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Thermodynamics, non-ideal gas

  1. Oct 25, 2016 #1
    1. The problem statement, all variables and given/known data

    A generalized [itex]TdS[/itex]-equation for systems of several types of "work-parts" and varying number of particles in multiple components, is given by

    [tex] dU = TdS + \sum_{i}y_idX_i+\sum_{\alpha =1}^{c}\mu_\alpha dN_{\alpha} [/tex]

    Thus, its natural to regard the internal energy [itex]U[/itex] (an extensive property), as a function of the extensive variables [itex] U, S, {X_i}, {N_{\alpha}}. [/itex] Here [itex] U_\alpha [/itex] is the chemical potential for component [itex] \alpha [/itex], and [itex] N_\alpha [/itex] is the number of particles in component [itex] \alpha [/itex] of the system (a number that can vary by [itex] dN_\alpha \neq 0 [/itex]). Thus we have

    [tex] U = U(S, X_i, N_\alpha) [/tex]

    Because [itex] (U, S, X_i, N_\alpha) [/itex] are all extensive properties, we have the following homogeneity condition

    [tex] U(\lambda S, \lambda {X_i}, \lambda {N_\alpha}) = \lambda U(S, {X_i}, {N_\alpha}) [/tex]

    2. Relevant equations

    My question regards a special case of this, namely a one-component gass system (not an ideal gass!) with the following internal energy

    [tex] U = U(S, V, N) = \frac{aS^3}{NV} [/tex]

    where [itex] a [/itex] is a a constant with dimension [itex] K^3m^3/J^2 [/itex].

    Problem: find the pressure [itex] p [/itex], the temperature [itex] T [/itex] and the chemical potential [itex] \mu [/itex] of this gas expressed by [itex] (S, V, N) [/itex]. And then find the heat capacities at constant volume [itex] C_V [/itex] and pressure [itex] C_p [/itex], expressed by [itex] (N, T, V ) [/itex] and [itex] (N, T, p) [/itex], respectively.

    3. The attempt at a solution

    The [itex] TdS [/itex]-equation becomes

    [tex] TdS = dU + pdV - \mu dN = C_vdT + \left[\left(\frac{\partial U}{\partial V}\right)_T + p\right]dV - \mu dN[/tex]

    But here Im completely at a loss.
     
  2. jcsd
  3. Oct 25, 2016 #2

    Charles Link

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    Homework Helper

    I can give you one answer. This whole problem, I think, is not very difficult. ## p=- (\frac{\partial{U}}{\partial{V}})_{S,N} ##. Now apply this to the function ## U ## that they gave you.
     
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