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## Ideal gas equation

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# Ideal gas equation example 1

## Video transcript

In the last video we hopefully
learned the intuition behind the ideal gas equation, that
pressure times volume is equal to the number of molecules we
have times some constant times the temperature. And that's all nice and it
hopefully it makes sense to you how all of these
fit together. That pressure should be inverse
to volume and that's why you're multiplying both
sides by each other. You could take volume
and put it on this side of the equation. Or that pressure should be
proportional to the number of particles and the temperature. But now let's apply it and
actually do some problems. Because just knowing this
isn't good enough. So let's say that I have a two
liter container, or let's say a two liter balloon, containing
hydrogen gas. And that's hydrogen as
a diatomic molecule. So each molecule has two
hydrogens in it. And let's say I'm measuring
it at 30 degrees Celsius. My brain is really
malfunctioning. And let's say that the pressure
on the outside of the balloon, we've measured
at two atmospheres. So my question to you
is how many moles of hydrogen do we have? So let's apply our ideal
gas equation. And since we're dealing with
liters and atmospheres, we have to make sure
we use the right proportionality constant. So our pressure is given
in atmospheres. Let me write down all
the units, actually. So we have 2 atmospheres times
our volume is 2 liters, is equal to n. n is the number of particles
we care about, and we care about it in moles, but let's
just write n there for now. Is equal to n times R. I'll do R in a second. Times T. Now you might be tempted to just
put 30 degrees in there. But in all of these problems--
in fact in general, whenever you're doing any of these gas
problems or thermodynamics problems, or any time you're
doing math with temperature-- you should always convert
into Kelvin. And just as a bit of review as
to what Kelvin is, it's just a different scale. So for example, the lowest
possible temperature that that can be achieved in the universe,
when you think about it in Celsius, let me draw a
little temperature scale here. So if that's the temperature
scale. I'll draw two, one for Celsius
and one for Kelvin. So the lowest possible
temperature that can be achieved in the universe, and
when we say the lowest possible temperature that means
that the average kinetic energy of the molecules
or the atoms are zero. They're just not moving. They're just stationary. So in Celsius, it's minus
273.15 degrees Celsius. So zero might be some
place over here. Zero, that's where
water freezes. And then 100 degrees, that's
where water boils. And you can immediately see,
the whole Celsius scale was made based on the freezing
point and the boiling point of water. So look at this and you say, if
I have something that's 5 degrees and I have another thing
that's 10 degrees, when you look at the Celsius scale,
you're like, oh, maybe the 10 degree thing it has twice
as much energy as the 5 degree thing. It has twice the temperature. But when you look at it from the
absolute distance to zero. Let me see if I can draw this. So the 10 degree is all the way
over here and the 5 degree is almost as far. So the 10 degrees Celsius is
only a slight increment over 5 degrees Celsius, if you were
to divide the two. It's not twice as hot. And that's why they came up
with the Kelvin scale. Because in the Kelvin scale,
absolute zero is defined as 0. So this right here is
zero degrees Kelvin. And so zero degrees Kelvin
is absolute zero. So what is zero degrees
Celsius? And the increments
are the same. One degree change in
Celsius is one degree change in Kelvin. So at least they keep it,
it's just a shift. So this is going to be plus
273 degrees Kelvin. And then 5 degrees would be plus
278; ' 10 degrees would be plus 283 Kelvin. And then you see that 5 and 10
degrees really aren't that different from each other. But in general, if you want to
convert from Celsius to Kelvin you just add 273 degrees. So 30 degrees Celsius is what? Well, this 5 and 10 I drew
too close to 100. But let's say it's
sitting here. It would be 303 degrees
Kelvin. All right, so now for our
temperature, that's what we were worried about. We wanted to put in the
temperature there. So now we can put in our
303 degrees Kelvin. Now we have to figure out what
constant to use here. And I've written a couple
of down here. Remember, we're dealing with
atmospheres and liters. So I wrote down a couple of
versions of R right here. Let's see we're dealing with
atmospheres and liters. And in the denominator we're
always dealing with mole and Kelvin no matter what. So those are always
going to be there. So we should use this
proportionality constant. R is equal to 0.082 liter
atmospheres per mole Kelvin. Let me write that down. So let me rewrite our whole
equation actually. So I have 2 atmospheres times 2
liters is equal to n times, I have a bad memory, 0.082 liter
atmospheres per mole Kelvin, times 303
degrees Kelvin. So let's see what we can do. Let's see if all of the
units work out. So we can always, when you do
dimensional analysis, you can treat units like numbers. So if you divide both sides of
this equation by atmospheres, the atmospheres cancel out. Divide both sides of this
equation by liters, liters cancel out. You have a Kelvin in the
numerator, Kelvin in the denominator, that cancels out. And so we have 2 times
2 is equal to n times 0.082 times 303. And then we have just a per mole
and a 1 over the mole. So to solve for n, or the number
of moles, what we do is we divide both sides of this
equation by all of this stuff. So we get 2 times 2 is 4. 4 divided by 0.082
divided by 303. I'm just taking this and putting
it on the left-hand side, dividing both
sides by it. And when you divide by a per
mole, if you put a 1 over a mole here, that's the
same thing as multiplying by a mole. So it's good, the units
all worked out. We're getting n in
terms of moles. And so we just have to get the
calculator out and figure out how many moles we're
dealing with. So we have 4 divided
by 0.082 divided by 303 is equal to 0.16. If we wanted to go more
digits, .161, but we'll just round. So this is equal to
0.16 moles of H2. I am telling you actually here,
the exact number of hydrogen molecules. But if you wanted a number,
you'd just multiply this times 6.02 times 10 to the 23 and then
you would have a number in kind of the traditional
sense. And of course, if you wanted
to know what is the mass of the hydrogen you have. You'd
say, OK well one mole of H2 has a mass of what? The mass of one hydrogen is
one atomic mass unit. The mass of two hydrogen when
it's in its molecular form, is two atomic mass units. So a mole of it is going
to be 2 grams. So in this case, we
have 0.16 moles. So if we wanted to convert that
to grams, this in the case of these hydrogen gas
molecules would be 0.32 grams. And I just multiplied it by 2
because each mole is 2 grams. Anyway, I hope you found
that useful. I'm going to do a bunch more
of these problems. Because I think the math is
pretty straightforward here. The thing that always makes it
daunting, I think, is the units and making sure you're
using the right units. What is they are using meters
cubed instead of liters, or kilopascals instead
of atmospheres. So I'll try to do a bunch of
examples where we use all the different units and you're able
to pick our constants appropriately.