Now that we understand what the heat capacity is,
as well as some basic deviation that we just discuss from the ideal gas law
and what those look like.
Let's now discuss the final topic that we'll touch on gases
which is partial pressure as well as Dalton's law of partial pressure.
The partial pressure is actually quite simple.
It simply define as the total pressure of a gas
times the mole fraction of a particular type of molecule in that gas.
We'll call that mole fraction X for this equation.
What this is referring to is
suppose for example you have a system with more than one type of gas in it.
So for example, I have here in this piston,
not just nitrogen but also some carbon dioxide.
Suppose I have 5 moles of nitrogen and I have 15 moles of carbon dioxide
and I know the total pressure of the system is one atmosphere.
How could we define the partial pressure of just the nitrogen
or the partial pressure of just the carbon dioxide?
We do that very simply by saying for example the partial pressure of the nitrogen
is equal to the total pressure in our piston
times the mole fraction of the nitrogen that's present.
In other words, how many moles of nitrogen are there,
divided by how many moles of gas are there in total?
So we divide 5 for the 5 moles of nitrogen
divided by the total number of moles which is 5 for nitrogen and 15 for carbon dioxide.
And we get a ratio of fraction of the total amount of pressure
that's present in this piston of 0.250 atmospheres as the partial pressure for nitrogen.
We could do the exact same thing for the carbon dioxide
and say that the partial pressure of the carbon dioxide
is equal to the total pressure, one atmosphere times the fraction of moles
that are made up by carbon dioxide
which in this case is 15 moles divided by the total number of moles, 15 plus 5.
This time arriving at an answer of .750 atmospheres.
Dalton's Law of partial pressure is very simple.
It simply says that by definition of all these partial pressures,
.750 and .250 in our case, when you add them together
gets you back to the total pressure of your system
which in this case of course works, adding .750 plus .25,
we arrive back at the total pressure of our system of one atmosphere.
Let's do a quick example of how we use a partial pressure in Dalton's law.
Suppose you have 10 moles of hydrogen gas
with 5 moles of oxygen gas
and then they react to completion to form steam
basically making some water molecules.
We could ask if the pressure in the system is the same before as after.
What is the ratio of the total pressure of the oxygen and the
I'm sorry the partial pressure of the oxygen,
the O2 in the final configuration
after we've finish reacting to the partial pressure of oxygen
in the initial configuration before we reacted?
So just use the definition of partial pressure here
as well as some basic stoichiometry
to make sure you get the numbers of the hydrogen and oxygen correct.
And see if you can find this ratio of the partial pressures after to before.
If you've done this problem which I highly recommend,
it should look something like this.
First of all we have to understand that we have an equation of hydrogens
reacting with oxygens and this is going to form steam.
And the reason, this is a good review of the chemistry
that we have the numbers that we have here,
is that it takes 4 hydrogens with 2 oxygens to create 2 water molecules.
And again, you can compare these numbers here.
We have 2, hydrogen 2s, which makes 4 hydrogens,
which is the same as over here.
We also have 1 oxygen multiplied by 2
which is the same as the 2 oxygen atoms here.
What we have in this problem is, sorry not 4 moles but 10 moles of hydrogen
reacting with 5 moles of oxygen.
So that the entire initial number of moles is 15 moles.
The question we have to answer for ourselves is
after all this has reacted to completion,
how many moles total do we have at the end of the day
and also how many moles of oxygen do we have at the end of the day?
So let's do that, we have to be careful here because it's not as though,
all of these 10 will react with all of these 5.
Notice for that, for 1 oxygen molecule,
we need 4 hydrogen molecules which means that if we use all 5 moles of our oxygen
we would need 5 times 4 or 20 moles of hydrogen.
but we don't have 20 moles.
We only have 10 moles so that means we can only use half of our oxygen.
So keeping this in mind,
we remember that we have 1 mole of O2 for every 4 moles of hydrogen.
That means we will only have 10 over 4 or 2.5 moles of oxygen used.
This means that the end of the day,
we in fact have 2.5 moles of oxygen left over
because starting with 5 and only using 2.5.
We have 5 minus 2.5 or 2.5 moles of oxygen left.
We will also have 5 moles of H2O,
which you can see probably best by looking at the number of moles of hydrogen.
If we had 10 moles of hydrogen,
we're using all of those up, where we need 4 hydrogens per 2 H2O.
Then we only get half of 10 moles
since we get 2 H2O but needed 4 hydrogens,
giving us 5 moles of H2O, this is our final system.
The total number of moles in the system is 7.5.
And now we know exactly what we need to compare.
We're comparing this final system, 2.5 and 5 making 7.5
to this initial system, 10 and 5 making 15.
So let's write those down, we have the partial pressure of oxygen
in the initial system is equal to the total pressure of the initial system
times the mole fraction of oxygen which is 5 for the number of oxygens
over the total amount of oxygen which is 15 here.
These are both moles so this fraction is just a number
and since 5 over 15 is one third,
the partial pressure of the oxygen is one third of the total pressure.
We can do the same analysis for the final situation
and now use this box which tells us that we have the total pressure
times the mole fraction of oxygen here
which is 2.5 for the number of moles of oxygen
divided by the total number of moles which is 7.5 and our final configuration.
Again, the units of moles will cancel here.
In 2.5 divided by 7.5 so in fact also one third of the total pressure.
The ratio then which is what we're looking for here
of the final partial pressure of oxygen to the initial partial pressure of oxygen
is simply the ratio of these two quantities
which both happened to be one third of the total pressure.
The final important thing here to know is that these two pressures are identical
and this is told to us in the problem that after the reaction
we have the same total pressure that we had before we started this.
And so this in fact is equal to one.
So in this particular case the total reaction
does not change the partial pressure of the oxygen in the system
and in fact the partial pressure of the oxygen in the system stays the same.
This wraps up our discussion of partial pressure
which also wraps up our discussion of gases
and more complicated systems and some more complicated gas dynamics.
And so we finish with the gas chapter
and are ready to move on to some other more practical systems
that we'll be discussing in this course.
Thanks for listening.