Liquids are incompressible; gases are not incompressible.
When you decrease the volume of a gas by 50%, that's no problem.
It's impossible to do that for a liquid.
In liquids, the atoms and the molecules effectively touch each other, whereas in gases, they are very far apart, so that's why you can compress the gases.
If you take air at one atmospheres, the density is a thousand times less than the density of water.
What it tells you is that the molecules are much further apart.
It is an experimental fact that there is a simple relation between the pressure that you see there, the volume of a gas, the temperature of a gas in degrees Kelvin, and the number of molecules that you have.
Now, when you see the word "molecules," I may often mean "atoms." I realize that helium and neon and krypton and argon are atomic gases, and that O2 and H2 and CO2 are molecular gases.
So I will use that word "molecules" even when I mean "atoms," and maybe vice versa, just for simplicity.
The relation that exists between these quantities, PV equals nRT: pressure, volume, n is the number of moles--
I'll get back to that--
R is the universal gas constant, which is 8.3 joules per degree Kelvin, and T must be in degrees Kelvin.
So, what is a mole? A mole has always about 6.02 times ten to the 23 molecules, or atoms, in the case that you have helium, but I will call that molecules.
And this number is called Avogadro's number.
So that's the definition of a mole.
If you take a mole of helium, or a mole of oxygen, or CO2, or N2, it doesn't matter, it always has this number of molecules, approximately.
Now, each of these substances have very different masses.
If I take, for instance, carbon, then one mole of carbon would weigh very close to 12 grams.
If I take helium, one mole of helium would weigh very close to four grams.
And if I took oxygen two, O2, then one mole would be very close to 32 grams.
So the masses are very different in a mole but not the number of molecules or the number of atoms.
When I take a neutral atom, then we have a nucleus, and the nucleus contains protons and neutrons.
It has Z protons and it has N neutrons.
The protons are positively charged, and it has Z electrons if it is a neutral atom.
There is almost no weight in the electrons; you can almost ignore that.
Everything is in the protons and in the neutrons.
N plus Z is called A, and that's called the atomic mass number.
Let's look at carbon in a little bit more detail.
If we have carbon--
and I call it carbon 12 for now, you'll see shortly why--
then carbon has always six protons in the nucleus; otherwise it isn't carbon.
And when it has six neutrons, then A is 12.
That's why we call it carbon 12.
So the atomic mass number of carbon is 12, but if you had, for instance, carbon 14--
which happens to be radioactive--
again, six protons, otherwise it wouldn't be carbon, you would have eight neutrons now, and now you would have...
atomic mass number would be 14.
A mole is this number in grams, and so you see carbon...
is the atomic mass number in grams--
you see 12 there.
If you go to helium, it has two protons and two neutrons, so A is four--
that's why you see your four grams.
If you take oxygen, it has eight protons and eight neutrons, so A is 16, but you have O2 in gas form, so now your atomic mass number has to be doubled to 32.
And so a mole of O2 is therefore 32 grams.
In fact, Avogadro's number is defined through carbon 12.
If you take 12 grams of carbon 12, and you count the number of atoms that you have, then you find exactly Avogadro's number.
That's the definition of that number, and that's very close to what we have there, 6.02 times ten to the 23rd.
The mass of the proton and the mass of the neutron are nearly equal.
I wrote down m2 for the mass of the neutron; of course, that should have been m of n.
So the mass of a molecule, or an atom, whatever the case may be, would be this number A--
because that's the sum of the protons and neutrons--
times the mass of the neutrons and the protons.
And so this is A times--
approximately, I should put a wiggle here--
1.66 times ten to the minus 27 kilograms.
So that's now an individual mass of either an atom or a molecule, and all that information, you have there and that's, of course, on the Web.
So, let's do a trivial example.
I take gases, any kind of gas--
you choose whatever you want--
and I take one atmosphere.
So that means that the pressure is 1.03 times ten to the fifth pascal.
I do it at room temperature, so T is 293 degrees Kelvin.
And I take in all cases only one mole, so n is one.
And I'm asking you now, what will be the volume of that gas? Well, you take the gas law, and it tells you that V, the volume, equals nRT divided by P.
You know n is one.
You know R, 1.03...
excuse me, you know...
[laughs]: I'm a little bit ahead of myself.
You know R, which is 8.3, you know the temperature, which is 293, and you know the pressure, which is 1.03 times ten to the fifth.
And when you calculate that, you find something very close to 24 liters, and a liter is about a thousand cubic centimeters.
And it's independent of whether it's helium or oxygen or nitrogen or CO2.
As long as you have a gas, one mole at one atmosphere pressure and room temperature always has the same volume of about 24 liters.
If a gas obeys that law exactly, we call it an ideal gas.
That's why we call that the ideal-gas law.
And many gases come very close to that.
In fact, if you took oxygen, O2, and you take one mole of oxygen at room temperature and at one atmosphere pressure and you were to calculate its volume, the actual volume that you measure is only one-tenth of a percent smaller than what you would have found with the ideal-gas law.
If you do it at 20 atmospheres, it would still be only two percent smaller, so it's a very good approximation in many cases.
What is very surprising, that in this ideal-gas law, the mass of the atoms and the molecules do not show up at all.
And that is very puzzling--
you wouldn't expect that at all.
And I'll show you why you wouldn't expect that.
Let's take two different kinds of gases with very different masses of the molecules, but we have the same number of moles, we have the same volume, we have the same temperature and therefore, we must have the same pressure, according to the ideal-gas law.
But the masses of the molecules--
So, here we have some of these molecules and the number of... density is the same, because the number of atoms is the same and the volume is the same.
Now, these molecules are flying in all directions with different speeds.
I will just now, for simplicity, take some average speed, and I assume this is going in this direction.
It's heading for the wall of the container, this area.
It hits the wall, there's an elastic collision, and it comes back in exactly the same direction.
So there is momentum transfer, and the momentum transfer for one collision is 2mv, because it comes in with mv in this direction, it comes back with mv in that direction, so the momentum transfer is 2mv.
But I'm interested in the momentum transfer per second, not just for one molecule.
And now, of course, I have to multiply by the velocity, because if the velocity is high, you have a lot of bombardments per second on here.
For each bombardment, this is the momentum transfer, but if there are many, well, you have to multiply that, of course, then, by the speed.
So the momentum transfer per second is proportional, let's say, to mv squared.
mv comes from the momentum, from one particle, and v comes from the fact that...
the number that hit it per second.
Now, momentum transfer per second is clearly...
It's a force, proportional to the force, and that is proportional to the pressure.
And yet the pressure is not affected by the mass, notice? If these are the same, the pressure must also be the same.
And so there's only one conclusion that you can draw, which is very nonintuitive--
that the pressure can only be the same if, for a given temperature, this product, mv squared, is independent of the mass of the molecule.
How can mv squared possibly be independent of the mass of the molecule? There's only one way that that's possible--
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