The First Law and Heat Capacities

Two Ways to Change the Energy

The first law states that internal energy changes only through heat and work:

$$dU = \delta Q + \delta W = \delta Q - P\,dV$$

$U$ is a state function while $Q$ and $W$ are not, the whole content of the law is that this particular combination is path-independent. It rules out perpetual motion of the first kind: a machine producing work from nothing.

Heat capacity is the energy needed per degree, and it depends on what is held fixed:

$$C_V = \left(\frac{\partial U}{\partial T}\right)_V, \qquad C_P = \left(\frac{\partial H}{\partial T}\right)_P$$

where $H = U + PV$ is the enthalpy. At constant volume no work is done, so all the heat raises $U$. At constant pressure the gas also expands and does work, so more heat is needed for the same temperature rise, hence $C_P > C_V$ always. For an ideal gas the excess is exactly the work of expansion:

$$C_P - C_V = nR$$

The ratio $\gamma = C_P/C_V$ characterises the molecule: $5/3$ for a monatomic gas, $7/5$ for a diatomic one at room temperature. Equipartition explains why, each quadratic degree of freedom contributes $\tfrac{1}{2}k_BT$ per molecule, and a diatomic molecule has two rotational modes a monatomic one lacks.

Applied to the standard processes, the first law gives:

Common pitfall: assuming an adiabatic process is also isothermal. In an adiabatic expansion no heat enters, yet the gas does work, so its internal energy, and therefore its temperature, must fall. Adiabatic and isothermal are different curves, and the adiabat is the steeper of the two.