3.3 Specific heats and the ratio γ

How much does it take to warm a gas by one degree? The answer depends on whether the gas is held at constant volume or allowed to expand against a constant pressure, and the ratio of those two heat capacities, $\gamma$ , turns out to govern the adiabatic processes of the next lesson.

Two heat capacities

The heat capacity is the heat needed per degree of temperature rise. From the first law it takes two values depending on what is held fixed. At constant volume no work is done, so all the heat raises the internal energy:

C_V \;=\; \left(\frac{\partial U}{\partial T}\right)_V.

At constant pressure the gas also does work $p\,dV$ as it expands, so more heat is needed for the same temperature rise. The natural state function is the enthalpy $H = U + pV$ (developed in 3.6), in terms of which

C_p \;=\; \left(\frac{\partial H}{\partial T}\right)_p.

The constant-pressure heat capacity is always the larger of the two, because some of the heat is spent on expansion work rather than on warming the gas.

▶ Mayer's relation: c_p − c_v = R Derivation

For one mole of an ideal gas the internal energy depends only on temperature, so $c_v = dU/dT$ . The enthalpy is $H = U + pV = U + RT$ (using $pV = RT$ per mole), so

c_p \;=\; \frac{dH}{dT} \;=\; \frac{dU}{dT} + R \;=\; c_v + R.

Hence

c_p - c_v \;=\; R.

The difference is exactly the gas constant — the work per mole per degree that the gas does on expanding at constant pressure. ✓

γ from molecular structure

Define the ratio of specific heats

\gamma \;\equiv\; \frac{c_p}{c_v} \;=\; \frac{c_v + R}{c_v} \;=\; 1 + \frac{R}{c_v}.

Equipartition fixes $c_v$ from a count of molecular degrees of freedom: each quadratic degree of freedom contributes $\tfrac12 R$ per mole (from the $\tfrac12 k_B T$ per molecule of the kinetic theory chapter). With $d$ active degrees of freedom, $c_v = \tfrac{d}{2} R$ and

\gamma \;=\; 1 + \frac{2}{d} \;=\; \frac{d+2}{d}.

| Molecule | Active DOF $d$ | $c_v$ | $c_p$ | $\gamma$ | |---|---|---|---|---| | Monatomic (He, Ar) | 3 | $\tfrac32 R$ | $\tfrac52 R$ | $5/3 \approx 1.67$ | | Diatomic (N₂, O₂) | 5 | $\tfrac52 R$ | $\tfrac72 R$ | $7/5 = 1.4$ | | Triatomic (CO₂, H₂O) | 6+ | $\ge 3R$ | $\ge 4R$ | $\le 4/3$ |

So $\gamma$ is a direct readout of molecular structure: measuring the ratio of specific heats of a gas counts its active degrees of freedom. This single number controls the adiabatic compression curve and, through it, the speed of sound — the subject of the next lesson.