Internal Energy

Consider a monatomic gas such as helium. An individual helium atom can only store energy in its translational motion. As we saw in Section 5.3.4, the mean energy associated with this motion is $(3/2)\,k_B\,T$. Hence, the internal energy of a gas consisting of $\nu$ moles of helium atoms is

$$U = \nu\,N_A\,\frac{3}{2}\,k_B\,T = \nu\,\frac{3}{2}\,R\,T.$$ (5.188)

However, according to Equation (5.108),

$$U= \nu\,c_V\,T,$$ (5.189)

where $c_V$ is the molar specific heat capacity of the gas at constant volume. The previous two equations imply that the molar specific heat capacity of a helium gas (or any monatomic gas) is

$$c_V = \frac{3}{2}\,R,$$ (5.190)

in accordance with Equation (5.106).

Consider a diatomic gas such as hydrogen. An individual hydrogen molecule can store energy in its translational motion, but can also store energy in its rotational motion. In principle, a molecule has three principal axes of rotation about which it could rotate. (See Sections 1.7.2 and 1.7.3.) Hence, the net rotational kinetic energy is

$$\frac{L_x^{\,2}}{2\,I_{xx}} + \frac{L_y^{\,2}}{2\,I_{yy}}+ \frac{L_z^{\,2}}{2\,I_{zz}},$$ (5.191)

where $L_x$ is the angular momentum about the $x$-axis, $I_{xx}$ is the principal moment of inertia for rotation about the $x$-axis, et cetera. Note that the previous expression consists of three terms that are quadratic in a momentum component. Hence, according to the law of equipartition of energy, the mean rotational energy of the molecule should be

$$\frac{1}{2}\,k_B\,T +\frac{1}{2}\,k_B\,T +\frac{1}{2}\,k_B\,T = \frac{3}{2}\,k_B\,T.$$ (5.192)

In fact, this is not the case. The reason for the discrepancy is that one of the principal axes of rotation of a hydrogen molecule corresponds to the axis that passes through the nuclei of the two hydrogen atoms that constitute the molecule. The principal moment of inertia for rotation about this axis is much smaller than the principal moments of inertia for rotation about the other two principal axes. In fact, the former moment of inertia is of order $m_e\,d^{\,2}$, where $m_e$ is the mass of an electron, and $d$ the radius of a hydrogen atom (the contribution of the protons to the moment is negligible), whereas the latter two moments of inertia are of order $m_p\,D^{\,2}$, where $m_p$ is the mass of a proton, and $D$ the length of the atomic bond joining the two hydrogen atoms. Given that $m_e\ll m_p$, while $d\sim D$, the former moment of inertia is indeed much smaller than the latter two. It turns out that quantum mechanical considerations prevent a degree of rotational freedom with an anomalously small moment of inertia from contributing $(1/2)\,k_B\,T$ to the mean energy of the molecule. (See Section 5.5.8.) Hence, the mean rotational energy of a hydrogen molecule (or any diatomic molecule) is

$$\frac{1}{2}\,k_B\,T + \frac{1}{2}\,k_B\,T=k_B\,T.$$ (5.193)

According to the previous discussion, the mean energy of a hydrogen molecule is

$$\frac{3}{2}\,k_B\,T + k_B\,T = \frac{5}{2}\,k_B\,T,$$ (5.194)

where the former contribution is the molecule's mean translation kinetic energy, whereas the latter contribution is the molecule's mean rotational kinetic energy. Hence, the internal energy of a gas consisting of $\nu$ moles of hydrogen molecules is [see Equation (5.189)]

$$U = \nu\,N_A\,\frac{5}{2}\,k_B\,T = \nu\,\frac{5}{2}\,R\,T=\nu\,c_V\,T.$$ (5.195)

It follows that the molar specific heat capacity at constant volume of a hydrogen gas (or any diatomic gas) is

$$c_V = \frac{5}{2}\,R,$$ (5.196)

in accordance with Equation (5.107).