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Thermodynamic Proof

  1. Nov 14, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that for an isentropic compression/expansion process that [itex]Pv^k=constant[/itex]


    2. Relevant equations

    The usual thermodynamic potentials, maxwell relations


    3. The attempt at a solution

    The solution I am arriving at follows that of the solutions manual, but there is a substitution they use in the proof that is non-intuitive to me.

    [tex]s=s(P,v)[/tex]
    [tex]ds=\left(\frac{∂s}{∂P}\right)_vdP+\left(\frac{∂s}{∂v}\right)_Pdv[/tex]

    Since the process is isentropic, ds=0, and using maxwell substitutions we arrive at:

    [tex]dP-\left(\frac{∂P}{∂v}\right)_sdv=0[/tex]

    This is where our proofs are the same, they then diverge by making the substitution, and dividing by pressure:

    [tex]k=-\frac{v}{P}\left(\frac{∂P}{∂v}\right)_s[/tex]
    [tex]\frac{dP}{P}+k\frac{dv}{v}=0[/tex]
    [tex]lnP+klnv=constant[/tex]
    [tex]Pv^k=constant[/tex]

    I don't find their substitution very intuitive. They describe it as the isentropic expansion coefficient. I do not understand how they can treat it as a constant however in the differential equation. Likewise, I feel that by their solution, if k can be treated as a constant then I would expect the partial differential component could be treated as such as well and the portion I had completed could be used to arrive at an algebraically different equation that could be rearranged but I don't see it. Can anyone help me understand the reasoning behind all of this?

    NOTE: Later as a part of the problem it asks to prove that k reduces to the ideal gas ratio, for an ideal gas. I also tried working it from that angle but to no avail.
     
  2. jcsd
  3. Nov 14, 2012 #2
    Are you sure they are claiming this more generally than for an ideal gas?
    I remember that even for Van der Vaals Gases this expression has to be modified in a way similar to the equation of state!
     
  4. Nov 14, 2012 #3
    Yes I am sure it is. The latter part of the proof is shown to be proven through further maxwell relations coupled with the cyclic properties of partial derivatives. I just don't understand the choice made for k and setting it constant. It just seems a bit arbitrary to me.
     
  5. Nov 15, 2012 #4
    It has been a while since I last did thermodynamics. But why may I not write this:

    [itex] k \equiv -\frac{\frac{dp}{p}}{\frac{dV}{V}} = \frac{Vdp}{pdV}[/itex]

    and then, using that I have an isentropic process:

    [itex]k = \frac{dH}{dU} = \frac{C_p}{C_v}[/itex].

    If this is still valid, then [itex]k[/itex] can not be constant in a real polyatomic gas. With varying temperature the number of "frozen" degrees of freedom in the gas will change!
     
  6. Nov 19, 2012 #5
    Sorry for my absence! Although I am not familiar with your first expression, the second I am familiar with as a definition for k. That ratio is constant for an ideal gas which is what the reduction should go to, but if that is the case I cannot see how they would allow it to be considered constant in the first portion of the proof.
     
  7. Nov 19, 2012 #6
    Exactly.
     
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