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**Gibbs energy=chem potential (solved)**

my thermal book gives a hand-waving argument saying the followings:

firstly, Gibbs energy, defined by:

[tex]G\equiv U+PV-TS[/tex]

is an extensive quantity (proportional to N), and also

[tex]\left (\frac{\partial G}{\partial N}\right ) _{T,P}=\mu[/tex]

so far so good, but then it says:

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

Thus,

[tex]G=N\mu[/tex]

But why must [itex]\mu[/itex] be solely dependent on T and V? why can't [itex]\mu[/itex] 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,

[tex]V\sim N[/tex]

[tex]S\sim N[/tex]

[tex]U\sim N[/tex]

Thus,

[tex]\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}[/tex]

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?!

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