The Ideal Gas Laws

Pressure is force per unit area, $p = F/A$, measured in pascals, where $1\ \mathrm{Pa} = 1\ \mathrm{N\,m^{-2}}$. Atmospheric pressure is about $1.0\times10^{5}$ Pa.

A gas exerts pressure because its particles collide with the container walls. Each collision reverses a component of a particle’s momentum, and the rate of change of momentum over the whole wall is the force.

Every gas law can be explained by thinking about how often particles hit the walls and how hard.

$\textbf{Boyle’s law}$: for a fixed mass of gas at constant temperature, $pV$ is constant. Halve the volume and particles hit the walls twice as often, doubling the pressure.

$\textbf{Charles’s law}$: at constant pressure, $V \propto T$. Hotter particles move faster, so the gas must expand to keep the collision rate per area unchanged.

$\textbf{The pressure law}$: at constant volume, $p \propto T$. Faster particles hit harder and more often.

In each case $T$ must be in kelvin. A graph of $V$ or $p$ against Celsius temperature is a straight line, but it only passes through the origin when plotted in kelvin.

Plot pressure against Celsius temperature for a fixed volume of gas and you get a straight line. Extrapolate it backwards and it meets zero pressure at about $-273^{\circ}$C.

The same intercept appears for any gas and any starting pressure. That temperature is $\textbf{absolute zero}$, where the particles would have minimum kinetic energy and exert no pressure.

In practice real gases liquefy long before this, so the extrapolation describes ideal behaviour, not a measurement.

Combining the three laws and allowing for the amount of gas gives $pV = NkT$, where $N$ is the number of particles and $k = 1.38\times10^{-23}\ \mathrm{J\,K^{-1}}$ is the $\textbf{Boltzmann constant}$.

An $\textbf{ideal gas}$ obeys this equation exactly at all pressures and temperatures. Real gases approach ideal behaviour at low pressure and high temperature, when particles are far apart and move fast.

To investigate Boyle’s law, trap air in a sealed tube, change the pressure with a pump, and read volume and pressure after waiting for the temperature to settle. A graph of $p$ against $1/V$ is a straight line through the origin.