Ideal gasesAQA A-Level Physics: Revision notes

The gas laws are empirical relationships found from experiments on a fixed mass of gas:

• Boyle's law: at constant temperature, $pV$ = constant, so $p \propto 1/V$.
• Charles's law: at constant pressure, $V/T$ = constant, so $V \propto T$ (in kelvin).
• Pressure law: at constant volume, $p/T$ = constant, so $p \propto T$ (in kelvin).

The three combine to give $pV/T$ = constant for a fixed mass of gas. Always convert to kelvin: $T/\text{K} = \theta/{}^\circ\text{C} + 273$.

Absolute zero, 0 K or −273 °C, is the temperature at which the gas would have zero volume (or pressure) when extrapolated from graphs on the Celsius scale; it is the lowest possible temperature.

An ideal gas obeys the gas laws at all pressures and temperatures. The equation of state is

$pV = nRT \qquad\text{or}\qquad pV = NkT$

where $n$ is the number of moles, $N$ the number of molecules, $R = 8.31$ J mol⁻¹ K⁻¹ the molar gas constant and $k = 1.38 \times 10^{-23}$ J K⁻¹ the Boltzmann constant. They are linked by $N = nN_A$ and $k = R/N_A$, where $N_A = 6.02 \times 10^{23}$ mol⁻¹ is the Avogadro constant.

Use SI units: $p$ in Pa, $V$ in m³, $T$ in K.

Worked example: 0.80 mol at 300 K in 2.0 × 10⁻² m³ has $p = nRT/V = 0.80 \times 8.31 \times 300/0.020 = 1.0 \times 10^5$ Pa.

The molar mass $M$ is the mass of one mole, in g mol⁻¹ or kg mol⁻¹. The molecular mass $m$ is the mass of one molecule. They are related by $m = M/N_A$.

The mass of gas is $mass = nM = Nm$, and the number of moles is $n = N/N_A$.

Worked example: 62.2 mol of helium ($M = 4.00$ g mol⁻¹) has a mass of $62.2 \times 4.00 = 249$ g and contains $62.2 \times 6.02 \times 10^{23} = 3.75 \times 10^{25}$ atoms.

When a gas expands against a constant pressure, the work done by the gas is

$W = p\Delta V$

If the gas is compressed, work is done on the gas. For a gas at constant $p$ expanding from $V_1$ to $V_2$, $W = p(V_2 - V_1)$.

Worked example: at $1.0 \times 10^5$ Pa, an expansion from 2.0 × 10⁻³ m³ to 5.0 × 10⁻³ m³ does $1.0 \times 10^5 \times 3.0 \times 10^{-3} = 300$ J of work. The final temperature is found from $V/T$ = constant: $290 \times 5.0/2.0 = 725$ K.

Boyle's law (constant temperature): trap air in a sealed tube above oil, vary the pressure with a foot pump and a Bourdon gauge, and measure the volume. Wait after each change so the air returns to room temperature, because compression heats the gas. Plot $p$ against $1/V$: a straight line through the origin confirms $pV$ = constant.

Charles's law (constant pressure): trap dry air in a capillary tube by an oil or acid thread in a stirred water bath, and measure the length of the air column (proportional to volume) at temperatures measured with a thermometer. Plot length against temperature in °C. The line extrapolated to zero volume cuts the axis at about −273 °C, which is absolute zero.

Improve accuracy by waiting for thermal equilibrium, reading the ruler at eye level, and using dry air.