Factorization of the molecular partition function

In equation (12 8) the partition function Q was expressed as a product of the partition functions /, of each of the N molecules, together with the factor jN which allows for indistinguishability. As noted in the last section, the justification for this procedure is the separabiUty of the wave equation in the case of a perfect gas. In addition, there is also the separability which is expressed by equations (12 21) and (12-22), and this permits a factorization of / itself. Let 0 - and be the degeneracies of the th translational level and the kth internal level respectively. The degeneracy of the energy state 

This is because any of the 0) translational quantum states, aU of energy ej -, may be combined with any of the internal quantum states, all of energy to give a state of a given total energy. Substituting the above in (12 9) we obtain 

In speaking of, say, the translational and internal parts of the energy as being independent forms of energy, it is to be borne in mind that whenever we have an isolated system the total energy is constant. Thus any decrease in the translational energy of the molecules would result in an increase in the energy of the internal motions and vice 

The allocation of a fixed quantity of energy between the various forms of motion thus depends on the spreading of this energy in such a way as to maximize Cl, due allowance being made for the different spacing of the quantum levels. This point was mentioned previously in 1-17. 

The equilibration of the various forms of energy implies also that the statistical parameter T, the temperature, has the same value in all of the various parts, such as (12 26) and (12 27), of the total partition function. This follows from the same kind of argument as was used in 11 9, where it was shown that two bodies at equilibrium have the same value of /3. To demonstrate the equality of T or fi between, say, translational and internal states, we should set up equations similar to (11 4)-(11 6), noting on the one hand that the probability of a chosen translational state and a chosen internal state is multiplicative and, on the other hand, that the energies are additive. 

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Equation (25.4-5) leads to a factoring of the molecular partition function ... 

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. 

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