question_answer

Two gases - argon (atomic radius 0.07 nm, atomic weight 40) and xenon (atomic radius 0.1 nm, atomic weight 140), have the same number density and are at the same temperature. The ratio of their respective mean free times is closest to [JEE MAIN Held on 09-01-2020 Evening]

A) 1.83

B) 4.67

C) 2.3

D) 3.67

Correct Answer: A , B , C , D

Solution :

(1.09) Bonus \[\lambda =\frac{1}{\sqrt{2}\pi {{d}^{2}}n}\] Mean free time, \[\tau =\frac{\lambda }{v}\] \[\tau \propto \frac{\sqrt{M}}{{{d}^{2}}}\] \[\frac{{{\tau }_{1}}}{{{\tau }_{2}}}=\frac{\sqrt{{{M}_{1}}}}{{{d}_{1}}^{2}}\times \frac{{{d}_{2}}^{2}}{\sqrt{{{M}_{2}}}}\] \[=\sqrt{\frac{40}{140}}\times {{\left( \frac{0.1}{0.07} \right)}^{2}}=1.09\] None of the option matches

Solution :

(1.09) Bonus \[\lambda =\frac{1}{\sqrt{2}\pi {{d}^{2}}n}\] Mean free time, \[\tau =\frac{\lambda }{v}\] \[\tau \propto \frac{\sqrt{M}}{{{d}^{2}}}\] \[\frac{{{\tau }_{1}}}{{{\tau }_{2}}}=\frac{\sqrt{{{M}_{1}}}}{{{d}_{1}}^{2}}\times \frac{{{d}_{2}}^{2}}{\sqrt{{{M}_{2}}}}\] \[=\sqrt{\frac{40}{140}}\times {{\left( \frac{0.1}{0.07} \right)}^{2}}=1.09\] None of the option matches

Solution :

(1.09) Bonus \[\lambda =\frac{1}{\sqrt{2}\pi {{d}^{2}}n}\] Mean free time, \[\tau =\frac{\lambda }{v}\] \[\tau \propto \frac{\sqrt{M}}{{{d}^{2}}}\] \[\frac{{{\tau }_{1}}}{{{\tau }_{2}}}=\frac{\sqrt{{{M}_{1}}}}{{{d}_{1}}^{2}}\times \frac{{{d}_{2}}^{2}}{\sqrt{{{M}_{2}}}}\] \[=\sqrt{\frac{40}{140}}\times {{\left( \frac{0.1}{0.07} \right)}^{2}}=1.09\] None of the option matches

Solution :

(1.09) Bonus \[\lambda =\frac{1}{\sqrt{2}\pi {{d}^{2}}n}\] Mean free time, \[\tau =\frac{\lambda }{v}\] \[\tau \propto \frac{\sqrt{M}}{{{d}^{2}}}\] \[\frac{{{\tau }_{1}}}{{{\tau }_{2}}}=\frac{\sqrt{{{M}_{1}}}}{{{d}_{1}}^{2}}\times \frac{{{d}_{2}}^{2}}{\sqrt{{{M}_{2}}}}\] \[=\sqrt{\frac{40}{140}}\times {{\left( \frac{0.1}{0.07} \right)}^{2}}=1.09\] None of the option matches