Temperature, Gases and Thermal Expansion: Quick Revision

A brief note on Temperature, Gases and Thermal Expansion. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Heat flows from hot to cold; temperature decides the direction. Bodies in contact reach thermal equilibrium.
• Scales: $T_F = \tfrac95 T_C + 32$; $T = T_C + 273.15$. A change of $1\ \text{K}$ equals one of $1\ ^\circ\text{C}$.
• Ideal gas: $pV = nRT = Nk_BT$; Boyle and Charles are special cases. Use kelvin.
• Absolute zero, $0\ \text{K} = -273.15\ ^\circ\text{C}$, where every gas's pressure line meets zero.
• Expansion: $\Delta L = \alpha L\,\Delta T$; area $2\alpha$, volume $\gamma = 3\alpha$. Water is densest at $4\ ^\circ\text{C}$.

The facts at a glance

A worked example

A weather balloon holds $50$ moles of helium. Find its volume on the ground ($1.0 \times 10^{5}\ \text{Pa}$, $288\ \text{K}$) and $10\ \text{km}$ up ($2.6 \times 10^{4}\ \text{Pa}$, $223\ \text{K}$).

1. Ground. $V = 50\ \text{mol} \times 8.31\ \text{J mol}^{-1}\text{K}^{-1} \times \dfrac{288\ \text{K}}{1.0 \times 10^{5}\ \text{Pa}} = 1.20\ \text{m}^3$.
2. High up. $V = 50\ \text{mol} \times 8.31\ \text{J mol}^{-1}\text{K}^{-1} \times \dfrac{223\ \text{K}}{2.6 \times 10^{4}\ \text{Pa}} = 3.56\ \text{m}^3$.
3. So it swells to about three times its size, though the air there is colder.

Practice with new numbers

A program to try

The program works out the balloon at three heights.

Mistakes to avoid

• Celsius in the gas laws. Always use kelvin.
• Gauge pressure in the gas laws. Use absolute pressure.
• Litres in $pV = nRT$. Change to cubic metres.
• $3\alpha$ for a liquid's $\gamma$. A liquid has only a volume coefficient.
• A hole shrinking when a plate is heated. It grows with the plate.

Check yourself

1. What is $-40\ ^\circ\text{C}$ in Fahrenheit?

2. A gas at a steady volume goes from $27\ ^\circ\text{C}$ to $327\ ^\circ\text{C}$. By what factor does its pressure grow?

3. How many molecules are in $1.0\ \text{mol}$ of any gas?

4. A steel beam $12\ \text{m}$ long warms by $30\ \text{K}$. How much does it grow?

5. Why is a gap left at the joints of a concrete road?