Writing qC in terms of the vibrational modes (qv) C [ which is approximated as kBT / h  ] and all the other remaining modes [  C], we have for qC, |
| qC = (kBT / h ) C
|
(33.26) |
| Combining (33.26), (33.23) and (33.4), we have, |
| |
|
k2 = k K = ( RT / p0) K = ( k BT / h )[ ( NA C / q A qB) e  RT / p0 ) |
(33.27) |
| |
= kB T / h
 |
(33.28) |
Where is the second, square bracketed expression in eq (33.27) which is akin to an equilibrium constant. There are often factors not included in (33.27) and they are included through a transmission coefficient
 and the rate constant in the transition state theory becomes, |
| |
k 2 =
 ( kBT / h ) |
(33.29) |
| We want to express in terms of molecular partition functions. Let us obtain a formula for simple species of A and B where A and B are atoms. The partition function of atoms is simply the translational partition function (as rotations and vibrations are absent). The translational partition function was given earlier in Lecture 29 as |
| |
q T = ( 2
 mkBT / h2 ) 3 / 2 V |
(33.30) |
| and using the molar volume, the partition function for A becomes |
| q T,A = ( 2 mA kB T / h 2 ) 3 / 2 V 0m |
= V 0m /
 ,
 =
h / ( 2 mA kBT)1 / 2 |
(33.31) |
| |
Where V 0m is the standard molar volume, given by RT / p0 and
 A is called the de Broglie thermal wavelength. For the activated complex, the partition function is the product of translational, vibrational and rotational partition functions, because in our present model, AB is a diatomic. However, we have already considered the vibrational partition function in (33.24) and we need to consider C now which includes only translation and rotation . |
| This is given by |
C = V 0m /
 .( 2 I kB T /  2 ) |
(33.32) |
| |
