Now that we have an idea of how to derive some of the basic laws
and how these are come up with experimentally, certain proportionalities.
We're ready to derive one all-encompassing law
that incorporates all the laws we just discussed.
And that's the ideal gas law.
What we do with the ideal gas law
is take the laws that we just came up with which are here.
These experimental results are giving a certain proportionalities
and try to put all the variables that we discussed into one equation.
If we do that, we could say that the pressure times the volume
is proportional to the number, the number of molecules
times the temperature of your system.
This satisfies as you can tell by examination
and I do recommend that you look at this and really think about it and see
how the ideal gas law actually matches
and would give you all of these individual experimental results that we found already.
We could then ask ourselves if it's proportional to the pressure
times the volume is proportional to the number of particles times there temperature.
What is the proportionality constant?
In other words, I know that if I double the pressure,
I will have doubled the number times the temperature.
But I don't know what the actual numbers are,
so I would like to know what this proportionality constant is.
It turns out and you can find this experimentally
that is proportionality constant is a number called R
and this number is called the universal gas constant.
It has a value of a little more than 8 joules per Kelvin mole
and you can always look at the unit yourself to make sure
that the left-hand side of the equation
and the right-hand side of the equation have the same units.
When we're giving a constant values,
here's how you would solve a problem using the ideal gas law.
So we've listed one more time the ideal gas law above,
pressure times volume equals number times the gas constant times temperature,
PV equals nRT.
You take particular constant values in a problem
so most problems you could assume some things are constant,
either because that's told to you or because you can infer on your own.
What you do is simply collect all those constants together mathematically
and then everything that's left on the other side of your equation
must be conserved because they're equal to a constant.
Just to give this a concrete example and make sure it makes some sense.
Suppose we had a constant volume and temperature in some given problem.
What we could do is re-arrange the ideal gas law
so that we have all of our constants together in this case the temperature
and the volume are constant and the gas constant
is of course by definition of constant so if we put all these together,
we have something like this, R times T divided by the volume.
Everything else in the gas law which is now just P over n
is clearly just mathematically equal to a constant.
Which means that P over n, the pressure over the number of particles
is conserved because it's a constant.
Which means that in a given problem,
we can take the initial pressure and number
and relate it to the final pressure and number
because this ratio as we just saw has to be some constant value.
There's one other way the gas law is written
so it's good to go over that and see where it comes from
and how it relates to the gas law that we've just derived.
What you can notice is that if we look at the right-hand side of the gas law here.
We can look at that, that quantity,
the number times that gas at a constant.
The number in units, so let's look at the units of this,
we have the number of moles times the gas constant which is joules per Kelvin mole.
Instead of measuring in moles what we could do
is instead actually measure the number of molecules.
So cancelling the units of moles or re-writing in terms of number of molecules
we could say that we have instead the number of molecules
times a new constant and that constant is called K.
K is Boltzmann's constant and instead has units of joules per Kelvin
rather than joules per Kelvin mole like the universal gas constant did.
Sometimes this is called the microscopic form of the ideal gas law
because instead of measuring moles which again is a very big number.
It is the way to measure, you know actual physical quantities
that you might scoop up from the ground.
We're actually measuring the number of molecules there at themselves
and so this microscopic form of the ideal gas law
is better suited the case as what we're talking about
particular interactions and actually counting the number of molecules.
There's a few ways to describe how this law is derive.
So here's one more that might help you if it makes more sense.
We can just look at that term R times T and compare it to the term K times T.
Again the only difference is that in one case we're taking the number of moles
and multiplying by joules per mole.
And then the other case we're taking the number of molecules
and multiplying by joules per molecule.
So again you get the same numbers
everything else in the ideal gas law is the same.
There's no actual difference if you were doing a problem.
It's just a different way of measuring the number of molecules or the number of moles,
in whatever system that you have.