Gibbs energy=chem potential (not convinced)

[tim_lou]

Gibbs energy=chem potential (solved)

my thermal book gives a hand-waving argument saying the followings:
firstly, Gibbs energy, defined by:
$$G\equiv U+PV-TS$$

is an extensive quantity (proportional to N), and also
$$\left (\frac{\partial G}{\partial N}\right ) _{T,P}=\mu$$

so far so good, but then it says:

if P and T are held constant then $\mu$ is also constant, which implies whenever a particle is added to the system, G is increased by $\mu$.

Thus,
$$G=N\mu$$

But why must $\mu$ be solely dependent on T and V? why can't $\mu$ depend on.. let's say N? is there any algebraic prove for that?

edit: oh yeah I see... the book skips a very Very important reason of why it works!
since V, S and U are also extensive,
$$V\sim N$$
$$S\sim N$$
$$U\sim N$$

Thus,
$$\left (\frac{\partial G}{\partial N}\right ) _{T,P}=\mu=<br /> \frac{\partial U}{\partial N}+P\frac{\partial V}{\partial N}-T\frac{\partial S}{\partial N}$$

and each of the three partial derivatives is independent of N since V, S and U are directly related to N...

don't you just hate it when books make some non-rigorous arguments, left out the important details and act as if the things are obvious and trivial?!

 

[StatMechGuy]

But what they did is entirely correct. I can always rewrite the chemical potential as a function of other intensive/extensive variables because of the existence of equations of state.

 

[quetzalcoatl9]

you can prove it rigorously, without reference to the macroscopic thermodynamics, by finding $$<N>\mu$$ in the grand canonical ensemble.

 

[tim_lou]

really...? I'm interested... can you provide more details please? I would really love a rigorous argument on this problem.

so, how would you go from the definition of G and mu??

 

[StatMechGuy]

you can prove it rigorously, without reference to the macroscopic thermodynamics, by finding $$<N>\mu$$ in the grand canonical ensemble.

I'm intrigued, since I've never seen this done before. I've always seen, starting from the microcanonical ensemble, a derivation that leads to something that we recognize as F or some such, and then that's the connection to thermodynamics.