Henry Eyring. The Activated Complex in Chemi.pdf


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The Activated Complex in Chemical Reactions

UNIMOLECULAR REACTIONS
Suppose we have a non-linear molecule of n atoms decomposing unimolecularly. We then
write, cancelling out factors common to the initial and activated states:

1

k8 = c? σ/σ ? (A? B ? C ? /ABC) 3

3n−7
Y

−1

(1 − exp (−hvi? /kT ))

(8)

i=1

·

3n−6
Y

(1 − exp (−hvi /kT ))

i=1

· (kT /h) exp (−E0 /kT ) .
Quantities referring to the activated state in equation (8) are starred. Now in the particular
case where hvi  kT , i.e., all vibrational degrees of freedom approach a classical behavior
−1
we have (1 − exp (−hvi /kT )) = kT /hvi ; and equation (7) takes the form:
k9 = c? (σ/σ ? )

3n−6
Y 3n−7
Y
i=1

−1

(vi? )

1

− 31

(A? B ? C ? ) 3 (ABC)

exp (−E0 /kT ) .

(9)

i=1

c? has the same meaning as the c defined in connection with equation (2). We of course
come to this same result (9) directly if we integrate the appropriate classical expressions for
vibration over phase space. Thus for each vibrational degree of freedom:
Z ∞
Z ∞


2
(1/h) ·
exp −p / (2m1 kT ) dpi ·
exp −fi gi2 / (2kT ) dqi = kT /hvi
−∞

−∞
1

1
2
2 π (fi /mi ) .

if we use the relationship vi =
In using (8) it must be remembered that for
certain molecules some of the degrees of freedom treated as vibrations can better be treated
as internal rotations. In any particular case there is no particular difficulty in doing this.
equation (9) is sufficiently near to that found for unimolecular reactions at high pressures that
there seems no doubt of the wide applicability of both equation (8) and (9). A formula very
similar to equation (9) was obtained by an approximate method in a paper by Polanyi and
Wigner.10

10

M. Polanyi and E. Wigner, Z. Phys. Chem. A (Haber Band), 439 (1928).

10