Internal energy

Internal energy is the total kinetic energy and potential energy of all the particles (atoms and molecules) in a system. The kinetic energy comes from the particles moving and vibrating; the potential energy comes from the forces between them. Heating a system or doing work on it increases its internal energy, which either raises its temperature or changes its state.

What is internal energy?

Every object is made of particles that are always moving. In a solid they vibrate about fixed positions; in a liquid they move around each other; in a gas they move freely and quickly. Each particle has:

• Kinetic energy, because it is moving. The faster the particles move on average, the higher the temperature.
• Potential energy, because of the forces between it and its neighbours. This depends on how far apart the particles are, and it increases when they are pulled further apart.

Add up both kinds of energy for every particle and you get the internal energy. At A-level the definition adds one detail: internal energy is the sum of the randomly distributed kinetic and potential energies of the particles in a body. The particles don't all have the same energy; at any moment some are fast and some are slow.

At GCSE you will also see internal energy called the thermal energy store of an object. A hot object has more energy in this store than the same object when it is cold.

Internal energy and temperature

Temperature and internal energy are linked but not the same. Temperature depends on the average kinetic energy of the particles. Internal energy is the total energy of all of them, kinetic and potential. So:

• A bath of warm water has far more internal energy than a cup of boiling water, even though it is at a lower temperature, because it has many more particles.
• Ice at 0 °C and water at 0 °C have the same temperature, but the water has more internal energy, because its particles have more potential energy.

How internal energy changes

There are two ways to change the internal energy of a system:

1. Heating: energy is transferred because of a temperature difference. A kettle element heats the water; a hot drink heats your hands.
2. Doing work: energy is transferred by a force. Pumping up a bicycle tyre does work on the air in the pump, and the pump gets warm. Rubbing your hands together does work against friction.

Energy transferred to the particles does one of two things: it speeds them up, which raises the temperature, or it pulls them apart, which changes the state. It does not do both at once for a pure substance.

Changing temperature

When the temperature rises, the kinetic energy of the particles increases. The energy needed depends on the mass, the material and the temperature rise:

See the specific heat capacity guide for more on this equation.

Changing state

When a substance melts or boils, the energy supplied breaks the forces holding the particles together. The potential energy of the particles increases, but their kinetic energy, and so the temperature, stays the same until the change of state is complete:

The specific latent heat guide covers this equation in detail. A heating curve shows both kinds of change:

Here x is the time since heating began and y is the temperature in °C. On the sloping parts the kinetic energy of the particles increases and the temperature rises. On the flat parts, at 0 °C and 100 °C, the internal energy is still increasing, but all of it goes into potential energy as the state changes. Watch this happen to the particles in the States of Matter Simulator.

Check your understanding.

Worked examples

Example 1: heating water

A pan holds 0.50 kg of water at 20 °C. How much does its internal energy increase when it is heated to 80 °C? The specific heat capacity of water is 4200 J/kg °C.

Example 2: melting ice

How much energy is needed to melt 0.20 kg of ice that is already at 0 °C? The specific latent heat of fusion of water is 334 000 J/kg.

All 66 800 J goes into the potential energy of the particles. The temperature stays at 0 °C.

Example 3: a bicycle pump

Pushing a bicycle pump in quickly does 60 J of work on the air inside, and there is no time for any energy to escape by heating. The internal energy of the air increases by 60 J, so its particles move faster on average and its temperature rises. That is why the pump feels warm.

Internal energy at A-level: the first law of thermodynamics

The first law of thermodynamics is conservation of energy applied to internal energy. AQA A-level Physics writes it as:

Q is the energy transferred to the system by heating, ΔU is the increase in internal energy, and W is the work done by the system. Rearranged, ΔU = Q − W: the internal energy goes up by the energy heated in, minus the energy that leaves as work.

Example 4: a gas that expands

A gas is heated with 500 J of energy and expands, doing 200 J of work pushing a piston out. Its increase in internal energy is:

Internal energy of an ideal gas

In an ideal gas there are no forces between the particles except during collisions, so there is no potential energy. All of its internal energy is the kinetic energy of the particles, and it depends only on the temperature. The average kinetic energy of one molecule is:

For a monatomic ideal gas such as helium, the total internal energy of n moles is therefore U = (3/2)nRT. For example, 2.0 mol of helium at 300 K has U = 1.5 × 2.0 × 8.31 × 300 ≈ 7500 J. Double the kelvin temperature and the internal energy doubles.

Common mistakes

• Treating internal energy and temperature as the same thing. Temperature depends on the average kinetic energy; internal energy is the total kinetic and potential energy.
• Leaving out potential energy. The definition needs both kinetic and potential energy of the particles.
• Saying the energy disappears during melting. It goes into the potential energy of the particles, so the internal energy still rises.
• Saying heating always raises the temperature. During a change of state it doesn't.
• Forgetting work. Doing work on a gas, such as compressing it quickly, also increases its internal energy.
• Mixing up the sign of W in the first law. Check whether W is work done by the system or on it.

Practise with a short timed paper

Internal energy and changes of state come up in GCSE Physics, and thermal physics in A-level Physics. The daily GCSE Physics Higher and A-level Physics: Year 2 papers are short timed papers of 8 questions, with a new one every day.

Read next

Physics · Thermal physics and matterFirst law of thermodynamics: Q = ΔU + W explainedThe first law of thermodynamics, Q = ΔU + W: sign conventions, work done by a gas W = pΔV, isothermal, adiabatic and constant volume changes, pV diagrams.