Simple Transformations of Partial Derivatives

It is often helpful in thermodynamic manipulations to. be able to replace one partial derivative with an equivalent but more convenient expression. For example, the reciprocity condition (2.9) may be used to interchange variables in a partial derivative. Like the reciprocity relation, most such transformations derive quite simply from the properties of partial derivatives. As an example, consider equation (2.5) for the volume of an ideal gas. Equation (2.6) may be solved for (dV/dT)p by in effect dividing through by dT. For a system at constant volume, dV = 0, so 

Such derivatives may also be treated, within limits, as simple fractions. Thus 

Note that all three of the variables V, P, and T are interrelated interchangeably by these two expressions. Thus, experimental determination of any two of these partial derivatives would give the third. 

A new dependent variable, such as x(P, T) in the above example could be introduced by the relationship 

we might wish to evaluate the partial derivatives dV/dT) or dV/dP)x-These may be obtained from (2.6) by taking the appropriate total derivative with the restriction that x be held constant  

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