Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

7.2 Applications of transition state theory 159
7.2.4 Example 3: Deviations from Arrhenius behavior (T dependence of k (T ))*
I
Exercise 7.2: The temperature dependence of bimolecular gas phase reactions of the
type A + B → productscanusuallybeexpressedintheform
Using the TST expression for ( )
( )= × × exp (−∆ 0 ) (7.55)
( )=
‡ µ
exp
− ∆ 0
, (7.56)
determine the values of for the types of reactions in Table 7.3, assuming that the
vibrational degrees of freedom are not excited. ¤
I Table 7.3:
A + B → TS ‡
transl. rot.
atom atom linear TS 1 32 1
+05
32 32 1
atom linear molecule linear TS 1 −32 1
−05
1
atom linear molecule nonlin. TS 1 −32 32
00
1
atom nonlin. molecule nonlin. TS 1 −32 32
−05
32
linear molecule linear molecule linear TS 1 −32 1
−15
1 1
linear molecule linear molecule nonlin. TS 1 −32 32
−10
1 1
linear molecule nonlin. molecule nonlin. TS 1 −32 32
−15
1 32
nonlin. molecule nonlin. molecule nonlin. TS 1 −32 32
−20
32
32
I Solution 7.2: See Table 7.3. ¥
Using the above results, we can recast ( ) and determine the following expression for
the Arrhenius activation energy, with as above:
= ∆ 0 + + 2 ln ‡
− 2
X
Reactants
ln reactants
(7.57)
We see that the value for depends on the vibrational frequencies of the TS and the
reactants. Often, the frequencies of the reactants are very high so that reactant
=1.
However, the TS may have soft vibrations so that ‡
may approach the classical value
=[1− exp (− )] −1 →
for →∞ (7.58)
Thus, each “soft” vibrational mode in the TS increases by 1, each “soft” vibrational
mode in the reactants decreases by 1.