Ideal gases

15.1 The mole

Amount of substance is an SI base quantity, with base unit the mole (mol). One mole of any substance contains $N_A = 6.02 \times 10^{23}$ particles, the Avogadro constant.

Number of particles: $N = nN_A$, where $n$ is the amount in moles.

15.2 Equation of state

An ideal gas obeys $pV \propto T$ at all pressures, volumes and temperatures, where $T$ is the thermodynamic temperature, in kelvin. Real gases behave like this at low pressures and at temperatures well above their boiling point.

$pV = nRT$

where $R = 8.31$ J K⁻¹ mol⁻¹ is the molar gas constant. In terms of the number of molecules, $N$:

$pV = NkT$

where $k = R/N_A = 1.38 \times 10^{-23}$ J/K is the Boltzmann constant.

15.3 Kinetic theory of gases

Basic assumptions

1. A gas contains a very large number of molecules in rapid, random motion.
2. The volume of the molecules themselves is negligible compared with the volume of the gas.
3. There are no forces between the molecules, except during collisions.
4. Collisions are elastic, and they take a negligible time compared with the time between collisions.

How molecules cause pressure

A molecule hits the wall and rebounds, so its momentum changes, and the wall exerts a force on it. By Newton's third law, it exerts an equal force on the wall. The very many molecular collisions add up to a steady pressure.

Deriving the pressure equation

Take a cube of side $L$ containing $N$ molecules, each of mass $m$. Consider one molecule with velocity $c_x$ in the $x$ direction.

1. Change of momentum at a wall: $2mc_x$.
2. Time between collisions with that wall: $2L/c_x$.
3. Average force on the wall: $F = 2mc_x / (2L/c_x) = mc_x^2/L$.
4. Pressure from all $N$ molecules: $p = F/L^2 = Nm\langle c_x^2\rangle/L^3$. So $pV = Nm\langle c_x^2\rangle$.
5. In three dimensions, the motion is random, so $\langle c_x^2\rangle = \tfrac{1}{3}\langle c^2\rangle$.

$pV = \tfrac{1}{3}Nm\langle c^2\rangle$

Here $\langle c^2\rangle$ is the mean-square speed: square the speeds, then average them.

The root-mean-square speed is $c_{\text{r.m.s.}} = \sqrt{\langle c^2\rangle}$. It is the square root of the mean of the squares, which is not the same as the mean speed.

Mean kinetic energy and temperature

Equate the two expressions for $pV$: $\tfrac{1}{3}Nm\langle c^2\rangle = NkT$. So $m\langle c^2\rangle = 3kT$, and:

$\tfrac{1}{2}m\langle c^2\rangle = \tfrac{3}{2}kT$

At the same temperature, lighter molecules have the same mean kinetic energy as heavier ones, so they move faster. Hydrogen molecules move much faster than oxygen molecules in the same container.