Maxwell Relations Maxwell relations are a “family” of relations between derivatives of thermal parameters. Their list is quite long. We will show how to derive some we most often use. The procedure used for deriving them is always the same – first, you take a differential of U, or the differential of one of the “thermodynamic functions” that are obtained by performing a Legendre transformation of U. We have no time in this course to discuss in detail what a Legendre Transformation is – but the transformed functions, as you can see below, have rather simple forms: H ≡ U + pV – this is the enthalpy we already know; F ≡ U – TS – is called the Helmholtz Free Energy; G ≡ U + pV – TS – is called the Gibbs Function. The list is much longer, but these three are the most often used ones.

The Maxwell relation we obtain by processing U : We begin with writing the First Law in the form: It clearly suggests a functional relation: Therefore, one can write: From the two differential forms it follows that: Now, we simply apply the Euler Criterion and we get a Maxwell relation:

Now, let’s do the same trick using the Gibss Function: By comparing the two dG forms, we get: By applying the Euler Criterion to the dG form obtained, we get another useful Maxwell relation: We will soon use these two!

Heat capacities expressed in terms of entropy and enthalpy All essential parameters describing thermal systems are interrelated through the state equations. Therefore, one parameter can be expressed as a function of the other parameters in different ways. For instance, entropy can be though of either as a function of T and V: S=S(T,V), or as a function of T and p: S=S(T,p). In a constant V process dV=0, hence: meaning that:

A similar trick can be done to obtain the heat capacity Cp expressed in terms of entropy: In a p=const. process dp=0, and then: meaning that: In term of enthalpy H: we have already shown earlier that:

Heat capacity – summary: BUT !!!!

Joule-Thompson Process again: We further processed that derivative, but is was a very tedious job. Earlier we have shown (File 4, slide 5) that: Here is a much faster method. 1. Let’s use the cyclic chain rule: 2. Take the Gibbs function and the enthalpy: So: On the preceding slide we showed this is Cp

Continiued from Slide 7: Use the standard coefficient: to obtain a compact form: Putting everyting together, we obtain: which is exactly the same expression as that we have obtained earlier using a much more time-consuming method.