Number of Accessible States of Ideal Gas

Consider an ideal gas, made up of spinless monatomic particles, whose volume is $V$, and whose internal energy lies in the range $U$ to $U+\delta U$. Let ${\mit\Omega}(U, V)$ be the total number of microscopic states that satisfy these constraints. This is a particularly simple example, because, for such a gas, the particles possess translational, but no internal (e.g., vibrational, rotational, or spin), degrees of freedom. By definition, interatomic forces are negligible in an ideal gas. In other words, the individual particles move in an approximately uniform potential. It follows that the internal energy of the gas is just the total translational kinetic energy of its constituent particles, so that

$$U = \frac{1}{2\,m}\sum_{i=1,N}{\bf p}_i^{\,2},$$ (5.293)

where $m$ is the particle mass, $N$ the total number of particles, and ${\bf p}_i$ the vector momentum of the $i$th particle.

Consider the system in the limit in which the internal energy, $U$, of the gas is much greater than the ground-state energy, so that all of the quantum numbers are large. The classical version of statistical mechanics, in which we divide up phase-space into cells of equal volume, is valid in this limit. (See Section 5.4.1.) The number of states, ${\mit\Omega}(U, V)$, lying between the internal energies $U$ and $U+\delta U$ is simply equal to the number of cells in phase-space contained between these energies. In other words, ${\mit\Omega}(U, V)$ is proportional to the volume of phase-space between these two energies:

$${\mit\Omega}(U, V) \propto \int^{U+\delta U}_U d^{3}{\bf r}_1\cdots d^{3}{\bf r}_N\, d^{3} {\bf p_1}\cdots d^{3} {\bf p_N}.$$ (5.294)

Here, the integrand is the element of volume of phase-space, with

$$d^{3}{\bf r}_i \equiv dx_i\,dy_i\,dz_i,$$ (5.295)

$$d^{3}{\bf p}_i \equiv dp_{x\,i}\,dp_{y\,i} \,dp_{z\,i},$$ (5.296)

where $(x_i$, $y_i$, $z_i)$ and $(p_{x\,i}$, $p_{y\,i}$, $p_{z\,i})$ are the Cartesian coordinates and momentum components of the $i$th particle, respectively. The integration is over all coordinates and momenta such that the total internal energy of the system lies between $U$ and $U+\delta U$.

For an ideal gas, the total internal energy $U$ does not depend on the positions of the particles. [See Equation (5.293).] This implies that the integration over the position vectors, ${\bf r}_i$, can be performed immediately. Because each integral over ${\bf r}_i$ extends over the volume of the container (the particles are, of course, not allowed to stray outside the container), $\int d^{3}{\bf r}_i= V$. There are $N$ such integrals, so Equation (5.294) reduces to

$${\mit\Omega}(U, V) \propto V^{\,N} \chi(U),$$ (5.297)

where

$$\chi(U) \propto \int^{U+\delta U}_U d^{3} {\bf p_1}\cdots d^{3} {\bf p_N}$$ (5.298)

is a momentum-space integral that is independent of the volume.

The internal energy of the system can be written

$$U= \frac{1}{2\,m} \sum_{i=1,N} \sum_{\alpha=1,3} p_{\alpha\,i}^{\,2},$$ (5.299)

because ${\bf p}_i^{\,2} = p_{1\,i}^{\,2}+p_{2\,i}^{\,2}+p_{3\,i}^{\,2}$, denoting the $(x$, $y$, $z)$ components by (1, 2, 3), respectively. The previous sum contains $3N$ square terms. For $U=$ constant, Equation (5.299) describes the locus of a sphere of radius $R(U) = (2\,m \,U)^{1/2}$ in the $3N$-dimensional space of the momentum components. Hence, $\chi(U)$ is proportional to the volume of momentum phase-space contained in the region lying between the sphere of radius $R(U)$, and that of slightly larger radius $R(U+\delta U)$. This volume is proportional to the area of the inner sphere multiplied by $\delta R \equiv R(U+\delta U)- R(U)$. Because the area varies like $R^{3N-1}$, and $\delta R \propto \delta U/ U^{1/2}$, we have

$$\chi(U) \propto R^{3N-1}/U^{1/2} \propto U^{3N/2-1}.$$ (5.300)

Combining this result with Equation (5.297), we obtain

$${\mit\Omega}(U, V) \simeq B\,V^{N} U^{3N/2},$$ (5.301)

where $B$ is a constant independent of $V$ or $U$, and we have also made use of the fact that $N\gg 1$ for a typical ideal gas. Note that ${\mit\Omega}(U, V)$ is a very strongly increasing function of $U$ because $N\gg 1$.