Work (inexact differential)

For a thermodynamic system undergoing a process , the work increment $\delta W$ is the infinitesimal amount of energy transferred out of the system (or into it, depending on sign) through organized interactions that can be modeled as generalized forces acting through generalized displacements at the boundary or via a work reservoir .

Like heat , work is an inexact differential: there is no state function $W$ with $\delta W=dW$. Its integral depends on the path in state space, so $\delta W$ is a path function .

This knowl follows the work sign convention used throughout: $\delta W>0$ means work is done by the system on the surroundings .

Physical interpretation

Work represents energy transfer in a form that is, in principle, fully convertible into mechanical or other “organized” forms (lifting a weight, turning a shaft, charging a capacitor). It contrasts with heat, which is energy transfer driven by a temperature difference and typically associated with microscopic degrees of freedom.

Key relations

• First law (closed system): for a closed system ,

  $$dU = \delta Q - \delta W,$$

  where $U$ is internal energy .

• Pressure–volume work: for a simple compressible system with a moving boundary (piston),

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

  where $V$ is volume and $P_{\text{ext}}$ is the external pressure exerted by the surroundings. In a quasistatic process , $P_{\text{ext}}$ matches the system pressure along the path, and for a reversible process this gives the reversible work.

• Generalized work form: many work modes can be written schematically as

  $$\delta W = \sum_i Y_i\, dX_i,$$

  where $X_i$ are generalized displacements (often extensive variables ) and $Y_i$ are conjugate generalized forces (often intensive variables ). The precise list of terms depends on what exchanges are allowed across the boundary (compare open systems versus closed systems ) and on bookkeeping conventions (see chemical work conventions ).

• Cycles: in a cycle of a closed system, $\Delta U=0$, so

  $$\oint \delta W = \oint \delta Q,$$

  emphasizing that the net work over a cycle generally does not vanish and depends on the path taken in state space.