April, 19GO
EXTENSIOS OF
SLATER’S
HIGHP R E S S U R E
RATEEXPItESSIOY
~-SIMOLECT-LAR
473
AS EXTEXSIOS OF SLATER’S HIGH PRESSURE UMMOLECULAR RA4TE EXPRESSIOS TO SIi\iIULTANEOUS REACTION C06RDIXATES’ BY EVERETT THIELE AND DAVID J. VILSON Department of Chemistry, Universzty of Rochester, Rochester 10,?IIm York Receiued September 2, 195.9
Slater’s classical expression for the high pressure rate of a unimolecular reaction is extended to the case where reaction is assumed to occur when ttvo internal coordinates are simultaneously greater than a given pair of critical values. The preevponential factor is given by an energy-weighted average of the two Slater frequencies for the individual reaction coordinates. The results of this work are applied t o cyclobutsne.
Introduction It was shown recently by Srinivasan and Kellner2 that cyclobutane decomposes unimolecularly to ethylene with no migration whatsoever of hydrogen atoms. This suggests rather strongly that the reaction coordinate for cyclobutane involves either (1) the breaking of a single C-C bond to yield a diradical which subsequently decomposes, or (2) the simultaneous breaking of two C-C bonds on opposite sides of the ring. Slater’s theory3 leads one to expect a high pressure limit of the pre-exponential factor much smaller than the observed4 value of 4 x l o L 5set.-' if the first coordinate mentioned above is appli~able.~Slater’s theory as it stands is not applicable to the simultaneous cotirdinates mentioned in case 2, and the effect on the pre-exponential factor of the requirement that two bonds be broken simultaneously was not immediately obvious to us. We shall therefore extend Slater’s theory for the high pressure limit of a unimolecular reaction to molecules in which two reaction coordinates must simultaneously each exceed some critical value. This mill permit the further testing of Slater’s harmonic classical model against the rather precise unimolecular rate data on cyclobutane a t the high pressure limit. It might be pointed out that the failure to detect radicals during the decomposition of cyclobutane does not eliminate the possibility of the diradical mechanism mentioned above, since the diradical might well possess an extremely short lifetime and therefore not be detected by the methods employed.‘j As remarked above. however, Slater’s theory applied to the diradical mechanism leads to unsatisfactory rewlts. There follows a brief resume of Slater’s theory and an expression of one of Slater’s results in terms of a Dirac &function; this is used in the following section on the extension of Slater’s theory to models involving simultaneous reaction coordinates. The mathematiral details of this last section are relegated to the Appendix. Slater’s Formulation.-Slater’s theory3 is based on the classical model of a vibrating polyatomic
molecule in which reaction is assumed to occur n-hen a particular internal coordinate (the reaction coordinate) attains a critical value pi*. If one approximates the kinetic and potential energy as quadratic forms, the vibrational motion of any coordinate qT may be expressed as
2 ff,,d/,
q, =
cos 2 r ( v , t
z=1
+ 6,)
(1)
where vi and are the frequency and energy of the ithnormal mode, and the aTL can be determined by a normal mode analysis. Using an expression ( L ( c Y ~ ~ , Efor ~ , vthe ~ ) ) frequency of zeros in a trigonometric sum of the type pr - q?, Slater formulates the high pressure rate constant as an average of L over a Boltxmann distribution in the normal mode energies. Thus
where the factor of one-half arises from the fact that only zeros occurring when dq,/dt is positive are considered physically meaningful. Slater’s final expression is K , = i;, e ~ p [ - - ( q , ~ ) * / ~ u ~ ~ k T l n D,2
=
ff,12v,=/ffr2
(3)
*=1
2=1
The quantity (q?) 2 / ~ r 2 represents the minimum energy necessary for p,. to reach p?. The identification of p. with a physically plausible reaction coordinate then determines the theoretical preexponential (or frequency) factor a t the highpressure limit. Kac’s formula can be derived with the aid of a Dirac 6 function,’ a device which me shall also find useful in the present work. Formally we may write for N , the number of times any bounded and continuous function q. attains a value q? during a given interval 0 6 t 6 T A-
(1) This work was supported In part b y a grant from the National Science Foundation. (2) R. Srinirasan and S Kellner, J . Am. Chem. Soc , a i , 5891 (1959). (3) N. B. Slater, Proc Roy. SOC.(London), 194A, 112 (1948). (1) C T Geiiauu, r Kern and W D. Walters, J . Am. Chem. SOC.,
S 6(9,
L:
- QTo)ldg,l
6 ( 5 ) = 0, 5 # 0
$1-
(4)
S(z)f(.)dz = S(0)
7 5 , 019G (1953) ( 5 ) E. Thiele and D J. Wilson, Can. J . Chem., 37, 1035 (1959). ( 6 ) We are 1ndebtt.d to Professor W-.D. Walters for a discussion on
( 7 ) The authors wish t o thank Professor Frank P. Buff for pointing out this physically meaningful, and mathematically straightforward method of deriving Kac’s formula. >I. Kao, Amer. J . Math.. 66, 609
this point.
(1943).
where the summation is over the branches of the multiplfwalued function t ( q r ) . Sinre q r ib a function of t, the frequency of zeros in the interval 0 t I' is given by
0 relative values of ql@and q20. In examining the validity of this expression let us As a further comparison of eq. 10 with Slater's consider two ways in which a critiral configuration expression, let us consider the speciai case of equivalent simultaneous reaction coordinates, as might be expected in the decomposition of cyclobutane. That is, q10 equals q20, B~ equals ~ 2 and , cy12 equals ( ~ 2 .Equation 10 then reduces to
