Specific heat capacity

How much heat warms a body?

Put a kilogram of water and a kilogram of copper on two equal heaters. The copper is too hot to touch long before the water is warm. Different materials need different amounts of heat for the same rise in temperature.

The heat needed is

where $c$ is the specific heat capacity, the heat that warms $1\ \text{kg}$ by $1\ \text{K}$.

A body's heat capacity is $mc$, the heat for $1\ \text{K}$ of the whole body. The heat for one mole for $1\ \text{K}$ is the molar heat capacity.

You should find that $1\ \text{kg}$ of water at $1000\ \text{W}$ takes about $5.6$ minutes to go from $20\ ^\circ\text{C}$ to $100\ ^\circ\text{C}$, and copper about eleven times less.

Why water's large $c$ matters

Water stores a lot of heat for each kelvin. The sea warms and cools slowly, so coastal climates are mild. Water carries heat well in central heating and cools car engines. And a hot water bottle stays warm for hours.

Watch out. Specific heat belongs to a material; heat capacity to a body. A big iron pot and a small iron nail have the same $c$.

A worked example

An aluminium pan of mass $0.80\ \text{kg}$ holds $1.5\ \text{kg}$ of water. How much heat warms both from $20\ ^\circ\text{C}$ to $90\ ^\circ\text{C}$?

1. The pan. $0.80\ \text{kg} \times 900\ \text{J kg}^{-1}\text{K}^{-1} \times 70\ \text{K} = 50\,400\ \text{J}$.
2. The water. $1.5\ \text{kg} \times 4186\ \text{J kg}^{-1}\text{K}^{-1} \times 70\ \text{K} = 439\,530\ \text{J}$.
3. Total. $4.90 \times 10^{5}\ \text{J}$; the pan takes about a tenth.

Practise with new numbers

Try it in Python

The program works out the pan of the example.

Remember these

• $Q = mc\,\Delta T$; $c$ in $\text{J kg}^{-1}\text{K}^{-1}$.
• Heat capacity of a body $= mc$.
• Water's $c = 4186\ \text{J kg}^{-1}\text{K}^{-1}$ is large: slow to warm, slow to cool.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. How much heat warms $0.40\ \text{kg}$ of iron by $60\ \text{K}$?

2. The same heat is given to equal masses of copper and lead. Which warms more, and by how many times?

3. What is the heat capacity of $3.0\ \text{kg}$ of water?

4. Why does the sand on a beach get hot by noon while the sea stays cool?