We originally defined our variables in terms of U, but now that we have more expressions for the energy, we can devise new expressions of the variables as well.

For example, we originally defined the temperature as τ = . But we can also use the enthalpy identity to write τ = .

It isn't hard to formulate these definitions on your own. Find an energy identity in which the variable you wish to define is free, and then take the other two differentials constant and solve. Say we wish to look at the entropy in terms of the Helmholtz free energy. We see that V and N are in differentials in F, and so we write: σ = - .

Many other relations exist, but we will leave you to derive them on your own and in the problems at the end of the section. Again, understanding this fluidity and flexibility in definition will be key to solving problems efficiently.

Above we showed how to define the variables in terms of the energy, but we can bypass the energy by keeping it constant. For example, suppose the energy U were held constant, as were the number of particles. Though we will gloss over some mathematics here, it seems plausible that you could then write: p = τ.

Moreover, we can use the definitions of the other energies to obtain more complex formulations of the variables. Take F = U - τσ. We know that p = - . We can take the derivative to obtain:

This result is interesting because we can think of the first term as the pressure due to energy and the second as the pressure due to the entropy.

What started as a simple picture may be looking confusing now, but remember that underneath all of the new equations that are arising is the Thermodynamic Identity, and the definitions of the other energies from that. Everything else follows from those and can be rederived without much difficulty.

We can utilize yet another mathematical truth, that double derivatives commute, to derive a new set of relations known as the Maxwell Relations.

We know, for example, that = - σ and that = V, by the methodology just discussed. Now, though, we can take the partial derivative of the first equation with respect to p, holding τ constant, to obtain:

Similarly, we can take the partial derivative of the second equation with respect to τ, holding p constant, to obtain:

The mathematical truth useful here is that:

Therefore, we can say that:

Using a similar method, we can derive a total of 12 Maxwell Relations. These are useful in relating any two sets of variables, a set consisting of the intensive and extensive variables that are coupled in the identities.