- #1
paweld
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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:
[tex]
\mu_l(p,T)=\mu_g(p,T)
[/tex]
(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:
[tex]
\mu_g(p,T) = \mu_g^0(T) + RT \log p_v
[/tex]
where [tex] p_v [/tex] is a partial preassure of vapour. So we can express the equillibrium
condition as follows:
[tex]
\mu_l(p,T)=\mu_g^0(T) + RT \log p_v(p)
[/tex]
where p is the total preassure of the system, i.e. the sum of the vapour partial preassure
[tex] p_v [/tex]and the other gases total preassure [tex] p_0 [/tex].
(I've written partial vapour preassure as a function of total preassure).
If we change total preassure of the system from [tex] p [/tex] to
[tex] p + \Delta p[/tex] 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):
[tex]
\mu_l(p + \Delta p,T)\approx \mu_l(p,T) + \upsilon_l \Delta p
[/tex]
(partial derivative of chemical potential with respect to preassure at constant
temperature is molar volume [tex] \upsilon [/tex]).
So the following approximate equality for equillibrium holds:
[tex]
\mu_l(p,T) + \upsilon_l \Delta p \approx \mu_g^0(T) + RT \log p_v(p+\Delta p)\approx
\mu_g^0(T) + RT \log p_v(p) + \frac{RT}{p} \frac{d p_v}{dp} \Delta p
[/tex]
As a result of the above equality and equillibrium condidtion at total preassure p we
had:
[tex]
\frac{d p_v}{dp} \approx \frac{p \upsilon_l}{RT}
[/tex]
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?
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:
[tex]
\mu_l(p,T)=\mu_g(p,T)
[/tex]
(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:
[tex]
\mu_g(p,T) = \mu_g^0(T) + RT \log p_v
[/tex]
where [tex] p_v [/tex] is a partial preassure of vapour. So we can express the equillibrium
condition as follows:
[tex]
\mu_l(p,T)=\mu_g^0(T) + RT \log p_v(p)
[/tex]
where p is the total preassure of the system, i.e. the sum of the vapour partial preassure
[tex] p_v [/tex]and the other gases total preassure [tex] p_0 [/tex].
(I've written partial vapour preassure as a function of total preassure).
If we change total preassure of the system from [tex] p [/tex] to
[tex] p + \Delta p[/tex] 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):
[tex]
\mu_l(p + \Delta p,T)\approx \mu_l(p,T) + \upsilon_l \Delta p
[/tex]
(partial derivative of chemical potential with respect to preassure at constant
temperature is molar volume [tex] \upsilon [/tex]).
So the following approximate equality for equillibrium holds:
[tex]
\mu_l(p,T) + \upsilon_l \Delta p \approx \mu_g^0(T) + RT \log p_v(p+\Delta p)\approx
\mu_g^0(T) + RT \log p_v(p) + \frac{RT}{p} \frac{d p_v}{dp} \Delta p
[/tex]
As a result of the above equality and equillibrium condidtion at total preassure p we
had:
[tex]
\frac{d p_v}{dp} \approx \frac{p \upsilon_l}{RT}
[/tex]
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?