The curve in the P-V plane

Since during a reversible transformation the gas is always in equilibrium, the pressure $P$ and the volume $V$ have a well-defined value at every instant. This makes it possible to represent the whole transformation as a curve in the $(V, P)$ plane: every point on the curve is an equilibrium state passed through by the gas, and the curve as a whole tells the story of the process, not just its beginning and its end.

Each of the four fundamental transformations has a characteristic shape in this plane. The isochore (constant volume) is a vertical segment: the gas changes pressure without changing volume. The isobar (constant pressure) is a horizontal segment. The isotherm (constant temperature) follows the hyperbola $PV = nRT = \text{const}$. The adiabat (no heat exchange) follows $PV^\gamma = \text{const}$ and is steeper than the isotherm, because the exponent $\gamma > 1$ makes the pressure fall faster as the volume increases.

The four reversible processes in the $P$-$V$ diagram. The area under the curve represents the work done by the gas. The adiabat is steeper than the isotherm: $PV^\gamma$ against constant $PV$.

A practical advantage of this representation is that the work done by the gas can be read directly as the area under the curve. In the case of the isobar, for example, the area is simply that of a rectangle with height $P$ and base $\Delta V$:

$\ev{L = P\,\Delta V}$

Only reversible processes have a curve!

To repeat: only reversible processes produce a well-defined curve in the $P$-$V$ diagram. During a free expansion or an irreversible thermal equilibration, $P$ and $V$ are not defined at every instant, and it makes no sense to trace a path. Only the points $A$ and $B$, initial and final, can be marked.

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Topics: Thermodynamics Concepts: Thermodynamic transformations Skills: Reading graphs Methods: P-V and ln P - ln V diagram Objects: Ideal gas

Related exercises: Worked exercise — reversible vs irreversible isothermal expansion · Problem — Rectangular cycle in the p-V plane · Problem — Area of a p-V cycle