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Homework Help: Condition for condensation from adiabatic expansion

  1. Apr 15, 2017 #1
    1. The problem statement, all variables and given/known data
    I'm stuck on part (c) of this question.
    2. Relevant equations
    $$T\frac{d}{dT}\bigg(\frac{L}{T}\bigg) \equiv \frac{dL}{dT} - \frac{L}{T}.$$
    Clausius-Clapeyron equation:
    $$ \frac{dp}{dT} = \frac{L}{T\Delta V} \approx \frac{L}{TV_{vap}}.$$
    3. The attempt at a solution
    My approach has been to find the equations for the adiabatic expansion in the p-T plane, given by
    $$\ln\bigg(\frac{p}{p_0}\bigg) = \frac{C_{p_{vap}}}{R}\ln T$$
    and the equation for the liquid-vapour phase coexistence curve, given by
    $$\ln\bigg(\frac{p}{p_0}\bigg) = -\frac{L_0}{RT} + \frac{\Delta C_p}{R}\ln T.$$
    Then for condensation to occur, the curves need to intersect (I can worry about the inequality later). This gives $$\frac{L_0}{T} + C_{p_{liq}}\ln T = 0.$$

    Try as I might, I can't seem to turn this into the answer they want. I also don't see how any condition on intersection can be formed by considering the gradients. Is there something important I've missed/a slip in my algebra?
  2. jcsd
  3. Apr 16, 2017 #2
    I would start (c) by doing the following:
    $$dS=\left(\frac{\partial S}{\partial T}\right)_PdT+\left(\frac{\partial S}{\partial P}\right)_TdP$$
    So, at constant entropy,
    $$dS=\left(\frac{\partial S}{\partial T}\right)_PdT+\left(\frac{\partial S}{\partial P}\right)_TdP=0$$
    I would also use: $$\left(\frac{\partial S}{\partial P}\right)_T=-\left(\frac{\partial V}{\partial T}\right)_P$$
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