Pressure–volume work sign convention

Definition and physical meaning

In this blog we adopt the thermodynamic work sign convention in which $\delta W>0$ denotes work done by the system on the surroundings. For the mechanical work associated with motion of a system boundary under an external pressure, the pressure–volume work is defined by

• Pressure–volume work (by the system): $\delta W_{PV} := P_{\mathrm{ext}}\, dV$.

Here $V$ is the system volume and $P_{\mathrm{ext}}$ is the pressure exerted by the surroundings on the boundary. Physically:

• if the system expands ($dV>0$), it pushes back the surroundings and does positive work, $\delta W_{PV}>0$;
• if the system is compressed ($dV<0$), the surroundings do work on the system and $\delta W_{PV}<0$.

Because work depends on the path, $\delta W_{PV}$ is an instance of inexact work rather than a state function .

Appearance in the first law

With the first law written as $dU = \delta Q - \delta W$, the $P\,dV$ contribution implies (when pressure–volume work is the only mechanical work)

For a quasistatic and reversible process, $P_{\mathrm{ext}}$ equals the system pressure $P$, so one may write $dU=\delta Q-P\,dV$.

Key relations and diagnostics

• Closed-cycle work: over a cycle ,

  $$W_{PV} = \oint P_{\mathrm{ext}}\, dV,$$

  which geometrically is the signed area enclosed in the $P$–$V$ plane (positive for the usual clockwise “engine” cycle under this convention).

• Connection to enthalpy: using enthalpy $H=U+PV$ and $dU=\delta Q-P\,dV$ (reversible, $PV$-only),

  $$dH = \delta Q + V\, dP.$$

  In particular, at constant pressure ($dP=0$) one gets $dH=\delta Q$, which underlies the interpretation of heat at constant pressure.

• Translation to the opposite convention: some texts define work as done on the system, writing $dU=\delta Q+\delta W^{(\mathrm{on})}$. In that convention the same physics is represented by $\delta W^{(\mathrm{on})}_{PV}=-P_{\mathrm{ext}}\, dV$.