Introduction of the Partition Functions

Let us now put all of this together to obtain a numerical value of the rate constant k for the chemical reaction. Note that chemists always use the symbol k for the rate constant. Elsewhere in this chapter and in other chapters, we also use k for Boltzmann s constant. Whenever there is a possibility of confusion, we will use kB for Boltzmann s constant. The rate constant is defined by the relation 

Returning to Equation 4.136 we see the isotope effect on the rate constant is then given by 

The partition function ratios needed for the calculation of the isotope effect on the equilibrium constant K will be calculated, as before, in the harmonic-oscillator-rigid-rotor approximation for both reactants and transition states. One obtains in terms of molecular partition functions q 

Equation 4.139 has been written for the case where the isotopic substitution is on reactant molecule A only. Therefore the qs ratio in the numerator cancels. The partition function ratio qA2/qAi in the numerator can be replaced by isotopic ratios of translational, rotational, and vibrational partition functions as in Equation 4.76. However, in the denominator one has to be careful to remember that the isotopic partition ratio involves the q functions which contain only 3N -7 (for a linear 

Equation 4.140 is the exact analogue of Equation 4.76 for a stable molecule except there is one less vibrational degree of freedom. It must now be noted that the derivation of the Teller-Redlich Product Rule, applies equally well to a transition state as to a stable molecule. Thus, when the Teller-Redlich Product rule is introduced into the expression for q2/qi of a transition state, the ratio of vibrational frequencies includes the isotopic ratio of the imaginary frequencies in the transition state. One can then write for transition state isotopic ratios, analogously to Equation 4.78 

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