# Dependence of relative humidity on temperature and pressure

How is relative humidity determined by temperature and pressure? What is the equation and its derivation? The following is a specific scenario.

Suppose a cubic container encloses a solution consists of two liquids with known molar fractions and occupies a simply connected strict subdomain of the cube. The solution is stationary and uniformly mixed. The remaining space in the cubic container contains only the vapor from the two liquids. The vapor molecules of two liquids have no interaction with each other but completely elastic collision with the container wall. When the vapor and the liquid are in thermal equilibrium, and each is homogeneous, what is the relative humidity of the two vapors? Can we apply Raoult's law here and equating relative humidity with the fugacity coefficient? If not, what will be a good equation of state for relative humidity in this case?

In light of the comments from Chester Miller and David White, I will present the following scenario for air and water vapor.

Suppose we are given a large enclosed cubic container completely rigid and insulated thermally and radiatively from outside. Fill it first with completely dry, still and homogeneous air. Fill quickly a large portion in volume (so large that the mass of the water that would later evaporate is but a negligible portion of the total mass of the water) of the container volume with water with no macroscopic flow within the water. Now the air is still without any water vapor mixed in and is at a certain temperature and pressure. After the water vapor has evaporated into the air and the mix has reached the thermal equilibrium through adiabatic process and without outside work done to the whole closed system, what is the relative humidity of the water vapor? What is the equation of state that governs the process?

• I think there are mainly approximative formulas only. – peterh Jun 5 '17 at 1:48

I'll address one of the key questions you are asking and, I believe, you'll be able to figure the answers to other questions yourself:

After the water vapor has evaporated into the air and the mix has reached the thermal equilibrium through adiabatic process and without outside work done to the whole closed system, what is the relative humidity of the water vapor?

The answer to this question is 100%.

The dynamic equilibrium is achieved when the rate of evaporation is equal to the rate of condensation. Condensation happens when the relative humidity is 100%.

In other words, given enough time, the relative humidity in a closed container with water will become 100%.

This Hyperphysics article contains some empiric formulas and graphs that will help you determine the density of the saturated vapor as a function of temperature.

• Suppose in a sealed container there is less water molecules than the 100% relative humidity at the dynamic equilibrium would require. Is there still going to be liquid phase water present, or the water molecules are all in the gas phase even at the dynamic thermal equilibrium? – Hans Sep 16 '20 at 17:01

Relative humidity (percent) is defined by $$RH=\frac{p}{p_s(T)}\times 100$$where p is the partial pressure of water vapor in the air and $p_s(T)$ is the equilibrium saturation vapor pressure of water at the air temperature T. It is not a derived quantity. You can easily find all this simply by Googling Relative Humidity.

• I know the definition of relative humidity. I am surprised when reading the wikipedia entry en.wikipedia.org/wiki/Relative_humidity that it is determined by temperature and pressure as if the two latter quantities control the relative humidity. How do temperature and pressure determine the fraction of the liquid water molecules amongsts all the other vapors in all the volume above the bulk liquid? – Hans Jun 5 '17 at 1:20
• @Hans -- That's yet another poorly written wikipedia article. Relative humidity does not depend on atmospheric pressure. It depends solely on the partial pressure of water vapor in the air and the saturation pressure of water at that temperature. Saturation pressure is highly temperature dependent, but it does not depend on atmospheric pressure. – David Hammen Jun 5 '17 at 3:49
• I suspect that the author of the article was a bit "loose" with his wording. Relative humidity depends on partial pressure of water vapor in the air, and temperature (which affects the vapor pressure of water). – David White Jun 5 '17 at 3:54
• @DavidWhite: You say "It depends solely on the partial pressure of water vapor in the air and the saturation pressure of water at that temperature." That is the definition of relative humidity. My question is given a liquid solution, what determines the relative humidity of the vapor above it. I think I have found an idealized answer. That is the ideal solution and the related Raoult's law en.wikipedia.org/wiki/Raoult%27s_law, which states that the relative humidity is simply the mole fraction of the particular liquid in the liquid phase solution. Do you agree? – Hans Jun 5 '17 at 7:54
• I can't say whether or not I agree without you being much more specific regarding the conditions of the problem you are trying to solve. Are you dealing with water? What is dissolved in the water? What concentration? Is there any air flow involved? Is there a covered container for the liquid? Note that your problem may well involve non-ideal vapor-liquid equilibrium, which requires fugacity coefficients rather than Raoult's law. – David White Jun 5 '17 at 15:03