Relation Among Intensive Parameters

[Juliush]

Hello all, this is my first post! Hopefully I can gain some valuable insight.

Homework Statement

Find the relation among T, P and mu for the system with the given equation

U = b S⁴/NV²​

I let b equal the several constants stated in the problem.

Homework Equations

T=dU/dS
P=-dU/dV
mu=dU/dN
The Euler Equation for Thermodynamics U = TS - PV +mu*N
Gibbs Duhem Relation : mu = -sdT + vdP with s = S/N and v = V/N

The Attempt at a Solution

I guess my biggest issue is understanding what is meant by 'relation'. Do I find mu as a function of T and P? If so, I cannot find a way (using the Gibbs Duhem relation) to express 's' and 'v'. I have already found the partial derivatives (equations of state) of the system. Any help would be greatly appreciated!

P.S. I'm not familiar with Latex so I apologize for any misleading notation :)

 

[tia89]

My guess is that they ask to express all those quantities as functions of the others... which you can easily do using the definition. Notice also that after computing all of them you can use Euler Equation to check the result. Other than this, I really don't see what else you could find with what you provided!

 

[Juliush]

I talked to my professor and here's the solution: Once I have all of my partial derivatives, although there are three extensive parameters, really only the entropy and volume are at play here, such that if I express entropy and volume as per-mole quantities, the Ns disappear from all of my partial derivatives. Thus, I can rewrite mu/T and mu/P as functions of molecular entropy and volume and substitute those into the Gibbs Duhem equation dmu = -sdT + vdP by solving for s and v. From there it's straightforward integration.