= Pressure-temperature path-work identity
{title2=$W_c=(W_a+W_b)/2$}
Suppose a small <volume> change obeys $dV=-V_1\kappa_TdP+V_1\alpha dT$, with its displayed coefficients held constant. Work on the body is $\delta W=V_1\kappa_TP\,dP-V_1\alpha P\,dT$. 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.
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