Page 162 - Introduction to chemical reaction engineering and kinetics

144 Chapter 6: Fundamentals of Reaction Rates

Table 6.2 Forms for translational, rotational, and vibrational contributions to the molecular
partition function
Mode Partition function Model
Q./V = translational (2mnkBTlh2)3’2 particle of mass m in 3D box of
(per unit volume) volume V, increases if mass increases
Qmt = rotational (8r2ZkBTlh2)“2 rigid rotating body with moment of
inertia I per mode; increases if
moment of inertia increases
& = vibrational (1 - exp(-hcvlksT))-’ harmonic vibrator with frequency v
per mode; increases if frequency
decreases (force constant decreases)

Qvib = Qvib, mode lQvib, mode 2 . . .

with a factor for each normal mode of vibration. The A factor can then be evaluated by
calculating the individual ratios. For the translational, rotational, and vibrational modes
of molecular energy, the results obtained from simplified models for the contributions
to the molecular partition function are shown in Table 6.2.
Generally, Q,, > Qrot > Qvib, reflecting the decreasing freedom of movement in the
modes. Evaluating the partition functions for the reactants is relatively straightforward,
since the molecular properties (and the related thermodynamic properties) can be mea-
sured. The same parameters for the transition state are not available, except in a few
simple systems where the full potential energy surface has been calculated. The prob-
lem is simplified by noting that if a mode is unchanged in forming the transition state,
the ratio for that mode is equal to 1. Therefore, only the modes that change need to be
considered in calculating the ratio. The following two examples illustrate how estimates
of rate constants are made, for unimolecular and bimolecular reactions.

For the unimolecular reaction in Example 6-4, C,H,Cl -+ HCl + C,H,, the transition state
should resemble the configuration below, with the C-Cl and C-H bonds almost broken, and
HCl almost formed:

The ratio of translational partition functions (Qi,./Q,,) is 1 here, and for all unimolecular
reactions, because the mass and number of molecules of the reactants is the same as for
the transition state. The rotational ratio (Q&,/Q,,,) is given by the ratio of the moments
of inertia: (Z~Z~ZjlZlZ2Z,)1’2. The moments of inertia are probably slightly higher in the