Physics 9702 · Paper 4

Temperature Means Kinetic Energy

Two expressions for $pV$ now exist — one from experiment, one from theory. Setting them equal finally says what temperature is.

Combining the two pV equations

Theory (last lesson): $pV = \tfrac{1}{3}Nm\langle c^2 \rangle$ . Experiment: $pV = NkT$ .
Equate and cancel $N$ : $\tfrac{1}{3}m\langle c^2 \rangle = kT$ .
Multiply by $\tfrac{3}{2}$ to build the kinetic energy on the left: $\tfrac{1}{2}m\langle c^2 \rangle = \tfrac{3}{2}kT$ .

Left side: the average kinetic energy of one molecule. Right side: temperature times a constant. So thermodynamic temperature is a direct measure of average molecular kinetic energy — the promise made back in the Temperature chapter, now kept. The derivation itself is a regular 3–4 marker (9702/42/M/J/25 Q4(b)) (9702/41/O/N/23 Q3(a)(ii)).
At 300 K: $\tfrac{3}{2} \times 1.38\times10^{-23} \times 300 = 6.2 \times 10^{-21}\ \text{J}$ per molecule — tiny, but multiplied by $10^{23}$ molecules it becomes everyday energy.

Internal energy of an ideal gas

Ideal-gas molecules feel no forces between collisions, so they store no potential energy. The internal energy is the sum of the kinetic energies of the molecules — nothing else (9702/41/M/J/25 Q4(a)(ii)) (9702/41/M/J/24 Q3(a)(ii)).
Sum the average over all $N$ molecules: $U = N \times \tfrac{3}{2}kT = \tfrac{3}{2}NkT = \tfrac{3}{2}pV$ (9702/42/O/N/25 Q4(b)). Two immediate consequences: $U \propto T$ for a fixed sample, and at constant pressure a graph of $U$ against $V$ is a straight line through the origin (9702/41/M/J/24 Q3(c)).

Worked example

Internal energy of one mole

Find the internal energy of 1.0 mol of an ideal gas at 300 K.

$U = \tfrac{3}{2}nRT = \tfrac{3}{2} \times 1.0 \times 8.31 \times 300 = 3.7 \times 10^{3}\ \text{J}$ .

Answer

U \approx 3700\ \text{J}

Doubling $T$ doubles $U$ — when a question heats a fixed amount of ideal gas, the change in internal energy is always $\Delta U = \tfrac{3}{2}nR\,\Delta T$ .

Only true for ideal gases

For real substances, internal energy = kinetic plus potential energy of the molecules. That is why latent heat exists: melting and boiling change the potential part without touching the kinetic part. The clean “U is all kinetic” statement belongs to ideal gases only, and examiners test whether you attach that condition.

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