Temperature and internal energy
Key idea: Temperature is linked to particles' average kinetic energy. Internal energy is their total kinetic and potential energy, so it also depends on the amount of substance and its state.
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Guided steps
- Compare temperature with average particle kinetic energy.
- Explain what warming does to average particle kinetic energy.
- Compare the averages when sample amounts are unknown.
- Compare how mass, temperature and state affect internal energy.
- Compare the two copper blocks at the same temperature.
- Explain how melting changes energy at constant temperature.
- Find the energy increase and choose which particle energy changes.
- Compare each pair’s mass, temperature and state.
- Rank the three state mixtures at the same temperature.
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ContinueA cup of tea is hotter than a swimming pool, yet the pool holds far more energy. Temperature and internal energy measure different things: temperature is linked to the average kinetic energy of the particles, while internal energy is the total energy of all of them. Melting ice shows the difference most clearly, because its internal energy rises while its temperature stays the same.
A higher temperature means a greater average kinetic energy
The particles in a body are always moving, and at any moment some move faster than others. Each particle has kinetic energy, so the particles have a range of kinetic energies. Temperature is linked to their average.
When the temperature of a body rises, the average kinetic energy of all its particles increases.
- Energy is transferred to the body, for example by heating, and the body does not change state.
- Its particles move faster on average: their average kinetic energy increases.
- The temperature of the body rises.
The same chain runs backwards when a body cools: its particles move more slowly on average and its temperature falls. The particles are still moving, just more slowly.
Average and total are different. A 100 g sample and a 200 g sample of the same liquid are both at 25 °C. Their particles have the same average kinetic energy, because they are at the same temperature. The 200 g sample has twice as many particles, so the total kinetic energy of its particles is twice as large. More particles do not make a sample hotter.
Worked example 1
Heating water without changing state
Problem
A beaker of water on a hot plate warms from 20 °C to 45 °C. Describe what happens to its particles.
Show full solution
Heating water without changing state
Method
Identify the state and interpret the temperature rise.Reason
Temperature describes average particle kinetic energy, rather than the total energy of a sample.Working
Answer
- Energy is transferred from the hot plate to the water by heating.
- The water stays liquid the whole time, so this is not a change of state.
- The water particles move faster on average, so their average kinetic energy increases. This is what the rise from 20 °C to 45 °C shows.
- Check: the particles do not all have the same kinetic energy at 45 °C. Some move faster and some slower; it is the average that has increased.
Check your understanding 1
Two samples of the same gas are at different temperatures, but you do not know how much gas is in each. Which sample’s particles have the greater average kinetic energy? Can you also say which has the greater total kinetic energy?
Show answer
The hotter sample’s particles have the greater average kinetic energy. You cannot rank the total kinetic energies: the total also depends on how many particles each sample contains, and the temperatures alone do not tell you that.
Internal energy is the total energy of the particles
The internal energy of a body is an energy store made up of two parts:
- the total kinetic energy associated with the random motion of all its particles, and
- the total potential energy between its particles, which comes from the attractive forces between them and depends on how far apart they are.
Internal energy is a total: it adds up the energy of every particle. So it depends on three things:
| What changes | Effect on internal energy |
|---|---|
| Temperature rises (same mass, same state) | increases: the particles have more kinetic energy on average |
| Mass increases (same temperature, same state) | increases: there are more particles, each with its share of kinetic and potential energy |
| State changes from solid to liquid or liquid to gas (same mass, same temperature) | increases: the particles move out of their fixed positions (melting) or far apart (boiling) against their attraction, so their potential energy increases |
A higher temperature does not always mean more internal energy. A 200 g cup of tea at 80 °C is hotter than a swimming pool at 30 °C, so its particles have a greater average kinetic energy, but the pool contains vastly more particles, so it has far more internal energy.
Check your understanding 2
A 1 kg block and a 3 kg block of copper are both at 50 °C. Compare their temperature, average particle kinetic energy and internal energy.
Show answer
Their temperatures and average particle kinetic energies are equal. The 3 kg block has three times as many particles, so it has three times the internal energy.
Same temperature, different internal energy
Compare equal masses of ice and liquid water at 0 °C. Melting the ice requires energy even though the temperature stays constant. This energy lets particles leave their fixed positions against their attractions, increasing particle potential energy rather than average kinetic energy. The liquid therefore has more internal energy than the ice at the same temperature.
The same distinction applies to boiling: at fixed pressure, a pure substance absorbs energy to separate particles while its temperature remains constant. Freezing and condensation release energy as particle potential energy falls.
Check your understanding 3 Net energy enters while the thermometer stays still
A fixed sample of pure ice and water is melting at 0 °C at standard atmospheric pressure. A sensor records a net transfer of 600 J into the sample; some ice remains afterwards. Give the change in the sample’s internal energy and select the corresponding particle-energy change. Compare with the full explanation after checking.
Show answer
The internal energy increases by 600 J. The energy increases the particles’ potential energy as ice melts, while their average kinetic energy remains unchanged at 0 °C. Both ice and water are still present, so the phase change is not complete. No specific heat or latent-heat calculation is needed: the prompt already gives the net energy entering the fixed sample.
Worked example 2
Comparing internal energies
Problem
For each pair, decide which has more internal energy, or whether they are equal.
(a) 1 kg of water at 20 °C and 1 kg of water at 60 °C.
(b) A 250 g cup of water and a 100 kg bathtub of water, both at 40 °C.
(c) 50 g of ice at 0 °C and 50 g of liquid water at 0 °C.
Show full solution
Comparing internal energies
Method
Check the mass, temperature and state before ranking internal energies.Reason
Internal energy includes the total kinetic and potential energies of all particles in the sample.Working
Answer
Pair a.
- Same substance, same mass and same state, so the same number of particles.
- The water at 60 °C is at the higher temperature, so its particles have a greater average kinetic energy, and so a greater total kinetic energy.
- The water at 60 °C has more internal energy.
Pair b.
- Same temperature, so the particles in both have the same average kinetic energy. Neither is hotter.
- The bathtub has 100 kg ÷ 0.25 kg = 400 times the mass, so 400 times as many particles, each with kinetic and potential energy.
- The bathtub has about 400 times the internal energy of the cup.
Pair c.
- Same temperature, so the same average kinetic energy. Same mass, so the same number of particles and the same total kinetic energy.
- To turn the ice into water at 0 °C, energy must enter the sample, yet the temperature does not change. That energy moves the particles out of their fixed positions against their attraction, which increases their potential energy.
- The water at 0 °C has more internal energy than the ice at 0 °C.
Check your understanding 4 The same temperature can hide a different state mixture
Three fixed samples each contain 50 g of water substance at 0 °C at standard atmospheric pressure. A is all ice, B is half ice and half liquid water by mass, and C is all liquid water. Rank their internal energies and compare their average particle kinetic energies. Would the three thermometer readings alone let you identify which sample is which? Explain.
Show answer
Their internal energies rank C greater than B greater than A. Melting requires energy at the same temperature, so the more of the equal mass that has melted, the greater its particle potential energy and internal energy. Their average particle kinetic energies are the same, because their temperatures are the same. Three readings of 0 °C alone cannot identify the state mixtures: you also need evidence of how much ice and liquid are present. You cannot calculate the energy differences from the temperature readings alone.
Common mistakes
- Saying a larger sample is hotter because it has more particles. Temperature is linked to the average kinetic energy, not the total.
- Saying a hotter sample always has more internal energy. Compare the mass and the state as well as the temperature.
- Saying two samples at the same temperature have the same internal energy. Equal masses of ice and water at 0 °C have different particle potential energies.
- Saying the particles stop moving when a body cools. They move more slowly on average, but they are still moving.
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Syllabus and review details
- SEC G2 Science Physics component 2027 · 2027
Physics section, PDF pages 8–23
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