Modern Engineering Thermodynamics

18.9 Three Classical Quantum Statistical Models 749
where λ = c/v is the wavelength of the photon. De Broglie then extended the argument to mass particles (like
electrons) by postulating that, for them,
p = mV = ħ λ
where λ is the particle’s wavelength. This postulation was experimentally verified in 1927, when it was demonstrated
that electrons could be diffracted in a wavelike manner from a ruled diffraction surface. Thus, electrons
appeared to have both particlelike and wavelike behavior, and the duality principle of matter was established.
Once the wavelike character of matter was recognized, it became clear that the kinetic behavior of atomic particles
ought to be governed by the same equations that govern the propagation of waves in a continuum. In
1926, Erwin Schrödinger developed an unsteady wave equation appropriate to matter waves of the form
∇ 2 ψ = 2mðε − ε
pÞ ∂ 2 ψ
ε 2 ∂t 2
(18.39)
where ψ is the wave function (the wave amplitude), ε is the total energy of the particle, ε p is the potential energy
of the particle, m is the particle’s mass, and ∇ 2 is the differential operator defined by
∇ 2 ðÞ= ∂2
∂x
∂y
∂z 2 ðÞ
∂2 ∂2
ðÞ+ ðÞ+
2 2
Remarkably, the solutions to (18.39) are inherently quantitized (i.e., solutions exist for only discrete values of ε);
thus, it has become a fundamental equation in quantum mechanics.
18.9 THREE CLASSICAL QUANTUM STATISTICAL MODELS
Consider a system composed of N = ∑N i particles that are distributed in some manner among ε i energy levels.
Then, the total internal energy of the system is U = ∑N i ε i . The most probable distribution (N i ) mp of the N particles
is the one that corresponds to the macrostate with the maximum probability P, and the total internal energy
of that macrostate is U mp = ∑(N i ) mp ε i . Once an equation for W is found, the distribution N i that maximizes it
can easily be found by setting d(W) = 0 subject to the constraint that the total energy and total number of particles
in the system are constant (i.e., dN = dU = 0), and solving for N i = (N i ) mp .
A particle N i has a total energy ε i , which, in general, is made up of a number of energy modes. For example, we
could partition the total energy of the particle into kinetic energy, rotational energy, vibrational energy, and so
forth; and the particle’s total energy can be divided among these modes in many ways. The total number of
arrangements of a particle’s different energy modes that add up to a given energy level ε i is called the degeneracy
of that energy level and is given the symbol g i .
The following three classical statistical models have been developed to describe the basic particle-wave nature of
certain material particles, and their corresponding W and (N i ) mp equations can be found in Table 18.7.
1. The Maxwell-Boltzmann model. 6 Here, all the N i particles are assumed to be indistinguishable from each
other and distributed among various degenerate energy levels. This model accurately represents the behavior
of most simple gases at low pressures.
2. The Fermi-Dirac model. Here, the particles are assumed to be indistinguishable and are distributed among
various degenerate energy levels with only one particle per degeneracy (g i ) value. This model accurately
represents the behavior of electron and proton gases.
3. The Bose-Einstein model. Here, the particles are assumed to be indistinguishable and are distributed among
various degenerate energy levels with no limit on the number of particles per degeneracy. This model
accurately represents the behavior of photon and phonon gases.
The second law of thermodynamics states that S p or _S p ≥ 0, which implies that, at equilibrium, the entropy of a
closed system is a maximum. Also, since thermodynamic equilibrium corresponds to the system’s beinginits
most probable macrostate, it is logical to assume that a functional relation exists between the entropy S of the
system and the statistical probability of the most probable macrostate, W mp . We postulate that this relation has
the form:
S = f ðW mp Þ
6 This is called the “corrected” Maxwell-Boltzmann model in most statistical thermodynamics texts.