Monatomic Gases

12.6.1 Monatomic Gases

The specific heat capacity of a gas is not a single fixed number — it depends on whether the gas is heated at constant volume or constant pressure. For a monatomic gas, the kinetic theory gives us clean, exact predictions because the molecules have only translational motion, with no rotation or vibration.

The Internal Energy of a Monatomic Gas

From the kinetic theory, the average kinetic energy of a single molecule is $\frac{3}{2} k_B T$. For one mole of a monatomic gas, which contains Avogadro's number $N_A$ of molecules, the total internal energy $U$ is:

Here $R = N_A k_B$ is the universal gas constant. This is a key result: the internal energy of one mole of a monatomic ideal gas depends only on its temperature, and is proportional to $T$.

Molar Specific Heat at Constant Volume ($C_V$)

The molar specific heat at constant volume is defined as the heat required to raise the temperature of one mole of the gas by 1 K while keeping the volume fixed. At constant volume, no work is done ($\Delta W = 0$), so from the first law of thermodynamics:

Therefore, $C_V$ is simply the rate of change of internal energy with temperature:

For a monatomic gas, $U = \frac{3}{2} R T$, so:

Numerically, with $R = 8.314 \text{ J mol}^{-1} \text{ K}^{-1}$, this gives $C_V \approx 12.47 \text{ J mol}^{-1} \text{ K}^{-1}$.

Molar Specific Heat at Constant Pressure ($C_P$)

When the gas is heated at constant pressure, it expands and does work on its surroundings. The heat supplied must therefore provide both the increase in internal energy and the work done. From the first law:

For one mole of an ideal gas, $PV = RT$. At constant pressure, $P \Delta V = R \Delta T$. Also, $\Delta U = C_V \Delta T$. Therefore:

Since $C_P = \left( \frac{\Delta Q}{\Delta T} \right)_P$, we get:

This is Mayer's relation, valid for any ideal gas. For a monatomic gas, substituting $C_V = \frac{3}{2} R$:

Numerically, $C_P \approx 20.79 \text{ J mol}^{-1} \text{ K}^{-1}$.

The Ratio of Specific Heats ($\gamma$)

The ratio $\gamma = C_P / C_V$ is an important parameter in adiabatic processes. For a monatomic gas:

Summary of Results for Monatomic Gases