Hello, I'm pretty horrible at thermodynamics (one of my weak subjects I'll admit) so sorry if my questions are real basic. My main question is "How do we know or find out the 'proper' variables to use?". For example, we are usually given E=E(S,V,N) so that dE=TdS-pdV+μdN. How do we know/find out that S, V and N are the "proper" variables to use, and not, for example, T, V, and N? I am told that, if I am given E=E(T,V,N), then I don't have "the whole picture" (in that there will be some state variables or state functions I can't deduce from this), but I can't see why this is. I know that if we wanted to use T, V and N as our proper variables, we would instead use the Helmholtz free energy F=E-TS, but I've only ever seen this derived after already using dE=TdS-pdV+μdN. Another question. S, the entropy, is a state function. But I am also given that dS=dQ/T. Integrating in a cycle dQ/T=0 only if the process is reversible. On the other hand, it is required that for any state function W, the integral of dW in a cycle (going from an initial state to the same initial state) must result in 0 because the state function must depend only on endpoints, and not the path. So, what am I getting wrong or what am I missing? Thanks.