Heat Added During a Thermodynamic Process

Just as the work done depends on the path taken, so does the heat transferred to a system — even between the same two endpoint states. This is shown using two different routes by which an ideal gas, confined to a cylinder with a movable piston, can be taken from an initial volume $V_i$ to the same final volume $V_f$.

Route 1 — controlled isothermal expansion. The gas is heated slowly and carefully, in a controlled manner, so that it expands to the final volume $V_f$ while its temperature is held constant throughout (isothermally). Because the change is slow and controlled, the system absorbs a finite, well-defined amount of heat during the expansion, and correspondingly does a finite amount of work on the piston.

Route 2 — sudden, uncontrolled free expansion. Now imagine the same amount of gas, at the same initial temperature, confined instead to compartment $X$ (volume $V_i$) of an insulated cylinder that is divided by a thin, breakable partition from an empty (vacuum) compartment $Y$ (volume $V_i'$, with $V_i + V_i' = V_f$). The partition is suddenly broken. The gas rushes to fill the whole combined volume $V_f$ essentially instantaneously. Because the cylinder is insulated, no heat crosses its boundary at all ($Q = 0$); and because the gas is not pushing against any piston or other movable surface as it expands into the vacuum, no work is done either ($W = 0$). This uncontrolled process is called free expansion. …