Specific Heat Capacity

When the temperature of a gas is raised even slightly, both its volume and pressure can change considerably depending on the conditions under which heating occurs. Because of this, two distinct specific heats are defined for a gas: the specific heat at constant volume, $C_V$, and the specific heat at constant pressure, $C_P$. Section 3.9.1 derives Mayer's relation connecting the two; the rest of this section uses the law of equipartition of energy (Section 3.8) to compute $C_V$, $C_P$ and their ratio $\gamma = C_P/C_V$ for gases of increasing molecular complexity.

1. Monatomic gases. A monatomic gas held at temperature $T$ has each of its $N_A$ atoms (per mole) possessing only 3 translational degrees of freedom, so the average energy per atom is $\tfrac{3}{2}k_BT$, and the total internal energy of one mole is

   $E = \dfrac{3}{2}N_Ak_BT = \dfrac{3}{2}RT$

   The molar specific heat at constant volume is then

   $C_V = \dfrac{dE}{dT} = \dfrac{3}{2}R$

   and, using Mayer's relation $C_P - C_V = R$ (Section 3.9.1),

   $C_P = \dfrac{5}{2}R, \qquad \gamma = \dfrac{C_P}{C_V} = \dfrac{5}{3}$

2. Diatomic gases. For a gas of diatomic molecules (O$_2$, N$_2$, CO, HCl, ...) treated as rigid rotators, each molecule has 3 translational + 2 rotational degrees of freedom, so the internal energy of one mole is $E = \tfrac{3}{2}RT + \tfrac{2}{2}RT = \tfrac{5}{2}RT$, giving

   $C_V = \dfrac{5}{2}R, \qquad C_P = \dfrac{7}{2}R, \qquad \gamma = \dfrac{7}{5}$

   For a non-rigid, vibrating diatomic gas, the extra vibrational mode contributes a further $RT$ per mole (kinetic + potential, each $\tfrac{1}{2}RT$), so $E = \tfrac{3}{2}RT + \tfrac{2}{2}RT + RT = \tfrac{7}{2}RT$, giving

   $C_V = \dfrac{7}{2}R, \qquad C_P = \dfrac{9}{2}R, \qquad \gamma = \dfrac{9}{7}$

3. Polyatomic gases. A polyatomic gas (molecules with more than two atoms, e.g. ammonia, NH$_3$) always has 3 translational degrees of freedom. Linear polyatomic molecules have 2 rotational degrees of freedom (like a diatomic); all other (non-linear) polyatomic molecules can rotate about three mutually perpendicular axes through their centre of mass, giving 3 rotational degrees of freedom. In addition, a polyatomic molecule can have several distinct vibrational modes, the exact number $f$ depending on the molecule's geometric structure, with each vibrational mode contributing $2\times\tfrac{1}{2}k_BT = k_BT$ (kinetic + potential) just as for a diatomic vibration. For one mole of a general polyatomic gas, …