Molecular Translational Partition Function

To evaluate i trans, we assume that the molecule acts as a particle in a three-dimensional cubic box of dimension a where a is the length of one side of the cube. The energy levels for this elementary quantum mechanical system are given by 

Evaluating Eqs. (10.3) and (10.5) for a molar quantity of particles using Eq. (10.15) for the translational partition function gives 

Note that, because we are working under the assumption of an ideal gas, V /N in the term in brackets can be replaced by k T/P.  

A noteworthy aspect of Eqs. (10.16) and (10.18) is that they are altogether free of any requirement to carry out an electronic structure calculation. Equation (10.16) is well known for an ideal gas and is entirely independent of the molecule in question, and Eq. (10.18) can be computed trivially as soon as the molecular weight is specified. Note, however, that the units chosen for the various quantities must be such that the argument of the logarithm in Eq. (10.18) (i.e., the partition function), is unitless. 

The translational partition function is a function of both temperature and volume. However, none of the other partition functions have a volume dependence. It is thus convenient to eliminate the volume dependence of 5trans by agreeing to report values that use exclusively some volume that has been agreed upon by convention. The choices of the numerical value of V and its associated units define a standard state (or, more accurately, they contribute to an overall definition that may be considerably more detailed, as described further below). The most typical standard state used in theoretical calculations of entropies of translation is the volume occupied by one mole of ideal gas at 298 K and 1 atm pressure, namely, y° = 24.5 L. 

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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... 

The standard entropies of monatomic gases are largely determined by the translational partition function, and since dris involves the logarithm of the molecular weight of the gas, it is not surprising that the entropy, which is related to tire translational partition function by the Sackur-Tetrode equation,... 

Note that the 1 /N term is assigned to the translational partition function, since all gases have translational motion, but only molecular gases have rotational and vibrational degrees of freedom. The electronic partition function is usually equal to one unless unpaired electrons are present in the atom or molecule. 

This term is calculated in a completely equivalent way to that used for in vacuo calculations. The only additional consideration is that the vibrational frequencies and molecular geometries necessary for the calculation of the vibrational, rotational and translational partition functions of the solute are calculated in solution. 

For p we use Eq. (38) of Chapter 5 with U0 as the desorption energy and the molecular partition function written as a product of a translational partition function [Eq. (72) of Chapter 5] and qint, which accounts for the internal degrees of freedom of the molecule ... 

For a molecular entity freely moving in the potential box of the size am the translational partition function is ... 

The treatment of units is here the same as that employed in connection with the translational partition function in the problem in 16e m, h and h are in c.g.s. units, and if P is in atm., R in the In R term should conveniently be in cc.-atm. deg. mole The actual weight m of the molecule may be replaced by M/N, where M is the ordinary molecular weight. Making these substitutions and converting the logarithms, it is found that... 

Calculate the translational partition function and entropy of one mole of xenon at 1 atm pressure and 298 K. The relative molecular mass of xenon is 131,30. 

The derived density of states for the translations, rotations, and vibrations can be used in Eq. (6.41) to obtain the corresponding classical partition functions. This will yield an accurate translational partition function at all temperatures of chemical interest because the translational energy level spacings are so dense. It will also yield accurate rotational partition functions at room temperature because molecular rotational constants are typically between 0.01 and 1 cm k However, at the low temperatures achieved in molecular beams, the accuracy of the classical rotational partition function (especially for molecules with high rotational constants, such as formaldehyde or H2 (Bg = 60.8 cm )) is insufficient. The energy level spacing of vibrations (ca. 2000 cm ) are considerably larger than the room temperature of 207 cm " so that even at room temperature, the vibrational partition function must be evaluated by summation in Eq. (6.40). 

In the molecular-motion contribution, the molecular partition function is the product of translational, rotational, vibrational, and electronic partition functions. If the molecule in solution is assumed to have the entire volume of the solution available to it, the ratio of gas-phase and solution-phase translational partition functions equals one. Likewise, the electronic partition function ratio will be one. It is unclear what one should use for the rotational partition function in solution, but if this is assumed to have the same form as that in the gas phase, the rotational partition function ratio (which involves the moments of inertia) will be very close to one, since structural changes from gas to solution are slight. Significant contributions to the vibrational partition function are made only by the low-frequency vibrational normal modes, and these modes sometimes show substantial changes in frequency on going from the gas phase to solution. If a vibrational calculation is done in the gas phase and in solutitm, one can calculate AG°oiv m, but the most common procedure is to omit it, assuming that its contribution is negligible. 

The translational partition function for a molecular confined within volume V is given by Eq. (A27). If we assume this to be the partition function for an ideal gas molecule with v - K/A we derive the correct values for many of the thermodynamic properties with the notable exception of the entropy. The entropy, however, is lower by k per molecule. This discrepancy arises because the gas is not really a system in which the molecules can properly be... 

In this equation k is called the transmission coefficient and is taken to be equal to unity in simple transition-state theory calculations, but is greater than imity when tunneling is important (see below), c° is the inverse of the reference volume assumed in calculating the translational partition function (see below), m is the molecularity of the reaction (ie, m = 1 for unimolecular, 2 for bimolecular, and so on), is Boltzmann s constant (1.380658 x 10 J molecule K ), h is Planck s constant (6.6260755 x 10 J s), Eq (commonly referred to as the reaction barrier) is the energy difference between the transition structure and the reactants (in their respective equiUbriiun geometries), Qj is the molecular partition function of the transition state, and Qi is the molecular partition function of reactant i. 

In the previous section we obtained a general formula for the translational partition function. In this section we obtain formulas for the other factors in the molecular partition function for dilute gases and carry out example calculations of partition functions. 

Wehave already determined that the molecular partition function for a dilute monatomic gas is the product of a translational partition function and an electronic partition function. We obtained a formula for the translational partition function in Eq. (25.3-21) ... 

In the harmonic oscillator-rigid rotor approximation polyatomic molecules obey the same separation of their energy into four independent terms as in Eq. (25.4-5). In this approximation the molecular partition function of a polyatomic substance factors into the same four factors as in Eq. (25.4-6). The translational partition function is given by... 

As previously stated, the classical molecular partition function has units of kg s raised to some power, so a divisor with units must be included to make the argument of the logarithm dimensionless. If a divisor of lkgm s is used, values are obtained for the entropy and the Helmholtz energy that differ from the experimental values. However, when the classical canonical translational partition function is divided by h A and Stirling s approximation is used for ln(iV ), the same formulas are obtained as Chapter 26. For a dilute monatomic gas the corrected classical formula is... 

The energy states associated with intermolecular translation and rotation are not only numerous, but also so irregularly spaced that it is impossible to derive them directly from molecular quantities. It is consequently not possible to construct the partition function explicitly. Nevertheless, we may derive formal expressions for U and A from eqs. (16.1) and (16.2). 

But molecular gases also have rotation and vibration. We only make the correction for indistinguishability once. Thus, we do not divide by IV l to write the relationship between Zro[, the rotational partition function of N molecules, and rrol, the rotational partition function for an individual molecule, if we have already assigned the /N term to the translation. The same is true for the relationship between Zv,h and In general, we write for the total partition function Z for N units... 

We now have equations for the partition functions for the ideal gas and equations for relating the partition functions to the thermodynamic properties. We are ready to derive the equations for calculating the thermodynamic properties from the molecular parameters. As an example, let us calculate Um - t/o.m for the translational motion of the ideal gas. We start with... 

Table 6.2 Forms for translational, rotational, and vibrational contributions to the molecular partition function...

The result (Equation 4.90) could have been derived more simply. It has been emphasized that the quantum mechanical contribution to the partition function ratio arises from the quantization of vibrational energy levels. For the molecular translations and rotations quantization has been ignored because the spacing of translational and rotational energy levels is so close as to be essentially continuous (As/kT 1). 

The ratio of symmetry numbers s s° in equation 11.40 merely represents the relative probabilities of forming symmetrical and unsymmetrical molecules, and ni and nf are the masses of exchanging molecules (the translational contribution to the partition function ratio is at all T equal to the power ratio of the inverse molecular weight). Denoting as AX, the vibrational frequency shift from isotopically heavy to light molecules (i.e., AX, = X° — X ) and assuming AX, to be intrinsically positive, equation 11.40 can be transated into... 

Barrer (3) makes similar calculations for the entropies of occlusion of substances by zeolites and reaches the conclusion that the adsorbed material is devoid of translational freedom. However, he uses a volume, area or length of unity when considering the partition function for translation of the adsorbed molecules in the cases where they are assumed to be capable of translation in three, two or one dimensions. His entropies are given for the standard state of 6 = 0.5, and the volume, area or length associated with the space available to the adsorbed molecules should be of molecular dimensions, v = 125 X 10-24 cc., a = 25 X 10-16 cm.2 and l = 5 X 10-8 cm. When these values are introduced into his calculations the entropies in column four of Table II of his paper come much closer together, as is shown in Table I. The experimental values for different substances range from zero to —7 cals./deg. mole or entropy units, and so further examination is required in each case to decide... 

Some important systems, which certainly do not fulfill the assumptions of harmonic transition state theory are gas phase reactions. In the gas phase, there are zero-modes such as translation and rotation, and these lead to totally different configuration integrals than those obtained from a normal mode analysis. For these species one can in a simple manner modify the terms going into the HTST rate by incorporating the molecular partition functions [3,119]. 

We have thus reduced the problem from finding the ensemble partition function Q to finding the molecular partition function q. In order to make further progress, we assume that the molecular energy e can be expressed as a separable sum of electronic, translational, rotational, and vibrational terms, i.e.,... 

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