Classical statistical mechanics dilute gases

The composite partition functions are identical with the quantum versions of Chapter 25. This illustrates the fact that classical statistical mechanics does not provide any advantage in treating a dilute gas. 

The classical translational and rotational contributions to thermodynamic functions must be corrected by choosing the correct divisors. This yields the same results as in quantum statistical mechanics. The classical formulas for vibration are numerically inadequate, even with the correct divisors, and we do not attempt to use classical statistical mechanics for electronic motion. There is nothing to be gained by using classical statistical mechanics for a dilute gas. 

Substances at high dilution, e.g. a gas at low pressure or a solute in dilute solution, show simple behaviour. The ideal-gas law and Henry s law for dilute solutions antedate the development of the fonualism of classical themiodynamics. Earlier sections in this article have shown how these experimental laws lead to simple dieniiodynamic equations, but these results are added to therniodynaniics they are not part of the fonualism. Simple molecular theories, even if they are not always recognized as statistical mechanics, e.g. the kinetic theory of gases , make the experimental results seem trivially obvious. 

Macroscopic states involve variables that pertain to the entire system, such as the pressure P, the temperature T, and the volume V. For a fluid system of one substance and one phase, the equilibrium macrostate is specified by only three variables, such as P, T, and V. If we assume that classical mechanics is an adequate approximation, the microstate of such a system is specified by the position and velocity of every particle in the system. If quantum mechanics must be used for a dilute gas, there are several quantum numbers required to specify the state of each molecule in the system. This is a very large number of independent variables or a very large number of quantum numbers. Statistical mechanics is the theory that relates the small amount of information in the macrostates and the large amount of information in the microstates. 

---