Temperature and Pressure in Statistical Mechanics
From the viewpoint of statistical mechanics, a collection of gas molecules is a system of particles satisfying the Schrödinger equation of quantum mechanics. The system can exist in any of a very large number of quantum states. Denoting the number of states by Γ, The entropy S of the system is defined by
where k is Boltzman's constant. If the total kinetic energy of the system is E, and the particles occupy a volume V, temperature and pressure are defined to be
These definitions appear to be quite different from the molecular model definitions used in this web site. The purpose of this section is to show that the definitions are in fact equivalent for most circumstances, and to discuss the differences that do exist. We employ a derivation similar to one given in the text by Goodstein, but omitting many details. For simplicity we assume monatomic molecules.
The set of particles under study is imagined to be a subsystem embedded in a much larger medium. The subsystem plus the medium is the total system, which consists of N0 particles occupying a volume V0, and with total kinetic energy E0. These three parameters are fixed constants. The subsystem consists of N particles occupying a volume V and with total energy E. All of these subsystem parameters are assumed to be much smaller than those for the medium. We assume here that N is constant, but that V and E can vary, meaning the subsystem and medium can exchange energy and volume with each other.
It is further assumed that the subsystem and medium are at the same temperature and pressure, and that these values are constant. Therefore we can calculate the temperature and pressure of the medium to obtain the values for the subsystem.
Density of States
In any specific quantum state the energy is a constant. However it is assumed that the subsystem and medium undergo fluctuations between states, and that the energy varies by a small amount dE. That is, if the subsystem gains dE, the medium loses dE, and vice versa. Rather than directly enumerating the states, it can be shown that the number of states within this range of energy fluctuation is given in terms of the density of states ρ(E), which for the medium is
where m is the particle mass, and h is Planck's constant. This equation is based on an approximation, but is quite accurate for the purposes of this section. Using the fact that N0 and V0 are very large, and much larger than N and V, the factorials can be approximated using Stirling's formula. N is constant, and the energy increment dE is assumed to be independent of the variables E and V. The result is that the entropy of the medium is
E is the energy in the subsystem, so a positive change in E is a negative change in the energy in the medium. Then for the medium
Under equilibrium conditions the total kinetic energy in terms of the variance of molecular velocity σ2, and the mass density of the medium ρ0 (not to be confused with the density of states) are
Then using equations (5) and (6), the results for temperature and pressure from equation (2) are
which are identical to the molecular model definitions.
The statistical mechanics definitions assume that the gas is at equilibrium; the molecular model definitions apply to non-equilibrium conditions as well. The statistical mechanics definitions apply globally to the entire volume of the gas; i.e. the E in equation (2) is the total energy in the volume V. The molecular model definition applies locally; i.e. pressure is defined in terms of the force on a surface, which only depends on conditions near the surface, and it doesn't matter what the molecules elsewhere in the volume are doing. An example where this makes a huge difference is the calculation of molecular noise. The molecular model analysis looks at the variance of force on a surface; the result depends on the area of the surface but is independent of the volume. Goodstein calculates a pressure variance using the statistical mechanics definitions, and in this case the result depends on volume.
The analysis elsewhere on this web site actually does not need the concepts of temperature and pressure at all; the statistics of molecular velocity are the basic variables. But since the rest of the world uses these terms it is appropriate to relate to them, and to their standard statistical mechanics definitions.
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