# Why is Avogadro's law always true?

Why is Avogadro's law always true? How and why do equal volumes of gases at equal pressure and temperature contain equal number of molecules? I know it is a fundamental principle in chemistry but I wonder how it works.

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I'll attempt an explanation without any equations.

In kinetic theory, the pressure exerted by an ideal gas (point-like, non-interacting particles, elastic collisions) depends on the rate at which momentum is exchanged with the walls of any container as the molecules bounce off them.

The momentum is of course proportional to the molecule's speed and mass, but the rate at which molecules hit a wall additionally depends on how fast they are going and how many molecules there are in the volume of gas.

If you put that together you find that pressure depends on the number of molecules in a volume of gas and how much kinetic energy each molecule has.

The final step is to understand that the (average) kinetic energy of molecules in an (ideal) gas depends only on their temperature.

Thus a gas of a given temperature and number of molecules per unit volume will always exert the same pressure (in ideal circumstances). Or to turn this around, a gas of a given pressure and temperature will contain the same number of molecules per unit volume.

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ok that means lighter molecules travel fast but their mass is less and heavier molecules travel slow but their mass is high..so that both lighter and heavier molecules exert same average momentum change p/t (force) and hence the pressure at walls.. or pressure becomes 'mass' independent and only number dependent.. –  xyzu Aug 2 '14 at 10:25
Yes, I think you have it. It ends up just depending on the kinetic energy of each particle, which does not depend on their mass, just their temperature. –  Rob Jeffries Aug 2 '14 at 11:29

Well, it's not always true — it's an approximation that breaks down at high pressures or at low temperatures (where many gases turn into liquids or solids). But if the mean distance between gas molecules is many, many, many times longer than the size of the molecules themselves, so that the molecules are effectively "noninteracting," then it no longer matters what species the molecules are or how they would interact if they could.

The number density of a gas at STP is approximately one amagat, which is 45 mol/m3 or $2.7\times10^{25}\,\mathrm{atoms/m^3}$. Flipping that over we find that each atom is alone in a volume of roughly 37 nm3, a cube about 3.3 nm on a side. But atoms in a solid are typically arranged in cells about 0.3–0.5 nm on a side, and atoms in covalent bonds are typically even closer than that. Imagine that I was putting on a concert you wanted to go to, and you had heard that tickets cost \$30–\$50. But when you actually got to the box office, you found out that the tickets actually cost \$300! You'd go home, no matter who you thought I was. All noninteractions are alike, so all noninteracting gases can obey the same rules. - Nice answer! Maybe a small addition that is useful for the OP. A noninteracting gas is called an ideal gas. That is also where the ideal gas law gets its name – Michiel Aug 2 '14 at 9:46 ok. i meant the case of ideal gases only. and my focus in the question was different molecules has different volume, but still same amount contain in same volume.. – xyzu Aug 2 '14 at 10:13 An interesting thing to note is that interstellar number densities approach$10^6$atoms/m$^3$, nearly 20 orders of magnitude less than terrestrial gases. – Kyle Kanos Aug 2 '14 at 11:53 Not sure how this argument holds up in the center of a star. Definitely an ideal gas, but number densities of$10^{32}$/m$^3\$. What size do we compare the particle separation to? Not atoms, because completely ionised - Wigner-Seitz sphere maybe? –  Rob Jeffries Aug 2 '14 at 13:03
@RobJeffries The difference there is temperature — the thermal collision interaction is so much stronger than the inter-atom interaction that the latter can just be neglected. –  rob Aug 2 '14 at 13:41

It holds exactly only for ideal gases, of which the constituent particles do not interact and the internal energy is merely kinetic.

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+1. This is the short and sweet correct answer. Strictly speaking, Avogadro's law is never true. It is approximately true for some gases, most notably the noble gases. But even the noble gases exhibit non-ideal behavior to some extent. –  David Hammen Aug 2 '14 at 20:43

## protected by Qmechanic♦Aug 2 '14 at 20:29

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