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

7.3 Thermodynamic interpretation of transition state theory 167
Kinetic isotope effects occur in particular in the case of H/D atom transfer reactions.
The reason is that the threshold energy ∆ ‡ 0 depends on the vibrational zero-point
energies.
We consider the reactions
where
A+H—R → A—H—R ‡ → products (7.117)
A+D—R → A—D—R ‡ → products (7.118)
∆ ‡ 0 (H—R) = ∆ ‡ + ‡ (A—H—R) − (H—R) (7.119)
∆ 0(D—R) ‡ = ∆ ‡ + ‡ (A—D—R) − (D—R) (7.120)
with
y
=
1
2
X
(7.121)
∆ ‡ 0(DH) = ∆‡ 0(D—R) − ∆ 0(H—R) ‡ (7.122)
=+ ‡ (A—D—R) − ‡ (A—H—R) − ( (D—R) − (H—R))
(7.123)
y
à !
(D—R)
(H—R) =exp − ∆‡ 0(D/H)
(7.124)
I
Example 7.2: R-H/R-D substitution. The vibrational frequencies depend on the force
constants and reduced masses :
= 1 µ 12
(7.125)
2
For H/D substitution:
(HR)
(DR) ≈ 1 (7.126)
2
y
µ 12
(HR) (HR)
(DR) = ≈ √ 2 (7.127)
(DR)
With (HR) = 3000 cm −1 and (DR) = 2100 cm −1 , we find ∆ =
1
2 (3000 − 2100) cm−1 =450cm −1 , by which the DR reactant is stabilized compared
to the HR reactant. For =300K:
µ
(DR)
(HR) =exp − 450
≈ −225 ≈ 01 (7.128)
200
For more realistic estimates, one has to take into account the change of all vibrational
frequencies in the reacting molecules and the TS.
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