Internal Energy

The internal energy of a system is the sum of the random kinetic and potential energies of its molecules. It depends only on the state of the system (its temperature, volume, phase), not on how it got there. A rise in temperature means more random kinetic energy. During a change of state the potential energy changes while the temperature stays constant. For an ideal gas there are no intermolecular forces, so internal energy is all kinetic: $U=\tfrac32NkT$ .

Cambridge 9702 Paper 4Edexcel YPH11 WPH15AQA 7408 Paper 2≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Define internal energy in terms of molecular kinetic and potential energies.
Explain that internal energy is a function of state.
Relate temperature rise to increased random kinetic energy.
Explain why an ideal gas's internal energy depends only on temperature.

Pause and think

A ball is thrown fast across a room. Its molecules are all moving quickly. Has its internal energy increased?

Section 2 of 14

Core idea

Start hereMelting ice: energy in, temperature steady

Heat ice at 0 °C and it melts without warming. The energy is going somewhere: into the potential energy of molecules pulling apart. Internal energy has two parts, and only one shows up on a thermometer.

The big idea

$U$ = sum of random KE + PE of molecules. Ideal gas: no intermolecular forces, so $U=\frac32NkT=\frac32nRT$ (monatomic).

1. Two parts

Kinetic: set by temperature.
Potential: set by separation; changes during melting and boiling.

Internal energy of an ideal gas is proportional to T

For a monatomic ideal gas,

U=\tfrac32nRT

: double the kelvin temperature, double the internal energy.

Internal energy of an ideal monatomic gas

$\langle E_k\rangle=\tfrac32kT$
Per molecule (kinetic theory).
$U=N\cdot\tfrac32kT=\tfrac32nRT$
No PE: no intermolecular forces.

Worked exampleHeating helium

2.0 mol from 300 K to 400 K at constant volume.

Goal: $\Delta U$ ; heat supplied.
Strategy: $\Delta U=\frac32nR\Delta T$ .

$\Delta U=\tfrac32(2)(8.31)(100)=2490\text{ J}$
Formula.
$W=0\Rightarrow q=2490\text{ J}$
Fixed volume: no work.

Answer

2490

✓ Check

\Delta U

depends only on

\Delta T

for an ideal gas ✓.

Common misconceptionIce and water at 0 °C Same temperature, different

U

: water has more PE.

Summary

$U$ = random KE + PE.
Ideal gas: $U\propto T$ .

Internal energy: Total random molecular KE + PE.

HistoryJoule's paddle wheel. In the

1840

s James Joule let falling weights turn a paddle wheel in water and measured a tiny temperature rise. Doing work and heating had the same effect on the water: both increase its internal energy. That idea became the first law of thermodynamics.

Fun factJoule's honeymoon. Legend says Joule spent his honeymoon in the Alps trying to measure whether water at the bottom of a waterfall was warmer than at the top. Falling

100

m only warms water by about

0.23

K, so his thermometer probably wasn't up to it.

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