The big idea: Warm a metal spoon in hot soup and it soon feels hot too — inside, its particles are jiggling faster and tugging harder on each other.
Internal energy: The total energy stored in all that jiggling and tugging is the spoon's internal energy.
It is made of two parts
- Their random kinetic energy (KE) — the energy of the particles jiggling and moving about (this sets the temperature).
- Their intermolecular potential energy (PE) — the energy stored in the forces between the particles (this depends on how far apart they are).
Internal energy — the definition: Internal energy = total random KE of the particles + total intermolecular PE of the particles.
What each part means
- Random KE = energy of the particles' motion.
- Intermolecular PE = energy stored in the bonds/forces holding particles together or apart.
- For a real gas both parts count.
- An ideal gas is a model with no forces between particles, so its internal energy is the random KE only.
Spot it: Temperature tells you about the average random KE of the particles — not the PE.
So heating something up always raises its KE part; the PE part changes mainly when the state changes (solid ↔ liquid ↔ gas).
Kelvin and Celsius: Physics uses the kelvin scale: T / K = θ / °C + 273. Room temperature 20 °C is 293 K; 0 K (−273 °C) is absolute zero.
One kelvin is the same size as one degree Celsius, so a temperature change is the same number on both scales: warming from 20 °C to 50 °C is a rise of 30 °C and of 30 K.
- average random kinetic energy of a particle (J)
- Boltzmann constant, 1.38 × 10⁻²³ J K⁻¹
- temperature (K)
At 27 °C = 300 K: Ē_k = 1.5 × 1.38 × 10⁻²³ × 300 = 6.2 × 10⁻²¹ J per particle. Double the kelvin temperature and the average kinetic energy doubles.
Which way does thermal energy flow?: The net thermal energy transfer is always from the higher temperature to the lower temperature, until both are at the same temperature (thermal equilibrium).
It is the temperature difference that decides, not the internal energy: a red-hot nail dropped into a bath has far less internal energy than the bath water, yet energy flows from the nail to the water.
Solid or liquid at the melting point?: Ice at 0 °C and water at 0 °C have the same temperature, so their particles have the same average kinetic energy. Melting the ice needed energy, and that energy went into intermolecular potential energy. So the water has more internal energy.
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The particle model pictures matter as particles with random KE (motion) and intermolecular PE (forces between them). It explains the three states of matter and why solids are usually denser than liquids.
| State | Particle spacing | Particle motion | Shape & volume |
|---|---|---|---|
| Solid | Very close, in a fixed pattern | Vibrate about fixed positions | Fixed shape, fixed volume |
| Liquid | Close, but free to move past each other | Move/slide around | No fixed shape, fixed volume |
| Gas | Far apart | Move fast in all directions | No fixed shape, fills the container |
Why solids are denser than liquids: Density = mass packed into a volume. In a solid the particles are held in a tight, regular pattern — packed closer together than in the same liquid.
More particles in the same space → more mass per volume → higher density. So melting a solid usually makes it less dense (the particles spread out a little).
Density links a substance's mass to its volume. It is a simple quotient (one quantity divided by another), and it is given in the data booklet:
- density (kg m⁻³)
- mass (kg)
- volume (m³)
m = mass, ρ = density, V = volume. Cover the one you want: two side by side → multiply; one above the other → divide.
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200 g of solid wax fills 2.0 × 10⁻⁴ m³. When melted, the same 200 g fills 2.2 × 10⁻⁴ m³. Find the density in each state and say which is denser.
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How far apart are the atoms?: Picture each atom sitting in its own small cube. If one atom occupies a volume V, the spacing between neighbours is about ∛V.
In solid aluminium each atom occupies 1.66 × 10⁻²⁹ m³, so the atoms are about ∛(1.66 × 10⁻²⁹) ≈ 2.6 × 10⁻¹⁰ m apart.
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How this is tested — this micro appears as short explain / identify questions:
Paper 1A
- A quick MCQ — identify what makes up the internal energy of a real gas (random KE + intermolecular PE).
Paper 2
- Explain why most substances are denser as solids than as liquids.
- Read a density–temperature graph for water to identify the temperature of maximum density.
The classic trap: Saying internal energy is only the kinetic (temperature) part, and forgetting the intermolecular PE.
Water is the odd one out: Most liquids get steadily denser as they cool. Water does not.
Water is densest at about 4 °C. Cool it further and it gets less dense, and ice is less dense still — so ice floats.
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The words to know
| Word | What it means |
|---|---|
| Internal energy | Total random kinetic energy + total intermolecular potential energy of the particles |
| Temperature (K) | A measure of the average random kinetic energy of the particles |
| Thermal equilibrium | Same temperature, so no net thermal energy transfer |
| Density | Mass per unit volume, ρ = m/V (kg m⁻³) |
The graph shows how the density of a sample of water changes as it is warmed from 0 °C. From the graph, identify the temperature at which the water has its maximum density, and state what this tells you about ice compared with liquid water.
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