The "cyclic relation" (sometimes called Euler's chain relation) is a calculus identity which is very useful in thermodynamics. This relation can appear in several different forms all of which are equivalent. The form that we will find most useful is,

We will use this in many different contexts, but one example would be the calculation of the derivative,
using the van der Waals equation of state. The derivative
is difficult to calculate directly because solving for *V* as a function of *p* and *T* requires solving a cubic equation in *V*. One can obtain this derivative from the cyclic relation by calculating,

It is relatively easy to solve the van der Waals equation of state for *p* in terms of *V* and *T* and to take the appropriate derivatives shown.

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