The ideal-gas equation of state may be written as $pV = NkT$. State the meaning of each symbol in this equation: $p$, $V$, $N$, $k$, $T$.
Use the relation in (a) to show that the average translational kinetic energy $E_{K}$ of a molecule in an ideal gas is $E_{K} = \frac{3}{2}kT$.
An oxygen molecule has mass $5.31 \times 10^{-26}\,\text{kg}$. Take oxygen to be an ideal gas. Use the equation in (b) to find the root-mean-square (r.m.s.) speed $u$ of an oxygen molecule at $23\,^{\circ}\text{C}$.
A fixed mass of oxygen gas is trapped at initial pressure $P$ in a cylindrical container by a movable piston at one end, as illustrated in Fig. 3.1. The gas temperature is $23\,^{\circ}\text{C}$. The piston is gradually pushed into the cylinder so that the oxygen gas is compressed. At every stage, the gas and the container are in thermal equilibrium with the surroundings. On Fig. 3.2, sketch how the r.m.s. speed of the oxygen molecules varies with pressure as the pressure increases.