Question

If the rate of diffusion of A is 5 times that of B, what will be the density ratio of A and B?
(A) $\dfrac{1}{{25}}$
(B) $\dfrac{1}{5}$
(C) 25
(D) 5

Answer 1

Hint : Graham's law says that a gas's rate of diffusion or effusion is inversely related to its molecular weight squared. If one gas has four times the molecular weight of another, it will diffuse through a porous plug or escape through a tiny puncture in a vessel at half the pace (heavier gases diffuse more slowly). Years later, the kinetic theory of gases offered a comprehensive theoretical explanation of Graham's law. Graham's law establishes a foundation for diffusing isotopes, a process that was important in the creation of the atomic bomb.

Complete Step By Step Answer:
Thomas Graham, a Scottish physical chemist, proposed Graham's law of effusion (also known as Graham's law of diffusion) in 1848. Graham discovered that a gas's rate of effusion is inversely related to the square root of its particles' molar mass. This formula is worded as follows:
$\dfrac{{{\text{Rat}}{{\text{e}}_1}}}{{{\text{Rat}}{{\text{e}}_2}}} = \sqrt {\dfrac{{{M_2}}}{{{M_1}}}}$
For molecular effusion, which includes the passage of one gas at a time through a hole, Graham's law is the most exact. Because these processes include the movement of more than one gas, it is only approximate for diffusion of one gas into another or in air. The molar mass is related to the mass density at the same temperature and pressure. As a result, the square roots of the mass densities of various gases are inversely related to their diffusion rates.
${\text{r}} \propto \dfrac{{\text{1}}}{{\sqrt d }}$
Given ${{\text{r}}_{\text{A}}} = 5{{\text{r}}_{\text{B}}},$ the rate of diffusion of a gas is inversely proportional to the square root of its' density. $\dfrac{{{r_A}}}{{{r_B}}} = \sqrt {\dfrac{{{d_B}}}{{{d_A}}}}$
$\dfrac{{5{r_{\text{B}}}}}{{{{\text{r}}_{\text{B}}}}} = \sqrt {\dfrac{{{{\text{d}}_{\text{B}}}}}{{{{\text{d}}_{\text{A}}}}}}$
Squaring we get
$\Rightarrow \dfrac{{{{\text{d}}_{\text{B}}}}}{{{{\text{d}}_{\text{A}}}}} = \dfrac{{25}}{1}$
Or
$\dfrac{{{d_A}}}{{{d_B}}} = \dfrac{1}{{25}}$
Hence option (A) is correct.

Note :
Graham became interested in gas diffusion after hearing about German chemist Johann Döbereiner's discovery that hydrogen gas diffused out of a tiny break in a glass container quicker than surrounding air diffused in to replace it. Graham used plaster plugs, very tiny tubes, and small orifices to determine the rate of diffusion of gases. He delayed the process in this way so that it could be analysed statistically. He discovered that the rate of effusion of a gas is inversely proportional to the square root of its density in 1831, and that this rate is also inversely proportional to the square root of its molar mass in 1848.