We will now go through a series of laws
regarding gases that can all be 
derived from the ideal gas law,
and each of them is a situation where two 
of the variables in the ideal gas law
are allowed to vary where the 
other two are not allowed to vary.
Of course, R (the universal gas constant)
never varies. it's always a constant.
So our first relationship is called Boyle's law,
and it occurs whenever pressure 
and volume are allowed to vary.
And because pressure and volume 
are the two that are allowed to vary,
then that means that the number of moles [n]
and T (temperature) are not allowed to vary,
so this whole right side of the 
ideal gas law is held constant.
An example of this is shown in this picture
where we have a container 
with a certain volume of gas
and we have a piston on the top of it 
and there is a pressure applied to it,
and therefore, there is pressure inside of it,
and then you change the situation 
where you apply more pressure
so the pressure inside the 
container is larger as well,
and therefore, the volume is smaller,
so when pressure increases,
volume decreases.
Going back to the ideal gas law, if you say 
that this right-hand side of the ideal gas law
is constant, then that means also 
the left side has to be constant.
So pressure (P) times volume (V) is a constant,
and if pressure times volume is a constant,
then that means that pressure times volume before
is going to equal pressure times 
volume after any given change
to the same number of moles of
gas and at the same temperature,
Therefore, we can say P sub1 V sub1
equals P sub2 V sub2,
so pressure in situation 1 
times volume in situation 1
will equal pressure in situation 2
times volume in situation 2.
So you need some kind of a container that 
can vary volume. You need variable volume.
Examples that are frequently used are pistons
and cylinders, like we show in this example.
Another container that is used is simply 
a balloon. A balloon has a variable volume.
So problems involving Boyle's law will often
be in pistons and cylinders or with balloons.
Let's do an example problem of Boyle's law,
and this one is a balloon.
So we'll say a certain balloon, which has 
a maximum volume of 4 liters (4 L)
has a volume of 1.7 L at sea level.
When that balloon is carried up 
Mount Everest, where the pressure is—
Do you remember what the pressure 
is on the top; of Mount Everest?
It's 30.0 kilopascal (kPa). Will the balloon pop?
So we start at 1.7 L, we carry it
up to Mount Everest,
we want to know if it's going to exceed 4 L.
So let's do the problem with Boyle's law
relationship: P sub1 V sub1 = P sub2 V sub2
The first situation is down at sea level, 
where the pressure is—
If we want to use the same units 
(kilopascals), it's about 101.325 kPa.
The volume at sea level is 1.7 L, and then P sub2 
we know is 30.0 kPa and we don't know V sub2.
So then we can solve for V sub2 . . .
and our kilopascals will cancel and this is
all liters, and we'll get a volume of 5.7 L.
So does it survive the trip up 
Mount Everest? No, it does not.
It pops because anything greater
than 4 L will pop, so it pops.
Notice that when you use this equation,
you don't have to use the pressure
units in the universal gas constant
because as long as you use the same one on both sides, 
the units will cancel (and the same thing with volume).
Our next relationship is when we allow
volume and temperature to change
and we hold the other quantities constant.
So in this case, the number of moles
is constant and the pressure is constant.
So what is constant? Pressure, number of
moles, and always the universal gas constant.
So what is an example of this?
This is an example where you have—
you also need a volume that 
can change, like a balloon,
so you have a balloon at a certain temperature
and you change the temperature 
and you see what happens to the balloon.
So if you warm up a balloon,
it will get larger.
Let's solve for Charles's law by putting everything 
that is constant on one side of the equation.
What needs to go on one side? 
We need to have [n] and R and P.
We have to divide by P, and that means 
that T is going to go over to the other side,
and we get V over T.
Now we have everything constant 
on this side of the equation,
which means that whatever is on the other
side of the equation must also be constant.
So the quotient, V over T, must be constant . . .
and that gives us our Charles's law relation,
which is V sub1 over T sub1 
equals V sub2 over T sub2,
and that is an expression of Charles's law.
Now make sure that when you
do a problem in Charles's law
(or any other ideal gas law-related example)
that you have to convert your
temperatures to absolute,
and you can see why in this example
because you have 0° C, which would
be dividing by 0, which you can't do,
so be sure to convert to Kelvin.
(I'll just write that there to remind you.)
The next law is when you allow pressure and
temperature to vary and everything else is constant
and that is known as Gay-Lussac's law.
We're going to let pressure and temperature
vary, and keep everything else constant.
So what is constant? Volume is constant
and R is always constant, and [n] is constant.
So in this case, we don't need
a volume that can change.
In fact, we want one that won't. 
We want it to be constant.
So we use a solid container and we're
varying temperature and pressure.
So one way to do that is take a
container full of gas and heat it,
and what happens is, all the molecules go faster; 
therefore, they collide with the walls more.
And since they collide with the 
walls more, that's more pressure.
Again, put everything constant 
on one side of the equation.
We'll put nR over V on one side and
that leaves P over T on the other side.
So this is the constant. This side, which means 
P over T, is a constant, and that gives us
P sub1 over T sub1 equals P sub2 over T sub2,
and that is our expression for Gay-Lussac's law.
So we've just done three laws.
We've done every combination of two
of pressure, volume, and temperature,
So when you vary P and V, it's Boyle's law;
when you vary V and T, it's Charles's law;
and when you vary P and T, it's Gay-Lussac's law.
We haven't varied [n] at all.
We're going to show one example of varying [n]
and that is known as Avogadro's law.
In Avogadro's law, you vary [n] and V.
What's an example of varying only [n] and V?
Well, if you have a volume that's going to change,
you need a balloon or some other volume that can move.
So we use a balloon, and a really good example
of Avogadro's law is simply blowing up a balloon.
As you are blowing into it, 
you're adding more moles of gas.
You're increasing [n], and therefore,
the volume increases. So, simple enough.
You've known this since you were a little kid,
that if you put more air in the 
balloon, volume will get larger.
So what are we holding constant?
We're holding pressure constant, we're holding 
R always, and we're holding temperature constant.
So again, we put everything constant on one side
RT over P, which gives us V over [n] is also constant,
so our Avogadro's law is V over [n] 
(v sub1 over [n] sub1) for one situation
equals V over [n] (v sub2 over [n] sub2)
for a second situation.
And we'll do an example problem of Avogadro's law.
We have a 5.00 L balloon of neon (Ne) gas with 0.965 mol 
(moles) of neon gas at room temperature and pressure.
If the amount of gas is increased 
to 1.8 mol, what's the new volume?
Let's write down what we know.
V sub1 is 5.00 L; V sub2 we don't know;
[n] sub1 is 0.965 mol, and [n] sub2 is 1.8 mol.
Let's set up our equation. V sub1 over [n] sub1 
equals V sub2 over [n] sub2.
So we get 5.00 liters over 0.965 mol equals—
We don't know V sub2; and that's over 1.8 mol.
You can solve for V sub2 with a little algebra
and get 9.33 L for the second volume.
So there are your 4 laws. We had Boyle's law, Charles's 
law, Gay-Lussac's law, all of which varied P, V, and T.
Then we finally have Avogadro's law,
which varies V and [n], all of which 
can be derived from the ideal gas law
by holding some things constant and 
allowing some things to change, two at a time.
sro
