# Gibbs energy=chem potential (not convinced)

by tim_lou
Tags: convinced, energychem, gibbs, potential
 P: 688 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, $$\left (\frac{\partial G}{\partial N}\right ) _{T,P}=\mu= \frac{\partial U}{\partial N}+P\frac{\partial V}{\partial N}-T\frac{\partial S}{\partial N}$$ 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!?!!
 P: 701 you can prove it rigorously, without reference to the macroscopic thermodynamics, by finding $$\mu$$ in the grand canonical ensemble.
 Quote by quetzalcoatl9 you can prove it rigorously, without reference to the macroscopic thermodynamics, by finding $$\mu$$ in the grand canonical ensemble.