Pressure-temperature path-work identity ($W_c=(W_a+W_b)/2$)

Suppose a small volume change obeys $dV=-V_1{\kappa }_{T}dP+V_1\alpha dT$, with its displayed coefficients held constant. Work on the body is $\delta W=V_1{\kappa }_{T}PdP-V_1\alpha PdT$. Two rectangular pressure-temperature paths between the same endpoints differ by $W_a-W_b=V_1\alpha \Delta P\Delta T$, where path a changes temperature first. The straight interpolation samples the mean pressure during its temperature change, giving $W_c=(W_a+W_b)/2$. Variable material coefficients need not preserve that average relation.