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Adiabatic Expansion Question

  1. Jan 14, 2008 #1

    nicksauce

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    1. The problem statement, all variables and given/known data
    Show that when an ideal gas expands adiabatically, the temperature and pressure are related by the differential equation:
    [tex]\frac{dT}{dP} = \frac{2T}{5P} [/tex]

    2. Relevant equations
    Ideal gas law


    3. The attempt at a solution
    [tex]PV = nkT[/tex]

    [tex]T = \frac{PV}{nk}[/tex]

    [tex]\frac{dT}{dP} = \frac{1}{nk}(V + P\frac{dV}{dP})[/tex]

    So what is dV/dP?

    [tex]PV^\gamma = C[/tex]

    [tex]V = C^{\frac{1}{\gamma}} P^{\frac{-1}{\gamma}}[/tex]

    [tex]\frac{dV}{dP} = C ^ {\frac{1}{\gamma}} \frac{-1}{\gamma}P^{\frac{-1}{\gamma} - 1}[/tex]

    [tex]\frac{dV}{dP} = \frac{-1}{\gamma} VP^{-1} [/tex]
    so

    [tex]\frac{dT}{dP} = \frac{1}{nk}[V - \frac{V}{\gamma}][/tex]

    But this gives dT/dP ~= 1/3 T/P (using gamma~=3/2), instead of 2/5 T/P. Can anyone see where I went wrong?
     
  2. jcsd
  3. Jan 14, 2008 #2

    nicksauce

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    Err. never mind, gamma should be 5/3, not 3/2, in which case this does give the correct answer.
     
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