Question

A gas mixture contains an equal number of molecules of ${N_2}$ and $S{F_6}$​, some of it is passed through a gaseous effusion apparatus. Calculate how many molecules of ${N_2}$​ are present in the product gas for every $100$ molecules of $S{F_6}$
A.$323$
B.$223$
C.$423$
D.$523$

Answer 1

Graham’s law can be used to solve this question .It is an empirical relationship which states that the ratio of the rates of diffusion or effusion of two gases is equal to the square root of the inverse ratio of their molar masses. The relationship is based on the assumption that all gases at the same temperature have the same average kinetic energy.

Complete step by step answer: We know, molar mass of ${N_2} = 28$g, we call it ${M_1}$
Molar mass of $S{F_6} = 140$g, we call it ${M_2}$
Mathematically, the ratio of a sample of that compound and the amount of substance in that sample is the molar mass of that substance. It is generally measured in moles.
Rate of effusion of gas 1/rate of effusion of gas 2 ​$= \sqrt {\dfrac{{{M_2}}}{{{M_1}}}}$
Let ${r_1} =$rate of effusion of gas 1
And ${r_2}$ $=$rate of effusion of gas 2​​​
then $\dfrac{{{r_1}}}{{{r_2}}} =$ $= \sqrt {\dfrac{{{M_2}}}{{{M_1}}}}$
${M_1} = 28$
${M_2} = 140$

On substituting the values we will get:
$\dfrac{{{r_1}}}{{{r_2}}} = \sqrt {\dfrac{{140}}{{28}}} = \sqrt 5 \\\ \\\$
$\dfrac{{{r_1}}}{{{r_2}}} = \dfrac{{{x_1}}}{{{x_2}}}$
where ${x_1} =$ molecules of ${N_2}$​
and ${x_2} =$molecules of $S{F_6}$
Given, ${x_2} = 100$

Now substituting the value of ${x_2}$ in equation i), we get the value of ${x_1}$ as calculated below:
Now substituting the value of ${x_2}$ in equation i), we get the value of ${x_1}$ as calculated below:
$\dfrac{{{x_1}}}{{100}} = \sqrt 5 \\\ \\\$
Further solving,
${x_1} = 100\sqrt 5$
On solving, we get
$x_{_1}^{} = 223$
And hence the Correct answer is B.

Note:
Diffusion is described as the gradual mixing of gases due to the motion of their particles even in the absence of mechanical agitation such as stirring. The result is the gas mixture with uniform composition. Diffusion is also a property of the particles in liquids and liquid solutions and, to some extent, of solids and solid solutions. The related process, effusion, is described as the escape of gaseous molecules through a small (usually microscopic) hole, such as a hole in a balloon, into an evacuated space.