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## Ideal gas equation

Current time:0:00Total duration:15:09

# Partial pressure example

## Video transcript

We have a situation here
where we have a four meter cubed container. Let's say that's a balloon
of some sort. And instead of having just one
type of molecule of gas in this container, we have three
molecules of gas. We have some oxygen molecules,
some hydrogen molecules, and some nitrogen molecules. And what the problem tells
us is, we have 2.1 total kilograms of gas. An of that, by math, 30.48% is
oxygen, 2.86% is hydrogen molecules, and 66.67%
is nitrogen. So what we need to figure,
and it's all at standard temperature at least. It's zero
degrees Celsius, which we know is 273 Kelvin. But we need to figure out what
is the total pressure in the container, or being exerted on
the surface of the container? And then, this is a new concept,
we want to figure out the partial pressures of
each of these gases. Essentially, how much are each
of these gases contributing to the total pressure? And you can imagine, if this
is a container and each of these are the three types of
gases, some of the pressure is going to be from the blue, maybe
oxygen is the blue gas, from the blue gas bumping
into the walls. Some of the pressure is going
to be from the hydrogen bumping into the walls, maybe
that's this yellow gas, and some of the pressure is going
to be from the nitrogen bumping into the walls. Let's say that's
the brown gas. So the partial pressure due to
nitrogen, that's the pressure just due to the brown particles
bumping into walls. Let's see if we can
figure this out. So the first thing to figuring
the total pressure is, we have to figure out the total moles of
molecules we have. And the easiest way I can figure out to
figure out the total number of moles is to figure
out the moles of each of these molecules. So if we have 2.1 kilograms of
gas-- let me write this down-- If we want to know moles
of nitrogen. Let me do it in the
nitrogen color. We know that 66.67% of this, we
could say 2.1 kilograms or 2100 grams, we know that
that's nitrogen. So let's do it in grams. Because
when we talk about molecular mass it's always in
grams. It doesn't have to be. But it makes it a lot simpler to
convert between atomic mass units and mass in our world. So this is 2/3 of 2100, that's
1400 grams of N2. Now what's the molar mass of
this nitrogen molecule? Well we know that the atomic
mass of nitrogen is 14. So this molecule has
two nitrogens. So its atomic mass is 28. So one of these molecules
will have a mass of 28 atomic mass units. Or one mole of N2 would have
a mass of 28 grams. So one mole is 28 grams. We have
1400 grams-- or we say grams per mole, if we want
to keep our units right. So if we say 1400 total grams
so divided by 28 grams per mole we should get the
number of moles. So 1400 divided by 28
is equal to 50. That worked out nicely. So we have 50 moles of N2. We could write that
right there. All right. Let's do oxygen next. So we do the same process
over again. 30% is oxygen. So let's do oxygen
down here, O2. So we take 30%. Remember, these percentages
I gave you, these are the percentages of the total
mass, not the percentage of the moles. So we have to figure out
what the moles are. So 30.48% of 2100 grams
is equal to about 640. Let's round off. So this is equal to 640 grams. And then what is the
mass of one mole of the oxygen gas molecule? The atomic mass of one
oxygen atom is 16. You can look it up on the
periodic table, although you should probably be pretty
familiar with it by now. So the atomic mass of
this molecule is 32 atomic mass units. So one mole of O2 is going to
be 32 grams. We have 640 grams. So how many
moles do we have? 640 divided by 32
is equal to 20. We have 20 moles of oxygen. Let me write that down. We have 20 moles. Now we just have to figure
out the hydrogen. 2.86% of all that is hydrogen. So let's see, if we do 2100
grams, remember I want to do everything in grams, so
I just want to do a unit conversion there. 2100 grams times 2.86% is
equal to about 60 grams. So hydrogen, this 2% of that
2100 grams is 60 grams. And then what's the molar mass
of one hydrogen. That's H2. So we know that the hydrogen
atom by itself has a mass of 1, doesn't have a neutron
in most cases. So the atomic mass
of this is 2. Or the molar mass of
this is 2 grams. So one mole of H2 is equal to
two grams. We have 60 grams. So we clearly have 60 divided
by 2, we have 30 moles. So this is interesting, even
though hydrogen was a super small fraction of the total mass
of the gas that we have inside of the container, we
actually have more actual particles, more actual molecules
of hydrogen than we do of oxygen. That's because each molecule of
hydrogen only has an atomic mass of 2 atomic mass units,
while each molecule of oxygen has 32 because there's two
oxygen atoms. So already we're seeing we actually have more
particles due to hydrogen than due to oxygen. And the particles are what
matter, not the mass, when we talk about part pressure
and partial pressure. So the first thing we can think
about is how many total moles of gas, how many
total particles do we have bouncing around? 20 moles of oxygen, 30 moles
of hydrogen, 50 moles of nitrogen gas. Add them up. We have 100 moles of gas. So if we want to figure out the
total pressure first, we can just apply this 100 moles. Let me erase this. I want to keep the
problem statement there the whole time. There you go. And I can erase some stuff
that you're not seeing off the screen. And now I'm ready. So we have 100 moles. So we just do our PV
is equal to nRT. We're trying to solve for P. P times 4 meters cubed
is equal to n. n is the number of moles. We have 100 moles. Is equal to 100 moles times R. I'll put a blank there for R,
because we have to figure out which R we want to use. Times temperature, remember we
have to do it in Kelvin. So 0 degrees Celsius
is 273 Kelvin. And then which R do we use? I always like to write
my R's down here. So we're dealing with meters
cubed, we're not dealing with liters, so let's use this one. 8.3145 meters cubed pascals
per mole Kelvin. The units there, I think I
should keep, these are in meters cubed pascals divided
by moles Kelvin. And then our temperature
was 273 Kelvin. Now let's do a little
dimensional analysis to make sure that we're doing
things right. These meters cancel out
with those meters. We divide both sides of the
equation by meters. These moles cancel
with these moles. The moles of the numerator, the
moles of the denominator. Kelvin in the numerator, Kelvin
in the denominator. And all we're left
with is pascals. Which is good because that
is a unit of pressure. So if we divide both sides
of this equation by 4. I'll just divide the 100 by 4. 25 times 8.3145 times 273. And the only unit that we were
left with was pascals. Which is nice, because that's
a unit of pressure. So, let's do the math, 25 times
8.3145 times 273 is equal to 56,746 pascals. And that might seem like
a crazy number. But the pascal is actually a
very small amount of pressure. It actually turns out that
101,325 pascals is equal to one atmosphere. So if we want to figure out how
many atmospheres this is, we could just divide that. Let me look it up
on this table. Yes, 101,325. So if we wanted to, we could
write this in kilopascals. That's 56.746 kilopascals. Or if we wanted it in
atmospheres we just take 56,746 divided by 101,325. It equals 0.56 atmospheres. So that's the total pressure
being exerted from all of the gases. I deleted that picture. So this is the total pressure. So our question is, what's
the partial pressure? We could use either of these
numbers, they're just in different units. What's the partial pressure of
just the oxygen by itself? Well, you look at the moles,
because we don't care about the actual mass. Because we're assuming that
they're ideal gases. We want to look at the
number of particles. Because remember, we said
pressure times volume is proportional to the number of
particles times temperature. And they're all at the
same temperature. So the number of particles
is what matters. So oxygen represents 20%
of the particles. 20/100. So, the partial pressure of
oxygen, let me write that as pressure due to oxygen,
due to O2. It's going to be 20% of
the total pressure. 20% times, let me write
56.746 kilopascals. I just took this pressure
measurement. If I wanted atmospheres,
times 0.56 atmospheres. So the partial pressure of
oxygen-- well, I already have the 0.56 written there. So times 0.2 is equal to
0.112 atmospheres. It's just 20% of this. How did I get 20%? We have a total of 100 mole
molecules in our balloon. 20 moles of them are oxygen. So 20% are oxygen. So 20% of the pressure is going
to be due to oxygen. So it's this many atmospheres. If I took 20% times
the 56,000. 0.2 times 56, you get roughly
11.2 kilopascals. I'm just multiplying 20% by
any of these numbers. And the numbers are going to
change depending on the units. So you do the same process. What is the partial pressure
due to nitrogen? Well even though 2/3 of the mass
is nitrogen, only 50% of the actual particles
are nitrogen. So 50% of the pressures is due
to the nitrogen particles. Remember, you have to convert
everything to moles. Because we only care about
the number of particles. So if you want to know the
partial pressure due to the nitrogen molecules,
it's 50% of these. So it's 28,373 pascals. That's roughly. Or if you took half of this,
approximately 28.4 kilopascals. Or approximately 0.28
atmospheres. And then finally, if you want
to figure out the partial pressure due to the hydrogen
atoms. The partial pressure due to the hydrogen atoms.
Hydrogen, even though it's a very small part of the mass,
it actually represents 30% percent of the molecules. And it's the molecules that
are bumping into things. We don't care so much
about the mass. So 30% of the molecules. And remember, when we talk
about kinetic energy, if something with a small mass has
the same kinetic energy, it's actually moving faster. So when we talk about
temperature, that's average kinetic energy. So maybe in this we could
imagine that the hydrogen might be moving faster than the
nitrogen or the oxygen. But we don't have to think about
that too much right now. But the partial pressure due to
hydrogen is just 30% of any of these numbers. Pick one. Let's do it in atmospheres. 0.3 times 0.56 is equal
to 0.168 atmospheres. And so the total pressure
should be equal to the pressure of each of the
partial pressures of each of the gases. Plus the partial pressure of
oxygen, plus the partial pressure of hydrogen. And so this one, we figured
out was 0.28 atmospheres. The oxygen was 0.112
atmospheres. And this was 0.168. And if you add these together,
you will see indeed, that they add to 0.56 atmospheres. Which was the total pressure
of our system. Anyway, this is a very long
and hairy problem. But the key takeaway is that
every molecule in the system contributes to the total
pressure in proportion to the number of moles it has as a
percentage of the total number of moles in the system. I hope that didn't confuse
you too much.