Question

Tetrafluoroethylene, ${C_2}{F_4}$ , effuses through a barrier at a rate of $8 \times {10^{ - 6}}mol/hr$ . An unknown gas ($X$ ) consisting only of boron and hydrogen; effuses at the rate of $10 \times {10^{ - 6}}mol/hr$ under the same conditions. The molecular mass of $X$ is [At. Weight of $F = 19amu$ ].

Answer 1

Hint: Graham's law of effusion (also called Graham's law of diffusion) was formulated by Scottish physical chemist Thomas Graham in 1848. Graham found experimentally that the rate of effusion of a gas is inversely proportional to the square root of the mass of its particles or the molar mass of the gas.

Complete step by step answer:
Graham's law states that the rate of diffusion or of effusion of a gas is inversely proportional to the square root of its molecular weight. Thus, if the molecular weight of one gas is four times that of another, it would diffuse through a porous plug or escape through a small pinhole in a vessel at half the rate of the other (heavier gases diffuse more slowly). Mathematically, Graham’s law can be represented as:
$R \propto \dfrac{1}{{\sqrt {{M_w}} }}$
Where, $R = $ rate of effusion of gas
${M_w} = $ molecular weight of the gas
Thus, for tetrafluoroethylene, ${C_2}{F_4}$, and the unknown gas, the mathematical relation can be represented by:
 $\dfrac{{{R_{{C_2}{F_4}}}}}{{{R_X}}} = \dfrac{{\sqrt {{M_X}} }}{{\sqrt {{M_{{C_2}{F_4}}}} }}$…(i)
Where, ${R_{{C_2}{F_4}}} = 8 \times {10^{ - 6}}mol/hr$ = rate of effusion of tetrafluoroethylene gas
${R_X} = 10 \times {10^{ - 6}}mol/hr$= rate of effusion of unknown gas
${M_{{C_2}{F_4}}} = (2 \times 12) + (4 \times 19) = 100amu$ = molecular mass of tetrafluoroethylene gas
${M_X} = ?$ = molecular mass of unknown gas
Substituting the values in the equation (i), we have:
$ \Rightarrow \dfrac{{8 \times {{10}^{ - 6}}}}{{10 \times {{10}^{ - 6}}}} = \sqrt {\dfrac{{{M_X}}}{{100}}} $
Thus, on solving, we have:
$ \Rightarrow \dfrac{{64}}{{100}} = \dfrac{{{M_X}}}{{100}}$
Hence, ${M_X} = 64amu$ .

Note:
A complete theoretical explanation of Graham's law was provided years later by the kinetic theory of gases. Graham's law provides a basis for separating isotopes by diffusion which is a method that came to play a crucial role in the development of the atomic bomb.