Calculate relative rate of effusion of SO₂ to CH₄, if the mixture obtained by effusing out a mixture with molar ratio $\frac{{\mathrm{n}}_{{\mathrm{SO}}_2}}{{\mathrm{n}}_{{\mathrm{CH}}_4}}=\frac{8}{1}$for three effusing steps.

Let, the ratio of initial no. of moles = $\frac{{\mathrm{n}}_{{\mathrm{SO}}_2}^{\mathrm{o}}}{{\mathrm{n}}_{{\mathrm{CH}}_4}^{\mathrm{o}}}=\frac{8}{1}$

The ratio of no. of moles after three effusing steps = ${\left(\frac{{\mathrm{n}}_{{\mathrm{SO}}_2}^{\text{'}}}{{\mathrm{n}}_{{\mathrm{CH}}_4}^{\text{'}}}\right)}^3$

$\begin{array}{r}{\left(\frac{{\mathrm{n}}_{{\mathrm{SO}}_2}^{\mathrm{\prime }}}{{\mathrm{n}}_{{\mathrm{CH}}_4}^{\mathrm{\prime }}}\right)}^3=\left(\frac{8}{1}\right) \left(\sqrt{\frac{{\mathrm{M}}_{{\mathrm{CH}}_4}}{{\mathrm{M}}_{{\mathrm{SO}}_2}}}\right)=\frac{8}{1}\times \sqrt{\frac{16}{64}}=1\\ \end{array}$

Now,   $\frac{{\mathrm{r}}_{{\mathrm{SO}}_2}}{{\mathrm{r}}_{{\mathrm{CH}}_4}}={\left(\frac{{\mathrm{n}}_{{\mathrm{SO}}_2}^{\text{'}}}{{\mathrm{n}}_{{\mathrm{CH}}_4}^{\text{'}}}\right)}^3\sqrt{\frac{{\mathrm{M}}_{{\mathrm{CH}}_4}}{{\mathrm{M}}_{{\mathrm{SO}}_2}}}=1\sqrt{\frac{16}{64}}=\frac{1}{2}$