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# Worked example: Vapor pressure and the ideal gas law

Vapor pressure example using the ideal gas law. Created by Sal Khan.

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• It seems to me that you must be assuming that this room has no water molecules in it before this experiment begins. I live in Florida, and I know that air already has moisture of water molecules in it. Maybe, this experiment was conducted in New Mexico, where the humidity is much less.
• Great point. The interesting conclusion to your good reasoning (that the room had to as dry as a desert), is that by knowing how much water actually evaporates from the bucket we can know the amount of moisture that was previously in the air – how many molecules of water were there.
***
• where did the number 760 mm Hg come from?
• 1 mmHg = 1 torr, a unit named after Evangelista Torricelli, the Italian scientist who invented the first barometer, the instrument which measures pressure. He created it using a column of mercury (open at the bottom, closed at the top), sitting in a bath of mercury. When sitting in the bath, the atmospheric pressure pushes down on the bath of mercury and forces the mercury to rise up in the column. When atmospheric pressure is high, the column rises higher. When the pressure is low, the column does not rise as high. At standard atmospheric pressure, the column rises to a height of 760mm. You can convert between other common units of pressure using this conversion:

760 torr = 760 mmHg = 1 atmosphere = 101.3 kilopascals
• At 25 degrees Celsius, shouldn't none of the water evaporate because the boiling point of water is 100 Celsius?
• Yes, and this happens not only with water.
For example: there are many reported cases of mercury intoxication due to breathing the metal’s vapor. (Interestingly, ingesting it doesn’t cause much harm).
The point is the same as water: the mercury boiling point is around 357° C, but evaporation can happen all the time, slowly, on the surface, as the particles get enough energy to separate themselves from the forces of attraction that hold them together.
• Imagine you have a closed box partially filled with liquid. Initially, you can imagine, that there is a vaccuum above the liquid. Of course, the liquid starts to evaporate and there will be some gas above it (remember the box is closed so that the gas can't escape). At some point, the evaporation will stop and we say that the liquid is in an equillibrium with the gas phase......And finally, if you now measure the pressure of that gas, that;s your vapour pressure.

In short, the vapour pressure is the pressure of the gas molecules of the given liquid in equillibrium with that liquid. It depends very much on temperature, when you increase the temperature, more liquid will evaporate and the vapour pressure will be higher.
• How do you measure mercury's pressure in millimeters?
• If you take two separate cups of mercury and hold them side by side. Then connect a tube from the bottom of one to the bottom of the other. The two liquids will level themselves.
Now put a cap with a pressure gage on top of one of the cups. Add one millimeter of extra mercury height to the open cup. The pressure in the capped cup should have increased by 1 mm Hg.
• If the temperature of the room were to be 15 degrees Celsius instead of 25, then n = 56.3 mol, which means that more water would evaporize. How could this be when, intuitively, more water should evaporize at a higher temperature?
• You are completely correct that vapor pressure changes with the temperature. The problem is the logic that you used. At 15 degress celsius, you need more moles (56.3 moles instead of the 54.4 moles) to reach 23.8 mm Hg because the molecules have lower kinetic energy. That does not mean, that at 15 degrees, the equilibrium point (vapor pressure) is still 23.8 mm Hg. At 15 degress the vapor pressure of water is 12.8 mm Hg. So that does agree with your intuition. At 15 degrees, the vapor only creates a pressure of 12.8 mmg Hg while at 25 degrees, the vapor creates a pressure of 23.8 mm Hg. Meaning, at 25 degrees, more water vaporizes to create a greater pressure.
• This scenario seems to be taking a non-ideal vessel (a dorm room) and applying ideal characteristics to it. Had this experiment been done in a carefully prepared (within the limits of a student's freedom to do so) dorm room, wouldn't all of the water evaporate? My reasoning is that the air temperature may be stable, but only because the heat losses 'to' balance the gains 'from' outside sources. Essentially, if the air was at some maximum humidity, I suspect that there would be condensation on the windows, leaving the water from the test vessel in another location. Leaving out the water in the plumbing waste traps, starting humidity, and hygroscopic substances (cereal) commonly found in the room, if the extra complication was put into the formula to make the vessel a "dorm room", then it can not simply be treated like an ideal vessel anymore.

I imagine that this would not be what a teacher is looking for on a quiz, but my question remains. Wouldn't the water continue to leave the vessel that is at the average room air temperature to make up for condensing water on cooler areas?
• I had some of the same thoughts. I was imagining water condensing on the windows (leading to water running off and pooling on window sills), any mirrors or glass objects (more runoff), the paint, the floor (sometimes tile in a dorm room), and even slightly dampening the bed sheets and such. I was also thinking about how unlikely it would be for significant amounts of vapor to return to the original 2L container before condensing. In practice, I'd expect all of the water to evaporate, and for the student to return to a damp room, given enough time. It's still an interesting problem, though!
• So, if I understand the principles behind this video, as temperature rises so does pressure, meaning that as temperature rises the amount of water able to be held in the air falls- that is, the equilibrium point mentioned in the video (where just as much water is condensing to liquid as is evaporating to gas) will happen at a smaller amount of water. Water is denser at higher temperatures, so any given volume of cold air will be able to hold more mols of water as a gas than an identical volume of warm air. Is this correct?

How does this apply to winter-time, when it "feels" drier? Does the low temperature "overcome" the water vapor and freeze it out of the air to lie on the ground as frost?
• The density of liquid water at any given temperature has little to no effect on how much water can exist in gaseous solution at that temperature. Thus, it is not correct to associate the density of liquid water with water vapor in air -- these are different phases of water and governed by different physical properties.
• Sal uses 0 degrees C = 273 Kelvins. I know it doesn't affect the final answer because of significant figures, but shouldn't we use 273.15? It's more accurate, so......