Calorimetry and the method of mixtures

Heat lost equals heat gained

Drop a hot spoon into cold water. Heat flows from the spoon to the water until both are at one temperature. If no heat escapes, the energy is simply passed across:

For two bodies: $m_1c_1(T_1 - T) = m_2c_2(T - T_2)$, which gives the final temperature $T$.

The calorimeter

A calorimeter is a metal can, usually copper, inside an insulating jacket, with a lid, a stirrer and a thermometer. The jacket stops heat escaping; the can itself takes part in the exchange.

The can's water equivalent is the mass of water with the same heat capacity: $\dfrac{m_cc_c}{c_w}$. Adding it to the water's mass accounts for the can.

You should find that equal masses settle at the average, and that the final temperature moves toward whichever water there is more of.

Watch out. Write each side as a positive heat: hot $(T_1 - T)$, cold $(T - T_2)$. Mixing up the signs gives a final temperature outside the starting range, which is impossible.

A worked example

A $0.20\ \text{kg}$ iron bolt at $300\ ^\circ\text{C}$ is dropped into $1.0\ \text{kg}$ of water at $20\ ^\circ\text{C}$ in a $0.15\ \text{kg}$ aluminium can. Find the final temperature.

1. Heat capacities. Bolt: $0.20\ \text{kg} \times 450\ \text{J kg}^{-1}\text{K}^{-1} = 90\ \text{J/K}$. Water and can: $4186\ \text{J/K} + 0.15\ \text{kg} \times 900\ \text{J kg}^{-1}\text{K}^{-1} = 4321\ \text{J/K}$.
2. Balance. $90\ \text{J/K}(300\ ^\circ\text{C} - T) = 4321\ \text{J/K}(T - 20\ ^\circ\text{C})$.
3. Answer. $T = \dfrac{27\,000\ \text{J} + 86\,420\ \text{J}}{4411\ \text{J/K}} = 25.7\ ^\circ\text{C}$.

Think first. The water has nearly fifty times the bolt's heat capacity, so the temperature barely moves.

Practise with new numbers

Try it in Python

The program works out the bolt of the example.

Remember these

• Heat lost $=$ heat gained, if none escapes.
• $m_1c_1(T_1 - T) = m_2c_2(T - T_2)$.
• Water equivalent of a calorimeter $= \dfrac{m_cc_c}{c_w}$.

Check yourself

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

1. $1.0\ \text{kg}$ of water at $90\ ^\circ\text{C}$ and $3.0\ \text{kg}$ at $10\ ^\circ\text{C}$ are mixed. What is the final temperature?

2. What is the water equivalent of a $0.30\ \text{kg}$ aluminium can?

3. Why is a calorimeter made of thin copper, polished and inside a jacket?

4. Can a mixture end up hotter than its hottest part?