Specific heats of gases

Monatomic gas: three translational degrees of freedom, so (3/2)k_BT per molecule and U = (3/2)k_BT × N_A = (3/2)RT per mole. Then C_v = dU/dT = (3/2)R.

For any ideal gas C_p − C_v = R, so a monatomic gas has C_p = (5/2)R and γ = C_p/C_v = 5/3.

Rigid diatomic gas (a dumbbell): 3 translational + 2 rotational = 5 degrees of freedom, U = (5/2)RT, C_v = (5/2)R, C_p = (7/2)R, γ = 7/5.

Diatomic gas that also vibrates: U = (5/2 k_BT + k_BT)N_A = (7/2)RT, so C_v = (7/2)R, C_p = (9/2)R, γ = 9/7.

Polyatomic gas with 3 translational, 3 rotational degrees of freedom and f vibrational modes: U = ((3/2)k_BT + (3/2)k_BT + f k_BT)N_A, so C_v = (3 + f)R, C_p = (4 + f)R and γ = (4 + f)/(3 + f).

C_p − C_v = R holds for every ideal gas, monatomic, diatomic or polyatomic.

Table 12.1 lists the predictions with vibration left out (J mol⁻¹ K⁻¹): a monatomic gas has C_v = 12.5, C_p = 20.8, γ = 1.67; a diatomic gas C_v = 20.8, C_p = 29.1, γ = 1.40; a triatomic gas C_v = 24.93, C_p = 33.24, γ = 1.33. In every row C_p − C_v = 8.31.

Table 12.2, measured values (same order): He 12.5, 20.8, 8.30, 1.66; Ne 12.7, 20.8, 8.12, 1.64; Ar 12.5, 20.8, 8.30, 1.67; H₂ 20.4, 28.8, 8.45, 1.41; O₂ 21.0, 29.3, 8.32, 1.40; N₂ 20.8, 29.1, 8.32, 1.40; H₂O 27.0, 35.4, 8.35, 1.31; CH₄ 27.1, 35.4, 8.36, 1.31.

Prediction and measurement agree well for these gases, so equipartition is well confirmed at ordinary temperatures. For gases such as Cl₂ and C₂H₆ the measured values are usually higher than Table 12.1, and including vibrational modes brings them closer.

Example 12.8: a fixed 44.8 L cylinder of helium at STP holds 2 mol (1 mol fills 22.4 L at 273 K and 1 atm). Fixed volume means C_v = (3/2)R applies, so heating it by 15.0 °C needs 2 × 1.5R × 15.0 = 45R = 45 × 8.31 = 374 J.