Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...
Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ... Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...
7.1 Foundations of transition state theory 151 • −→ ‡ = −→ ‡ is the mean speed of the molecules crossing the DS in the forward direction. (6) We now use the equilibrium constant to determine ‡ via ‡ = ‡ ABC A BC (7.18) y ‡ = ‡ ABC = ‡ A BC (7.19) y −→ −→ ‡ ‡ = ‡ A BC = A BC (7.20) with −→ ‡ = ‡ (7.21) (7) From statistical thermodynamics (see Appendix G), we have the following expression for the equilibrium constant ‡ ( ) in terms of the molecular partition functions: ‡ ( )= ∗ = µ exp − ∆ 0 (7.22) • The ’s are the respective molecular partition functions (per unit volume). • ∗ is written with respect to the zero-point level of the reactants. • The exp (−∆ 0 ) factor arises subsequently, because we now express the value of partition function for the TS relative to the zero-point level of the TS ( ∗ = exp (−∆ 0 )). (8) Separating the motion along the reaction coordinate ‡ from the other degrees of freedom, we write = ‡ ‡ (7.23) where ‡ is the 1-D (translational) partition function describing the translational motion of the molecules along ‡ across the TS and ‡ is the partition function for all 3 − 7 other (rotational-vibrational-electronic) degrees of freedom. Here, • the 1-D translational partition function is ‡ = µ 2 2 12 × (7.24) • the mean speed along ‡ for crossing the DS in the forward direction is given by R −→ ∞ ‡ −2 2 µ 12 0 = R ∞ −∞ −2 2 = (7.25) 2
7.1 Foundations of transition state theory 152 (9) Collecting and inserting the results into −→ ‡ ( )= ( ) (7.26) we obtain ( )= 1 µ 12 µ 12 µ 2 × × ‡ exp − ∆ 0 (7.27) 2 2 y ( )= ‡ µ exp − ∆ 0 (7.28) I Result: In order to account () forsomerecrossingoftheTSand() for quantum mechanical tunneling, we add an additional factor called the transmission coefficient. For well-behaved reactions, ≈ 1. Thus, we end up with the fundamental equation of TST in the form: ( )= ‡ µ exp − ∆ 0 Note that ‡ is the partition function of the TS excluding the RC. Equation 7.29 allows us to calculate ( ) using the following data: (7.29) • The threshold energy for the reaction ∆ 0 (or, per mol, ∆ 0 ). ∆ 0 has to be obtained from ab initio quantum chemistry calculations, and • structural information on the reactants and the TS (i.e., bond lengths and angles, vibrational frequencies). I Figure 7.4: Illustration of ∆ ‡ , ∆ 0 ,and .
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7.1 Foundations of transition state theory 152<br />
(9) Collecting and inserting the results into<br />
−→<br />
<br />
‡<br />
( )=<br />
( ) (7.26)<br />
we obtain<br />
( )= 1 µ 12 µ 12 µ<br />
2 <br />
×<br />
× ‡ <br />
exp − ∆ <br />
0<br />
(7.27)<br />
2<br />
2<br />
<br />
y<br />
( )= ‡ µ<br />
<br />
exp − ∆ <br />
0<br />
(7.28)<br />
<br />
I Result: In order to account () forsomerecrossingoftheTSand() for quantum<br />
mechanical tunneling, we add an additional factor called the transmission coefficient.<br />
For well-behaved reactions, ≈ 1. Thus, we end up with the fundamental equation<br />
of TST in the form:<br />
( )= <br />
<br />
‡ µ<br />
<br />
exp − ∆ <br />
0<br />
<br />
Note that ‡ <br />
is the partition function of the TS excluding the RC.<br />
Equation 7.29 allows us to calculate ( ) using the following data:<br />
(7.29)<br />
• The threshold energy for the reaction ∆ 0 (or, per mol, ∆ 0 ). ∆ 0 has to be<br />
obtained from ab initio quantum chemistry calculations, and<br />
• structural information on the reactants and the TS (i.e., bond lengths and angles,<br />
vibrational frequencies).<br />
I Figure 7.4: Illustration of ∆ ‡ , ∆ 0 ,and .
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