Ideal gas laws
Vapor Pressure Example Vapor pressure example using the Ideal Gas Law
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- This exercise is from
- chapter 12 of the Kotz, Treichel and Townsend
- and Chemistry and Chemical Reactivity book,
- and I'm doing it with their permission.
- So they tell us you place 2 liters of water
- in an open container in your dormitory room.
- The room has a volume of
- 4.25 times 10 to the fourth liters.
- You seal the room and wait for the water to evaporate.
- Will all of the water evaporate
- at 25 degrees Celsius?
- And then they tell us at 25 degrees Celsius,
- the density of water is 0.997 grams per milliliter.
- And its vapor pressure is 23.8 millimeters of mercury.
- And this is actually the key clue
- to tell you how to solve this problem.
- And just as a bit of review,
- lets just think about what vapor pressure is.
- Let's say it's some temperature,
- and in this case we're dealing at 25 degrees Celsius.
- I have a bunch of water,
- and let me do that in a water color.
- I have a bunch of water molecules
- sitting here in a container.
- At 25 degrees Celsius,
- they're all bouncing around in every which way.
- And every now and then
- one of them is going to have enough kinetic energy
- to kind of escape the hydrogen bonds
- and all the things
- that keep liquid water in its liquid state
- and it will escape.
- It'll go off in that direction,
- and then another one will.
- And this'll just keep happening.
- The water will naturally vaporize in a room.
- But at some point,
- enough of these molecules have vaporized over here
- that they're also bumping back into the water.
- And maybe some of them can be captured back
- into the liquid state.
- Now, the pressure at which this happens
- is the vapor pressure.
- As you can imagine,
- as more and more these water molecules vaporize
- and go into the gaseous state,
- more and more will also create pressure,
- downward pressure.
- More and more will also be
- colliding with the surface of the water.
- And the pressure at which
- the liquid and the vapor states are in equilibrium
- is the vapor pressure.
- And they're telling us right now.
- It is 23.8 millimeters of mercury.
- Now, what we need to do to figure out this problem is
- say, OK, if we could figure out
- how many molecules need to evaporate
- how many molecules of water need to evaporate
- to give us this vapor pressure,
- we can then use the density of water
- to figure out how many liters of water that is.
- So how do we figure out how many molecules--
- let me write this down
- --how many molecules of water need to evaporate
- to give us the vapor pressure of
- 23.8 millimeters of mercury?
- So what, I guess, law or formula--
- and I never like to just memorize formulas,
- but we've given this formula in the past
- and it's probably one of the top most useful formulas
- in chemistry, or really all of science--
- what formula or law deals with pressure?
- They give us the volume of the room
- because that's where the pressure will be inside of.
- So we have pressure, the equilibrium vapor pressure.
- We have a volume of a room right over here.
- We know the temperature of the room right over there.
- And we're trying to figure out the number of molecules
- that need to evaporate for us to get that pressure
- in that volume at that temperature.
- So what deals with pressure, volume,
- number of molecules, let's say in moles,
- so I'll write a lower case n
- number of molecules, and temperature?
- Well, we've seen this many, many times.
- It's the Ideal Gas Law.
- Pressure times volume is equal to
- the number of moles of our idea gas
- in this case we're going to use water as our ideal gas
- or vapor as our ideal gas
- times the universal gas constant times temperature.
- And this should never seem like some bizarre formula to
- you because it really, really makes sense.
- If your pressure goes up, then that means that either the
- number of molecules have gone up, and we're assuming the
- volume is constant.
- That means either the number of molecules have gone up,
- which makes sense-- more things bouncing onto the side
- of the container.
- Or your temperature has gone up-- the same number of
- things, but they're bumping with higher kinetic energy.
- Or if your pressure stays the same and your volume goes up,
- then that also means that your number of molecules went up,
- or your temperature went up.
- Because you now have a bigger container.
- In order to exert the same pressure you need either more
- molecules or more kinetic energy for the molecules you
- have.
- And you could keep playing around with this, but I just
- want to make it clear this isn't some mysterious formula.
- The first time I was exposed to this I kind of did view it
- as some type of mysterious formula.
- But it's just relating pressure, volume, number of
- molecules and temperature.
- And then this is just the universal gas constant.
- So let's just get everything into the right units here.
- And then what we're trying to solve for, we want to figure
- out the number of molecules of water.
- So we want to solve for n.
- And if we know the number of moles of water, we can figure
- out the number of grams of water.
- And then given the density of water we can figure out the
- number of milliliters of water we are dealing with.
- So let's just rewrite the Ideal Gas Law by dividing both
- sides by the universal gas constant and temperature.
- So that you get n is equal to pressure times volume, over
- the universal gas constant times temperature.
- Now, the hardest thing about this is just making sure you
- have your units right and you're using the right ideal
- gas constant for the right units, and we'll
- do that right here.
- So what I want to do, because the universal gas constant
- that I have is in terms of atmospheres, we need to figure
- out this vapor pressuree- this equilibrium pressure between
- vapor and liquid-- we need to write this down in terms of
- atmospheres.
- So let me write this down.
- So the vapor pressure is equal to 23.8
- millimeters of mercury.
- And you can look it up at a table if you don't have this
- in your brain.
- One atmosphere is equivalent to 760 millimeters of mercury.
- So if we wanted to write the vapor pressure as
- atmospheres-- let me get my calculator out, get the
- calculator out, put it right over there-- so it's going to
- be 23.8 times 1 over 760, or just divided by 760.
- And we have three significant digits, so
- it looks like 0.0313.
- So this is equal to 0.0313 atmospheres.
- That is our vapor pressure.
- So let's just deal with this right here.
- So the number of molecules of water that are going to be in
- the air in the gaseous state, in the vapor state, is going
- to be equal to our vapor pressure.
- That's our equilibrium pressure.
- If more water molecules evaporate after that point,
- then we're going to have a higher pressure, which will
- actually make them favor more of them going into the liquid
- state, so we'll go kind of past the equilibrium, which is
- not likely.
- Or another way to think about it-- more water molecules are
- not going to of evaporate at a faster rate than they are
- going to condense beyond that pressure.
- Anyway, the pressure here is 0.0313 atmospheres.
- The volume here-- they told us right over here-- so that's
- the volume-- 4.25.
- 4.25 times 10 to the fourth liters.
- And then we want to divide that by-- and you want to make
- sure that your universal gas constant has the right units,
- I just looked mine up on Wikipedia-- 0.08-- see
- everything has three significant digits.
- So let me just allow that more significant digits and we'll
- just round at the end.
- 0.082057, and the units here are liters atmospheres per
- mole at kelvin.
- And this makes sense.
- This liter will cancel out with that liter.
- That atmospheres cancels out with that atmospheres.
- I'm about to multiply it by temperature
- right here in kelvin.
- We'll cancel out there.
- And then we'll have a 1 over moles in the denominator.
- A 1 over moles in the denominator will just be a
- moles because you're going to invert it again.
- So that gives us our answer in moles.
- And so finally our temperature-- and you've got
- to remember you've got to do it in kelvin.
- So 25 degrees Celsius-- let me right it over here-- 25
- degrees Celsius is equal to, you just add 273 to it, so
- this is equal to 298 kelvin.
- So times 298 kelvin.
- And now we just have to calculate this.
- So let's do that.
- So let me clear this out.
- So we have-- let me use my keyboard-- so 0.0313
- atmospheres times 4.25 times 10 to the fourth.
- That e just means times 10 to the fourth.
- That's just the way that it works on this calculator.
- And then divided by 0.082057 divided by-- actually, just to
- make it clear, let me show you that I'm dividing by this
- whole thing, so let me insert some parentheses right here.
- So in the denominator we also are multiplying by 298.
- And let me close the parentheses.
- And then we get 54.4.
- We only have three significant digits.
- So this is equal to 54.4 moles.
- And we could see this liters cancels out with that liters.
- Kelvin cancels out with kelvin.
- Atmospheres with atmospheres.
- You have a 1 over mole in the denominator.
- So then 1 over 1 over moles is just going to be moles.
- Now, this is going to be 54.4 moles of water vapor in the
- room to have our vapor pressure.
- If more evaporates, then more will condense-- we will be
- beyond our equilibrium.
- So we won't ever have more than this amount
- evaporate in that room.
- So let's figure out how much liquid water that actually is.
- Let me do it over here.
- So 54.4 moles-- let me write it down-- moles of H2O.
- That's going to be in its vapor form and
- its going to evaporate.
- But let's figure out how many grams that is.
- So what is the molar mass of water?
- Well, it's roughly 18.
- I actually figured it out exactly.
- It's actually 18.01 if you actually use the exact numbers
- on the periodic table, at least one that I used.
- So we could say that there's 18.01 grams of H2O for every 1
- mole of H2O.
- And obviously, you can just look up the atomic weight of
- hydrogen, which is a little bit over 1, and the atomic
- weight of oxygen, which is a little bit below 16.
- So you have two of these.
- So 2 plus 16 gives you pretty close to 18.
- So this right here will tell you the grams of water that
- can evaporate to get us to that equilibrium pressure.
- So let's get the calculator out.
- So we have the 54.4 times 18.01 is equal to 970-- well,
- we only have three significant digits-- so 900, if your round
- this 0.7, it becomes 980.
- So this is 980 grams of H2O needs to evaporate for us to
- get to our equilibrium pressure,
- to our vapor pressure.
- So let's figure out how many milliliters of water this is.
- So they tell us the density of water right here.
- 0.997-- let me do this in a darker color-- 0.997 grams per
- millileter.
- Or another way you could view this is for everyone 1
- milliliter you have 0.997 grams of water
- at 25 degrees Celsius.
- So for every milliliter-- this is grams per milliliter-- we
- want milliliters per gram because we want this and this
- to cancel out.
- So we're essentially just going to divide 980 by 0.997.
- So what is that?
- Get the calculator out.
- So we have 980-- not cover up our work-- divided by 0.997 is
- equal to 980-- we'll just round this-- 983.
- So this is equal to 983.
- This and this canceled out, or that and that canceled out.
- So 983 milliliters of H2O.
- So we've figured out, using the Ideal Gas Law, that at 25
- degrees Celsius, which was 298 kelvin, that 983 milliliters
- of H2O will evaporate to get us to our
- equilibrium vapor pressure.
- Nothing more will evaporate, because beyond that if we have
- higher pressure than that, then you'll also have more
- vapor going to the liquid state.
- Because you'll have more stuff bouncing here.
- So if this much volume of water evaporates, we'll have
- the state where just as much is evaporating as just as much
- is condensing.
- So you will never get to a higher pressure than that at
- that temperature.
- So going back to the question, we figured out that 983
- milliliters of water will evaporate.
- The question was is that we placed 2 liters of water in an
- open container.
- So we just figured out that only 983 milliliters of that--
- so that's a little bit less than a liter.
- So this is a little bit less than 1,000 milliliters, and
- this is 1 liter.
- So a little bit less than half of this will evaporate for us
- to get to our vapor pressure.
- So to answer our question-- will all of the water
- evaporate at 25 degrees Celsius?
- No-- if we're assuming the room is sealed-- well, no, all
- of it will not.
- Only a little bit less than half of it will.
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