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 \( 32\times10^{18}/s \) then collision frequency in gas A is__________/s.

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(B) \( 4\times10^{18} \)

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Collision frequency (z) \( =\sqrt{2}\pi d^{2}N\sqrt{\frac{8RT}{\pi M}} \)
Temp, N are same
\( Z\propto\frac{d^{2}}{\sqrt{M}} \)
\( d_{A}=\frac{d_{B}}{2} \)
\( M_{A}=4M_{B} \)
\( \frac{Z_{A}}{Z_{B}}=\frac{d_{A}^{2}}{\sqrt{M_{A}}}\times\frac{\sqrt{M_{B}}}{d_{B}^{2}}=(\sqrt{\frac{M_{B}}{M_{A}}})(\frac{d_{A}}{d_{B}})^{2} \)
\( =(\sqrt{\frac{1}{4}})(\frac{1}{2})^{2} \)
\( \frac{Z_{A}}{Z_{B}}=\frac{1}{2}\times\frac{1}{4}=\frac{1}{8} \)
\( \Rightarrow Z_{A}=\frac{32\times10^{8}}{8}=4\times10^{18}/s \)

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