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 $32 \times 10\frac{18}{\text{ }s}$ then collision frequency in gas A is
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Solution
Collision frequency $(z) = \sqrt{2}\pi d^{2}N\sqrt{\frac{8RT}{\pi M}}$ Temp, N are same
$${Z \propto \frac{d^{2}}{\sqrt{M}} }{d_{A} = \frac{d_{B}}{2} }{M_{A} = 4M_{B} }{\frac{Z_{A}}{Z_{B}} = \frac{d_{A}^{2}}{\sqrt{M_{A}}} \times \frac{\sqrt{M_{B}}}{d_{B}^{2}} = \left( \sqrt{\frac{M_{B}}{M_{A}}} \right)\left( \frac{d_{A}}{d_{B}} \right)^{2} }{= \left( \sqrt{\frac{1}{4}} \right)\left( \frac{1}{2} \right)^{2} }{\frac{Z_{A}}{Z_{B}} = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} }{\Rightarrow Z_{A} = \frac{32 \times 10^{8}}{8} = 4 \times 10^{8}/s}$$
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