Liquid-Vapour Interface: Adiabatic Expansion

[unscientific]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

Finished parts (a) and (b).

Part (c)

Starting:
$$\frac{d}{dT} = \left(\frac{\partial}{\partial T}\right)_P + \left(\frac{dp}{dT}\right)\left(\frac{\partial}{\partial p}\right)_T$$

$$= \frac{d}{dT}(\frac{L}{T}) = (\frac{\partial \Delta S}{\partial T})_P + (\frac{dP}{dT})(\frac{\partial \Delta S}{\partial P})_T$$

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

$$= \frac{\Delta C_p}{T} - (\frac{dp}{dT})\left(\frac{\partial}{\partial T}(V_v - V_l)\right)_P$$

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

$$= \frac{\Delta C_p}{T} - (\frac{R}{P})(\frac{LP}{RT^2})$$

$$= \frac{\Delta C_P}{T} - \frac{L}{T^2}$$

Therefore:

$$C_{P,liq} + T\frac{d}{dT}(\frac{L}{T}) = C_{P,vap} - \frac{L}{T_{vap}}$$

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 $(\frac{\partial P}{\partial T})_S = C_{P,vap} - \frac{L}{T_{vap}}$

 

[unscientific]

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