Work sign convention

A work sign convention specifies whether work is counted as positive when it is done by the system or on the system. Because heat and work appear together in the first law of thermodynamics , the sign convention must be fixed to interpret equations and plots consistently.

Convention used in these knowls:

• $\delta W>0$ means work is done by the system on the surroundings .
• $\delta Q>0$ means heat flows into the system.

With this choice, the first law for a closed system is

where $U$ is internal energy and $\delta W$ is work .

Physical interpretation

Under this convention, $\delta W>0$ corresponds to energy leaving the system in an organized form (pushing a piston, turning a shaft), while $\delta Q>0$ corresponds to energy entering because of a temperature difference. The sign convention is not physics; it is bookkeeping. Physics enters through consistent application and through inequalities such as the Clausius inequality .

Key consequences and examples

• Pressure–volume work: for boundary work against an external pressure,

  $$\delta W_{PV} = P_{\text{ext}}\, dV.$$

  Since $dV>0$ for expansion, expansion gives $\delta W_{PV}>0$ (the system does work on the surroundings). Compression gives $\delta W_{PV}<0$. A more focused discussion can be found under pressure–volume work sign convention .

• Reversible simple compressible system (fixed particle number): combining $\delta Q_{\mathrm{rev}} = T\, dS$ from entropy with $\delta W_{\mathrm{rev}} = P\, dV$ yields

  $$dU = T\, dS - P\, dV,$$

  consistent with the fundamental relation (energy representation) for such a system.

• Alternative convention (common in some chemistry texts): some authors define $\delta W_{\text{on}}>0$ for work done on the system. Then $\delta W_{\text{on}} = -\delta W$ (with $\delta W$ as defined here) and the first law becomes

  $$dU = \delta Q + \delta W_{\text{on}}.$$

  Either convention is valid if used consistently, including in definitions of thermodynamic potentials such as Helmholtz free energy and Gibbs free energy and in any conventions for particle exchange (see particle number and chemical potential together with chemical work conventions ).