Notes on Maxwell relations

If you're confused about Maxwell relations, here is something of an additional explanation. Consider the derivative:

(dN/dT)_(V,mu)

The first thing you want to do is identify from which thermodynamic potential this derivative stems. To do that, we look at the independent variables. These are the derivative variable (T) as well as the constant condition variables (V,mu). Thus, we want a function whose natural variables are T,V,mu.

We don't actually know offhand such a function, but we can construct one. Look at the Helmholtz free energy,

dA = -S dT -P dV + mu dN

Here we have independent variables T,V, and N. We need to swap mu for N. To do this, we just perform a Legendre transform to get some new potential Y:

Y = A - mu N
dY = -S dT - P dV - N dmu

Now we can use Y to develop our Maxwell relation. We see that our original function is just

-d2Y/dT dmu = -d(dY/dmu)/dT = dN/dT

If we switch the order of the derivatives, we get

-d2Y/dmu dT = -d(dY/dT)/dmu = dS/dmu

Therefore, we find that:

(dN/dT)_(V,mu) = (dS/dmu)_(T,V)

So the general rule of thumb is:

1. Determine which thermodynamic potential underlies the derivative of interest.
2. Construct that potential via Legendre transforms, if necessary.
3. Find the second derivative giving rise to your derivative of interest, and evaluate it in two different cases where the order of the derivatives is switched.

There is one point on which you must be careful. Not every derivative of a thermodynamic property can be related by a Maxwell relation. For example, consider:

(dT/dS)_(V,N)

This derivative comes from the second derivative of the energy:

d2E/dS2

Note that this is not a cross-derivative, between two independent variables. Therefore (dT/dS)_(V,N) cannot be related to another derivative.

MSS