Paths in the P-V plane (heating processes)

Problem

In the $P$-$V$ plane, sketch, starting from the same initial state $A$, the qualitative path of (a) heating at constant volume and (b) heating at constant pressure. Indicate the direction of travel. State whether it is possible to draw a curve also for (c) a free expansion into a vacuum, justifying your answer.

Solution

(a) Constant volume (isochore). $V$ stays fixed while $T$ increases; from the gas law $P = nRT/V$ increases with $T$. The path is a VERTICAL segment going up.

(b) Constant pressure (isobar). $P$ stays fixed while $T$ increases; the volume $V = nRT/P$ increases. The path is a HORIZONTAL segment going right.

(c) Free expansion. It is NOT possible to draw a curve: during a free expansion the gas passes through non-equilibrium states in which $P$ and $V$ are not well defined across the whole gas. Only the initial and final states are points in the plane, but the path between them does not exist as a curve.

$\ev{(a)\ \text{vertical isochore},\ (b)\ \text{horizontal isobar},\ (c)\ \text{cannot be drawn}}$

Links

Topics: Thermodynamics Concepts: Thermodynamic processes Methods: P-V and ln P - ln V diagram Objects: Ideal gas