What is the ratio of he velocities of ${\mathrm{CH}}_{4 } \mathrm{and} {\mathrm{O}}_2$ molecules so that they are associated with debrogile waves of equal wave length?

 $$\begin{gathered} \mathrm{From} \mathrm{the} \mathrm{debrogile} \mathrm{relationship} \\ \mathrm{\lambda }=\frac{\mathrm{h}}{\mathrm{mv}} \\ \mathrm{for} {\mathrm{CH}}_4, {\mathrm{\lambda }}_{{\mathrm{CH}}_4}=\frac{\mathrm{h}}{{\left(\mathrm{m}\times \mathrm{v}\right)}_{{\mathrm{CH}}_4}}----\left(\mathrm{i}\right) \\ \mathrm{for} {\mathrm{O}}_{2, {\mathrm{\lambda }}_{{\mathrm{O}}_2}}=\frac{\mathrm{h}}{{\left(\mathrm{m}\times \mathrm{v}\right)}_{{{ }_{\mathrm{O}}}_2}}----\left(\mathrm{ii}\right) \\ \mathrm{wavelength} \mathrm{of} {\mathrm{CH}}_4 \mathrm{and} {\mathrm{O}}_2 \mathrm{are} \mathrm{equal} , \mathrm{hence} \\ \frac{{\mathrm{\nu }}_{{\mathrm{CH}}_4}}{{\mathrm{\nu }}_{{\mathrm{O}}_2}}=\frac{{\mathrm{m}}_{{\mathrm{O}}_2}}{{\mathrm{m}}_{{\mathrm{CH}}_4}}=\frac{32}{16}=2 \\ {\mathrm{\nu }}_{{\mathrm{CH}}_4}=2{\mathrm{\nu }}_{{\mathrm{O}}_2} \\ \mathrm{The} \mathrm{velocity} \mathrm{of} {\mathrm{CH}}_4 \mathrm{molecule} \mathrm{is} \mathrm{two} \mathrm{times} \mathrm{the} \mathrm{velocity} \mathrm{of} {\mathrm{O}}_2 \mathrm{molecule}. \end{gathered}$$