- #1
tim_lou
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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?!
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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