Thermodynamics Part 5

Concept:Collision frequency $Z$ for an ideal gas is $Z = \sqrt{2} \pi d^2 v_{\text{avg}} n$, where $d$ is molecular diameter, $v_{\text{avg}}$ is average speed, and $n$ is number density.Explanation:Given that temperature, pressure, and number density are same for gases $A$ and $B$.Thus $Z \propto \frac{d^2}{\sqrt{m}}$, because $v_{\text{avg}} \propto \frac{1}{\sqrt{m}}$.Let $d_A = \frac{1}{2} d_B$ (size of $A$ is half of $B$).Let $m_A = 4 m_B$ (mass of $A$ is four times $B$).Then $\frac{Z_A}{Z_B} = \left(\frac{d_A}{d_B}\right)^2 \times \sqrt{\frac{m_B}{m_A}} = \left(\frac{1}{2}\right)^2 \times \sqrt{\frac{1}{4}} = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$.Given $Z_B = 32 \times 10^{18} \ \mathrm{s}^{-1}$,So $Z_A = \frac{Z_B}{8} = \frac{32 \times 10^{18}}{8} = 4 \times 10^{18} \ \mathrm{s}^{-1}$.Answer:$4 \times 10^{18} \ \mathrm{s}^{-1}$ (Option D in the list, but note the power of 10: the options use $10^8$; the correct value is $4 \times 10^{18}$).