Behaviour of a Gas

The macroscopic description

A gas enclosed in a container is characterised by its pressure $P$, volume $V$ and temperature $T$. Together with the internal energy $E$, these are the quantities that specify the state of the gas, and the relation connecting them is called the equation of state. For an ideal gas that equation is

$$PV = nRT = Nk_{B}T$$

obtained by combining Boyle's law ($V \propto 1/P$ at constant $T$), Charles' law ($V \propto T$ at constant $P$) and Gay-Lussac's law ($P \propto T$ at constant $V$), each of which applies to a fixed mass of enclosed gas.

Why a molecular description is needed, and why it must be statistical

A stone thrown upwards rises to a certain height and falls back, and its whole motion is described by Newton's laws applied to one particle. A gas cannot be treated that way. The particles of a gas are in constant motion, and their number is enormous — of the order of $10^{23}$ particles per m$^{3}$. Any attempt to relate the macroscopic parameters $P$, $V$, $T$ and $E$ to the motion of individual particles would therefore be futile: one would need $10^{23}$ equations of motion, and their solutions would tell us nothing directly about pressure or temperature.

The approach of kinetic theory

Kinetic theory replaces the impossible problem by a tractable one in three steps.

• Certain assumptions are made about the particles (molecules) of the gas — that they are small hard spheres, that they move randomly, that they collide elastically, and that they exert force on one another only during collisions.

• Averages of physical quantities are taken over the very large number of particles involved, for instance the mean square speed $\overline{v^{2}}$ rather than the speed of any one molecule.

• These averages are finally related to the macroscopic parameters of the gas. The result $P = \dfrac{1}{3}\dfrac{N}{V}m\overline{v^{2}}$ is exactly such a bridge: a measurable macroscopic quantity on the left, an average over molecular motion on the right.

Picture of the gas

The molecules are uniformly dispersed throughout the volume of the container and execute random motion. When one molecule approaches another there is a repulsive force between them, so the molecules behave as small hard spherical particles and the collisions are elastic; both the speed and the direction of motion change abruptly at each collision. The molecules also collide with the walls of the container — this is the origin of pressure. Because force acts only during collisions, between two successive collisions a molecule moves along a straight path with constant velocity.

Standard results

• Equation of state of an ideal gas: $PV = nRT = Nk_{B}T$, with $R = N_{A}k_{B} = 8.314$ J mol$^{-1}$ K$^{-1}$.

• Number density in a gas is of the order of $10^{23}$ particles per m$^{3}$, which is why the treatment must be statistical.

• Kinetic theory = assumptions about molecules $\rightarrow$ averages over many molecules $\rightarrow$ macroscopic parameters.