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Equation for S from state func and C

  1. Aug 10, 2008 #1
    Heat capacity of a liquid is [tex]C=T^4[/tex] and the state function is [tex]V(T,P) = Aexp(aT-bP)[/tex]
    Derive an equation for entropy. Use the relevant Maxwell relations.

    [tex]dU = T dS - PdV[/tex]
    [tex]\frac{\partial U}{\partial T}_V = C = T^4 \Rightarrow U = \frac{T^5}{5} + f(V)[/tex]
    Since it's a liquid, and there're no separate [tex]C_V[/tex] and [tex]C_P[/tex], I assumed that expansion can be ignored, so [tex]dU \approx TdS[/tex] and
    [tex]dS = \frac{dU}{T} = T^3 dT[/tex]

    but it's unlikely to be true since I haven't used the state function or Maxwell relation at all. My assumption is probably wrong. Anyone solve the problem?
  2. jcsd
  3. Aug 11, 2008 #2
    Solved it.That equation I wrote will have an integration factor [tex]f(V)[/tex].
    Using [tex]\frac{\partial S}{\partial V}_T = \frac{\partial P}{\partial T}_V[/tex], we have the solution for S, this time with an integration factor of T. By compraison of these two statements of S, it's perfectly defined.
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