How Does Adiabatic Expansion Lead to Condensation?

[albega]

Homework Statement

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+ΔC_p-L/V_vap(∂V_vap/∂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.

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!

 

[Chestermiller]

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

 

[albega]

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

Clausius Clapeyron equation: dp/dT=L/TV_vap
Adiabatic expansion of an ideal gas: p^1-yT^y=constant where y is the adiabatic index
Not too sure where to go though...

 

[Chestermiller]

Clausius Clapeyron equation: dp/dT=L/TV_vap
Adiabatic expansion of an ideal gas: p^1-yT^y=constant where y is the adiabatic index
Not too sure where to go though...

If you substitute the ideal gas law for V_vap 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

 

[albega]

If you substitute the ideal gas law for V_vap 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

Answers in order:

dp/dT=pL/RT²
Also if I treat it as an ideal gas I have L=L₀+ΔC_pT, giving p=p₀exp[(ΔC_plnT-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...

 

[Chestermiller]

Answers in order:

dp/dT=pL/RT²
Also if I treat it as an ideal gas I have L=L₀+ΔC_pT, giving p=p₀exp[(ΔC_plnT-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...

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

 

[albega]

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

Gradient of adiabat<gradient of phase boundary, or (∂p/∂T)_S<dp/dT at the crossover point?

 

[Chestermiller]

Gradient of adiabat<gradient of phase boundary, or (∂p/∂T)_S<dp/dT at the crossover point?

I get just the opposite.

Chet

 

[albega]

I get just the opposite.

Chet

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 C_p,liq+Td(L/T)/dT<0, but I don't see any way of getting there.

 

[Chestermiller]

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.

Can you provide your schematic of the CC plot and the adiabat plot?

Despite this, the actual mathematical condition I have to find is C_p,liq+Td(L/T)/dT<0, but I don't see any way of getting there.

I don't either. This looks like it has something to do with the second derivative of the log of the saturation pressure.

Chet