Internal Energy

• The internal energy of a system is the sum of the kinetic and potential energies of its molecules, associated with their random motion. The sum is one mark; the word random is the other (9702/42/O/N/23 Q3(a)) (9702/42/M/J/24 Q3(a)) (9702/41/M/J/25 Q4(a)(i)).
• Why random matters: throw the stone and every molecule gains ordered kinetic energy together — that is the stone's kinetic energy, not its internal energy. Internal energy only counts the disordered part: molecules vibrating and darting in all directions, plus the potential energy stored in their separations.
• Two dials therefore change a system's internal energy from inside: temperature (the kinetic part — recall $\tfrac{1}{2}m\langle c^2 \rangle = \tfrac{3}{2}kT$ from the Ideal Gases chapter) and molecular separation (the potential part — the idea behind latent heat in the Temperature chapter).
• Special case worth two marks on its own: an ideal gas has no intermolecular forces, so its molecular potential energy is zero and $U$ is purely kinetic — which is why $U \propto T$ (9702/41/M/J/24 Q3(a)(ii)) (9702/42/F/M/24 Q2(a)).

• A whole family of 3-markers asks what happens to internal energy in some process, “with reference to the kinetic and potential energies of the molecules”. The recipe: deal with KE via temperature, then PE via separation, then conclude about $U$.

• Each row is a real mark scheme: the heated gas and the stretched wire were one question's two halves (9702/42/M/J/24 Q3(b)), and the boiling-water row is the standard explanation of why temperature stays constant during vaporisation (9702/41/O/N/25 Q2(b)(ii)).

Internal energy = sum of random KE + PE of the molecules. Analyse any process in two steps: KE follows temperature, PE follows separation.