The distributions laws of statistical mechanics, of which Boltzmann’s is one, are concerned with the distribution of energy within a system of molecules. Knowledge of the true distribution function is of fundamental importance and permits evaluation of the thermodynamic properties of the system from statistical mechanics. Boltzmann’s distribution law refers specifically to a system of noninteracting molecules in a state of thermodynamic equilibrium. It was postulated before the discovery of quantum mechanics and, in its original form, relied upon a classical description of molecular energies. In this description, the distribution function for a system of structureless molecules is specified by the probability P that a molecule will, at any instant, be located within the element of volume dxdydz and have velocity components in the ranges u to u + du, v to v + dv, and w to w + dw. According to Boltzmann’s distribution law, this probability is given by:
where ε is the total (kinetic + potential) energy of the molecule, k is a positive constant known as Boltzmann’s constant, and the integral is performed over all possible positions and velocities of the molecule. This law states that the probability of a molecule acquiring an energy ε declines with increasing energy in proportion to the Boltzmann factor exp(−ε/kT). For molecules with rotational and vibrational modes of motion, the classical distribution law may be extended, but a quantum-mechanical generalization is to be preferred.
In quantum mechanics, molecular energies are restricted to discreet levels ei and the counterpart of Eq. (1) is:
Here, Pi is the probability that, at any instant, a given molecule will be found in a quantum state having energy εi, gi is the degeneracy of that energy level (i.e., the number of quantum states with the same energy εi), and q is the molecular partition function given by:
Since the energy levels of the molecule depend on the volume of the system, q is a function of both T and V. It turns out that the Helmholtz free energy (Free Energy) A of N noninteracting molecules is related to q by the simple formula:
Thus, all of the thermodynamic properties of the perfect gas may be determined from q(T,V) (see Hill, 1960).
An important simplification arises when we assume that the energy of each molecule may be written as the sum of independent terms for translations (t), rotations (r), vibrations (v) and other internal degrees of freedom. In this case, q factorizes into the product qt qr qv g0 … ext(−ε0/kT). Here, g0 and ε0 are the degeneracy and energy of the lowest energy level and all other energies are measures relative to ε0. Each the of factors qt, qr, qv, etc., in q is given by an equation similar to (3), but with the summation running only over energy levels of the specified kind. The separation of molecular energies and consequent factorization of q is exact if rotations are treated as those of a rigid body and molecular vibrations are assumed simple harmonic. These are fair approximations for many molecules. For a diatomic molecule of mass m, moment of inertia I, vibration frequency ν, and symmetry number s, the factors of q are:
Here, s = 2 for homonuclear molecules, s = 1 otherwise and the zero-point energy ε0 is taken to be the energy of the molecule in its lowest vibrational state. A similar but more complicated treatment is applicable to polyatomic molecules (Hill, 1960). It should be noted that the formula for qr is based on a semi-classical treatment, but that a full quantum treatment may be needed at low temperatures for molecules with small moments of inertia (e.g., H2).
Although Eq. (2) is based on quantized energy levels, it is not strictly consistent with quantum-mechanical restrictions. The correct distribution function depends upon whether the molecules in question contain an even or an odd number of elementary particles (protons, neutrons and electrons): the first case leads to the Bose–Einstein distribution, while the second gives the Fermi-Dirac distribution. However, both of these reduce to the Boltzmann distribution at the limit where the number of available quantum states greatly exceed the number of molecules. This limit is approached with increasing molecular mass and increasing temperature; in practice, Boltzmann statistics apply with great accuracy, except for the isotopes of hydrogen and helium at temperatures below 10 K.
It is possible to make a generalization of Boltzmann’s distribution law applicable to a system of N interacting molecules. We now define a new probability function ∏i as the probability that, at a given instant, the entire system will be in a quantum state with energy Ei. This probability too is proportional to a Boltzmann factor, exp(−Ei/kT), and we have:
Here, Ωi is the degeneracy of Ei, Q is called the canonical partition function and the summation in Eq. (7) is carried out over all possible energy levels. According to statistical thermodynamics, the Helmholtz free energy and all of the other thermodynamic properties of the system may be obtained from Q(N,T,V) (see Hill, 1960).
Hill, T. L. (1960) An Introduction to Statistical Thermodynamics, Addison-Wesley Publishing, Reading (Mass.).