Partial Pressures and Vapour Pressure

[WWCY]

Homework Statement

Two systems in diffusive equilibrium have equal chemical potentials. We can use this fact to solve the following problem. We begin with a closed system consisting of a liquid such as water in diffusive equilibrium with its vapour. At the start, only the liquid and its vapour are present. Then we pump in an inert gas (that is, a gas that doesn’t react chemically with the liquid or vapour) to increase the pressure in the container (we’re also assuming that everything is at the same temperature, so no heat flow occurs). Assuming that the inert gas doesn’t dissolve in the liquid, and that the liquid and its vapour remain in diffusive equilibrium, the chemical potentials of the liquid and vapour must change by the same amount: dµ`= dµv. What happens to the partial pressure Pv of the vapour?

I had a look at solutions but there are a few pretty important concepts that I couldn't get, and need help with:

1) I have learned Dalton's law, that states that for ideal gases, the total pressure of a system is the sum of the partial pressures of its component gases.

So if I start with Gas $A$ in a volume $V$, it has a pressure of $P_A = \frac{N_A KT}{V}$.
If I stick in another Gas, Gas $B$, then the partial pressures are now (I believe ) $P_A = \frac{N_A KT}{V}$ and $P_B = \frac{N_B KT}{V}$,
that is to say the Partial Pressure of Gas $A$ "doesn't change".

Why then, would the partial pressure of water vapour $P_v$ in the problem above vary with the addition of additional gas / total pressure?

2) If I wanted to find the chemical potential of the liquid as a function of some pressure, what pressure would it be a function of?

i.e. What is the $P_?$ in $\mu (T,P_?)$ and why? I understand that the identity $\mu = G/N$ allows me to avoid this issue but it's something I would prefer to clarify.

Thanks in advance!

Homework Equations

The Attempt at a Solution

 

[Chestermiller]

Are you familiar with the Poynting effect of pressure on the free energy of a liquid?

 

[WWCY]

Are you familiar with the Poynting effect of pressure on the free energy of a liquid?

Hi, thanks for the response.

Unfortunately I'm not. I have given it a quick read on wikipedia, and it was stated that "In thermodynamics, the Poynting effect generally refers to the change in the vapor pressure of a liquid when a non-condensable gas is mixed with the vapor at saturated conditions". Is this it?

 

[Chestermiller]

Hi, thanks for the response.

Unfortunately I'm not. I have given it a quick read on wikipedia, and it was stated that "In thermodynamics, the Poynting effect generally refers to the change in the vapor pressure of a liquid when a non-condensable gas is mixed with the vapor at saturated conditions". Is this it?

This is it, but the explanation isn't very good. How much does the chemical potential of liquid water change when we increase the total pressure on it from the equilibrium vapor pressure to P? How much does the chemical potential of water vapor change when we increase its partial pressure from the equilibrium vapor pressure to p*? How do these changes in chemical potential have to compare if the water vapor in the gas phase is in equilibrium with that of the liquid water in the final state?

 

[WWCY]

This is it, but the explanation isn't very good. How much does the chemical potential of liquid water change when we increase the total pressure on it from the equilibrium vapor pressure to P? How much does the chemical potential of water vapor change when we increase its partial pressure from the equilibrium vapor pressure to p*? How do these changes in chemical potential have to compare if the water vapor in the gas phase is in equilibrium with that of the liquid water in the final state?

Unfortunately I don't think I can answer any of the above, are all these necessary to know to answer the two points I raised?

This problem was from Schroeder's book, which I believe doesn't go into the Poynting effect (up to this chapter at least).

 

[Chestermiller]

Unfortunately I don't think I can answer any of the above, are all these necessary to know to answer the two points I raised?

This problem was from Schroeder's book, which I believe doesn't go into the Poynting effect (up to this chapter at least).

He's trying to get you to derive it. $$\Delta \mu_L=V_L(P-P_{sat})$$
$$\Delta \mu_V=RT\ln{(p^*/P_{sat})}$$
where $V_L$ is the molar volume of the liquid and p* is the partial pressure of the vapor in the gas phase at the new equilibrium.