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This exercise is from

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chapter 12 of the Kotz, Treichel and Townsend

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and Chemistry and Chemical Reactivity book,

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and I'm doing it with their permission.

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So they tell us you place 2 liters of water

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in an open container in your dormitory room.

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The room has a volume of

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4.25 times 10 to the fourth liters.

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You seal the room and wait for the water to evaporate.

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Will all of the water evaporate

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at 25 degrees Celsius?

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And then they tell us at 25 degrees Celsius,

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the density of water is 0.997 grams per milliliter.

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And its vapor pressure is 23.8 millimeters of mercury.

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And this is actually the key clue

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to tell you how to solve this problem.

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And just as a bit of review,

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lets just think about what vapor pressure is.

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Let's say it's some temperature,

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and in this case we're dealing at 25 degrees Celsius.

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I have a bunch of water,

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and let me do that in a water color.

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I have a bunch of water molecules

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sitting here in a container.

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At 25 degrees Celsius,

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they're all bouncing around in every which way.

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And every now and then

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one of them is going to have enough kinetic energy

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to kind of escape the hydrogen bonds

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and all the things

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that keep liquid water in its liquid state

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and it will escape.

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It'll go off in that direction,

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and then another one will.

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And this'll just keep happening.

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The water will naturally vaporize in a room.

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But at some point,

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enough of these molecules have vaporized over here

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that they're also bumping back into the water.

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And maybe some of them can be captured back

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into the liquid state.

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Now, the pressure at which this happens

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is the vapor pressure.

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As you can imagine,

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as more and more these water molecules vaporize

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and go into the gaseous state,

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more and more will also create pressure,

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downward pressure.

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More and more will also be

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colliding with the surface of the water.

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And the pressure at which

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the liquid and the vapor states are in equilibrium

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is the vapor pressure.

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And they're telling us right now.

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It is 23.8 millimeters of mercury.

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Now, what we need to do to figure out this problem is

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say, OK, if we could figure out

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how many molecules need to evaporate

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how many molecules of water need to evaporate

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to give us this vapor pressure,

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we can then use the density of water

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to figure out how many liters of water that is.

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So how do we figure out how many molecules--

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let me write this down

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--how many molecules of water need to evaporate

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to give us the vapor pressure of

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23.8 millimeters of mercury?

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So what, I guess, law or formula--

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and I never like to just memorize formulas,

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but we've given this formula in the past

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and it's probably one of the top most useful formulas

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in chemistry, or really all of science--

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what formula or law deals with pressure?

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They give us the volume of the room

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because that's where the pressure will be inside of.

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So we have pressure, the equilibrium vapor pressure.

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We have a volume of a room right over here.

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We know the temperature of the room right over there.

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And we're trying to figure out the number of molecules

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that need to evaporate for us to get that pressure

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in that volume at that temperature.

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So what deals with pressure, volume,

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number of molecules, let's say in moles,

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so I'll write a lower case n

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number of molecules, and temperature?

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Well, we've seen this many, many times.

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It's the Ideal Gas Law.

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Pressure times volume is equal to

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the number of moles of our idea gas

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in this case we're going to use water as our ideal gas

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or vapor as our ideal gas

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times the universal gas constant times temperature.

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And this should never seem like some bizarre formula to

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you because it really,
really makes sense.

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If your pressure goes up, then
that means that either the

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number of molecules have gone
up, and we're assuming the

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volume is constant.

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That means either the number
of molecules have gone up,

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which makes sense-- more things
bouncing onto the side

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of the container.

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Or your temperature has gone
up-- the same number of

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things, but they're bumping with
higher kinetic energy.

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Or if your pressure stays the
same and your volume goes up,

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then that also means that your
number of molecules went up,

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or your temperature went up.

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Because you now have
a bigger container.

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In order to exert the same
pressure you need either more

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molecules or more kinetic energy
for the molecules you

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have.

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And you could keep playing
around with this, but I just

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want to make it clear this isn't
some mysterious formula.

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The first time I was exposed to
this I kind of did view it

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as some type of mysterious
formula.

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But it's just relating pressure,
volume, number of

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molecules and temperature.

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And then this is just the
universal gas constant.

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So let's just get everything
into the right units here.

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And then what we're trying to
solve for, we want to figure

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out the number of molecules
of water.

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So we want to solve for n.

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And if we know the number of
moles of water, we can figure

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out the number of
grams of water.

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And then given the density of
water we can figure out the

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number of milliliters of water
we are dealing with.

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So let's just rewrite the Ideal
Gas Law by dividing both

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sides by the universal gas
constant and temperature.

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So that you get n is equal to
pressure times volume, over

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the universal gas constant
times temperature.

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Now, the hardest thing about
this is just making sure you

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have your units right and you're
using the right ideal

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gas constant for the right
units, and we'll

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do that right here.

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So what I want to do, because
the universal gas constant

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that I have is in terms of
atmospheres, we need to figure

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out this vapor pressuree- this
equilibrium pressure between

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vapor and liquid-- we need to
write this down in terms of

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atmospheres.

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So let me write this down.

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So the vapor pressure
is equal to 23.8

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millimeters of mercury.

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And you can look it up at a
table if you don't have this

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in your brain.

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One atmosphere is equivalent to
760 millimeters of mercury.

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So if we wanted to write
the vapor pressure as

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atmospheres-- let me get my
calculator out, get the

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calculator out, put it right
over there-- so it's going to

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be 23.8 times 1 over 760,
or just divided by 760.

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And we have three significant
digits, so

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it looks like 0.0313.

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So this is equal to 0.0313
atmospheres.

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That is our vapor pressure.

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So let's just deal with
this right here.

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So the number of molecules of
water that are going to be in

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the air in the gaseous state,
in the vapor state, is going

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to be equal to our
vapor pressure.

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That's our equilibrium
pressure.

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If more water molecules
evaporate after that point,

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then we're going to have a
higher pressure, which will

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actually make them favor more of
them going into the liquid

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state, so we'll go kind of past
the equilibrium, which is

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not likely.

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Or another way to think about
it-- more water molecules are

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not going to of evaporate at
a faster rate than they are

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going to condense beyond
that pressure.

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Anyway, the pressure here
is 0.0313 atmospheres.

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The volume here-- they told us
right over here-- so that's

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the volume-- 4.25.

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4.25 times 10 to the
fourth liters.

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And then we want to divide that
by-- and you want to make

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sure that your universal gas
constant has the right units,

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I just looked mine up on
Wikipedia-- 0.08-- see

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everything has three
significant digits.

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So let me just allow that more
significant digits and we'll

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just round at the end.

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0.082057, and the units here
are liters atmospheres per

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mole at kelvin.

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And this makes sense.

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This liter will cancel
out with that liter.

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That atmospheres cancels out
with that atmospheres.

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I'm about to multiply
it by temperature

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right here in kelvin.

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We'll cancel out there.

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And then we'll have a 1 over
moles in the denominator.

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A 1 over moles in the
denominator will just be a

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moles because you're going
to invert it again.

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So that gives us our
answer in moles.

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And so finally our temperature--
and you've got

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to remember you've got
to do it in kelvin.

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So 25 degrees Celsius-- let
me right it over here-- 25

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degrees Celsius is equal to,
you just add 273 to it, so

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this is equal to 298 kelvin.

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So times 298 kelvin.

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And now we just have
to calculate this.

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So let's do that.

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So let me clear this out.

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So we have-- let me use my
keyboard-- so 0.0313

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atmospheres times 4.25 times
10 to the fourth.

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00:10:21,340 --> 00:10:24,340
That e just means times
10 to the fourth.

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00:10:24,340 --> 00:10:27,220
That's just the way that it
works on this calculator.

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00:10:27,220 --> 00:10:39,310
And then divided by 0.082057
divided by-- actually, just to

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make it clear, let me show you
that I'm dividing by this

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whole thing, so let me insert
some parentheses right here.

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So in the denominator we also
are multiplying by 298.

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00:10:53,300 --> 00:10:55,285
And let me close the
parentheses.

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And then we get 54.4.

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00:11:02,270 --> 00:11:03,710
We only have three significant
digits.

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00:11:03,710 --> 00:11:10,850
So this is equal
to 54.4 moles.

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00:11:10,850 --> 00:11:13,580
And we could see this liters
cancels out with that liters.

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00:11:13,580 --> 00:11:15,300
Kelvin cancels out
with kelvin.

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00:11:15,300 --> 00:11:16,880
Atmospheres with atmospheres.

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00:11:16,880 --> 00:11:18,880
You have a 1 over mole
in the denominator.

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00:11:18,880 --> 00:11:24,040
So then 1 over 1 over moles
is just going to be moles.

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00:11:24,040 --> 00:11:29,730
Now, this is going to be 54.4
moles of water vapor in the

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00:11:29,730 --> 00:11:32,580
room to have our
vapor pressure.

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00:11:32,580 --> 00:11:36,070
If more evaporates, then more
will condense-- we will be

216
00:11:36,070 --> 00:11:37,800
beyond our equilibrium.

217
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So we won't ever have more
than this amount

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00:11:40,610 --> 00:11:41,890
evaporate in that room.

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00:11:41,890 --> 00:11:47,440
So let's figure out how much
liquid water that actually is.

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00:11:47,440 --> 00:11:48,550
Let me do it over here.

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00:11:48,550 --> 00:11:59,170
So 54.4 moles-- let me write
it down-- moles of H2O.

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00:11:59,170 --> 00:12:00,386
That's going to be in
its vapor form and

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00:12:00,386 --> 00:12:01,780
its going to evaporate.

224
00:12:01,780 --> 00:12:04,270
But let's figure out how
many grams that is.

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00:12:04,270 --> 00:12:08,560
So what is the molar
mass of water?

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00:12:08,560 --> 00:12:10,110
Well, it's roughly 18.

227
00:12:10,110 --> 00:12:11,910
I actually figured
it out exactly.

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00:12:11,910 --> 00:12:16,600
It's actually 18.01 if you
actually use the exact numbers

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00:12:16,600 --> 00:12:18,880
on the periodic table, at
least one that I used.

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00:12:18,880 --> 00:12:28,120
So we could say that there's
18.01 grams of H2O for every 1

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00:12:28,120 --> 00:12:30,490
mole of H2O.

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00:12:30,490 --> 00:12:33,580
And obviously, you can just look
up the atomic weight of

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hydrogen, which is a little
bit over 1, and the atomic

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weight of oxygen, which is
a little bit below 16.

235
00:12:38,955 --> 00:12:40,250
So you have two of these.

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00:12:40,250 --> 00:12:43,530
So 2 plus 16 gives you
pretty close to 18.

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00:12:43,530 --> 00:12:47,990
So this right here will tell
you the grams of water that

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00:12:47,990 --> 00:12:51,760
can evaporate to get us to that
equilibrium pressure.

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00:12:51,760 --> 00:12:53,580
So let's get the
calculator out.

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00:12:53,580 --> 00:13:05,090
So we have the 54.4 times 18.01
is equal to 970-- well,

241
00:13:05,090 --> 00:13:07,650
we only have three significant
digits-- so 900, if your round

242
00:13:07,650 --> 00:13:10,380
this 0.7, it becomes 980.

243
00:13:10,380 --> 00:13:17,180
So this is 980 grams of H2O
needs to evaporate for us to

244
00:13:17,180 --> 00:13:19,160
get to our equilibrium
pressure,

245
00:13:19,160 --> 00:13:20,540
to our vapor pressure.

246
00:13:20,540 --> 00:13:24,180
So let's figure out how many
milliliters of water this is.

247
00:13:24,180 --> 00:13:27,090
So they tell us the density
of water right here.

248
00:13:27,090 --> 00:13:33,500
0.997-- let me do this in a
darker color-- 0.997 grams per

249
00:13:33,500 --> 00:13:34,920
millileter.

250
00:13:34,920 --> 00:13:39,370
Or another way you could view
this is for everyone 1

251
00:13:39,370 --> 00:13:49,780
milliliter you have 0.997
grams of water

252
00:13:49,780 --> 00:13:52,810
at 25 degrees Celsius.

253
00:13:52,810 --> 00:13:55,200
So for every milliliter-- this
is grams per milliliter-- we

254
00:13:55,200 --> 00:13:58,470
want milliliters per gram
because we want this and this

255
00:13:58,470 --> 00:13:59,600
to cancel out.

256
00:13:59,600 --> 00:14:04,260
So we're essentially just going
to divide 980 by 0.997.

257
00:14:04,260 --> 00:14:05,185
So what is that?

258
00:14:05,185 --> 00:14:06,860
Get the calculator out.

259
00:14:06,860 --> 00:14:16,760
So we have 980-- not cover up
our work-- divided by 0.997 is

260
00:14:16,760 --> 00:14:21,580
equal to 980-- we'll just
round this-- 983.

261
00:14:21,580 --> 00:14:25,930
So this is equal to 983.

262
00:14:25,930 --> 00:14:29,230
This and this canceled out, or
that and that canceled out.

263
00:14:29,230 --> 00:14:32,620
So 983 milliliters of H2O.

264
00:14:32,620 --> 00:14:38,140
So we've figured out, using the
Ideal Gas Law, that at 25

265
00:14:38,140 --> 00:14:44,090
degrees Celsius, which was 298
kelvin, that 983 milliliters

266
00:14:44,090 --> 00:14:47,520
of H2O will evaporate
to get us to our

267
00:14:47,520 --> 00:14:50,730
equilibrium vapor pressure.

268
00:14:50,730 --> 00:14:53,590
Nothing more will evaporate,
because beyond that if we have

269
00:14:53,590 --> 00:14:57,540
higher pressure than that, then
you'll also have more

270
00:14:57,540 --> 00:14:58,930
vapor going to the
liquid state.

271
00:14:58,930 --> 00:15:01,640
Because you'll have more
stuff bouncing here.

272
00:15:01,640 --> 00:15:07,950
So if this much volume of water
evaporates, we'll have

273
00:15:07,950 --> 00:15:11,020
the state where just as much is
evaporating as just as much

274
00:15:11,020 --> 00:15:11,700
is condensing.

275
00:15:11,700 --> 00:15:16,350
So you will never get to a
higher pressure than that at

276
00:15:16,350 --> 00:15:18,190
that temperature.

277
00:15:18,190 --> 00:15:21,320
So going back to the question,
we figured out that 983

278
00:15:21,320 --> 00:15:23,990
milliliters of water
will evaporate.

279
00:15:23,990 --> 00:15:26,940
The question was is that we
placed 2 liters of water in an

280
00:15:26,940 --> 00:15:28,360
open container.

281
00:15:28,360 --> 00:15:31,920
So we just figured out that only
983 milliliters of that--

282
00:15:31,920 --> 00:15:34,200
so that's a little bit
less than a liter.

283
00:15:34,200 --> 00:15:40,460
So this is a little bit less
than 1,000 milliliters, and

284
00:15:40,460 --> 00:15:41,960
this is 1 liter.

285
00:15:41,960 --> 00:15:47,050
So a little bit less than half
of this will evaporate for us

286
00:15:47,050 --> 00:15:48,430
to get to our vapor pressure.

287
00:15:48,430 --> 00:15:50,370
So to answer our question--
will all of the water

288
00:15:50,370 --> 00:15:52,390
evaporate at 25 degrees
Celsius?

289
00:15:52,390 --> 00:15:55,730
No-- if we're assuming the room
is sealed-- well, no, all

290
00:15:55,730 --> 00:15:56,870
of it will not.

291
00:15:56,870 --> 00:16:00,980
Only a little bit less
than half of it will.