What is meant by an elastic collision?
State two further assumptions of the kinetic theory of gases.
An ideal gas molecule has mass $m$ and is inside a cube of side length $L$. The molecule moves with velocity $u$ towards the face of the box shaded in Fig. 3.1. It collides elastically with the shaded face and then the opposite face alternately. Deduce an expression, in terms of $m$, $u$ and $L$, for the magnitude of the change in momentum of the molecule on colliding with a face.
Deduce an expression, in terms of $m$, $u$ and $L$, for the time between consecutive collisions of the molecule with the shaded face.
Deduce an expression, in terms of $m$, $u$ and $L$, for the average force exerted by the molecule on the shaded face.
Deduce an expression, in terms of $m$, $u$ and $L$, for the pressure on the shaded face if the force in (iii) is exerted over the whole area of the face.
For the model in (b) extended to three dimensions and to a gas containing $N$ molecules, each of mass $m$, moving with mean-square speed $\langle c^2 \rangle$, it can be shown that $pV = \frac{1}{3}Nm\langle c^2 \rangle$, where $p$ is the pressure exerted by the gas and $V$ is the volume of the gas. Use this expression, together with the equation of state of an ideal gas, to show that the average translational kinetic energy $E_k$ of a molecule of an ideal gas is given by $E_k = \frac{3}{2}kT$, where $T$ is the thermodynamic temperature of the gas and $k$ is the Boltzmann constant.
The mass of a hydrogen molecule is $3.34 \times 10^{-27}\,\text{kg}$. Use the expression for $E_k$ in (c) to determine the root-mean-square (r.m.s.) speed of a molecule of hydrogen gas at $25\,^{\circ}\text{C}$.