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Liquid-Vapour Interface: Adiabatic Expansion

  1. Apr 16, 2014 #1
    1. The problem statement, all variables and given/known data


    Part(a): Show dL/dT can be expressed as:
    Part(b): Show L = L0 + ΔCT for an indeal gas
    Part(c): Show the following condition holds for an adiabatic expansion, when some liquid condenses out.

    2. Relevant equations

    3. The attempt at a solution

    Finished parts (a) and (b).

    Part (c)

    [tex]\frac{d}{dT} = \left(\frac{\partial}{\partial T}\right)_P + \left(\frac{dp}{dT}\right)\left(\frac{\partial}{\partial p}\right)_T[/tex]

    [tex]= \frac{d}{dT}(\frac{L}{T}) = (\frac{\partial \Delta S}{\partial T})_P + (\frac{dP}{dT})(\frac{\partial \Delta S}{\partial P})_T [/tex]

    Where ##\Delta_S = S_v - S_l## and using maxwell relation from ##dG = -sdT + VdP##:

    [tex]= \frac{\Delta C_p}{T} - (\frac{dp}{dT})\left(\frac{\partial}{\partial T}(V_v - V_l)\right)_P[/tex]

    Using ideal gas equation ##PV = RT## and Clausius-Clapeyron: ##\frac{dP}{dT} = \frac{L}{TV_v} = \frac{LP}{RT^2}##:

    [tex]= \frac{\Delta C_p}{T} - (\frac{R}{P})(\frac{LP}{RT^2})[/tex]

    [tex]= \frac{\Delta C_P}{T} - \frac{L}{T^2}[/tex]


    [tex] C_{P,liq} + T\frac{d}{dT}(\frac{L}{T}) = C_{P,vap} - \frac{L}{T_{vap}}[/tex]

    Condition for condensation: ##(\frac{\partial P}{\partial T})_S < 0 ## (Gradient must be less than zero for cooling effect).

    Now what remains is to show that ##(\frac{\partial P}{\partial T})_S = C_{P,vap} - \frac{L}{T_{vap}}##
  2. jcsd
  3. Apr 18, 2014 #2
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