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jek8214
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Homework Statement
I'm stuck on part (c) of this question.
Homework Equations
$$T\frac{d}{dT}\bigg(\frac{L}{T}\bigg) \equiv \frac{dL}{dT} - \frac{L}{T}.$$
Clausius-Clapeyron equation:
$$ \frac{dp}{dT} = \frac{L}{T\Delta V} \approx \frac{L}{TV_{vap}}.$$
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?