Voiceover: So continuing the history
of the ideal gas equation here.
We get to the 19th century with an Italian chemist
name Amedeo Avogadro.
Actually his name was Lorenzo Romano
Amedeo Carlo Avogadro di Quaregna e di Cerreto,
but we're going to call him Amedeo.
Amedeo spent a bit of his time experimenting
with tiny particles.
In honor of his experiments,
the number of particles in one mole of something
was named Avogadro's Number
which is approximately 6.02 times 10 to the 23.
One thing that Avogadro postulated
was that equal volumes of gas
at the same temperature and pressure
contain equal number of particles
whether those particles are atoms or molecules.
For instance, if you filled up four balloons
to exactly one liter at 25 degrees Celsius
with different gases,
so let's have a green balloon,
and we can say that this green balloon is argon,
and we'll have a pink balloon,
and I'll try to make it the exact same size
as the previous one, and we'll say it's nitrogen.
Then we've got a blue balloon
that we'll say is hydrogen.
Again, I'm off a little bit in the sizes,
but we're saying that all of these
have exactly one liter volume.
Then our fourth balloon will be yellow,
and we'll say it's filled with methane.
What Avogadro was saying was that
at one liter each one of these four balloons
would have the same number of particles.
I'm using five particles,
but the point is that all of these balloons
would have the same number of particles.
At one liter, each of these balloons
would contain .041 moles.
If we use Avogadro's Number here,
that's the same thing as saying
that each of these balloons would be
filled with 2.5 times 10 to the 22 particles.
If you think about it,
this is the exact same thought behind our value
for the molar volume at STP.
We said that for any gas, one mole at STP
would take up a volume of about 22.4 liters.
The fact that we know this is true for any gas
not just kind of a particular gas,
is credited to the work of Amedeo Avogadro.
Avogadro used this idea and some intuition
to develop his law which is V is equal to A,
a constant A, times the number of moles
which means that the moles of gas,
the number of particles of gas in the system
varies directly with the volume.
Or the quotient of V and N is equal to a constant.
I'm sure that you've used this same intuition
when you've blown up a balloon.
As you put in more air particles,
as you blow more air into the balloon,
the balloon gets larger.
For example, if we put in one mole of air particles,
the balloon would be about 22.4 liters.
As we blew more air into it,
say another third of a mole,
we would increase the volume to 29.88 liters.
Another third of a mole of air gets us
up to 29.88 liters.
Our volume is getting bigger.
If we were to blow even more air
into this system,
we would increase the volume even more.
So another third of a mole of air
would bring our volume up to 37.35 liters.
Our volume has expanded proportionally
to the amount of air that we've increased
in this system.
Avogadro's law validates the V,N part
of the ideal gas equation because the volume
and the moles are directly proportional
to each other.
Let's put this principle to work in an example.
When .15 moles of helium gas are added
to a piston containing .82 miles of another gas,
by what percent does the total volume increase
assuming isothermal and isobaric conditions?
Isothermal is same temperature,
and isobaric is same pressure.
We have same temperature, same pressure,
and we're looking for the percent
of volume increase related to the molar increase.
This is a perfect opportunity to use Avogadro's Law.
Let's start with V1 divided by N1 is equal
to V2 divided by N2.
We know this is true because Avogadro's Law
says that the quotient of the volume
and the number of moles is constant for an ideal gas.
So the initial conditions would equal
the final conditions.
We have a ratio set up here,
and we can rearrange this ratio
to say that V1 over V2 is equal to N1 over N2.
Here we've got kind of an interesting part
that we can actually start using to solve
this problem because what we see
is that the increase in volume
is directly proportional to the increase in moles.
For that reason, the percent change in volume
looks like the final volume minus the initial volume
divided by the initial volume.
This is just the formula kind of for percent change
which is pretty intuitive.
The final volume minus the initial volume,
the change, divided by the initial volume
is going to give us the percent change.
We know that this is equal to the
percent change in the number of moles
so we can say N2 minus N1 divided by N1.
This really gets us to a point where we can use
the values in the formula because it gives us
an initial number of particles, an initial moles
and it gives us the change in moles.
We had the final moles as well.
To find the final moles because it just says
that .15 moles are added to .82.
We'll need to add those together
so .82 plus .15 is equal to .97.
So .97 is our final number of moles.
That would be our N2.
Let's substitute these values in.
Our final moles is .97 minus our initial moles
which is .82 divided by our initial moles.
That's going to give us .15 divided by .82.
Now we're at a point where we can solve this,
and we can just kind of evaluate this expression
and that would give us the total,
or the percent of the total volume increase,
but I want to take a second and
I want to think about how we could
think about this kind of with some common sense
and use rounding to give us an approximate answer.
So .15 divided by .82 is between .15 divided by 1,
and .15 divided by .75.
And .15 divided by 1 is 15%,
and .15 divided by .75 it goes into .75
five times so that would give us 20%
because one-fifth is 20%.
If we had to think about this expression
in a hurry, say on a test, and we didn't have
a calculator,
we could use rounding to give us
a pretty approximate answer.
We know that the answer is between 15 and 20%.
It turns out that that actual answer
if you do use a calculator is really close
to 18.3%.
So when .15 moles of helium gas
are added to a piston containing .82 moles
of another gas,
by what percent does the total volume increase?
It's just about exactly 18.3%.
We were able to solve this problem
with a little bit of critical thinking
based on the principles of Avogadro's Law.