Ideal Gases in Isohoric Processes

Definition: Isohoric Process

An isohoric process is a process in which the volume of the gas does not change.

Theorem: Work in an Isohoric Process

An ideal gas does no work in an isohoric process.

$$W=0$$

PROOF

According to this theorem

$$W={\int }_V^{V}p\mathrm{dV}=0$$

Theorem: Molar Heat Capacity in an Isohoric Process

The molar heat capacity of an ideal gas in an isohoric process is denoted by $C_V$ and is

$$C_V=\frac{f}{2}R$$

where $f$ is the number of degrees of freedom of a single gas particle and $R$ is the molar gas constant.

PROOF

The internal energy of the gas is

$$U=\frac{f}{2}nRT$$

Since the gas does no work in an isohoric process, the first law of thermodynamics tells us that the change in the internal energy depends only on the heat which enters / exits the gas.

$$\mathrm{d}U=\mathrm{d}Q$$

From this follows

$$\mathrm{d}Q=\frac{f}{2}nR\mathrm{dT}⟹\frac{\mathrm{d}Q}{\mathrm{d}T}=\frac{f}{2}nR$$

The left-hand side is just the gas’s heat capacity $C$.

$$C=\frac{\mathrm{d}Q}{\mathrm{d}T}=\frac{f}{2}nR$$

To obtain the molar heat capacity, we need to divide $C$ by the number of moles $n$.

$$C_V=\frac{C}{n}=\frac{f}{2}R$$