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Thermodynamics: PV diagram: High temperature adiabat

  1. Jul 18, 2012 #1
    1. The problem statement, all variables and given/known

    Hi, this is my first post on here.

    Does an adiabat get steeper at higher temperatures? I think it does, but I wanted to make sure I'm thinking about it in the right way. Also, what is the constant in the PV relations?

    2. Relevant equations

    PV=const, for constant temperature.

    PVγ=const, for zero heat transfer.

    γ=(f+2)/f, where f=degrees of freedom.

    3. The attempt at a solution

    Using the equations for total thermal energy, the first law, and the ideal gas law, we can derive relationships between pressure and volume that show us what the PV diagram will look like. Since an isotherm comes from
    PV=const
    and an adiabat comes from
    PVγ=const, where γ=(f+2)/f,
    we can readily see that the adiabat will be steeper than the isotherm.

    However, as temperatue increases, so do the degrees of freedom for a multi-atom molecule. As degrees of freedom increase, so too does the exponent on the V for the adiabatic case. This would lead to a steeper adiabat for non-monatomic molecules at higher temperatures. Furthermore, because the maximum total degrees of freedom possible=3N, where N=number of atoms comprising the molecule, our adiabat would get steeper and steeper for larger and larger molecules at higher temperatures.

    Is that correct?

    As for the constant, I've looked all over my textbook, but it doesn't describe it at all. Is it some kind of characteristic constant dependent on the kind of gas we're talking about? Are the constants in the isothermal case and adiabatic case the same?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jul 18, 2012 #2

    Andrew Mason

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    Science Advisor
    Homework Helper

    What do you mean by "steeper"? Is this a P-V diagram (P on the vertical axis, V on the horizontal)? What part of the curve are you talking about?
    This is not necessarily true. It depends on the temperature. The number of degrees of freedom are limited by geometry. So when all modes are activated, there is no increase in the number of degrees of freedom with further temperature increases.

    The adiabatic condition assumes an ideal gas obeying PV=nRT. As you increase the number of atoms in the molecule, the atoms tend to lose their ideal gas behaviour.

    AM
     
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