1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Entropy expression

  1. May 25, 2014 #1


    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data

    From the 2nd TD law:
    [itex] TdS= d(\rho V) + P dV - \mu d(nV) [/itex]

    find that:
    [itex] S= \frac{V}{T} (\rho+P- \mu n)[/itex]

    2. Relevant equations

    [itex] \frac{dP}{dT}= \frac{P+\rho - \mu n}{T}[/itex]

    3. The attempt at a solution
    [itex] TdS= d(\rho V) + P dV - \mu d(nV) [/itex]

    [itex] TdS= d[(\rho+ P- \mu n) V] - V dP + nV d \mu [/itex]

    or I can write:

    [itex] dS=\frac{1}{T} d[(\rho+ P- \mu n) V] - \frac{V}{T} dP + \frac{nV}{T} d \mu [/itex]

    Now I write that (using the given formula):
    [itex] dP= \frac{P+\rho - \mu n}{T} dT [/itex]

    [itex] dS=\frac{1}{T} d[(\rho+ P- \mu n) V] - V (P+\rho - \mu n) \frac{dT}{T^{2}} + \frac{nV}{T} d \mu [/itex]

    My problem is that this result gives the correct formula I'm looking for the entropy, except for the last [itex] \frac{nV}{T} d \mu [/itex]

    For the last I tried to take:

    [itex] \frac{dS}{dT}= \frac{dS}{d \mu} \frac{d \mu}{dT}= \frac{nV}{T} \frac{d \mu}{dT} [/itex]

    [itex] \frac{d^{2}S}{d(nV)dT}= \frac{1}{T} \frac{d \mu}{dT} [/itex]

    [itex]\frac{dS}{d(nV)}= -\frac{ \mu }{T}[/itex]

    [itex]\frac{d^{2}S}{dT d(nV)}= \frac{ \mu }{T^{2}}[/itex]

    So that I have:

    [itex] \frac{1}{T} \frac{d \mu}{dT} =\frac{ \mu }{T^{2}}[/itex]
    That means:
    [itex] d \mu = \frac{ \mu }{T} dT [/itex]

    Inserting in the expression for the entropy at last:

    [itex] dS=\frac{1}{T} d[(\rho+ P- \mu n) V] - V (P+\rho - \mu n) \frac{dT}{T^{2}} + (nV \mu) \frac{dT}{T^{2}} [/itex]

    Which can't be written as:

    [itex] dS= d[(\rho+ P- \mu n) \frac{V}{T}] [/itex]
    due to the last term...

    Any help?
  2. jcsd
  3. May 25, 2014 #2
    Start with:

    [tex]dU = TdS - pdV + \mu dN[/tex]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted