Air conditioners not only cool air, but dry it as well. A...

Air conditioners not only cool air, but dry it as well. A room in a home measures 6.0 $\mathrm{m} \times 10.0 \mathrm{m} \times 2.2 \mathrm{m} .$ If the outdoor temperature is $30^{\circ} \mathrm{C}$ and the partial pressure of water in the air is 85$\%$ of the vapor pressure of water at this temperature, what

Key Concepts

Relative Humidity

Relative humidity is a measure of how much water vapor is present in the air compared to the maximum amount the air can hold at that temperature. This concept is crucial for understanding processes like air conditioning, where changes in temperature directly affect the air’s capacity to hold moisture, leading to condensation when the air is cooled below its dew point.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid (or solid) form at a given temperature. It represents the maximum concentration of water vapor that the air can hold at that temperature, and is key to determining when condensation will occur as the air is cooled.

Partial Pressure

Partial pressure refers to the pressure contributed by a single component of a gas mixture. In the context of humidity, it is the pressure due to water vapor alone. Understanding partial pressure is important for calculating how much water vapor is present initially and how its reduction relates to dehumidification.

Condensation

Condensation is the process by which water vapor in the air turns into liquid water when the air is cooled below its dew point. Air conditioners remove moisture from the air through this process, which is essential for both cooling and drying indoor environments.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas. In the study of air moisture content, it provides a framework for linking the partial pressure of water vapor to its density and mass. This relationship is useful when calculating how much water is condensed out of the air during cooling.

Transcript

00:01 So before we solve this problem, we need to know two basic things, vapor pressure and the ideal gas law.

00:07 So vapor pressure is essentially the pressure exerted by a vapor when it's at thermodynamic equilibrium.

00:16 So when it's at a constant temperature.

00:20 Now this is the amount of pressure that it would exert in its environment, in a solution.

00:24 So say you have air, there's water vapor in the air.

00:27 That vapor is going to exert a certain pressure on its environment when it's in the air.

00:35 And the ideal gas law, the equation associated with it, is pv equals nrt.

00:45 This is used to determine the behavior of a substance or a gas in this case under specific conditions.

00:55 Okay, set with that out of the way, let's begin to solve the problem.

00:58 So let's write down what we know.

01:00 So we know the room dimensions.

01:04 That's useful.

01:05 So our room dimensions are 6 meters times 10 meters times 2 .2 meters.

01:16 Multiplying these three dimensions gives you the volume.

01:19 So you just multiply those together and you would get 132 meters cubed, which would be about 132 ,000 liters.

01:42 Okay? so that's the volume.

01:45 And we also know the temperature of the room, which is 30 degrees celsius.

01:50 Now when it comes to the ideal gas law, you always want to convert the temperature into kelvin.

01:55 So in order to convert celsius to kelvin, we just have to add 273.

02:02 So 273 plus 30 gives us 303 kelvin.

02:10 And we also know the vapor pressure, which is at this temperature, 85 % of what it would be at room temperature.

02:19 So all we need to do is just take our vapor pressure, which we know to be 31 .8 tor, and we just multiply it by 0 .85, or 85 over 100 if you prefer fractions.

02:41 You multiply that, and we get 27 .03 tor.

02:48 All right, so we have the volume, the temperature, and the pressure.

02:57 Now all these three values are applicable to our ideal gas law...

Please give Ace some feedback