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Gibbs energy=chem potential (not convinced)

  1. Mar 3, 2007 #1
    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!?!!
     
    Last edited: Mar 3, 2007
  2. jcsd
  3. Mar 4, 2007 #2
    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.
     
  4. Mar 4, 2007 #3
    you can prove it rigorously, without reference to the macroscopic thermodynamics, by finding [tex]<N>\mu[/tex] in the grand canonical ensemble.
     
  5. Mar 5, 2007 #4
    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??
     
  6. Mar 5, 2007 #5
    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.
     
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