What is internal energy?
All matter is made of tiny particles (atoms or molecules). They never stop moving.
- Moving particles have kinetic energy (KE).
- Particles pull on each other. Because of these forces, they also store potential energy (PE), a bit like a stretched spring.
Internal energy (U) is the total kinetic energy plus potential energy of all the particles in a system. Its unit is the joule (J).
A bigger amount of the same substance at the same temperature has more particles, so it has more internal energy.
Temperature and internal energy are different
Temperature tells us the average kinetic energy of the particles: how fast they move on average. It is measured in °C or kelvin (K).
Internal energy is the total energy of all the particles.
| Temperature | Internal energy | |
|---|---|---|
| Depends on | average KE per particle | KE + PE of all particles |
| Depends on amount? | No | Yes |
| Unit | °C or K | J |
So a spark at 1000 °C barely hurts, but a bath at 40 °C holds far more energy.
Two ways to change internal energy: heating and work
You can raise the internal energy of a system in two ways:
- Heating: energy flows from a hotter object to a colder one (flame under a pan).
- Doing work: a force moves something (rubbing hands, pumping a bicycle tyre, hammering a nail).
Heating changes internal energy in one of two ways:
- it raises the temperature (particles move faster, KE goes up), or
- it changes the state (solid → liquid → gas). Bonds break, PE goes up, and the temperature stays the same.
For older students: the first law of thermodynamics writes this as ΔU = Q + W, where Q is heat given to the system and W is work done on it.
Specific heat capacity: Q = m c ΔT
The specific heat capacity (c) of a substance is the energy needed to raise the temperature of 1 kg of it by 1 °C. Unit: J/(kg °C) or J/(kg K).
Q = m × c × ΔT
Q = energy (J), m = mass (kg), ΔT = change in temperature (°C).
- Water: c ≈ 4200 J/(kg °C), very large. That is why seas warm slowly and water is used in car radiators and hot-water bottles.
- Aluminium: about 900. Copper: about 385. Metals heat up fast.
Specific latent heat: Q = m L
During melting or boiling, energy goes in but the temperature does not rise. This hidden energy is called latent heat.
The specific latent heat (L) is the energy needed to change the state of 1 kg of a substance without changing its temperature.
Q = m × L
- Latent heat of fusion (melting) of ice: about 334 000 J/kg.
- Latent heat of vaporisation (boiling) of water: about 2 260 000 J/kg.
On a heating graph, the flat parts are where the state is changing.
Try it: feel internal energy change
1. Rub your palms together fast for 10 seconds. They get warm: your work became internal energy.
2. Put an ice cube in a cup of water with a kitchen thermometer (or just touch). The water stays near 0 °C until the ice is gone, just like step 4 in the 3D.
3. In the free-play step, give the same 20 kJ to water and to copper. Predict first: which warms more?
Key formulas and definitions
- Internal energy U = total kinetic energy + total potential energy of all particles
- Q = m × c × ΔT (heating without change of state)
- Q = m × L (change of state, temperature constant)
- ΔU = Q + W (heat in + work done on the system)
- Water: c ≈ 4200 J/(kg °C); ice L_f ≈ 3.34 × 10⁵ J/kg; water L_v ≈ 2.26 × 10⁶ J/kg
Worked examples
1. How much energy is needed to warm 2 kg of water from 20 °C to 70 °C? (c = 4200 J/(kg °C))
ΔT = 70 − 20 = 50 °C. Q = m c ΔT = 2 × 4200 × 50 = 420 000 J = 420 kJ.
2. A 0.5 kg copper block (c = 385 J/(kg °C)) gets 9625 J of heat. By how much does its temperature rise?
ΔT = Q ÷ (m c) = 9625 ÷ (0.5 × 385) = 9625 ÷ 192.5 = 50 °C.
3. How much energy melts 0.2 kg of ice at 0 °C? (L = 334 000 J/kg)
Q = m L = 0.2 × 334 000 = 66 800 J. The temperature stays at 0 °C the whole time.
4. A heater gives 30 000 J to 1.5 kg of a liquid and its temperature rises by 8 °C. Find c.
c = Q ÷ (m ΔT) = 30 000 ÷ (1.5 × 8) = 30 000 ÷ 12 = 2500 J/(kg °C).
5. How much energy turns 0.1 kg of ice at 0 °C into water at 30 °C?
Step 1, melt: Q₁ = m L = 0.1 × 334 000 = 33 400 J. Step 2, warm: Q₂ = m c ΔT = 0.1 × 4200 × 30 = 12 600 J. Total = 46 000 J.
6. A gas gets 500 J of heat and 200 J of work is done on it. What is the change in its internal energy?
ΔU = Q + W = 500 + 200 = 700 J. Its internal energy rises by 700 J.
Common mistakes
- Thinking temperature and internal energy are the same. Temperature is the average KE per particle; internal energy is the total of all particles and depends on mass.
- Thinking the temperature rises while ice melts. During a change of state the temperature stays the same; the energy breaks bonds.
- Using grams in Q = mcΔT. Mass must be in kg when c is in J/(kg °C): 250 g = 0.25 kg.
- Forgetting that work also changes internal energy. Rubbing, squashing a gas or bending a wire also warms it.