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Phase change during expansion

  1. Mar 17, 2015 #1
    1. The problem statement, all variables and given/known data
    In an earlier part of the question, I derived the temperature dependence of the latent heat of vapourisation of a liquid as
    dL/dT=L/T+ΔCp-L/Vvap(∂Vvap/∂T)p
    I am asked to find the condition that upon expanding the gas adiabatically, we get condensation to occur, by considering dp/dT and (∂p/∂T)S.

    3. The attempt at a solution
    So I think I should consider the p-T plane here, and there will be a phase boundary between liquid and vapour, and we're currently below the boundary in the vapour region, and we want to get above the boundary somehow.

    We are on an adiabat so the gradient of our path has to be (∂p/∂T)S. The gradient of the phase boundary itself will be dp/dT.

    So I was maybe thinking (∂p/∂T)S>dp/dT so we cross the line - however this seems a bit restrictive - surely it could be less than it at some stage, then be greater than it and condensation would still occur. Even so, I don't see how to use that to get anywhere.

    Any clues? Thanks!
     
  2. jcsd
  3. Mar 17, 2015 #2
    Please write down the Clausius Clapeyron equation. Also, please write down the relationship between pressure and temperature for the adiabatic expansion of an ideal gas.

    Chet
     
  4. Mar 18, 2015 #3
    Clausius Clapeyron equation: dp/dT=L/TVvap
    Adiabatic expansion of an ideal gas: p1-yTy=constant where y is the adiabatic index
    Not too sure where to go though...
     
    Last edited: Mar 18, 2015
  5. Mar 18, 2015 #4
    If you substitute the ideal gas law for Vvap into your CC equation, what do you get?

    At the initial state of the gas (at an initial temperature T0) where condensation has not taken place yet, how does the pressure calculated from the adiabatic expansion equation have to compare with the pressure calculated from the CC equation (is it higher or lower)?

    Draw a schematic graph of ln p versus T, showing both the CC prediction and the adiabatic expansion prediction. How do the slopes of these plots have to compare in order for the adiabatic expansion line to intersect the CC line at some pressure below the starting pressure?

    Chet
     
  6. Mar 18, 2015 #5
    Answers in order:

    dp/dT=pL/RT2
    Also if I treat it as an ideal gas I have L=L0+ΔCpT, giving p=p0exp[(ΔCplnT-L0)/T/R], which may be helpful?

    Ok so if you look at the phase boundary in the p-T plane, we have a liquid region at higher pressures and a gas region at lower pressures separated by the phase boundary defined by the above expression for p as a function of T. So the pressure calculated from the adiabatic expansion must be lower.

    I obviously have lnp from above for the phase boundary from the CC equation. The adiabat gives me lnp=yln(AT)/y-1 where A is some constant I don't know. Ah, now I see p has to decrease (I was thinking it could increase before, but obviously we are expanding the gas). I'm stuck in comparing the two graphs because I don't know what some of the constants are and how they compare... Even so I don't see where I would go...
     
  7. Mar 18, 2015 #6
    You are really close to having the answer. In the schematic diagram I have drawn, the CC plot of ln p vs T is up and to the left, and the adiabatic expansion line is down and to the right. And, in order for condensation to occur, we need to be able to move to the left and down along the adiabatic expansion line until it intersects the CC line. So, at the intersection temperature T, the CC ln p will have to match the adiabatic expansion ln p. But, there is an additional constraint that also needs to be satisfied. What if the two lines are parallel? Then, they will never cross. Also, if the slope of the adiabatic line is too steep, it will not be able to intersect the CC line by moving down and to the left. So there is an inequality involving the slopes that needs to be satisfied. Hope this makes sense. What is that inequality (at the crossover point)?

    Chet
     
  8. Mar 18, 2015 #7
    Gradient of adiabat<gradient of phase boundary, or (∂p/∂T)S<dp/dT at the crossover point?
     
  9. Mar 18, 2015 #8
    I get just the opposite.

    Chet
     
  10. Mar 19, 2015 #9
    Well I originally put (∂p/∂T)S>dp/dT in the original post but that was because I was imagining p,T increasing along the adiabat whereas they would fall in an expansion... So I'm not quite seeing it.

    Despite this, the actual mathematical condition I have to find is Cp,liq+Td(L/T)/dT<0, but I don't see any way of getting there.
     
  11. Mar 19, 2015 #10
    Can you provide your schematic of the CC plot and the adiabat plot?
    I don't either. This looks like it has something to do with the second derivative of the log of the saturation pressure.

    Chet
     
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