Thermodynamic Properties of Fluids

The goal of this section is to show that the thermodynamic properties of fluids (e.g., the internal energy, the virial coefficient, etc.) can be computed in terms of the corresponding properties of an ideal gas plus a correction term that is determined by the radial distribution function of the fluid. This is illustrated by computing the internal energy of a classical fluid.

The ensemble average of the internal energy $E$ is

$$\langle E \rangle = \langle K(p^N) \rangle + \langle V(r^N) \rangle,$$ (401)

where $K(p^N)$ is the total kinetic energy. The simplest model for the potential energy $V(r^N)$ is the pairwise additive potential

$$V(r^N)=\sum_i \sum_{j<i} u(\vert r_i -r_j\vert),$$ (402)

where $u(r)$ is, for example, a Lennard-Jones potential

$$u(r)=4\epsilon [ (\frac{\sigma}{r})^{12}-(\frac{\sigma}{r})^6],$$ (403)

represented by the following diagram:

According to Eq. (402),

$$\langle E \rangle = N \langle \frac{p^2}{2m} \rangle + \sum_{j=1}^N \sum_{k\neq j} \frac{1}{2} \langle u(\vert r_j - r_k\vert) \rangle,$$ (404)

therefore,

$$\langle E \rangle = N \frac{3}{2} k T + \int dr_1 \int dr_2 u(\ve......N-1)}{2} \frac{\int dr^{N-2} e^{-\beta V(r^N)}}{\int dr^N e^{-\beta V(r^N) } }.$$ (405)

Eq. (405) can be simplified, according to Eqs. (388) and (389), as follows:

$$\langle E \rangle = \frac{3}{2} N k T + \frac{1}{2} \int dr_1 \int dr_2 u (\vert r_1 -r_2\vert) \rho^{2/N} (r_1, r_2),$$ (406)

or

$$\langle E \rangle =\frac{3}{2} N k T + \frac{V \rho^2}{2} \int dr u(r) g(r).$$ (407)

Therefore, the ensemble average internal energy per particle

$$\frac{\langle E \rangle}{N} = \frac{3}{2} k T + \frac{\rho}{2} \int dr u(r) g(r),$$ (408)

is the sum of the internal energy per particle in an ideal gas (i.e., 3/2 k T) plus a correction term that can be obtained in terms of the radial distribution function g(r).