Specific Heat, Calorimetry and Change of State: Quick Revision

A brief note on Specific Heat, Calorimetry and Change of State. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Specific heat: $Q = mc\,\Delta T$; a body's heat capacity is $mc$. Water's $c$, $4186\ \text{J kg}^{-1}\text{K}^{-1}$, is large.
• Mixtures: heat lost $=$ heat gained, if none escapes. Add the calorimeter's water equivalent.
• Latent heat: during melting or boiling the temperature stays steady and $Q = mL$.
• Pressure raises boiling points; evaporation cools.

The facts at a glance

A worked example

A cup of $0.25\ \text{kg}$ of tea at $85\ ^\circ\text{C}$ gets $0.050\ \text{kg}$ of milk at $5\ ^\circ\text{C}$, then a $10\ \text{g}$ ice cube at $0\ ^\circ\text{C}$. Treat everything as water.

1. Milk. $T = \dfrac{(0.25 \times 85 + 0.050 \times 5)\ \text{kg}\,^\circ\text{C}}{0.30\ \text{kg}} = 71.7\ ^\circ\text{C}$.
2. Ice. $0.30\ \text{kg} \times 4186\ \text{J kg}^{-1}\text{K}^{-1} \times (71.7\ ^\circ\text{C} - T) = 0.010\ \text{kg} \times 3.34 \times 10^{5}\ \text{J/kg} + 0.010\ \text{kg} \times 4186\ \text{J kg}^{-1}\text{K}^{-1}\,T$.
3. Answer. $T = 66.8\ ^\circ\text{C}$: the small cube cools the tea by about $5\ \text{K}$, mostly through its latent heat.

Practice with new numbers

A program to try

The program works out the cup of tea.

Mistakes to avoid

• The plain average for unequal masses. Weight each temperature.
• $mc\,\Delta T$ during melting. Use $mL$: the temperature does not change.
• Forgetting that melted ice must warm up.
• Leaving out the calorimeter. It takes heat too.
• Not checking whether all the ice melts. If not, the answer is $0\ ^\circ\text{C}$.

Check yourself

1. How much heat warms $0.80\ \text{kg}$ of water by $25\ \text{K}$?

2. $0.5\ \text{kg}$ of water at $90\ ^\circ\text{C}$ and $1.5\ \text{kg}$ at $20\ ^\circ\text{C}$ are mixed. What is the final temperature?

3. How much heat freezes $0.50\ \text{kg}$ of water at $0\ ^\circ\text{C}$, and where does it go?

4. Why does a pressure cooker cook faster?

5. Why do we feel cold when we step out of a swimming pool on a windy day?