A classical gas of N point particles occupies volume V at...

A classical gas of $N$ point particles occupies volume $V$ at temperature $T$. The particles interact pairwise, $\phi\left(r_{i j}\right)$ being the potential between particles $i$ and $j, r_{i j}=\left|\mathbf{r}_i-\mathbf{r}_j\right|$. Suppose this is a "hard sphere" potential $$ \phi\left(r_{i j}\right)= \begin{cases}\infty, & r_{i j}<a, \\ 0, & r_{i j}>a .\end{cases} $$ (a) Compute the constant volume specific heat as a function of temperature and specific volume $v=\frac{V}{N}$. (b) The virial expansion for the equation of state is an expansion of $\frac{p V}{R T}$ in inverse powers of $V$ : $$ \frac{p V}{R T}=1+\frac{A_1(T)}{V}+\frac{A_2(T)}{V^2}+\ldots . $$ Compute the virial coefficient $A_1$.

A classical gas of $N$ point particles occupies volume $V$ at temperature $T$. The particles interact pairwise, $\phi\left(r_{i j}\right)$ being the potential between particles $i$ and $j, r_{i j}=\left|\mathbf{r}_i-\mathbf{r}_j\right|$. Suppose this is a "hard sphere" potential
$$
\phi\left(r_{i j}\right)= \begin{cases}\infty, & r_{i j}<a, \\ 0, & r_{i j}>a .\end{cases}
$$
(a) Compute the constant volume specific heat as a function of temperature and specific volume $v=\frac{V}{N}$.
(b) The virial expansion for the equation of state is an expansion of $\frac{p V}{R T}$ in inverse powers of $V$ :
$$
\frac{p V}{R T}=1+\frac{A_1(T)}{V}+\frac{A_2(T)}{V^2}+\ldots .
$$

Compute the virial coefficient $A_1$.

Key Concepts

Hard Sphere Potential

This concept refers to a model where particles are considered as perfectly rigid spheres that cannot overlap. The interaction potential is infinite when the distance between two particle centers is less than a certain diameter and zero otherwise. This simple model captures the essence of excluded volume effects, which are key in understanding the behavior of dense fluids and gases where repulsive interactions dominate.

Equipartition Theorem

The equipartition theorem is a fundamental principle in classical statistical mechanics stating that, at thermal equilibrium, the energy is equally distributed among all quadratic degrees of freedom. It is essential for calculating thermodynamic quantities like the specific heat, as each degree of freedom contributes a fixed amount to the internal energy, independent of the details of interparticle interactions when those interactions do not depend on temperature.

Specific Heat at Constant Volume

Specific heat at constant volume is a thermodynamic property that measures how much energy is required to raise the temperature of a system at a fixed volume. In the case of a classical ideal gas or gases with hard sphere interactions, the internal energy is typically determined solely by the kinetic energy, leading to a specific heat that is independent of temperature. The equipartition theorem provides the basis for computing this quantity by assigning a set energy per mode of motion.

Virial Expansion

The virial expansion is a method for expressing the equation of state of a gas as a power series in the density (or inverse volume). Each term in the series accounts for interactions among molecules, with the coefficients (virial coefficients) quantifying the contributions from two-body, three-body interactions, etc. This expansion is particularly useful for understanding how non-ideal behavior arises from interparticle forces in a gas.

Virial Coefficients

Virial coefficients are the terms in the virial expansion that represent the corrections to the ideal gas law due to intermolecular interactions. The first virial coefficient is unity by definition, while the second virial coefficient quantifies the impact of pairwise interactions, such as those from the hard sphere potential. These coefficients are calculated using integrals over the pair potential, encapsulating how the finite size and repulsive effects of the particles alter the pressure.

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