Partition function of translation

It is instructive to compare Eq. (22) vith the situation of a particle in a box, because this automatically yields a useful expression for the partition function of translation (Fig. 3.3). 

Starting with the partition function of translation, consider a particle of mass m moving in one dimension x over a line of length I with velocity v. Its momentum Px = mVx and its kinetic energy = Pxllm. The coordinates available for the particle X, px in phase space can be divided into small cells each of size h, which is Planck s constant. Since the division is so incredibly small we can replace the sum with integration over phase space in x and Px, and so calculate the partition function. By normalizing with the size of the cell h the expression becomes... 

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... 

Statistical thermod5rnatnics enables us to express the entropy as a function of the canonical partition function Zc (relation [A2.39], see Appendix 2). This partition function is expressed by relation [A2.36], on the basis of the molecular partition functions. These molecular partition functions are expressed, in relation [A2.21], by the partition functions of translation, vibration and rotation. These are calculated on the basis of the molecule mass and relation [A2.26] for a perfect gas, the vibration frequencies (relation [A2.30]) of its bonds and of its moments of inertia (expression [A2.29]). These data are determined by stud5dng the spectra of the molecules - particularly the absorption spectra in the iirffared. Hence, at least for simple molecules, we are able to calculate an absolute value for the entropy - i.e. with no frame of reference, and in particular without the aid of Planck s hypothesis. 

The entropy consists, like the partition function, of translational, rotational, vibrational and electronic contributions ... 

Similarly, for a binary mixture of two species A and B, we can write the canonical partition function of translation in the form ... 

It is demonstrated that for one degree of freedom of translation, the partition function of translation of a molecule of mass m per unit of volume is ... 

If the molecule has n degrees of freedom of translation and if the volume V is available for the system, the partition function of translation is given by ... 

In calculating the isotope effect, Bigeleisen considers all partition functions of translation and rotation, vibrational excitation, and vibrational zero-point energies. Complete analysis of a kinetic isotope effect may be carried out if the frequencies of all (3N-6) modes of the reagents and (3N-7) modes of the transition state are known (in the transition state, the stretching of the bond being broken is not a true vibration, but an anomalous one, with an imaginary frequency v ). 

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... 

Partition functions are very important in estimating equilibrium constants and rate constants in elementary reaction steps. Therefore, we shall take a closer look at the partition functions of atoms and molecules. Motion, or translation, is the only degree of freedom that atoms have. Molecules also possess internal degrees of freedom, namely vibration and rotation. 

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... 

Under ivhat conditions are the partition functions for translation, rotation, and vibration of an adsorbed molecule (a) dose to unity, (b) moderate, and (c) large ... 

The first term on the right is the formula for the chemical potential of component a at density pa = na/V in an ideal gas, as would be the case if interactions between molecules were negligible, fee is Boltzmann s constant, and V is the volume of the solution. The other parameters in that ideal contribution are properties of the isolated molecule of type a, and depend on the thermodynamic state only through T. Specifically, V/A is the translational contribution to the partition function of single a molecule at temperature T in a volume V... 

For translation motion, the partition functions of each degree of freedom is given by... 

In Equation 4.71 the individual qvib s have been specified qVib(v ) to indicate that these partition functions depend on the normal mode frequencies. It is interesting to note that the partition function for translation, which is usually considered in terms of the problem of the particle in a three dimensional rectangular box, is, itself a product of three partition functions one for motion in the x dimension, one for y, etc. 

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... 

We see that the partition function of a molecule is the product of the contributions of the translational, rotational, vibrational, and electronic partition functions, which we can calculate separately, as discussed next. We will see in Section 8.5 that any thermodynamic quantity of interest can be derived from the molecular partition function. Thus it is important to be able to evaluate q. 

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. 

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