The Partition Function of an Ideal Gas

The theoretical entropy is calculated from the partition function of an ideal gas... 

The problem, thus, of evaluating the partition function of an ideal gas has been reduced to that of determining the molecular partition function. We will see next that this is further simplified through the factoring of the molecular partition function. 

In an ideal gas there are no interactions between the particles and so the potential ener function, 1 ), equals zero. exp(- f (r )/fcBT) is therefore equal to 1 for every gas partic in the system. The integral of 1 over the coordinates of each atom is equal to the volume, ai so for N ideal gas particles the configurational integral is given by (V = volume). T1 leads to the following result for the canonical partition function of an ideal gas ... 

Expressions for the partition function can be obtained for each type of energy level in an atom or molecule. These relationships can then be used to derive equations for calculating the thermodynamic functions of an ideal gas. Table 11.4 or Table A6.1 in Appendix 6 summarize the equations for calculating the translational, rotational, and vibrational contributions to the thermodynamic functions, assuming the molecule is a rigid rotator and harmonic oscillator.yy Moments of inertia and fundamental vibrational frequencies for a number of molecules are given in Tables A6.2 to A6.4 of Appendix 6. From these values, the thermodynamic functions can be calculated with the aid of Table 11.4. 

But if Qn is the partition function for an ideal gas of N identical molecules, we have the relation... 

It is useful to keep the classical way of expressing the partition function and extend its application to more complex situations. First of all, one may write an expression for the canonical partition function of an ideal gas in terms of the partition function for each molecule. On the basis of the total Hamiltonian for each molecule (equation (2.2.18)), Hj, the canonical partition function is... 

In the statistical-mechanical evaluation of the molecular partition function of an ideal gas, the translational energy levels of each gas molecule are taken to be the levels of a particle in a three-dimensional rectangular box see Levine, Physical Chemistry, Sections 22.6 and 22.7. 

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... 

Thus far, we have taken only formal steps of rewriting the same quantity Q in a new form. The latter suggests, however, a new point of view. We notice that (5.75) has the form of a partition function of an ideal gas mixture of two components A and B, where and Nb are the number of A and B molecules, respectively. 

To derive the partition function for an ideal gas as defined in Equation 5.13 we should know the discrete spectrum, at least at a low energy. Imagine the molecules to be moving freely in a cube with the side equal to L. In this three-dimensional box model, the potential is equal to zero inside the cube and infinitely large outside the cube. Since P = 0 outside the cube, all wave functions tend to zero at the edge of the cube. Inside the cube, the Schrddinger equation (SE) is... 

Here /i j3 is the chemical potential of the ideal gas at the standard pressure. It will be seen subsequently that qi for an ideal gas depends linearly on the volume V, so fif is a function only of the temperature. It does of course depend on the distribution of energy levels of the ideal gas molecules. The form of Equation 4.59 for the chemical potential of an ideal gas component is the same as that previously derived from thermodynamics (Equation 4.47). The present approach shows how to calculate m through the evaluation of the molecular partition function. Furthermore, the... 

Equation 5.17 may be simplified somewhat by finding expressions for the absolute activity A./ and the individual particle partition function qjj in terms of experimentally measured or fitted parameters. To achieve such a simplification, we first consider the chemical potential of an ideal gas and its relation to the particle partition function. 

The denominator in this equation has been given a special name, partition function, often symbolized by Z, which is derived from the German Zustandsumme (sum over states). The successive terms in the partition function describe the partition of the configurations among the respectives states available. One can express the thermodynamic state functions of an ideal gas in terms of the molecular partition function Z as follows ... 

As seen earlier ( 12k), Aflo for a reaction may be evaluated from thermal measurements, including heat capacities at several temperatures. However, instead of using experimental heat capacity data to derive AHq from A/f values, the results may be obtained indirectly from partition functions (cf. 16c). The energy content of an ideal gas is independent of the pressure, at a given temperature hence, E — Eo in equation (16.8) may be replaced by E — JSo, so that... 

The treatment of the canonical ensemble for the special case of an ideal gas is now outlined. First of all the partition function for the macrosystem is related to the properties of the individual molecules. At the same time, the energy of a given state in the ensemble 6) can be related to the energy of an individual molecule e,. In the case of a gas or a liquid the particles in the system are indistinguishable because of random motion. Furthermore, if the particles do not interact with one another as in an ideal gas, it is easily shown that... 

A classical gas at low temperatures is probably like an ideal gas composed of clumps of molecules a clump will again be called an /-cluster when the number of molecules in the clump is /. In other words clusters move freely in the container, intercluster force being neglected in comparison with the intracluster force. The partition function of an /-cluster is then given by... 

It is convenient, for purposes of graph theory, to imagine a system with the same temperature, volume, and number of particles as the system of interest, but in which the particles do not interact with each other. Such a system is an ideal gas or ideal gas mixture, and its thermodynamic properties have a rather simple form. (Note, however, that if the original system is at high density, the imaginary system is a dense ideal gas. ) The partition function for this ideal gas will be denoted Oig(r, V, Ni,..., its Helmholtz free energy is Ajg(r, V, Ni,..., N ), and the relationship between these two quantities is the same as between A and Q. We now define the quantity pi, p2, .., p ), where... 

Combining the definition of chemical potential p = (dF/dN)r, with Equation (11.46) gives the chemical potential of an ideal gas in terms of its partition function ... 

Although an attempt to calculate the actual value of the partition function for a system with N = mole of particles is absurd, this simple and elegant formula can be used to determine theoretically the thermodynamic behavior of an ideal gas. 

The ideal gas law and the thermodynamic properties of an ideal gas are completely derived from the canonical ensemble partition function. This is a remarkable illustration of how statistical mechanics explains macroscopic observables in terms of microscopic properties. 

A consequence of writing the partition function as a product of a real gas and an ideal g part is that thermod)mamic properties can be written in terms of an ideal gas value and excess value. The ideal gas contributions can be determined analytically by integrating o the momenta. For example, the Helmholtz free energy is related to the canonical partitii function by ... 

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... 

The statistical thermodynamic approach to the derivation of an adsorption isotherm goes as follows. First, suitable partition functions describing the bulk and surface phases are devised. The bulk phase is usually assumed to be that of an ideal gas. From the surface phase, the equation of state of the two-dimensional matter may be determined if desired, although this quantity ceases to be essential. The relationships just given are used to evaluate the chemical potential of the adsorbate in both the bulk and the surface. Equating the surface and bulk chemical potentials provides the equilibrium isotherm. 

We can now utilize some of the statistical mechanics relationships derived in Chapter 8 to find expressions for the free energy and the equilibrium constant in term of the molecular partition functions. From the definition of the free energy (Eq. 9.1) the expression for the enthalpy of an ideal gas (Eq. 8.121), and recalling that Ho = Eq (for an ideal gas), we obtain... 

The internal motion partition function of the guest molecule is the same as that of an ideal gas. That is, the rotational, vibrational, nuclear, and electronic energies are not significantly affected by enclathration, as supported by spectroscopic results summarized by Davidson (1971) and Davidson and Ripmeester (1984). 

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