(a)[4]
From the kinetic theory of ideal gases, it can be shown that, if every molecule travels with speed $v$ at right angles to one face of the box, the pressure $p$ on that face is given by $pV = Nmv^2$ (equation 1). This result then leads to $p = \frac{1}{3}\rho \langle c^2 \rangle$ (equation 2) for the pressure $p$ of an ideal gas, where $\rho$ is the gas density and $\langle c^2 \rangle$ is the mean-square speed of the molecules. Explain how each of the following parts of equation 2 is obtained from equation 1: $\rho$, $\frac{1}{3}$, $\langle c^2 \rangle$.
(b)[4]
An ideal gas has the volume, pressure and temperature shown in Fig. 2.1. The gas has a mass of $20.7\,\text{g}$. Calculate the mass of one molecule of the gas.