Average Energy of a Monatomic Ideal Gas

Let a molecule be considered as a small system in thermal equilibrium with a heat reservoir at temperature \(T\). According to the canonical ensemble, the probability that the molecule occupies the \(i\)-th energy state having energy \(\varepsilon_i\) is given by the Boltzmann distribution:

\div \[ P_i = \frac{e^{-\beta\varepsilon_i}} {\displaystyle\sum_i e^{-\beta\varepsilon_i}} \tag{1} \]

where \[ \beta = \frac{1}{k_BT}, \] \(k_B\) is the Boltzmann constant and \(T\) is the absolute temperature.

The average energy of a molecule is given by

\[ \bar{\varepsilon} = \sum_i P_i\varepsilon_i. \]

Substituting Eq. (1), we obtain

\[ \bar{\varepsilon} = \frac{\displaystyle\sum_i \varepsilon_i e^{-\beta\varepsilon_i}} {\displaystyle\sum_i e^{-\beta\varepsilon_i}}. \tag{2} \]

Partition Function

The canonical partition function is defined as

\[ Z = \sum_i e^{-\beta\varepsilon_i}. \tag{3} \]

Differentiating Eq. (3) with respect to \(\beta\), we get

\[ \frac{\partial Z}{\partial\beta} = -\sum_i \varepsilon_i e^{-\beta\varepsilon_i}. \tag{4} \]

Using Eqs. (3) and (4) in Eq. (2),

\[ \boxed{ \bar{\varepsilon} = -\frac{1}{Z}\frac{\partial Z}{\partial\beta} } \tag{5} \]

or, equivalently,

\[ \boxed{ \bar{\varepsilon} = -\frac{\partial\ln Z}{\partial\beta} } \]

Translational Partition Function of a Monatomic Ideal Gas

The classical translational partition function for one molecule in a volume \(V\) is

\[ Z = \frac{V}{h^3} \left(\frac{2\pi m}{\beta}\right)^{3/2}. \tag{6} \]

Taking the logarithm of the partition function,

\[ \ln Z = \ln V + \frac{3}{2}\ln(2\pi m) - \frac{3}{2}\ln\beta - 3\ln h. \]

Differentiating with respect to \(\beta\),

\[ \frac{\partial\ln Z}{\partial\beta} = -\frac{3}{2\beta}. \]

Substituting this result into Eq. (5),

\[ \bar{\varepsilon} = -\left(-\frac{3}{2\beta}\right) = \frac{3}{2\beta}. \]

Since

\[ \beta = \frac{1}{k_BT}, \]

Average energy of a monatomic ideal gas molecule

\[ \boxed{\bar{\varepsilon}=\frac{3}{2}k_BT} \]

Thus, the average energy of a monatomic ideal gas molecule depends only on the absolute temperature and is independent of the volume of the container.

Total Energy of the Gas

If the gas contains \(N\) molecules, the total internal energy is

\[ U = N\bar{\varepsilon}. \]

Substituting the average energy per molecule,

\[ \boxed{U=\frac{3}{2}Nk_BT} \tag{7} \]

Using \(N=nN_A\) and \(R=N_Ak_B\), the equation can also be written as

\[ \boxed{U=\frac{3}{2}nRT}. \]

This result is valid for a classical monatomic ideal gas, in which intermolecular potential energy is neglected.

Average Energy of a Polyatomic Ideal Gas

An ideal gas may also be polyatomic, such as \(\mathrm{O_2}\), \(\mathrm{N_2}\), \(\mathrm{CH_4}\), or \(\mathrm{Cl_2}\), in which two or more atoms are present in a molecule.

A polyatomic molecule can possess translational, rotational, vibrational, and electronic energy. The total average energy per molecule may be written as

\[ \bar{\varepsilon} = \bar{\varepsilon}_{\mathrm{trans}} + \bar{\varepsilon}_{\mathrm{rot}} + \bar{\varepsilon}_{\mathrm{vib}} + \bar{\varepsilon}_{\mathrm{elec}}. \tag{8} \]

The translational contribution is the same as that of a monatomic ideal gas:

\[ \bar{\varepsilon}_{\mathrm{trans}} = \frac{3}{2}k_BT. \]

The remaining contributions arise from molecular rotation, vibration, and electronic excitation. At ordinary temperatures, electronic excitation is usually negligible, and vibrational modes may not be fully excited. Consequently, the total energy depends on the degrees of freedom that are thermally accessible.

According to the classical equipartition theorem, if a molecule has \(f\) active quadratic degrees of freedom, its average energy is

\[ \boxed{\bar{\varepsilon}=\frac{f}{2}k_BT}. \]

Therefore, the average energy of a polyatomic ideal gas depends on temperature and on the number of active degrees of freedom.