Thermodynamics Finding a fundamental equation.

[corr0105]

Thermodynamics! Finding a fundamental equation.

This is the question:
"While he Gibbs free energy G is the fundamental function of the atral variables (T,p,N), (T=temperature, p=pressure, N=number of molecules), growing biological cells often regulte not the numbers of molecules N, but the chemical potentials μ. That is, they control concentrations. What is the fundamental function Z of natural variables (T,p,μ)?

I know a few equations that deal with Gibbs free energy:
G=H-TS
G=$$\sum$$μN

Basically, I have no idea where to start with this problem. If anyone can give me a push in the right direction that would be much appreciated!

 

[Mapes]

You need to apply a Legendre transform (so check out what this is). An example of of Legendre transform: we use G instead of H because we wish to use T as a natural variable instead of S. Now you've learned that you'd also like to use μ instead of N. This should be enough to get you started.

 

[corr0105]

Thanks so much, I looked up how to use Legendre transforms for this type of problem and eventually figured it out (not to mention learned something!). Thanks for your help!