Legendre transform of the energy with respect to temperature for the classical ideal gas
Starting with the fundamental equation in the entropy representation (that is, the entropy of the classical ideal gas-which you have memorized):
1. Derive the fundamental relation in the energy representation (U = U(S, V, N)).
2. Derive the temperature as a function of entropy, volume, and number of particles (one of the three equations of state).
3. Find the entropy as a function of temperature, volume and number of particles.
Is this an equation of state or a fundamental relation?
4. Derive the Helmholtz free energy of the classical ideal gas, F(T,V,N) = U - T S.