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Deriving thermodynamic relations

  1. Jul 21, 2011 #1
    1. The problem statement:

    Show that

    a) (∂H/∂T)V = CV(1 - βμ/κ)

    b) (∂H/∂V)T = μCP/Vκ

    c) (∂T/∂V)H = μ/(V(μβ - κ))


    2. Relevant equations:

    i) β = (1/V)(∂V/∂T)P

    ii) κ = -(1/V)(∂V/∂P)T

    iii) β/κ = (∂P/∂T)V

    iv) CV = (∂U/∂T)V

    v) CP = (∂H/∂T)P

    vi) CP - CV = TVβ2

    vii) η = (∂T/∂V)U = (1/CV)(P - Tβ/κ)

    viii) μ = (∂T/∂P)H = (V/CP)(βT - 1)


    3. The attempt at a solution:

    a) H = U + PV
    (∂H/∂T)V = (∂U/∂T)V + V(∂P/∂T)V

    Using (iv) and (iii):
    (∂H/∂T)V = CV + Vβ/κ --> stuck


    b) H = U + PV
    (∂H/∂V)T = (∂U/∂V)T + P + V(∂P/∂V)T

    The change in internal energy with respect to volume at constant temperature for an ideal gas is 0, and using (ii):
    (∂H/∂V)T = P - 1/κ --> stuck


    c) I have no idea how to get this one.
     
  2. jcsd
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