Specific heat capacity equation: ΔE = m c Δθ
Change in thermal energy = mass × specific heat capacity × temperature change, or ΔE = m c Δθ. ΔE is in joules (J), m in kilograms (kg), c in joules per kilogram per degree Celsius (J/kg °C) and Δθ in degrees Celsius (°C). Every board's exam sheet includes it, and the specifications list it as a given equation.
Equation
ΔE = m c Δθ
| Symbol | Quantity | Unit |
|---|---|---|
ΔE | change in thermal energy | J |
m | mass | kg |
c | specific heat capacity | J/(kg·°C) |
Δθ | temperature change | °C |
What each specification says
| Exam board | Status | Equation |
|---|---|---|
| AQA Physics (8463) | Given on the sheet | ΔE = m c Δθ |
| AQA Combined Science (8464) | Given on the sheet | ΔE = m c Δθ |
| Edexcel Physics (1PH0) | Given on the sheet | ΔQ = m c Δθ |
| Edexcel Combined Science (1SC0) | Given on the sheet | ΔQ = m c Δθ |
| OCR Gateway Physics A (J249) | Given on the sheet | ΔE = m c Δθ |
| OCR Twenty First Century Physics B (J259) | Given on the sheet | ΔE = m c Δθ |
Calculator: ΔE = m c Δθ
What specific heat capacity means
Specific heat capacity, c, is the energy needed to warm 1 kg of a material by 1 °C. Water has a high value, about 4200 J/kg °C. Copper is much lower, at under 400. So the same energy warms a copper pan far more than the water inside it.
This is why a hot water bottle stays warm for hours. The water holds a lot of energy for each degree, and it gives that energy out slowly as it cools.
The Greek letter θ (theta) stands for temperature. Δθ, said delta theta, is the change in temperature: the final value minus the starting value. A change of 1 °C is the same size as a change of 1 K, so either unit works for Δθ.
How AQA, Edexcel and OCR show it
- AQA (8463 and 8464): ΔE = m c Δθ.
- Edexcel (1PH0 and 1SC0): ΔQ = m c Δθ, with Q for the thermal energy.
- OCR Gateway (J249) and OCR Twenty First Century (J259): ΔE = m c Δθ.
All six courses sort it into the select and apply group. Questions normally tell you the value of c for the material as well. You can check the full Edexcel list on the Edexcel equation list page.
Rearranging for c, m or Δθ
- c = ΔE ÷ (m Δθ)
- Δθ = ΔE ÷ (m c)
- m = ΔE ÷ (c Δθ)
In a common class experiment, you heat a metal block with an electric heater. The heater's energy comes from E = P t, and then you rearrange for c.
When a substance melts or boils, its temperature stops rising, so this equation no longer applies. Use specific latent heat for that part. The physics equation sheet page shows both side by side.
Worked examples
E
- 500 g = 0.5 kg
- Δθ = 80 − 20 = 60 °C
- ΔE = m c Δθ
- ΔE = 0.5 × 4200 × 60
- ΔE = 126 000 J
126 000 J
c
- 18 kJ = 18 000 J
- c = ΔE ÷ (m Δθ)
- c = 18 000 ÷ (2 × 10)
- c = 900 J/kg °C
900 J/kg °C
Δθ
- Δθ = ΔE ÷ (m c)
- Δθ = 5000 ÷ (0.25 × 2000)
- Δθ = 5000 ÷ 500
- Δθ = 10 °C
10 °C
Common mistakes
- Putting the final temperature in for Δθ. Water heated from 20 °C to 80 °C changes by 60 °C, not 80 °C.
- Keeping the mass in grams. Specific heat capacity is per kilogram, so 500 g must go in as 0.5.
- Using this equation while ice melts or water boils. The temperature is flat then, and the energy goes into the change of state.
- Dividing by m only when finding c. You must divide by the product m × Δθ.
Rearranging three-term equations like this gets quicker with the Equation Rearranger.
Questions people ask
Why does the sea warm up more slowly than the beach?
Water has a much higher specific heat capacity than sand. Each kilogram of sea needs far more energy for the same rise in temperature.
Does Δθ ever come out negative?
Yes, when something cools. Then ΔE is negative too, which shows energy leaving the object.