If saturated vapor does not obey ideal gas laws then how is mass of the saturated vapor proportional to the saturated vapor pressure?

My textbook mentions that unsaturated vapor obeys ideal gas law but saturated vapor does not. I could understand that air/vapor are not ideal so they won't obey ideal gas laws. But how can unsaturated vapor obey ideal gas laws?

Again while deriving the equation of relative humidity my textbook mentioned that mass of the saturated vapor is proportional to the vapor pressure? Why are they proportional? Shouldn't temperature play a vital role in increasing/decreasing mobility of vapor molecules and increase the pressure in addition to pressure due to weight of the vapor for which mass and pressure aren't proportional?

Thank you

• " .. mass of the saturated vapor pressure .. " ??? Commented Nov 8, 2021 at 14:43
• Corrected it. Thank you for notifying
– MSKB
Commented Nov 8, 2021 at 14:46
• Are you asking about, say, water vapor in air? The air remains pretty ideal in behavior, the water bit does odd stuff as you approach saturation because, well, it starts thinking about condensing into liquid. But the saturation concentration of water in air at normal room temperatures is really not that much water overall. Commented Nov 8, 2021 at 15:47
• But how is mass of the saturated air vapor proportional to the saturated vapor pressure?
– MSKB
Commented Nov 8, 2021 at 15:57

1 Answer

Since water is normally the chemical species that condenses out of air, assume water for the discussion immediately following. Vapor pressure is a function of temperature only, and is easily calculated with the Antoine equation, as shown here.

When the partial pressure of water vapor in air reaches the vapor pressure of water at the current temperature, the air is saturated with water vapor, and is at 100% humidity. The concentration of water vapor at that point will be low for "normal" temperatures (e.g., 20 deg C), and the ideal gas law may be used to calculate the mass of water vapor in a given volume of air. Thus,

$$PV=nRT$$

$$n=\frac{PV}{RT}$$

$$m=MW\frac{PV}{RT}$$

where

$$P$$ = partial pressure of water calculated from the Antoine equation

$$V$$ = the volume that you are considering

$$n$$ = the number of moles of water vapor in the given volume

$$R$$ = the universal gas constant

$$T$$ = the equilibrium temperature of the water vapor

$$m$$ = the mass of water vapor in the given volume

$$MW$$ = the molecular weight of water (aka 18.02)

Note that the mentioned textbook is ambiguous per your statement that saturated water vapor does not obey the ideal gas law while unsaturated water vapor does obey that law. Due to this, you can think of this problem as the mass of water vapor in a given volume, in the limit, as you approach saturated conditions.