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Van der Waal expansion and delivered work

  1. Aug 3, 2017 #1
    1. The problem statement, all variables and given/known data
    Assume that one mole of an ideal van der Waals fluid is expanded isothermally, at temperature [itex]T_h[/itex] from an initial volume [itex]V_i[/itex] to a final volume [itex]V_f[/itex]. A thermal reselvoir at temperature [itex]T_c[/itex] is available. Apply [tex]
    dW_{RWS} = \left ( 1 - \frac{T_{RHS}}{T} \right ) (-dQ) +(-dW) [/tex]
    to a differential process and integrate to calculate the work delivered to a reversible work source (RWS). RHS is reversible heat source. Corroborate by overall energy and entropy conservation.

    Hint: remember to add the direct work transfer [itex]pdV[/itex] to obtain the total work delivered to the reversible work source.

    2. Relevant equations
    Van der Waal equations:
    [tex]
    u + a/v = cRT
    [/tex]
    where [itex]u, a, v, c, R, T[/itex] are, respectively, energy per mole, constant, volume per mole, another constant, temperature.
    [tex]
    p = \frac{RT}{v-b} - \frac{a}{v^2}
    [/tex]

    The entropy is
    [tex]
    S = NR\log [ (v-b)(cRT)^c] + Ns_0
    [/tex]
    where [itex]N[/itex] is the number of moles and [itex]b[/itex] is another constant.

    3. The attempt at a solution

    Using the first equation with [itex]T=T_h, T_{RHS} = T_c[/itex] and [itex]-dW = pdV[/itex] and integrating we get
    [tex]
    W_{RHS} = - \left ( 1- \frac{T_c}{T_h} \right) Q + RT_h \log \left ( \frac{v_f-b}{v_i - b} \right) + \frac{a}{v_f} - \frac{a}{v_i}
    [/tex]
    Also energy conservation gives
    [tex]
    \Delta u + W + Q =0
    [/tex]
    and entropy conservation
    [tex] R \log \left ( \frac{v_f-b}{v_i - b} \right) + \frac{Q}{T_c} = 0
    [/tex]
    Finally the energy change is
    [tex]
    \Delta u = \frac{a}{v_i} - \frac{a}{v_f}
    [/tex]
    Everything seems to work out except that fraction [itex] T_c/T_h[/itex] is the wrong way round and I see no way of dealing with this. Help would be appreciated.
     
    Last edited: Aug 3, 2017
  2. jcsd
  3. Aug 3, 2017 #2
    I see my problem: there are two Qs. The first equation was the heat from the subsystem and the energy and entropy the heat RHS.
     
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