Problem 7.

One day the relative humidity is 90% and the temperature is 25 degrees Celsius. How many grams of water will condense out of each cubic meter of air if the temperature drops to 15 degrees Celsius? How many energy does the condensation from each cubic meter release?

Solution:

An air contains water vapor. The amount of water vapor, which a cubic meter of air can contain, cannot be more than a maximum value. We characterize the amount of water vapor in one cubic meter of air by its mass: . Then we can tell that the mass of the water vapor in one cubic meter of air should be less than the maximum value:

The maximum mass of the water vapor depends on the temperature: . If the vapor has the maximum mass then such vapor is called saturated vapor.

The humidity of the air is determined as the ratio of the mass of the vapor to the maximum possible vapor mass:

We know at 25 degrees Celsius the humidity is 90 %. It means that

From this expression we can find the mass of the vapor in the air:

The temperature of the air drops to 15 degrees Celsius. At this temperature the maximum possible mass of the vapor is . When the temperature of the air decreases the original mass of the vapor remains the same [ ]. If this mass is larger than the maximum mass then part of the vapor condenses into water. The mass of the water is

Therefore to find the mass of the water we need to find the maximum mass of the vapor.

We assume that the vapor is an ideal gas. If we introduce the maximum pressure of the vapor ( ) then the ideal gas equation takes the form (we write this equation for volume of 1 cubic meter)

Where - molar mass of water.

Then

The maximum pressures at 15 and 25 degrees Celsius are

Then the maximum masses of the vapor are

Now we can find the mass of the water

The second question: How many energy does the condensation from each cubic meter release?

The energy released from the condensation of the water is determined by the specific latent heat of vaporization:

The specific latent heat of vaporization (in SI units) is 2260000 J/kg. Then