The ideal gas equation and kinetic theoryEdexcel International A Level Physics: Revision notes

For an ideal gas the pressure p, volume V, number of molecules N and absolute temperature T are linked by

$pV = NkT$

where $k = 1.38 \times 10^{-23}$ J K⁻¹ is the Boltzmann constant. T must be in kelvin, p in pascals and V in m³.

It is equivalent to pV = nRT, with N = nN_A and R = N_Ak.

Worked example: a 2.0 × 10⁻³ m³ flask at 1.0 × 10⁵ Pa and 300 K holds N = pV/kT = 4.8 × 10²² molecules.

At a fixed N, rearranging gives pV/T = constant, so:

• constant T: pV is constant (Boyle's law), so $p_1V_1 = p_2V_2$
• constant V: p is proportional to T
• constant p: V is proportional to T

The ratio of two states needs only ratios of volumes or pressures, so unit conversions cancel, but T must still be in kelvin.

To investigate the relationship between pressure and volume at constant temperature:

• Trap a fixed mass of air in a syringe connected to a pressure gauge.
• Change the volume slowly with the plunger and wait before each reading so that the air returns to room temperature.
• Record p and V, and repeat.
• Plot p against 1/V. A straight line through the origin shows that pV is constant.

Compressing quickly warms the gas, and leaks change the mass of gas, so these are the main sources of error.

Kinetic theory models the pressure of a gas as the result of many molecular collisions with the walls, giving

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

where ⟨c²⟩ is the mean square speed.

Derivation: equate this with NkT, divide by N and multiply by 3/2:

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

The left side is the mean kinetic energy of one molecule. It depends only on the absolute temperature, so doubling T doubles the mean kinetic energy.

Find c_rms for nitrogen (molecule mass 4.65 × 10⁻²⁶ kg) at 300 K.

Mean kinetic energy = (3/2)kT = 6.2 × 10⁻²¹ J.

⟨c²⟩ = 2 × 6.2 × 10⁻²¹ / 4.65 × 10⁻²⁶ = 2.7 × 10⁵ m² s⁻²

c_rms = 5.2 × 10² m s⁻¹.

At the same temperature all gases have the same mean kinetic energy per molecule, so lighter molecules have a larger c_rms.