Implicit Functions

Along a contour of constant p, when T=f(V) or V=g(T) so that the locus of acceptable V and T results in p staying constant, i.e., P(T,V) = C,

$$\Delta p = p(T+\Delta T, V+\Delta V) - p(T,V) = \left (\frac... ... T +\left (\frac{\partial P}{\partial V}\right )_T\Delta V = 0$$

since p is constant. Then,

$$\left (\frac{\partial P}{\partial T}\right )_V\frac{\Delta T}{\Delta V} = - \left (\frac{\partial P}{\partial V}\right )_T$$

so

$$\left (\frac{\partial T}{\partial V} \right )_p = - \frac{\le... ...V}\right )_T} {\left (\frac{\partial P}{\partial T}\right )_V}$$

after algebraic manipulations and appropriate limit taking.