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?