The ideal gas equation and absolute temperature

The gas laws

Pump up a bicycle tyre and you squeeze air into a smaller space at a higher pressure. Experiments on gases at low density give two simple rules.

• Boyle's law: at a steady temperature, $pV$ is constant. Halve the volume, double the pressure.
• Charles's law: at a steady pressure, $V \propto T$, the temperature in kelvin.

The ideal gas equation

Together, for $n$ moles:

where $R = 8.31\ \text{J mol}^{-1}\text{K}^{-1}$. Counting molecules instead, $N = nN_A$ and $pV = Nk_BT$, with Boltzmann's constant $k_B = \dfrac{R}{N_A} = 1.38 \times 10^{-23}\ \text{J/K}$.

For a fixed amount of gas: $\dfrac{p_1V_1}{T_1} = \dfrac{p_2V_2}{T_2}$.

You should find that halving the volume doubles the pressure (Boyle), and that at a fixed volume the pressure grows in step with the kelvin temperature.

Watch out. In every gas law, use kelvin. $100\ ^\circ\text{C}$ is not twice as hot as $50\ ^\circ\text{C}$.

Absolute zero

Hold a gas at a steady volume and plot its pressure against Celsius temperature. The points lie on a straight line. Extend the line down: it reaches zero pressure at $-273.15\ ^\circ\text{C}$, and it does so for every gas. That point is absolute zero, $0\ \text{K}$, the base of the kelvin scale.

A worked example

A car tyre holds $25$ litres of air at $27\ ^\circ\text{C}$ and an absolute pressure of $2.5 \times 10^{5}\ \text{Pa}$. How many moles of air is that, and what is the pressure after a drive warms it to $52\ ^\circ\text{C}$ (volume unchanged)?

1. Moles. $n = \dfrac{2.5 \times 10^{5}\ \text{Pa} \times 0.025\ \text{m}^3}{8.31\ \text{J mol}^{-1}\text{K}^{-1} \times 300\ \text{K}} = 2.51\ \text{mol}$.
2. Warmed. $p_2 = 2.5 \times 10^{5}\ \text{Pa} \times \dfrac{325\ \text{K}}{300\ \text{K}} = 2.71 \times 10^{5}\ \text{Pa}$.

Think first. That is why tyre pressures should be checked cold.

Practise with new numbers

Try it in Python

The program finds absolute zero from two readings of a gas thermometer, and works out the tyre.

Remember these

• Boyle: $pV$ constant; Charles: $V \propto T$.
• $pV = nRT = Nk_BT$; $R = 8.31\ \text{J mol}^{-1}\text{K}^{-1}$.
• Kelvin in every gas law; absolute zero is $-273.15\ ^\circ\text{C}$.

Check yourself

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

1. A gas at a steady temperature is squeezed from $6.0$ to $1.5$ litres. By what factor does its pressure grow?

2. At a steady pressure, a gas is warmed from $0\ ^\circ\text{C}$ to $273\ ^\circ\text{C}$. What happens to its volume?

3. How many moles are in $0.50\ \text{m}^3$ of gas at $1.0 \times 10^{5}\ \text{Pa}$ and $250\ \text{K}$?

4. Why can there be no temperature below absolute zero?