Avogadro's principle

[robinpike]

Avogadro's principle states that at equal temperatures and equal pressures, equal volumes of gases contain the same number of molecules.

But why is this true? For wouldn't a heavier gas not need as many molecules to exert the same pressure as a lighter molecule?

Is the reason because a mixture of different gas molecules, such as hydrogen and oxygen, the oxygen molecules move at a slower speed than the hydrogen molecules? If so, why is it that they move at different speeds for the same temperature? And if so, then why is it that the heavier but slower moving molecule occupies the same volume as the lighter but faster moving molecule?

[buckleymanor1]

Must be to do with the size of heavy and lighter molecules.

[StuartG]

From what I remember Avogadro used a good bit of logic to arrive at His 'hypothesis', as was, when I was told. It revolves around the fact that...
You have a red marble and a green marble, the reds are twice as heavy as the greens. It then follows that if a number, say 30, of reds are put in a box and then 30 greens in another, the reds are twice as heavy as the greens.
So with that in mind, we find a box of reds and a box of greens and we weigh both and find that the reds are twice as heavy as the greens, without knowing the numbers involved, that there are the same number of marbles in each box.
That's the logic bit.
StuartG

[StuartG]

I've been looking for a chemical reaction that I can understand...
It is found by experiment that 1 volume of Hydrogen combine with 1 volume of Chlorine to form 2 volumes of Hydrogen Chloride.
so
x molecules of hydrogen + x molecules of chlorine give 2x molecules of hydrogen chloride,
or 1 molecule of hydrogen + 1 molecule of chlorine give 2 molecules of hydrogen chloride
The hydrogen molecule must have twice as many atoms of hydrogen as the hydrogen chloride molecule, because an hydrogen chloride molecule cannot less than one atom of hydrogen.
Therefore:
Hydrogen Chloride may be HClx, H2Clx, H3Clx...
If the firsat one is correct then the hydrogen molecule contains 2 atoms, so will be H2, if the second, then H4 and so on [we know now that it is H2 see en.wikipedia.org/wiki/Hydrogen
"Hydrogen gas (now known to be H2) was first artificially produced in the early 16th century," ]
StuartG

[buckleymanor1]

Jun 6, 2011 20:42:34 GMT 1 StuartG said:

From what I remember Avogadro used a good bit of logic to arrive at His 'hypothesis', as was, when I was told. It revolves around the fact that...
You have a red marble and a green marble, the reds are twice as heavy as the greens. It then follows that if a number, say 30, of reds are put in a box and then 30 greens in another, the reds are twice as heavy as the greens.
So with that in mind, we find a box of reds and a box of greens and we weigh both and find that the reds are twice as heavy as the greens, without knowing the numbers involved, that there are the same number of marbles in each box.
That's the logic bit.
StuartG

How do you arrive at equal volumes of marbles in each box are they the same size.

[StuartG]

"How do you arrive at equal volumes of marbles in each box are they the same size."
No, the boxes are.
StuartG

[buckleymanor1]

Jun 6, 2011 23:11:56 GMT 1 StuartG said:

"How do you arrive at equal volumes of marbles in each box are they the same size."
No, the boxes are.
StuartG

So you have 30 reds in one box and 30 greens in another.

equal volumes of gases contain the same number of molecules.

I don't get it how do you have equal volumes of gases and equal numbers of molecules filling equal sized boxes without them being the same size.
Different weight ok. but the volume I don't see.

[StuartG]

"equal volumes of gases and equal numbers of molecules " "filling equal sized boxes"
You appear to have mixed 'thought trains', the gases are in 'volumes', it's the marbles that are in 'boxes'. Anyway, try this, if it doesn't look right first time, read it again and carry on till the ideas come to You.
This is a lead to a book called 'Descriptive Chemistry', it's old [1903] but the content is about right, certainly for this discussion, download it and go to Page 166, and read on...
"Avogadro's Hypothesis. In 1811 an Italian physicist proposed an hypothesis to account for the similar behavior of gases. At that time the properties of gases were not generally known, and the views of Avogadro were overlooked until about 1860. Since then the hypothesis has been helpful in explaining many facts, and it is generally accepted by chemists as a very probable assumption. It may be stated thus: "There is an equal number of molecules in equal volumes of all gases at the same temperature and pressure." This statement cannot be proved directly by experiment, but there is much physical, chemical, and mathematical evidence in harmony with it.
According to Avogadro's hypothesis a liter of hydrogen and a liter of oxygen at the same temperature and pressure contain the same number of molecules, though we do not know how many. Suppose, however, that each liter contained 1000 molecules. A liter of hydrogen
weighs 0.0896 gm. and a liter of oxygen at the same temperature and pressure weighs 1.43 gm. But 0.0896 and 1.43 are in the same ratio as i and 16. Therefore, since a thousand molecules of oxygen weighs 16 times more than a thousand molecules of hydrogen, a single molecule of oxygen must weigh 16 times more than a single molecule of hydrogen. Therefore, in general, in order to find how much heavier any gaseous molecule is than a hydrogen molecule, it is only necessary to compare the weights of equal volumes of hydrogen and the gas under examination....."

DESCRIPTIVE
CHEMISTRY
BY
LYMAN C. NEWELL, PH.D. (JOHNS HOPKINS)
INSTRUCTOR IN CHEMISTRY, STATE NORMAL SCHOOL, LOWELL, MASS.
AUTHOR OF "EXPERIMENTAL CHEMISTRY"
BOSTON, U.S.A.
D. C. HEATH & CO., PUBLISHERS
1903
ia600408.us.archive.org/2/items/descriptivechemi00newerich/descriptivechemi00newerich.pdf
StuartG

[speakertoanimals]

"There is an equal number of molecules in equal volumes of all gases at the same temperature and pressure.

Lets start with gas molecules. Idea is that the SIZE of the actual molecules doesn't matter, because they are so teeny-tiny compared with spaces between them in the gas.

So, we have these very small gas molecules banging about, and hitting (as well as each other), the sides of the box.

So, lets take a thought experiment. We'll have ONE molecule banging about from left to right and back again inside a box of width L, and cross-sectional area A.

So, what about the pressure on the end walls? Depends on how FAST the molecule is moving. Also depends on the mass, since a heavier molecule hitting at the same speed gives more of a kick. The pressure also depends on hown often the molecules hits the wall. A small L means lots of hits per second, whilst if the box is much wider, greater spacing between hits, hence lower pressure.

The width of the box gives the link to volume. The temperature is linked to the kinetic energy of the molecule, hence mass and speed.

To see why the mass drops out, we need a little algebra.

Kinetic energy E = 1/2 mv^2

Momentum: mv

So each time the molecule hits the left-hand wall, the 'hit' is 2mv (net change of momentum assuming rebounds with equal speed.

It takes a time t = 2L/v before it comes back and hits again.

Rate of hits is v/2L, each hit is 2mv.

So, multiplying togther to get pressure (pressure increases with bigger hits, or more hits over the same time), we get an expression of the form:

mv^2/L

mv^2 is just the piece we had in kinetic energy, L is proportional to volume, hence we have found that pressure varies as the kinetic energy, and inversely with volume.

Equate kinetic energy with temperature, and we find that:

pressure is proportional to temperature/volume for our one molecule, whatever its mass.

Add a second molecule with the SAME volume and SAME average kinetic energy, and the pressure doubles.

Hence now we have pressure equals number of molecules times temperature/volume of box

Hence if pressure, temperature and volume are the same for two different boxes, so is number of molecules.

The real link is that the temperature decides the average kinetic energy per molecule, which is the piece of the puzzle that gets the mass coming in.

[speakertoanimals]

To answer the earlier questions, what happens when we have light and heavy molecules in the same box? The point is that they collide with each other, and exchange energy. What happens is that at a fixed temperature, the kinetic energy of each molecule is the same (on average), which means that the heavier molecules in a mixture will have a slower speed on average.

Having a slower speed, they will collide with the walls less often (that's the time to travel a distance 2L in my simple example), BUT their mass also comes in via the momentum mv when it comes to how hard a knock they give the walls when they do collide.

For wouldn't a heavier gas not need as many molecules to exert the same pressure as a lighter molecule?

Except the having more mass is countered by the fact that they are going slower. Hence (as my simple example shows), the effect of the mass cancels out as long as the relation between temperature and average kinetic energy is true.

[robinpike]

Thanks STA, I would like to clarify the part that is not clear to me: how in a mixture of gases, the different types of molecules remain moving at different speeds.

For example, a box contains both hydrogen and oxygen. Let’s simplify this by treating the speed of the molecules as a simple flat speed (as opposed to an average of a collection of speeds) and say that the hydrogen molecules are travelling at 1,600 metres per second, and the oxygen molecules are travelling at 400 meters per second (being that an oxygen molecule has 16 times the mass of a hydrogen molecule, therefore for the two to have the same kinetic energy – and therefore the same temperature - the oxygen molecule needs to be moving at a ¼ of the speed of the hydrogen molecule).

The bit that I don’t understand is how, as the faster moving hydrogen molecules collide with the slower moving oxygen molecules, that they don’t cause the oxygen molecules to gain an increase in speed?

[speakertoanimals]

as the faster moving hydrogen molecules collide with the slower moving oxygen molecules, that they don’t cause the oxygen molecules to gain an increase in speed

In collisions, you have to remember that both momentum (mv) and energy (mv^2) are conserved. This limits what exchange can occur. Then you have to remember that theyn can rebound at various angles as well..............

In technical terms, its called the equipartition theorem, the fact that at equilibrium, every part of a system (like each molecule in a box of gas) has (on average) the SAMe energy (per degree of freedom, but we're just talking about translational motion here). Hence lighter molecules tend to be moving faster than heavier ones.

It may seem counter-intuitive, so lets take a simple example. We have speedy light molecules, and heavy, sluggish ones. Suppose a fast one travelling left to right collides head-on with a slow one moving right to left. We might expect that light one would fly off moving right to left, and to balance momentum, heavy one now moving left to right. That is certainly a big change of velocity for each, now moving in opposite direction, but need not be a big change in speed!

I don't know a simple argument, but the energy is equally shared seems a reasonable result -- collisions do occur, some molecules get speeded-up, others get slowed down, but the net result is all molecules have (on average) the same energy -- as democratic as you could be really!

I think when it comes to slow oxygen hit by fast hydrogen, need to think that the hydrogen can hit from any angle, from behind or the side as well as a head-on smash. Somtimes the oxygen gains speed other times is looses it, depending on the exact circumstances and the exact way they fly off again.

[buckleymanor1]

Lets start with gas molecules. Idea is that the SIZE of the actual molecules doesn't matter, because they are so teeny-tiny compared with spaces between them in the gas.

Right then lets start there.If the SIZE of the actual molecule doesn't matter because of there small size compared with spaces between them in the gas.If they ain't the same SIZE, molecule plus gas won't equal the same volume.
Unless you wan't to ignore there small size and just wan't to consider the spaces between them.
Which does not seem to hold with the original principal.

[principled]

bucklymanor,
I'll let STA give you the definmitive answer, but on reading you post I think I can see your "error".

If they ain't the same SIZE, molecule plus gas won't equal the same volume.

The molecules are the gas. Between the gas molecules there is nothing.
However, there is an interesting point that you allude to and that is the space that is occupied by the molecules and why molecules that are larger do not need more space at equal levels of energy than those that are smaller? For example, let's take two identical containers each containig a different gas. We increase the pressure to an infinite level by forcing more gas into each container. Temperature and volume are the same. I assume the answer is that the larger molecules travel slower and so "occupy" the same space as the smaller, faster moving molecules.
P
P

[speakertoanimals]

The size of the actual molecules doesn't have an effect, to the first approximation, because their size is so small compared to the average distance between molecules.

You can however correct for the finite size of molecules, but this only becomes appreciable at high pressure and high density. Then the volume they have to move in is the box volume minus the volume occupied by the actual finite-size molecules. This gives us the van der waals equation, which is for finite-size molecules that are slightly sticky. It models some gases and liquids better than the ideal gas laws (zero size molecules and not sticky).