The translational partition function

The summations are over all available translational or internal energy levels and q(tr) and q(int) are the translational and internal partition functions respectively. 

The validity of the last expression is limited by the fact that the rotational and vibrational motions are not independent the centrifugal force from the rotational motion stretches the molecule, thus effecting its vibrational motion and the change in nuclear distances due to vibration, effects the moment of inertia and thus the rotation. 

Evaluation of the internal p.f. is rather complex, involving a combination of spectroscopic measurements and theory (see, for example, Reed and Gubbins), and will not be considered here. On the other hand, evaluation of the translational p.f. is simpler and is presented next. 

Consider an ideal gas molecule moving inside a box, which is in the origin of the coordinates system with dimensions a, b and c along the x, y, and z axes respectively (Section 16.4). According to the separability of the translational energy, Eq. 16.4.2, and following the reasoning of the previous Section  

To evaluate this summation, we note that use of reasonable values of T and a gives very small values for g. This, in turn, leads to terms that, starting with a value of practically ual to one - for = 1, decrease extremely slowly. We can replace thus this summation with the integral 00 

Let it be supposed that the gas is confined in a rectangular box whose sides have lengths a, b and c parallel to the a , y and z axes respectively. In accordance with the probability interpretation (equation 12 11), must therefore be zero at aU values of x which lie outside the range of length a. This is because we specify that the molecules are certainly inside the box. Thus xjrg is zero for all values of X less than zero and greater than a, and it must approach these zero values quite smoothly within the box, as otherwise would not be a continuous function and the probability interpretation 

It will be noticed from (12 37) that to each possible value of the energy e, there is only a single choice of the quantum number g. The levels are therefore not degenerate and the same applies also to y and g. The corresponding eigenfunctions are obtained by substituting (12 36) and (12 37) in (12 35)  

On the other hand, the total translational energy is given by 

Thus if a, b and c are commensurate, it will occur that many of the energy levels e are degenerate. 

The translational part of the molecular partition function is given by (12-26) /to =SwJ exp[-eJV 5T]. 

---

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,... 

Equations (5-38) and (5-40) can also be derived by making use of the translational partition function. 

We assign the 1 /N term to the translational partition function because all gases have translational motion, so that... 

This is the translational partition function for any particle of mass m, moving over a line of length I, in one dimension. Please note that this result is exactly the same as that we calculated from quantum mechanics for a particle in a one-dimensional box. For a particle moving over an area A on a surface, the partition function of translation is... 

Hence, we conclude that the translational partition function of a particle depends on its mass, the temperature, the dimensionality as 3vell as the dimensions of the space in vhich it moves. As a result, translational partition functions may be large numbers. The translational partition function is conveniently calculated per volume, which is the quantity used, for example, when the equilibrium conditions are determined, as we shall see later. The partition function can conveniently be written as... 

Since qi are proportional to V (this stems from the translational partition function), (ji/Vonly depends on T. Nevertheless, if... 

The translational partition functions at T = 1000 K are easily calculated for both... 

Most students are introduced to quantum mechanics with the study of the famous problem of the particle in a box. While this problem is introduced primarily for pedagogical reasons, it has nevertheless some important applications. In particular, it is the basis for the derivation of the translational partition function for a gas (Section 10.8.1) and is employed as a model for certain problems in solid-state physics. 

Consider a bimolecular reaction, A + B -> products. Confining two molecules A and B to be together in the transition state in a bimolecular reaction always produces a loss of entropy. This is dominated by the ratio of the translational partition functions ... 

The translational partition function Qtnms is equal to the classical one at all temperatures... 

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

The superscript o indicates that a standard state is being referred to (see below).) Conventionally, however, the last two terms in the final line of Eq. (10.9), i.e., those deriving from Stirling s approximation, are typically assigned to the translational partition function as well. As they have no temperature dependence, this has no impact on f/transl however, the entropy of translation becomes... 

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. 

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 translational partition function trans is evaluated in the high-7" continuum limit as... 

The simplest QCE model incorporates environmental effects of cluster-cluster interactions by (1) approximate evaluation of the excluded-volume effect on the translational partition function >trans (neglected in Section 13.3.3) and (2) explicit inclusion of a correction A oenv) for environmental interactions in the electronic partition function qiQiec. Secondary environmental corrections on rotational and vibrational partition functions may also be considered, but are beyond the scope of the present treatment. 

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. 

We could have restricted our discussion to positive velocities. The average velocity would then be twice as big however, the translational partition function would then take only half of the value given in Eq. (6.3). That is, the final result is unchanged. 

The ratio involving the translational partition functions is easily evaluated ... 

We will now calculate the rate constant at T = 300 K using Eq. (6.9) and the data in Table 6.2. The relevant ratio between the translational partition functions is... 

For the translational partition function, we first consider a particle in a onedimensional box of length /. The energy levels, with the zero of energy as the zero-point level, are En = (n2 — l)h2/(8ml2), with n = 1,2,..., and degeneracy wn = 1. The partition function takes the form... 

It is noted that the right-hand side is the ratio of the translational partition functions of products and reactants times the Boltzmann factor for the internal energy change. In the derivation of this expression we have only used that the translational degrees of freedom have been equilibrated at T through the use of the Maxwell-Boltzmann velocity distribution. No assumption about the internal degrees of freedom has been used, so they may or may not be equilibrated at the temperature T. The quantity K(fhl, ij) may therefore be considered as a partial equilibrium constant for reactions in which the reactants and products are in translational but not necessarily internal equilibrium. 

The second situation is where the adsorption is delocalized the molecule can move in the neighborhood of the site, and the partition function for the movement in the z-direction in the adsorption site is the translational partition function [109] ... 

The diffusion coefficients for mobile adsorption can also be calculated by introducing in Equation 5.56 the expression for x, described in Equation 5.55, and using Equation 5.103 for Z, the translational partition function ... 

PROBLEM 5.3.9. Show that in a CE the translation partition function for a ... 

---