Partition function per unit volume

As an example, evaluate the molecular translational partition function per unit volume for Ar atoms at 1000 K. The mass of one Ar atom is 6.634 x 10-26 kg. So the translational partition function per unit volume is... 

Give the numerical values for the transition state, HC1 and Cl partition functions (per unit volume) at 300 K. 

Calculation of the partition functions for reactants is straightforward, but the partition function for the activated complex needs explanation. The activated complex has been shown to have the unique feature of a free translation along the reaction coordinate over the distance occupied by the activated complex. The statistical mechanical quantity for this free translation has already been factorized out from the total partition function for the activated complex in the derivation. This has been done simply because doing so allows cancellation of some awkward terms in the derivation of the rate constant equation. This is why the symbol has appeared along with the symbol f, this latter indicating that the process is one of forming the activated complex, often very loosely termed activation. Q/ is now a partition function per unit volume for the activated complex but with one crucial term missing from it, i.e. the term for the free translation. This is more fully explained in the section below. 

The TST was developed originally by Eyring and others on the basis of statistical mechanics [see, e.g., Lasaga (1983) or Moore and Pearson (1981)]. The fundamental result is a bimolecular rate constant for an elementary process expressed in terms of (1) the total molecular partition functions per unit volume iqi) for reactant species and for the activated complex species (q ), and (2) tlie difference in zero-point potential energies between the activated complex and reactants (Eq) ... 

Consequently the evaluation of the molar Gibbs free energy of formation reduces to the evaluation of q(n, T). The quantity q(n, T) has units of the partition function per unit volume. 

The second factor here, which will be denoted by is the reciprocal of the translational partition function per unit volume for a molecule of mass (see Section A.2.3). Since the moment of inertia of a system composed of the masses and m2 separated by a distance d such that nd = ais i = 1 12 12 /ti2 s + m2), the last factor in the above expression,... 

A completely analogous derivation leads to the rate coefficient for bimolecular reactions, where /are partition functions per unit volume ... 

The partition function of the system Q is related to the molecular partition function of the individual molecules in the system. In our development of rate constants we make use of the molecular partition functions. The molecular partition function per unit volume for an ideal gas is the product of the translational, rotational, vibrational and electronic energy states in the molecule... 

This equation says that is proportional to the proper quotient of partition functions per unit volume. As usual, we have expressed the equilibrium constant as a product of dimensionless ratios. 

The values of v and can be calculated if we write the equilibrium constant in terms of molecular partition functions per unit volume, qjV. [See Eq. (29.75).] Then... 

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