Dependance of saturated vapour preassure of external preassure

[paweld]

I'm interested in an equillibrium state between pure liquid (e.g. water)
and its vapour preassure mixed with other gases (e.g water vapour in the atmospheric gases). I would like to compute the dependence of vapour saturated preassure
on the total preassure of the system (vapour partial preassure + preassures of all
other gases).

The saturated vapour preassure fullfill the following condition at given temperature T:
$$\mu_l(p,T)=\mu_g(p,T)$$
(equality of chemical potentials of liquid and gaseous phases), where p is total preassure
of the system. If we assuem that vapour behaves almost like ideal gas we obtain the following
formula for its chemical potential:
$$\mu_g(p,T) = \mu_g^0(T) + RT \log p_v$$
where $$p_v$$ is a partial preassure of vapour. So we can express the equillibrium
condition as follows:
$$\mu_l(p,T)=\mu_g^0(T) + RT \log p_v(p)$$
where p is the total preassure of the system, i.e. the sum of the vapour partial preassure
$$p_v$$and the other gases total preassure $$p_0$$.
(I've written partial vapour preassure as a function of total preassure).
If we change total preassure of the system from $$p$$ to
$$p + \Delta p$$ we can use approximate formula to compute the change
in chemical potential of liquid phase (the chemical potential of vapour depends only on
partial preassure and temperature):
$$\mu_l(p + \Delta p,T)\approx \mu_l(p,T) + \upsilon_l \Delta p$$
(partial derivative of chemical potential with respect to preassure at constant
temperature is molar volume $$\upsilon$$).

So the following approximate equality for equillibrium holds:
$$\mu_l(p,T) + \upsilon_l \Delta p \approx \mu_g^0(T) + RT \log p_v(p+\Delta p)\approx<br /> \mu_g^0(T) + RT \log p_v(p) + \frac{RT}{p} \frac{d p_v}{dp} \Delta p$$
As a result of the above equality and equillibrium condidtion at total preassure p we
had:
$$\frac{d p_v}{dp} \approx \frac{p \upsilon_l}{RT}$$
So if we increase the total preassure, the saturated vapour preassure should also increase.
Have I made a mistake somewhere or my result is true. Is it obsereved in nature?

 

[GeorgeRaetz]

...I would like to compute the dependence of vapour saturated preassure on the total preassure of the system (vapour partial preassure + preassures of all
other gases).
So if we increase the total preassure, the saturated vapour preassure should also increase.
Have I made a mistake somewhere or my result is true. Is it obsereved in nature?

Result is true.
Start with a liquid of specific volume v_l in equilbrium with its vapor of specific volume v_v, at vapor pressure p_v. Now add a little bit of "foreign" non-interacting gas of partial pressure p_f.Suprisingly, the foreign gas will extract a liile bit of vapor out of the liquid. (not push some vapor back in).
Quick proof: (similar to yours, just abbrreviated)

d \mu_l= v_l dp
(p=p_v+p_f=total gas phase pressure=liquid pressure.)
d \mu_v= v_v dp_v
Equilibrium condition:
v_l dp =d \mu_l=d \mu_v= v_v dp_v
Therefore:
d p_v/dp = v_l/v_v
a small but positive number.
I don't know of experimental evidence but the effect is mentioned in the book "Thermal
Physics" by P.M. Morse. (chapter 9: Changes of Phase).