Path Dependence Of Heat And Work

Heat and work are path functions, meaning their values depend on the specific route taken between two states, whereas internal energy is a state function depending only on the endpoints. Since both paths share the same initial and final states, the change in internal energy $\Delta U$ is identical for them, fixed entirely by the temperature difference between start and finish. The heat absorbed differs because each path exchanges energy with the surroundings differently. The work done by the gas equals the area under each path on a PV diagram, and those areas plainly differ for an isothermal curve versus an isobaric-isochoric staircase. Therefore neither heat nor work alone, nor both together, is guaranteed equal. The First Law $$\begin{gathered} \Delta \\ cup=Q-W \end{gathered}$$ ties them together so that on each path $Q$ and $W$ adjust to give the same $\Delta U$. This path independence is the defining mathematical property of a state function, whose differential is exact, in contrast to heat and work whose infinitesimals are inexact and require a path to be specified. As a consistency check, the isothermal path keeps $T$ constant so $$\begin{gathered} \Delta \\ cup=0 \end{gathered}$$ there, and the two-step path must also return to that same temperature, confirming both routes share the identical internal energy change.