Question

A mixture of equal gas of ${O_2}$ and ${O_3}$ gases are allowed to effuse through an orifice, then what will be the ratio of, the rate of effusion of ${O_3}$ to ${O_2}$.

Answer 1

Hint: Effusion is the interaction wherein a gas escapes from a compartment through a hole of a diameter considerably smaller than the mean free path of the molecules. Such an opening is frequently portrayed as a pinhole and the getaway of the gas is because of the pressure difference between the container and the exterior. Under these conditions, basically, all particles that show up at the hole proceed and pass through the hole, since impacts between atoms in the hole are insignificant.
Formula used:
According to Graham’s law, the rate of effusion of a gaseous substance is inversely proportional to the square root of its molar mass.
Boyle’s law states that at constant temperature, the pressure of a fixed amount of gas varies inversely with its volume.
Rate of effusion of a mixture of gases A and B is at similar condition of temperature and pressure is:
 $\dfrac{{{r_1}}}{{{r_2}}} = \sqrt {\dfrac{{{M_2}}}{{{M_1}}}} $(As both are inversely proportional)
${r_1}$ Is the rate of diffusion of gas A and ${r_2}$ is the rate of diffusion of gas B.
${M_1}$ And ${M_2}$ is the molar mass of gas A and gas B respectively.

Complete answer:
Using the formula given above, we can say that the rate or moles can be calculated as:
   $\dfrac{{{r_{{O_3}}}}}{{{r_{{O_2}}}}} = \sqrt {\dfrac{{{M_{{O_2}}}}}{{{M_{{O_3}}}}}} $
The molar mass of oxygen is $32gm/mol$ and that of ozone is $48gm/mol$ .
So, the rate of effusion is:
  $\dfrac{{{r_{{O_2}}}}}{{{r_{{O_3}}}}} = \sqrt {\dfrac{{48}}{{32}}} $
So, the ratio of effusion of ${O_2}$ and ${O_3}$ is $\sqrt {\dfrac{2}{3}} $ .

Note:
According to Boyle’s Law, the gases become denser at high pressure and the expansion in gas volume is due to decrease in atmospheric pressure. Rate of diffusion for gases equal to the volume of gas diffused per unit taken. Density is directly proportional to the molar mass. At constant temperature, pressure and volume is constant.