1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Liquid-Vapour Interface: Adiabatic Expansion

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

    2hgxzl4.png

    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)

    Starting:
    [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]

    Therefore:

    [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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Liquid-Vapour Interface: Adiabatic Expansion
  1. Adiabatic expansion (Replies: 4)

  2. Adiabatic expansion (Replies: 5)

  3. Adiabatic expansion (Replies: 3)

  4. Adiabatic Gas expansion (Replies: 29)

Loading...