Consider two boxes containing ideal gases $A$ and $B$ such that their temperatures, pressures and nu

Question

Consider two boxes containing ideal gases $A$ and $B$ such that their temperatures, pressures and number densities are same. The molecular size of $A$ is half of that of $B$ and mass of molecule $A$ is four times that of $B$. If the collision frequency in gas $B$ is $32 	imes 10^{18} / \mathrm{s}$ then collision frequency in gas $A$ is $\_\_\_\_$ /s.

Professor PC Thomas Classes · Answer

The collision frequency is given by $
u = \sqrt{2} \pi d^2 n \bar{v}$, where $d$ is the molecular diameter, $n$ is number density, and $\bar{v} \propto \frac{1}{\sqrt{m}}$ is the average speed. Thus, $
u \propto \frac{d^2}{\sqrt{m}}$. We are given $d_A = \frac{1}{2}d_B$ and $m_A = 4m_B$. Taking the ratio gives $\frac{
u_A}{
u_B} = (\frac{d_A}{d_B})^2 \sqrt{\frac{m_B}{m_A}} = (\frac{1}{2})^2 \sqrt{\frac{1}{4}} = \frac{1}{4} 	imes \frac{1}{2} = \frac{1}{8}$. Given $
u_B = 32 	imes 10^{18} 	ext{ /s}$, we find $
u_A = \frac{1}{8} 	imes 32 	imes 10^{18} = 4 	imes 10^{18} 	ext{ /s}$ (the printed options contain a typo and omit the exponent $10$, rendering it as $4 	imes 10^8$).