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.
 A: 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.
A: It holds exactly only for ideal gases, of which the constituent particles do not interact and the internal energy is merely kinetic.  
A: 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.
If the "ideal" approximations break down, Avogadro's law may not be true.
