Heat (inexact differential)

For a thermodynamic system undergoing a thermodynamic process , the heat increment $\delta Q$ is the infinitesimal amount of energy transferred into the system across the system boundary because of a temperature difference (as opposed to energy transfer classified as work ).

The notation $\delta Q$ (rather than $dQ$) emphasizes that heat is an inexact differential: there is no state function $Q$ with $\delta Q = dQ$. Equivalently, the integral $\int_{\Gamma} \delta Q$ depends on the process path $\Gamma$, so heat is a path function .

Throughout these knowls, the sign of $\delta Q$ is coordinated with the work sign convention : $\delta Q>0$ means heat flows into the system.

Physical interpretation

Heat is not a “thing contained in the system”; it is a bookkeeping label for one channel of energy transfer. Once energy has entered the system as heat, it contributes to changes in state variables such as internal energy and entropy , but the amount “heat contained” is not defined independent of the process.

A diathermal wall permits $\delta Q\neq 0$ (heat exchange), while an adiabatic wall enforces $\delta Q=0$.

Key relations

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

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

  where $U$ is internal energy and $\delta W$ is work (inexact differential) .

• Cycles: in a cyclic process , $\Delta U=0$, so

  $$\oint \delta Q = \oint \delta W.$$

  This equality highlights that $\oint \delta Q$ is generally nonzero, reinforcing that $\delta Q$ is not exact.

• Reversible link to entropy: for a reversible process , the second law gives

  $$\delta Q_{\mathrm{rev}} = T\, dS,$$

  relating heat to temperature $T$ and thermodynamic entropy $S$. The factor $1/T$ acts as an integrating factor that turns $\delta Q_{\mathrm{rev}}$ into an exact differential.