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Non-ideal Gas

  1. Feb 11, 2006 #1
    For a van der Waals gas experiencing an adiabatic process derive the following expression:

    T(V-nb)^(R/Cv) = Constant

    I tried using PV^gamma = Constant with gamma = Cp/Cv
    and Cp - Cv = nR with PV = nRT but could not get it.

    Any hints?

    I would have to use Boyle's law to account for the factor of b but I'm not sure of its relavance to the problem.
     
    Last edited: Feb 11, 2006
  2. jcsd
  3. Feb 12, 2006 #2

    Astronuc

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  4. Feb 12, 2006 #3
    This problem is specifically non-ideal but the equations only apply to the ideal case.
     
  5. Feb 12, 2006 #4

    Astronuc

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    Well perhaps one can start with P (V-nb) = nRT, which ignores the +an2/V2 term, which correct pressure, i.e. (P + an2/V2).
     
  6. Feb 12, 2006 #5
    Okay so, it's non-ideal therefore V = V - nb
    It's adiabatic so the other ideal equations still hold.
    And this is a van der Waals gas.

    If I do start with P (V-nb) = nRT then I would have to eventually end up with T(V-nb)^R/Cv = Constant. But how would I ever get that power R/Cv?
     
  7. Feb 12, 2006 #6
    I prove nothing:

    P(V-nb)^R/Cv = nRT
    Cp-Cv = nR
    Cp = nR + Cv

    gamma = Cp/Cv = (nR+Cv)/Cv = 1+nR/Cv

    T(V-nb)^(gamma -1) = Constant
    TP^(1/gamma -1) = Constant

    T(V-nb)^(gamma -1) = TP^(1/gamma -1)

    gamma root((V-nb)) / (V-nb) = gamma root(P)/P

    P/(V-nb) = gamma root(P/(V-nb))

    (P/(V-nb))^gamma = P/(V-nb)

    (P/(V-nb))^(1+nR/Cv) = nRT

    This gets me nothing, hints?
     
  8. Feb 12, 2006 #7

    Gokul43201

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    1. Replace V by V-nb
    2. Write the new equation of state
    3. Write the new adiabatic equation
    4. Substitute and complete

    That will give you the result of post#1

    PS : If you have trouble, perform the above steps and we'll help from wherever you are stuck...
     
  9. Feb 14, 2006 #8
    So we know V = V-nb
    PV^gamma = cst

    gamma = Cp/Cv
    Cp-Cv = nR
    Cp = nR + Cv

    P(V-nb) ^ gamma = cst
    P(V-nb) ^ Cp/Cv = cst
    P(V-nb) ^ (nR+Cv/Cv) = cst
    P(V-nb) ^ (nR/Cv +1) = cst


    Adiatbatic process so Q = 0 and dU = -W = CvMdT
    Where does T supposed to come from in the T(V-nb)^R/Cv = cst?
    How do I cancel n in the power?
     
    Last edited: Feb 14, 2006
  10. Feb 14, 2006 #9
    Fine, let me try again.

    Non-ideal Gas

    --------------------------------------------------------------------------------

    For a van der Waals gas experiencing an adiabatic process derive the following expression:



    PV ^ gamma = cst.
    P=nRT/V

    nRTV^(gamma -1) = cst.
    TV^(gamma -1 ) = cst.

    (P + n(a/v)^2)(V-nb)^gamma = cst.
    (P + n(a/v)^2)(V-nb) = nRT

    Dividing those two we get

    (V-nb)^(Cp/Cv - 1) = cst/nRT

    T(V-nb)^(Cp/Cv - Cv/Cv) = cst/nR

    But cst/nR is a cst. so

    T(V-nb)^(nR/Cv) = cst

    There I'm close but I still have that n term in the power.

    What do I do eliminate it?
     
  11. Feb 15, 2006 #10
    Okay i got it. but what is the justification for not useing P:P + an2/V2)

    Please tell me before 8 hours from this post
     
  12. Feb 15, 2006 #11

    Gokul43201

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    Guess this is too late...but for what it's worth, Cp and Cv are the molar specific heats (heat capacity per mole of gas).

    This gives Cp - Cv = R
     
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