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Adiabatic expansion

  1. Feb 4, 2010 #1
    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: dT/dP = (2/f+2)(T/P).

    2. Relevant equations

    PV=NkT
    VT^(f/2) = constant
    V^(gamma)*P = constant

    3. The attempt at a solution

    I started off with the formula for ideal gases, PV=NkT.

    I rearranged to get T=(PV/Nk).

    At this point I don't know where to go. I don't see any equations I can use to make substitutions and I'm not sure if I should take the derivative at this point or not?
     
    Last edited: Feb 4, 2010
  2. jcsd
  3. Feb 4, 2010 #2

    kuruman

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    Use the ideal gas law to replace V in the second "relevant equation" that you posted.
    Solve for T in terms of p.
    Take the required derivative dT/dp.
     
  4. Feb 4, 2010 #3
    Okay. I'm getting a little tripped up at the derivative part. I have at this point:

    T = (c/Nk*P)^(2/f+2) , where c is a constant

    Taking the derivative will bring down the 2/f+2, but that leaves me with (2/f+2)-1 as the exponent plus the derivative of the inside.
     
  5. Feb 5, 2010 #4

    kuruman

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    A constant is a constant is a constant so you can write

    [tex]T=CP^{\frac{2}{f+2}}[/tex]

    Then you say that

    [tex]\frac{dT}{dP}=C\frac{2}{f+2}\:P^{\frac{2}{f+2}-1}[/tex]

    I am not sure what you mean by "the derivative of the inside." What do you get when you simplify the exponent? How is that related to the expression of T as a function of P?
     
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