Partial derivatives of thermodynamic state functions

[physlosopher]

I'm in a first-year grad course on statistical mechanics and something about multivariable functions that has confused me since undergrad keeps popping up, mostly in the context of thermodynamics. Any insight would be much appreciated!

This is a general question, but as an example imagine you're given an equation of state of a thermodynamic system, specifically its internal energy in terms of temperature, volume, and number of particles (say there's one species of particle).
$$E=E(T,V,N)$$
My ultimate questions are, is the partial derivative $(\frac {\partial E} {\partial V})_{T,N}$ something equilibrium thermodynamics can tell us about, and is it (generally) the case that $(\frac {\partial E} {\partial V})_{T,N}=-p$?

My immediate inclination is that if I want to say anything about pressure for this system, I should write down the exact differential of the internal energy using the First Law: $dE = TdS - pdV + \mu dN$. Please correct me if I'm wrong: this exact differential suggests that the internal energy is a natural function of the entropy, volume, and number, and so if I want to relate $-p$ to the energy by a partial derivative, i.e. $\frac {\partial E} {\partial V}=-p$, I really need to be holding the other natural variables constant - that is, I'm really interested in the partial $(\frac {\partial E} {\partial V})_{S,N}=-p$, where the subscripts denote the variables held constant.

But I have E as a function of T, so if I want the pressure given my equation of state, I'd want to first write temperature in terms of entropy in order to express E in its natural variables, or apply a Legendre transform from the internal energy to the Helmholtz free energy (call it A), in which case I'd have the relationship $(\frac {\partial A} {\partial V})_{T,N}=-p$, correct?

Thanks in advance for any help!

 

[mitochan]

As you say
A=E-TS
dA=-SdT-pdV+\mu dN
(\frac{\partial A}{\partial V})_{T,N}=-p
From here you can derive
(\frac{\partial E}{\partial V})_{T,N}=-p+T(\frac{\partial S}{\partial V})_{T,N}