 % for A'/L = 10~3, and within 1 % for A'/L = 10-2.

R e f e r e n c e s

Ginsburg, V. 1945 J. Phya. U.S.S.R. 9, 305.de Haas, W. J. & Casimir-Jonker, J. M. 1933 Physica, 1, 291.Landau, L. 1943 J. Phya. U.S.S.R. 7, 99.London, H. 1935 Proc. Roy. Soc. A, 152, 650.Meshkovsky, A. & Shalnikov, A. 1947 J. Phya. U.S.S.R. 11, 1.

Aspects of a theory of unimolecular reaction rates

By N. B. S l a t e r , University of Leeds

(Communicated by M. G. Evans, F.R.8.—Received 29 December 1947 )

A uniform gas of polyatomic molecules is treated as an assembly of classical vibrating systems, which dissociate when one internal co-ordinate q reaches a critically high value qQ which is related to the dissociation energy E0. The unimolecular velocity constant at temperature T is found to be

K — v exp (— EJkT),

where the ‘frequency factor’ v lies in the range of molecular vibration frequencies. The factor v may be interpreted (i) as a weighted average of the noi nal vibration frequencies, or (ii) as the ratio of the product of the normal frequencies to the product of the frequencies with q fixed, or (iii) in the case where dissociation is due to the rupture of an isolated bond, as the vibration frequency of an imaginary diatomic molecule consisting merely of the two bonded atoms connected by the original bond-force.

The dissociation rate is formulated in several ways, which represent different aspects of the physical picture. The main contrast is between (a) the rate as the average frequency with which the normal-mode vibrations come sufficiently into phase to carry q to the critical value q0, and (b) ‘transition state’ formulations, giving the rate as the product of the relative concentration of activated complexes (molecules with q near q0) and the mean transition frequency. The mathematical equivalence of these methods is shown by a study of the asymptotic distribution of values of sums of harmonic vibrations.

The present model is used to illustrate some concepts of transition state or activated complex theory, such as the ‘effective mass’ in the reaction co-ordinate, and the partition function of the activated complex. The relation of the model to Kassel’s theory is shown by calculating the dissociation rates of molecules of specified total energy.

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1. I n t r o d u c t io n

This theory of unimolecular reaction rates in gases is based on the model of a polyatomic molecule as a classical vibrating system which dissociates when one particular internal co-ordinate, such as a bond-extension, attains a critically high value. The general formula derived for the high-pressure rate a t absolute temperature T is

- ~ = K = v e x p ( -E J k T )s e o .- \ (1)

where N is the concentration, k is Boltzmann’s constant and E0 the ‘activation energy’ per molecule. The frequency factor v lies between the greatest and least of the vibration frequencies of the molecule.

Polanyi & Wigner (1928) and Pelzer (1933) obtained reaction rates of this nature for simple chain molecules. For the general molecule approximate, and later precise, formulae of this type have since been derived (Slater 1939, 1947a). Now the model giving this result is crude, in that it uses classical mechanics and ignores anharmonicity of the vibrations. Its simplicity, however, makes it suitable for comparing different approaches to rate problems, and also for illustrating some concepts of transition state theory. Using the same model of dissociation we shall therefore derive the rate K by several methods, and examine their relation to each other and to the general formalism of the transition state as expounded by Glasstone, Laidler & Eyring (1941). We summarize in (i) to (vi) below these different approaches, and then state the corresponding forms of v in the rate (1).

(i) In § 3 the internal co-ordinate q which is to break at a critical value q0 is treated as a sum of normal-mode vibrations:

q = F(t) = ^ a8ylescos27T(v3t + 8), (2)8 = 1

where e8 and v8 are the energy and frequency in the 5th mode. A formula of Kac (1943) gives the asymptotic frequency, 2 Lof zeros of F(t) — q0 in the case where the v8 are linearly independent numbers. The rate A, (1), is obtained as the average of L over an equilibrium distribution of the energies es. This formulation, and the significance of linear independence, is discussed in § 4.

(ii) In § 5 it is shown that the frequency 2 can be expressed in terms of m(q, q), the distribution function of a sum q, (2), and its derivative q. This suggests the formulation of A in § 6 as the mean of q, the velocity in the reaction co-ordinate, over the distribution m(q0, a) averaged over energy states.

(iii) In § 7, an extension of Pelzer’s method, K is formulated as the direct average of q over an equilibrium distribution in phase-space, where the kinetic and potential energies are the general quadratic forms

T = ^ L a r8qrqay V = %{q = qx). (3)

No use is made here of normal modes, but in § 8 the equivalence of this method to that of (i) is shown by a transformation of phase-space.

(iv) In transition state theory, a rate is calculated as the product of the population of ‘activated complexes’ (in our case, molecules with (/near q0) and of the mean

Aspects of a theory of unimolecular reaction rates 113

VoL 194. A. 8



114 N. B. Slater

transition velocity through the critical configuration. These concepts have been used in (ii) and (iii), but in § 9 the rate is formulated in direct imitation of Eyring’s partition function treatment. The physical significance of the transition velocity and of the partition function of the activated complex is discussed.

(v) In § 10 the model is illustrated by the potential energy diagrams used in transition state theory, with particular reference to the error of the assumption that the vibrations are exactly harmonic.

(vi) Finally in § 11 the dissociation rate is derived for molecules of specified total energy. The result is formally identical with the specific rate derived by Kassel (1932) on a different model of the dissociation process.

The constant v takes different forms in the derivations (i) to (iv) of the rate (1), namely

v (i), (ii); v 1 M n B y27T\ABn J (iii); (4)

v = vxv2...v j{ y 2...v'n). (iv)..

Here vs, as are as in (2): A, B denote the determinants ||arg ||, ||6rg || of (3) and B X1 the co-factors of axl ,bxx:v2, are the frequencies obtained with a fixed value of q in (3), and correspond to the ‘frequencies of the activated complex’. The equivalence of the forms (4) will be shown first in §2. In the important case where dissociation is due to the breaking of an isolated bond between two atoms, v is equal to the vibration frequency of an imaginary diatomic molecule consisting of these two atoms connected by the same bond-force as in the polyatomic molecule.

2. M o l e c u l a r v ib r a t io n s

Some formulae are derived here concerning the normal modes of a vibrating system, and the ‘average frequency’ v which will occur in the dissociation rate.

The internal kinetic energy T' (marked to distinguish it from the temperature T) and potential energy Vof a polyatomic molecule will be taken to be the classical positive definite quadratic forms:

r = i s*s V = 1 S S b„q,q,. (5)1 1

Here n is the number of internal degrees of freedom; rotation of the whole molecule is ignored. Typical co-ordinates qr are elongations of interatomic bonds, or changes in bond angles, from the equilibrium configuration; the inertia coefficients ars then depend on the atomic masses and equilibrium bond-lengths. By treating the ‘force constants’ brs as fixed we neglect anharmonicity of the vibrations.

Let Q i , Q n be normal co-ordinates, in terms of which

T ’ = i< 3 W , V = £ q(A, - 2™.), (6)1 1

where the normal mode frequencies jq ,..., will be assumed to be distinct. The solution of the motion is, from (6),

Qs = ylescos2n(vst + i/rs){s = 1



where es is the energy in the sth normal mode and is a phase constant. I f a = (ars)is the matrix of transformation from q to Q, so that

ft = 'L<*rsQa> (7)8

then the solution for one co-ordinate, for example qx, is

f t = 2 <*isVescos (8)8


We shall be interested mainly in qx, and shall usually omit the suffices 1 in (8). We refer to the ocu or <x8 as the ‘amplitude factors’ for the co-ordinate qx.

The ‘ average frequency’ v associated with co-ordinate qx will be defined first byn

v = A/27 t,A2 = a~2 2 a2Af,l

(9)

where a 2 = 2 oc8.l

(10)

Thus v2 is the mean of the v8, weighted by the amplitude factors a8 of co-ordinate qx. We shall now derive two alternative formulae for v.

Since (7) reduces the forms (5) to (6), in matrix notation

!&aa = A-1, |a b a = 1,

where a is the transpose of a, and A is the diagonal matrix with elements Af,..., A2. Multiplying these equations each by initial a -1 and final a -1 and inverting the results, we obtain

aAa = 2a.-1, aa = 2b -1.

The leading elements of these matrix equations are

2 A J A , 2 B J B ,

where A, B denote the determinants ||ars||, ||6rs || and A xx, Bxx the co-factors of axx, bxx. Hence in (9), (10)

a 2A2 = 2AXX/A,a 2 = 2Bn /B, (11)

and A 1 M u g\* 2n 27t\A B xx) ( 12)

which is the second form of v.To obtain a third form, we note that

B/A = Af Ai... A ,

since the Af are the roots in A2 of the secular equation

IlftsA2 —6rs|| = 0 = l,.. . ,n ) . (13)

Let Ag2, ..., A f be the roots of the equation obtained by omitting the first row and column of (13): then similarly

£ u/A i = a'2.. .a;2. (14)



N. B. Slater

Hence (12) may be writtenv = v 1v2. . .v j ( v ,2...v ’n) (15)

which is the third form. The v'8 are the normal frequencies for the system constrained to have qx fixed at zero (or at any other value), And so by a well-known theorem their magnitudes are interlaced with those of the original frequencies v8. I t follows then from (15), or directly from the first definition (9), that v lies between the greatest and least of the v8.

The ‘ activation energy’. By (8), q can just reach a prescribed positive value q0 ifthe energies es satisfy S | «, | V*. = «,.

The values e8 satisfying this and making the total energy E = Se8 a minimum are easily found to be * _ _ *« •« -•, (16)

where a2 is defined by (10). Hence the minimum total energy E0 for q to attain qQ is

116

= 2es0 = a2.

This may also be written, by (11),

Bo - \ 4 - i l

(17)

(18)

We shall later identify E0 with the ‘activation energy’A particular case. If the co-ordinate qx is the elongation of the distance between

two atoms of masses mx, m2, then it can be shown that, irrespective of the nature of the remaining co-ordinates q2... qn,

A /A xx = mxm2l(mx + m2) = m, (19)

which is the ‘reduced mass’ of the two atoms in relative motion. If, further, the potential energy contains no cross terms qxq8, so that in (5)

2V = 6u g! + EZ&rsgrVs>2

then EJBHence in this case, by (12), (18),

ii — ^u*

J_ Ih l2ttaJ m ’

( 20)

(21)

( 22)

Thus v is here the vibration frequency of an imaginary diatomic molecule with masses mvm2 and force constant bn , and E0 is its energy at extension q0. 3

3. T h e un im o le c t jl a r v e l o c it y c o n s t a n t

In this section the dissociation rate is formulated, and evaluated by using a formula of Kac for the frequency of zeros of a trigonometrical sum. The validity of the formulation will be discussed in § 4, and Kac’s formula in § 5.

We consider the dissociation at high pressure of a gas of polyatomic molecules represented by the vibrating systems treated in § 2. We assume that a molecule



dissociates when a particular internal co-ordinate = attains a large critical value q0; we assume also that the vibrations remain harmonic up to the point of rupture. In an undissociated molecule, the behaviour of q is given by (8), namely

q = F{t) = %at <xx2in(vat + f s), (23)i

with a8 = asVe8. (24)

The energies es and phase constants change only when the molecule collides. The molecule is said to be energized if q can reach q0, that is if the es satisfy

S |a , |a S |a . |V e .£ g 0- (25)A molecule energized by collision will not dissociate, however, unless the terms in (23) come sufficiently into phase for q to reach g0 before further collision removes the high energy. If, as we assume, the pressure is large enough, collisions will be much more frequent than the attainment of the dissociation configuration, and so will effectively maintain a steady population of energized molecules which is slowly depleted by dissociation. We assume, then, that

(а) the ‘dissociation frequency’, or fraction dissociating per second, of energized molecules with specified values (ex, ...,en) is equal to L, the average frequency with which F(t), (23), rises to the value g0; and

(б) the number of molecules, out of a total concentration N, which have energies in the ranges (ex, ex + dex)... (en, en + den) is effectively the equilibrium number

Nf(e)de1...deni /(e) = (&T)-ne x p (-Z e s/&T), (26)

where T is the absolute temperature and k is Boltzmann’s constant.These assumptions lead immediately to the dissociation rate; for as a fraction L

of the molecules (26) in any small energized range dissociate in unit time, the overall unimolecular velocity constant is

K = - - ~ = j...jL f(e )d e1...den, (27)

where the integral is over the ranges es >0 satisfying (25).Two types of definition may be suggested for the dissociation frequency L,

namely a long-time average L x and a phase-average L2:

L x = \ lim { M ( t )It }, (28)00

Li = “ ' J o M{r)d\lrx ...dylrn(29)

where M(r) is the number of zeros of F{t) — q0 [(23)] in the time 0 ^ r. (The factor | represents our selection of zeros for F(t) rising to q0.) We shall assume that the frequencies vx, ..., vnin (23) are linearly independent in the field of rational numbers, that is that there is no set of integers i8, not all zero, such that

= 0 .


i(30



118 N. B. Slater

In this case L x and L 2 are equal, and have the value (Kac 1943)

L = j i j J j n - n J„[a, + A*)] J dVdg.. (31)

We use this to evaluate (27). As L vanishes where (25) is not satisfied, each es in (27) may be given the range (0, oo). By an inversion of the order of integration, (27) then becomes

k = m . o) - n m , ?))

where I,{£, i/)= At [ < * » + AAYf] exp

= e x p { - p i ’aJ(^ + y1!Af)}> (32)

by the use of Weber’s first exponential formula (Watson 1922, p. 393). Thus by the definitions (9), (10) of A, v and a,

K = 4 ^ / _ ro 008 exp (“ P ^ a2£2) &Z J_ t1 “ exP ( - P T a 2A29/2)]

= vexp (— qlloc2JcT), (33)

by well-known formulae. In terms of the energy E0 of (17), the result takes the final form A = v exp ( - E JkT). (34)

Here E0 is the minimum energy for dissociation (that is, for q to reach q0), and for comparison with experimental rates is to be identified with the activation energy per molecule. As we mentioned in § 2, vlies in the range of the molecular vibration frequencies, and so is in effect confined to the range 104 * * * * * * * 12 * to 1014sec.-1. Its value may be calculated from (12), or from (22) in the case where dissociation is due to the rupture of an isolated bond.

4. N o t e s o n t h e fo r m u l a t io n

We examine the significance of some assumptions made in § 3. First we estimatethe relative magnitudes of the dissociation frequency L and of collision and vibrationfrequencies. Secondly we shall discuss the condition of linear independence of thevibration frequencies, and the correct definition of L.

For the distribution (26) of energies to be effectively maintained, it was assumedthat L is small compared with yCoii.> the frequency of collisions per molecule. Wetake the simple case of a molecule for which the amplitude factors txs (24) are approximately the same, and likewise the normal frequencies v8. If E0 > nkT, the average molecule with energy greater than E0 will have total energy approximately E0 + kT. The distribution most favourable to dissociation will be approximate equipartition:

e8^ ( E 0 + kT)ln (s = 1 ,...,w). (35)

Using my previous approximate formula (Slater 1939, equation (8)) for L, and the typical values E JkT _ „ _ 10Useo - t>



119

we find for the energies (35) that when n = 1 5 10 15,then L = 1013 109 104 10-1sec.-1.If the energy distribution is more uneven than in (35), then L will be smaller.

Collision frequencies per molecule, as typified by ethane at atmospheric pressure and temperature 500° K, are of the order

coii. = 109 sec.-1.Thus for polyatomic molecules, with n ^ 10, at normal pressures

L colL (36)so that the assumption that L is small compared with ycoll is valid. For simple molecules with small n, fco11 may greatly exceed L only at impossibly high pressures, so that a first-order reaction would not be observed: this agrees with experimental results.

Linear independence and the dissociation frequencyWhen the vibration frequencies ps are not linearly independent the limit L x of

(28) depends on the phase constants \{rs in (23), although the phase-average (29) still has the value (31). If the xjrs were after a collision randomly distributed over the range (0,1), then an average of the type (29) would correctly represent the dissociation frequency in all cases. Energizing collisions tend, however, to concentrate the phases towards the values J or f , just as a pendulum suddenly struck from small into large oscillation begins its new large swing relatively near the middle. Thus the initial phases are not entirely random. As, however, the inequalities (36) indicate, there is time after an energizing collision for many individual oscillations (of average frequency v) to occur and smooth out any initial phase-concentrations. These concentrations will recur quickly if the frequencies obey relations of the type (30) with small is, but not if the frequencies are linearly independent. This indicates that linear independence is of physical significance in the present classical treatment. I t will be remembered that independence is required in quantal theory if the normal modes are to be quantized separately.

A deeper analysis of the ‘dissociation frequency’ would require consideration of the evenness of spacing of zeros of F(t) — q0, and of the effect of an initial concentration of phase on the time to the next zero. This has not been attempted here: a problem of this type has been investigated by Rice (1945).

5. T h e d is t r ib u t io n o f v a l u e s o f a t r ig o n o m e t r ic a l s u m

In this mathematical interlude we shall sketch proofs of the basic formulae used in §§ 3 and 6. We derive first the asymptotic distributions of a sum F(t), (23), representing the co-ordinate q, and of dFfdt representing the velocity q. We then express 2 L x,(28), the asymptotic frequency of zeros of F(t) — q0, as an integral of q over the distribution of q and q, and indicate how this leads to Kac’s formula (31).

Let 0S denote the fractional part of v8t + i]rs, so that1 ( 5 = 1 , ...,w ).


(37)



120 N. B. Slater

We employ the following form of the Kronecker-Weyl approximation theorem: if vx,..., vnare linearly independent, and (dx> ...,6n) are considered as the rectangular co-ordinates of a point Pt in the ‘unit cube’ (37); and if ‘measT(i^in@)’ denotes the measure of t, out of 0 ;£ t < r,fo r Vtdiich Pt lies in a region 0 within the unit cube, then

lim {-measT(iJ in 0)> = extension of 0 . T->OOW J

(38)

Consider first the sum F(t) of (23), which in terms of the d8 may be written

F = 2 a8cos 277#s, (39)l

and let (40)MVo) dq0 m lim ( i measT (q0 < F(t) <q0+ )r -> oo \T )

be the ‘fraction of time’ for which F(t) lies in a short range dq0. By the theorem (38)

m{q0)dq0 = j . . .(d 0 1...ddn,(41)

where the integral is over values in (37) such that F (the function (39) with dx, ..., now treated as independent variables) lies in (q0, q0 + dq0). By (41)

fro pi pim(q0) exp {iq0 g) dqQ = ... exp d6n

J - 00 Jo Jo

= n Jo(a8£)>iby a formula for the Bessel function J0. The Fourier transform of (42) is

(42)

™(?o) = (27r)_1J _ ooexP (“ ^o ^ )n j o («.£)<*£

= (27T)-1 f cos n J0(asg) (43)J — 00

This result could also be inferred from a discussion by Wintner (1932). The present method is easily extended to give the distribution of F(t) and of the derivative

F'(t) = - 2asAs sin 2tt{v8 t + f 8) (As =

which in terms of the ds can be written

F ' ——'La8\ 8am2nd8. (44)

For, if the simultaneous distribution of F and F ’ is given by

w(?0>4)dq0dq = lim (im easT [q0< F{t) <q0 + dq0,q < F \t)< q + j , (45)r —>■ 00 v* ;

then by a simple extension we find

™(<hA)-■(2*r)-2 ... exv{ i{F -q 0)£+ i(F , -q)ij}dOx...d(46)— 00 J — 00 J 0 JO *



121Aspects of a theory of unimolecular reaction rates

where F, F ' denote (39), (44) as functions of the 0S. Hence

M<lo,4) = 4^2J ^ j w008 (?o£) c°s m f t V(£2 + * * * ! ) ] ( 4 7 )

For a given value q0 of F, the possible values q of F' are limited to a symmetrical range (—U,U) depending o n q0. Thus

4) = 0 for | | > (48) I t is to be inferred from the definitions that

r* rum(q0,q )d q = \ m(q0,q)dq (49)

J - 00 J - u

The frequency of zeros of — q0

Out of a long time r, the time for which F(t) and F'(t) lie simultaneously in specified ranges dq0, dq is by (45) asymptotically equal to

rm(q0>q)dq0dq.The time taken by F(t) to pass through the range (q0, q0 + dqQ) with is

<kol\4\•The ratio, rm(q0, q) \ q| dq, of these expressions therefore gives the number of ‘passages’, that is the number of zeros of F(t)—q0, with a given value q of F'. The total number of zeros in time r is therefore

M(t) ~ t jm(q0,q)\4\d4>

where by (48) the limits of integration (here and in the following integrals of dq) are ( — U, U) or (— oo, oo). Hence the asymptotic frequency of zeros (28) is

2 L x = lim M ( t ) J™(?o >4)\4\d4- (50)

I t is worth noting that if v0 denotes the average of | for a given value q0 of F, so that . . .

v0 = J 4)| 4| dq I m(q0, q) dq.

then by (49) we may write (50) as2LX = rn{qf)vQ,(51)

which expresses Lx in a manner typical of ‘ transition state ’ theory.To derive Kac’s formula (31), we insert (46) in (50) and integrate first for q and rj,

obtaining r « r i r i2L1 = (27r)-1 ... ex]>{i(F(52)

J — 00 J 0 JO

To proceed, we write r<a| F '| = n~l j y ~ \\ — cos (F'y)]

J — oo

and thence derive (31) by a process similar to the last stages of Kac’s derivation (Kac 1943).



122 N. B. Slater

6. T h e v e l o c it y c o n s t a n t in t e r m s o f d is t r ib u t io n f u n c t io n s

The argument leading to the formula (50) for suggests an analogous formulation of the velocity constant K. We again assume that the frequencies vs are linearly independent. The fraction of molecules with energies ...,eTO)which have the breaking co-ordinate q in the critical region (q0 — dqQ, q0) with also in a range (q,q + dq) may be represented by (45), where is given by (23), (24). Hencefor an equilibrium distribution (26), the number of molecules of all energies with q, q in these ranges is NM(q0,q)dq0dq, where

/• oo r oo

M(.qQ,q)=\••• m(53) Jo Jo

Inserting (26) and (47) for / and m(q0, q), we obtain

1 f f 00 n= 4 ^ 2 J J _ ^ c o s <^o^) c ° s (4v)nU£,V)dvd£,1 / E

7ia2AytTeXp\ kT

where we have used (32) and have introduced E0 from (17).The time taken to pass through the critical range ( — dq0, q0) to dissociation at

q0 is dqjq, if q is positive. Hence the proportion of molecules dissociating per second— the velocity constant—is

* = J o M (<lo><i)4dq = ^ ^M{q0,q)\q\dq. (55)

(The second form follows as M is an even function of q.) Using (54) we obtain

K — yexp (— EJkT)(v = A/2n), (56)

which agrees with the earlier result (34). This is to be expected, as (55), (53) are seen to be equivalent to the earlier basic formula (27), by reference to the form (50) of L v The interest of the present method lies in the use of the distribution m(q0, q) instead of using an explicit frequency L v

A ‘transition state ’ formulation

In transition state theory a dissociation rate is formulated as the product of the equilibrium population of * activated complexes ’ (molecules in the ‘ transition state ’ bordering on dissociation) and of the average rate of transition. The method just given can be translated into this form. Our transition state comprises all configurations with q increasing and in a short range (q0 — dq0, q0). The chance that a molecule with energies (e1?. en)is in this state may be represented by \m(qf) dq0, (40). Hence out of an equilibrium population (26), the concentration of activated complexes is N*, where

N*/N = %dq0 f ... f m{q0)f(e) ... den.Jo Jo

( 57)



123Aspects of a theory of unimolecular reaction rates

(The factor \ represents the choice of molecules with q increasing. Eyring and others transfer this factor to the mean velocity of transition.) Using (26) and (43), with (32) in the evaluation, we find

N* 1 dq0 N ~ 2ot<]exp { - E J k (58)

Now the average velocity v in co-ordinate q through the transition state is, by reference to the definition of M(q0, q) above equation (53),

i* oo / r 00v = M(qQ,q)\q\dqj\M(q0,q)dq.

J — 0 0 / J — CO

(59)

Hence using (54) we find v = (xX^kT (60)The average frequency of transition is vjdq0: so the velocity constant is

<6»as before. The equivalence of this method with the first approach of this section is seen by reference to (49). The ‘transition sta te ’ formulation appears less direct because of the separate averaging of vin (59).

7. G e n e r a l iz a t io n o f P e l z e r ’s c a l c u l a t io n

Pelzer (1933) calculated the unimolecular rate for a chain molecule, with the potential energy of the form V = , by a direct statistical method not involvingnormal vibrations. We shall extend his method to the general energies of (5). Transition state formalism will be used, as this does not here involve any complication.

In a population of N molecules in statistical equilibrium, the number with qs and qs in specified ranges dq8, dqs (s — 1,..., n)at any time is assumed to be Ndg, with

wheref

dg= C - 'e x p { - (T ' + V)lkT}dq i ...dqn,

C = J . . . J exp{ — ( V + V)/kT}...

(62)

(63)

and the energies T' and V are given by (5). The fraction of molecules ‘activated’, that is with q1 positive and qx in the transition region (q0 — dq0,q0), is

jV*x = ¥<h

J ... Jexp ( - VJkT) dq2... dqn

J . . .Jexp ( — V/kT)dq ... dqn(64)

where V0 denotes V with q1 = q0. The mean co-ordinate velocity, | j, a t the transition state is r r

... exp( - T'/kT\qx \dqx ..'dqn “ - JJ (65)

J .. .Jexp ( - T'lkT) ... dqn

t The limits ( — 00, +00) are to be understood in all integrals in §§ 7, 8, 9 where no limits are specified.



124 N. B. Slater

The unimolecular velocity constant is, as in § 6, the product of N*/N and the meantransition frequency: ^

K = -ktT -- (66)N dq0

This could be written by (64), (65), or derived directly in imitation of Pelzer, as

K = $C-1j . . . j e x p { - { T ' + V0)lkT}\q1\dq2...dqn dq1...dqn. (67)

We proceed to evaluate the factors (64), (65) of (66). In (64) we writeqx — xx, q8 = x8 + d^isxil-n(5 = 2> • • • > ( 68)

where Br8 is the co-factor of br8 in B = || bra||. This gives in (5)

2Vn

B xyB n + ZZ br8xrxa, 2

and in particular, when qx = qQ, the potential energy is, by (18),

V0 = E0 + \ (69)2

This identifies E0 again as the minimum energy at the transition state, since the sum in (69) is essentially positive. By (68), (64) becomes

N*JN = bdq0exp (— E0fkT) I fexp (— Bx\j2BlxkT) dxx

= |dg0e x p ( - E JkTThis agrees, by (11), with the result (58) of § 6.

To evaluate (65) we use the similar transformation

4 i = v v 4a = V3 + A la ^ llA l l (s = 2, ,where A rs is the co-factor of ar8 in = || ar81|. This gives in (5)

2 T ' = AvHAxx + ^ a rsvrv8.2

Thus (65) becomesv = J| vx| e~^dvxjje~^dvx (0 =

= *J(2JcTAlxlnA),which agrees with (60) of § 6 by virtue of (11).

The results (70), (71) for the factors of K give for (66)

K — v exp (— EJkT),

(70)

(71)

(72)

which by (12) is the same as the results (34) of §3 and (61) of §6. Here, however, v has not arisen as an ‘average frequency’, but as the combination (72) of determinants. The form (72) leads naturally to the ‘particular case’ discussed in § 2, and shows by reference to (22) that the unimolecular reaction rate for the breaking of an isolated bond between two atoms of a large molecule is the same as the rate for an imaginary diatomic molecule of these atoms with the same bond-strength (Pelzer 1947; Slater 19476).



Aspects of a theory of unimolecular reaction rates125

8. R elations of the methods

The parallelism of the transition state methods in § 7 and a t the end of § 6 was indicated by the identification of the corresponding expressions for (70) andv (71). The method of §6 is a bridge between the sharply contrasted methods of §§ 3 and 7. I t is of interest to complete the cycle by transforming the basic formula (67) of § 7 into that (27) of § 3.

Since in (67) 1 denotes V with qx = q0,

exp (-VJJcT) = ^ j j e x ^ - ^ + i f a - q j f ^ d q ^ Z .

This gives for (67)

K = 1(2*0-* f... fe x p { -(2 ” + V)lkT}exV{i(ql - q 0)£l\q1\dq1...dq(73)

We transform from (qr, qr) to normal mode energies e8 and phase angles 08, where (compare (8))

qr = 2<xrs cos 2tt68, qr - Sars Aa Sfes sin 2nds. (74)

As this transformation may be compounded of linear transformations together with the contact transformation from normal co-ordinates and momenta to action and angle variables, it has a constant Jacobian. Applying (74) to the integral in (73) and also to the integral (63) for C, we obtain

K = 1(2 77C')-1 Jd£ f ” ... exp {i(F - Q | F ‘ |exp ( - SeJhT), (75)

where F, F ' denote (39), (44) with a8 = ots*Je3 and

C' = I* dex...f dOx... exp (—Jo Jo

Hence by the formula (52) for the frequency 21^,

K = (kT)~nf . . . f L 1exip( — ?leslkT)de1...den, (76)Jo Jo

which by (26) is the basic formula (27) of § 3.This identification raises the question of the status, in the formulation § 7, of the

linear independence of the frequencies, imposed in § 3. I t was suggested in § 4 that(a) an energized molecule starts with phases concentrated towards £ or f , and that (6) if the frequencies are linearly independent this distribution will be smoothed out during the energized lifetime. The corresponding picture in the {q,q) or (q,p) phase space of § 7 (with pr = dT'jdqr,the generalized momentum) is that (a) an energized molecule starts with comparatively small s and large p ’s, and that(b) these values spread out over available phase-space. I t is known that if the frequencies considered as the ‘average motions’ of the system (compare Epstein 1936) are linearly independent, then any trajectory fills the ergodically available phase space uniformly. This corresponds to the distribution (62) of § 7, extended as in (63) over all phase space: this distribution would therefore not be justified



126 N. B. Slater

(with our initial concentration of q, p) if the vs were not linearly independent. There is thus a close correspondence between the role of linear independence in §§ 3 and 7. The transformation (74) preserved equality of weighting of elements of phase space, as is required if the correspondence is to hold.

9. T r a n s it io n v e l o c it y a n d p a r t it io n f u n c t io n s

The present model will now be used to illustrate some concepts of transition state theory, in particular the transition velocity and the partition function of the activated complex. This theor}^ has been fully expounded by Glasstone, Laidler & Eyring (1941), to whom we refer as ‘G.L.E.’.

In our model the transition velocity v, defined by (59) or (65), takes the form (71). This corresponds to the formula of G.L.E. (p. 187, first line)

x = |^/(2 IcT/n(77)

if we identify their m* with AfAu in (71). (The extra factor \ in (77), absent from (71), was included in the activated population (57), as we noted in §6.) G.L.E. call m* in (77) the ‘ effective mass of the activated complex in the reaction co-ordinate \ Now if the kinetic energy T' of (5) is expressed in terms of the momenta pr = the term in p\ is \p \A xxfA. Thus A jA lx is a mass-coefficient associated with the breaking co-ordinate qx. In the important case in § 2 where is the bond extensionbetween two atoms mx, m2, then by (19) .4/^4 u is the ‘reduced mass’ mxm j(m x + m,2). This shows that in this case the effective mass of the activated complex depends only on the adjacent atoms and not on the rest of the molecule.

We turn now to partition functions, and imitate the notation of G.L.E. (pp. 188-9) to facilitate comparison. The relative concentration N*/N ((57) or (64)) of ‘activated complexes’ equals the ratio F't/Fof the partition functions of activated and normal molecules, so that the rate (66) may be written

K v_ F[ dq0 F ’ (78)

The classical partition function F for the energies (5) is

F = h-nj . . . j e x p { - { T , + V)/JcT}dp1...dqn = n (7 9 )

where Planck’s constant h is introduced to equate classical to quantal weighting of phase space, and the vs are the normal frequencies. The partition function F't for the activated complex (qx increasing and in (q0 — dq0,q0)) is

Ft = lh~ndq0J ... Jexp { - ( T '+ V0)/JcT} dpx ... dpndq2...

where V0 denotes V with qx = q0, as in § 7. By using (68) we find

F’t = §h-ndq0(27TkT)n-lV(^/Bn )e x p ( - EJ(80)



Aspects o f a theory of unimolecular reaction rates 127This may be arranged in the form

F't — /<r(i)^texp (— (81)

where = d ^ ( 2 n k T A jA u ), (82)

F, = (fcriT/A V -V M u/Bu)nn (ir/K)> (83)

by the use in (83) of the formula (14) for the frequencies v$ = A'/27r of the system with fixed. Now by (71), (82),

v fi kT d q / ,r{1) ~ h '

(84)

Thus the rate (78) becomes, by (81), (84),

K X § exP ( (85)

which may also be written by (79), (83)

K = ly 2" ; > exp ( - (86)

This repeats, by (15), our earlier result v exp (— it also corresponds explicitlyto the classical approximation of G.L.E. (p. 193) if only vibrational motion is considered.

We shall examine more closely the factors of the previous form (85), which is a strict imitation of the general rate formula of G.L.E. (p. 189). The first factor JcT/h, called by them a ‘ universal constant ’ of rate processes at a given temperature, was constructed in (84) out of v/dq0, the mean transmission frequency, and which by (82) has the form of a translational partition function for motion in the reaction co-ordinate with A/Au as the ‘effective mass’. On our model kT/h appears to be at most a formal construct, just as the factor h, as noted under (79), plays only a formal role in the classical theory and does not affect the total rate vexp (—

Of the remaining factors of (85), i^exp (— is the partition function forthe molecule with q1 fixed at q0. For by (68), (69) the energies are then

v = E0 + l2 2 Ks^r - Qro) (Vs - &o)> T ' £ 2 2 ars4r4s> (87)2 2

where qr0 = B lrq0/B n gives the configuration of least energy (E0) a t the transition state. As the frequencies of the system (87) are the defined under equation (15), Ft (83) is the corresponding vibrational partition function, and exp (— E JkT) represents the difference in zero energy level between this system and the normal molecule.

This completes the parallel with G.L.E.’s treatment, showing the rate as a product (85) with factors kT/h and a modified partition function representing vibrations at the transition point. This analysis has an artificial character here, although formulae of the type (85) have been much used in the development of the general theory.



128 N. B. Slater

10. P o t e n t ia l e n e r g y d ia g r a m s

The dissociation model is illustrated here by the potential energy diagrams used in transition state theory. This leads to a discussion of the earlier assumption that the vibrations are strictly harmonic.

Graphical representations are more simply interpreted if we choose co-ordinates q8 such that the kinetic energy is a sum of squares (G.L.E. chapter h i). To preserve the identity of the breaking co-ordinate q we choose a transformation (with unit Jacobian) which gives in (5)

with «;■*,. (88)

This leaves the potential (5) in a general form

v = i (89)l

The internal motion of the molecule can now be represented by the motion of a particle P of mass AfAu in an w-dimensional space with rectangular co-ordinates (q'x,...,q'n) under the potential (89).

The equipotential curves are illustrated in figure 1 for the case = 2. The curves are similar ellipses, centred at the origin (the equilibrium configuration), and increasing in size with V. The principal axes represent the paths of P in one or other of the normal modes of vibration. I f the system has energies ex, e2 in the two normal modes, with for example e1 + e2 = 10, then P describes a Lissajous curve bounded by a rectangle (indicated in the figure by a broken line) inscribed in the ellipse = 10. I f the frequencies vx, v2 are linearly independent, the Lissajous curve is not closed,but covers in time the whole rectangle: this is the fundamental role of the condition of linear independence. Different values of e2 with the same sum 10 give motions in other rectangles inscribed in the same ellipse. The ‘transition state’ is a narrow strip qx = (q0 — dq0,q0), and the molecule dissociates when P crosses the line qx = qQ to the right. The ‘ activation energy ’ in figure 1 is = 20, since the ellipse V = 20 touches this line. The point A of contact is the transition point of least energy: its co-ordinates were given under equation (87).

This picture may be extended mentally from the case 2 to the large values of n characterizing polyatomic molecules. If is large, most energetic molecules will have energies not much larger than E0, so that their equipotentials will contain only a small region of the hyperplane qx = q0. Thus the Lissajous curves may perform many loops before crossing this plane. This illustrates the result of § 4 that the dissociation frequency L is small even for a favourable distribution of the typical high energy E0 -1- JcT, when n is large.

Returning to the illustrative case n — 2, we discuss finally the harmonicity of the vibrations. If the molecule is to dissociate a t must diminish as qx increasesbeyond q0, &o that the elliptic contours cannot continue into this region. If, moreover, the contours are elliptic right up to qx = q0, then dV/dqx is strictly positive to the left and is negative to the right of this line. Our model thus implies a discontinuity in the gradient of V. This may be a useful approximation, but for continuity we



129

require dV/dq1 = 0 on = qQ. Thus the elliptic contours, and the potential function (5) or (89), must be modified for qx < q0; the complete equipotentials must therefore be of the well-known type sketched in figure 2, showing the characteristic ‘col’ at the activation point A.

This ‘distortion’ of the contours for ql less than q0 implies that the vibrations cannot remain harmonic up to the point of rupture. The error of our harmonic model will, however, be small if, for example, V is modified only very near ql = q0, with a rapid change of gradient. In a one-dimensional diffusion model Kramers (1940) has compared passage over a rounded barrier with passage over a sharp peak of V. The present model might be modified by changing the form of V to round off its values near qx = q0;this modification could for example be introduced into the formula (67) for the dissociation rate. The general effect of such a modification has been discussed by Evans & Rushbrooke (1945) with reference to entropy changes.


F ig u r e 1. Elliptical equipotential curves.

F iguhe 2. Modified equipotential curves.

11. S p e c if ic d is s o c ia t io n r a t e s

In conclusion a brief account is given of the analysis of the dissociation rate into the contributions of molecules of specified total energy, and the results are compared with Kassel’s theory. The method of § 3 is used as a basis.

Molecules whose internal energy £es. lies in a short range 4- dE) will be called ‘^-molecules’. The number of these in an equilibrium distribution (26) is N fE dE, where » .

f E dE = ... \fle)del ...den = Jb 6 , . (90)J E < T .e t < E + d E J 1 W («^ )

If N K E dE is the number of ^/-molecules dissociating per second, then the overall rate K, (27), is . .

K = J K E dE = J ... j L

Hence, as E represents £es, we have by (26)

j*o K E e~EzdE = (lcT)~n...j^ Lexp j - ^ 2esj dex ...den

= K l + ^ ^ ) _ne x p J - ^ 2 + J , (91)

Vol. 194. A. 9



130 N. B. Slater

where v, E0 are defined by (9), (17) and the integration has imitated the process of § 3. Denoting the last expression in (91) by <ji{z), we have as the Laplace transform

l /•c+ ioo

K e = 2ni}c-ioo^ eE:ZdZ(C“ 0)‘

Evaluation givesve-E!kT^E _ E^n-l

T (w ) (

= 0

(E > E 0),-

(E < E 0).,(92)

The specific dissociation rate, or is by (90)

fraction of 2£-molecules dissociating per second,

IE — E^\n~x= •'(— g - 9) (E > E t). (93)

This represents the average of L (§ 3) over all distributions (ex, ..., en) of a total energy E, including those distributions for which (25) is not satisfied and L vanishes. A similar application of Laplace transforms to the results of §§ 6 or 7 gives'the number of ^-molecules with specified values of q, or of q and q. The average velocity of ^/-molecules through the transition state is found, for example, to be

T(n—l ) / 2(Jg— E j E m { rrA/An ) ' (94)

The specific rate (93) is noteworthy for its formal identity with the corresponding classical rate of Kassel (1932). He obtains the factor [(E — E0)/E]n~1 as the chance that, of a total energy E distributed between n equivalent oscillators, more than E0 is in one (breaking) oscillator. Instead of the precise factor v in (93), he has an arbitrary constant representing the frequency of interchange of energy between oscillators. The resemblance of this model to the present model of dissociation is thus not close, although the common formula (93) ensures the same form, pexp (— EJIcT), of high-pressure rate.

The formula (93) could be used to extend the theory to low-pressure dissociation, in exact imitation of Kassel’s method. On our model, however, this use of K'E would not be accurate. At low pressures the concentration of energetic molecules is depleted by dissociation, and for a given total energy E those distributions (els ...,en) giving the largest values of L are the most depleted, so that the use of an average K'E of L is incorrect. The low-pressure rate can, however, be formulated precisely as an integral involving Las a function of the energies e1, ...,en. Kac’s formula (31) for L is unsuitable for integration in this case, but the rate can be estimated by using my previous approximate formula (Slater 1939, equation (8)) for L.

I am indebted to Professor M. G. Evans and Dr G. S. Rushbrooke, who suggested the translation of the theory into ‘transition sta te’ form, and to Mr H. D. Ursell, for many helpful discussions. I am grateful also to Professor P. Stein for communicating to me some unpublished theorems on trigonometrical sums.




R e f e r e n c e s

Epstein, P. S. 1936 Article U (vol. 11) in Commentary Willard Gibbs. Yale UniversityPress.

Evans, M. G. & Rushbrooke, G. S. 1945 Trans. Faraday Soc. 41, 621.Glasstone, S., Laidler, K. J. & Eyring, H. 1941 The theory of rate processes. New York:

McGraw-Hill Book Co.Kac, M. 1943 Amer. J. Math. 65, 609.Kassel, L. S. 1932 Kinetics of homogeneous gas reactions, chapter v. New York: Chemical

Catalog Co.Kramers, H. A. 1940 Physica, 7, 284.Pelzer, H. 1933 Z. Elektrochem. 39, 608.Pelzer, H. 1947 Nature, 160, 576.Polanyi, M. & Wigner, E. 1928 Z. phys. Chem. A, 139, 439.Rice, S. O. 1945 Bell Syst. Tech. J. 24, 46.Slater, N. B. 1939 Proc. Camb. Phil. Soc. 35, 56.Slater, N. B. 19470 Nature, 159, 264.Slater, N. B. 19476 Nature, 160, 576.Watson, G. N. 1922 Theory of Bessel functions. Cambridge University Press.Wintner, A. 1932 Math. Z. 36, 618.

The transmutation of magnesium, aluminium and silicon by deuterons

By H. R. A l l a n , Trinity College, University of Cambridge a n d Ch r is t ia n e A . W il k in s o n , Nevmham College, University of Cambridge

( Communicated by J. D. Cockcroft, F.R.8.—Received 31 December 1947)

A study has been made of the protons emitted by magnesium, aluminium and silicon targets when bombarded by deuterons of energy between 700 and 1000 keV. The Q values of these (d, p) reactions have been accurately determined. Observations of the induced radioactivity in the targets and comparisons with the Q values in analogous nuclei permit tentative assignments to the isotopes responsible. The Q values and assignments are:

24Mg (d, p) 2SMg 26Mg (d, p) 26Mg* 2®Mg (d, p) 27Mg 27A1 (d, p ) 28A1

28Si (d, p)2»Si

“ Si (d, p)«Si

Q = 5 03 MeVQ = 4-45 MeV (to excited state)Q = 4 05 MeV Q = 5-46 MeV

= 4-46 MeVl= 3-98 MeV 1 (to excited states) = 3-31 MeV j

Q = 6-16 MeV.= 4-87 MeV) .. .. , . , .= 3-75 MeV j (to excited states)

Q = 4-16 MeVThe yield of the proton group attributed to 26Mg (d, p ) 26Mg* shows a resonance at 955 kV

bombarding voltage, as does also the yield of radioactive 26A1 formed in the reaction 26Mg (d, n) 26A1. This indicates a resonance level in the compound nucleus 27A1 at an excitation energy of 17 MeV. The maximum energy of the /?+ rays from 26A1 was measured as 2-8 MeV.

The Q values listed above lead to a set of nuclear mass values, but these mass values are not always in satisfactory agreement with the published Q values for certain other reactions.

9 - 2



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