5.0 Graham's law of effusion or diffusion

  Gaseous State

This law states that 'Rate of effusion(escaping of gas under pressure gradient through a fine hole) or diffusion (intermixing of gases without help of any external agent) of a gas is inversely proportional to square root of its density'. Mathematically, $$r \propto 1/\sqrt d $$ If $r_1$ and $r_2$ represent the rate of effusion of two gases having densities $d_1$ and $d_2$ respectively then,$${r_1}/{r_2} = \sqrt {{d_2}/{d_1}} $$ Rate can also be expressed in terms of vapour density and molecular weight of gas.$${r_1}/{r_2} = \sqrt {{(vapour\ density)_2}/{(vapour\ density)_1}} = \sqrt {{(molecular\ weight)_2}/{(molecular\ weight)_1}} $$ Further, if the pressure of gas is also varying then $$r \propto P/\sqrt d $$

Question 6. A straight glass tube of $200$ cm length has inlets $X$ and $Y$ at the two ends. $HCl$ gas through inlet $X$ and $NH_3$ through inlet $Y$ are allowed to enter in the tube at the same time and same pressure. white fumes first forms at a point $P$ inside the tube. Calculate the distance of $P$ from $X$.

Solution: $$\begin{equation} \begin{aligned} \frac{{{r_{HCl}}}}{{{r_{N{H_3}}}}} = \frac{{{v_{HCl}}}}{{{v_{N{H_3}}}}} = \sqrt {\frac{{{M_{N{H_3}}}}}{{{M_{HCl}}}}} = \sqrt {\frac{{17}}{{36.5}}} = 0.6825 \\ {\text{Let after time $t$ white fume forms at a distance $x$ from end $X$}}{\text{.}} \\ Now,{v_{HCl}}{\text{t + }}{v_{N{H_3}}}{\text{t = 200 }} \\ \Rightarrow x{\text{ + }}\frac{{{v_{HCl}}{\text{t}}}}{{0.6825}} = 200 \\ \Rightarrow 1.6825x = 0.6825 \times 200 \\ \Rightarrow x = 81.129{\text{ cm}}{\text{.}} \\\end{aligned} \end{equation} $$