Isothermal Process

An isothermal process is one in which a system's pressure and volume change while its temperature is held exactly constant, so $\Delta T = 0$. This requires the system to stay in very good thermal contact with an environment at essentially fixed temperature, and for any heat transfer to happen extremely slowly, so that thermal equilibrium (Section 4.2) is maintained at every instant throughout the change. Melting ice at its constant melting point is a familiar physical example of an isothermal process.

Equation of state. Since the temperature is fixed, the ideal gas equation $pV = nRT$ reduces to Boyle's Law:

so for the initial and final states of an isothermal change, $p_iV_i = p_fV_f = \text{constant}$.

Work done. Starting from the infinitesimal work $dW = p\,dV$ (Eq. 4.1) and substituting $p = nRT/V$ from the ideal gas equation (with $T$ constant, so it can be pulled outside the integral):

Heat exchanged. Because the internal energy of an ideal gas depends only on its temperature (Section 4.4.1), and the temperature is unchanged in an isothermal process, $\Delta U = 0$ throughout. Applying the First Law, $\Delta U = Q - W$, with $\Delta U = 0$:

So for an isothermal process, $Q = W = nRT\,\ln(V_f/V_i)$: all the heat absorbed by the gas is converted completely into the work it does in expanding (and, on compression, all the work done on the gas is released completely as heat). Note that although $Q = W$, neither individually needs to be zero — for an isothermal process, none of $Q$ and $W$ is zero (only $\Delta U$ is), and consequently the total internal energy content of the system does not stay 'constant' in any deeper sense — heat is continuously exchanged with the environment even while $U$ itself does not change. …