Abstract
The activated behavior of the reaction rate can be understood from a simple model of two colliding atoms. In this chapter, we discuss the connection to transition state theory, which takes into account the internal degrees of freedom of larger molecules and explains not only the activation energy, but also the prefactor of the Arrhenius law. We formulate transition state theory in a thermodynamic context and discuss kinetic isotope effects. Finally, we present quite general rate expressions based on the flux over a saddle point.
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Notes
- 1.
There is a factor of two since \(v^{2}=(-v)^{2}\).
- 2.
The quantity \(\pi d^{2}/4\) is equivalent to the collision cross section used in connection with nuclear reactions.
- 3.
3N-5 for a collinear molecule.
- 4.
The activated complex is treated as a normal molecule with the exception of the special mode.
- 5.
Asymptotically, the states on the product side will leave the reaction zone with positive momentum \(p_{r}\), so that we may replace \(\theta (q_{r})\) with \(\theta (p_{r})\). Again asymptotically these states are eigenfunctions of the Hamiltonian, so that \(P(t=\infty )\) and \(\exp (-\beta H)\) commute.
- 6.
This follows from time inversion symmetry: The trace has to be symmetric with respect to that operation. Time inversion changes \(p_{r}\rightarrow -p_{r}\). The operators \(\rho \) and \(q_{r}\) are not affected. So the trace has to be symmetric and antisymmetric at the same time.
- 7.
The particle moves to the left or right side with equal probability.
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15.1
Transition State Theory
Instead of using a vibrational partition function to describe the motion of the activated complex over the reaction barrier, we can also use a translational partition function. We consider all complexes lying within a distance \(\delta x\) of the barrier to be activated complexes. Use the translational partition function for a particle of mass m in a box of length \(\delta x\) to obtain the TST rate constant. Assume that the average velocity of the particles moving over the barrier isFootnote 7
15.2
Harmonic Transition State Theory
For systems such as solids, which are well described as harmonic oscillators around stationary points, the harmonic form of TST is often a good approximation, which can be used to evaluate the TST rate constant, which can then be written as the product of the probability of finding the system in the transition state and the average velocity at the transition state.
For a one-dimensional model assume that the transition state is an exit point from the parabola at some position \(x=x^{\ddagger }\) with energy \(\varDelta E=m\omega ^{2}x^{\ddagger 2}/2\) and evaluate the TST rate constant.
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Scherer, P.O.J., Fischer, S.F. (2017). Calculation of Reaction Rates. In: Theoretical Molecular Biophysics. Biological and Medical Physics, Biomedical Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55671-9_15
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DOI: https://doi.org/10.1007/978-3-662-55671-9_15
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