Gas laws

📚The physics

Counting molecules. One mole contains \( N_A = 6.02 \times 10^{23} \) particles. The number of moles is \( n = N/N_A \), and also \( n = m/M \) where \(M\) is the molar mass. Most gas-law errors are not physics errors at all — they are someone putting a mass in grams where the formula wants moles.

The ideal gas law ties everything together:

with \( R = 8.31 \) J mol\(^{-1}\) K\(^{-1}\) and the Boltzmann constant \( k = R/N_A \). Temperature must be in kelvin — always, without exception. Pressure in pascals, volume in cubic metres.

The familiar named laws are just special cases, which is worth seeing rather than memorising. Hold \(T\) constant and \(pV\) is constant (Boyle). Hold \(p\) constant and \( V \propto T \) (Charles). Hold \(V\) constant and \( p \propto T \) (Gay-Lussac). If you can rearrange \( pV = nRT \) you never need to recall which name goes with which.

Where pressure comes from. Molecules collide with the container walls; each collision reverses the molecule’s momentum component perpendicular to the wall, and by Newton’s second law a rate of change of momentum is a force. Pressure is that force per unit area, averaged over an enormous number of collisions. Raising the temperature makes molecules faster, so collisions are both harder and more frequent, and pressure rises. Reducing the volume makes collisions more frequent for the same speeds, and pressure rises again. Both results fall out of the same picture.

Kinetic theory makes that quantitative:

where \( \overline{c^{2}} \) is the mean square speed. Comparing this with \( pV = NkT \) gives the result that matters most:

Temperature is average molecular kinetic energy, in disguise. Note that this depends only on temperature, not on which gas — at the same temperature, helium and xenon molecules have the same average kinetic energy, so the heavier xenon molecules must be moving more slowly.

The ideal gas assumptions, which you should be able to list: the molecules are identical, small compared with the separation between them, and in constant random motion; collisions between molecules and with the walls are perfectly elastic; the duration of a collision is negligible compared with the time between collisions; and there are no intermolecular forces except during collisions.

When real gases stop cooperating follows directly from those assumptions. At high pressure the molecules are close together, so their own volume is no longer negligible. At low temperature they move slowly enough for intermolecular attractions to matter. Push far enough in either direction and the gas condenses — something an ideal gas can never do, because an ideal gas has no attractive forces to condense with.