Question

Using Graham's law of effusion, if $ C{O_2} $ takes 32 sec to effuse, how long will hydrogen take?

Answer 1

Hint: We are already provided with the compound. We know the Graham's law states that the rate of diffusion or effusion of a gas is inversely proportional to the square root of its molar mass. We shall use the ratio to find out the effusion time of hydrogen.

Formula used: $ {\text{rate of effusion}} \propto \dfrac{{\text{1}}}{{\sqrt {{\text{molar mass}}} }} $

Complete step by step solution:
We already know that Graham's Law of Effusion tells you that the rate of effusion of a gas is inversely proportional to the square root of the mass of the particles of gas.
This can be written as:
 $ {\text{rate of effusion}} \propto \dfrac{{\text{1}}}{{\sqrt {{\text{molar mass}}} }} $
Essentially, the rate of effusion of a gas will depend on how massive its molecules are. The heavier the molecules, the slower the rate of effusion.
Likewise, the lighter the molecules, the faster the rate of effusion.
Right from the start, you should look at your two gases, carbon dioxide, $ C{O_2} $ , and hydrogen gas, $ {H_2} $ , and be able to predict that it will take less time for the hydrogen gas to effuse when compared with the carbon dioxide.
This is a valid prediction because the molecules of hydrogen gas are smaller, i.e. lighter, than the molecules of carbon dioxide.
 For the given compounds
 $ {\text{rat}}{{\text{e}}_{C{O_2}}} \propto \dfrac{1}{{\sqrt {44.01{\text{gmo}}{{\text{l}}^{{\text{ - 1}}}}} }} $
Similarly,
 $ {\text{rat}}{{\text{e}}_{{H_2}}} \propto \dfrac{1}{{\sqrt {2.016{\text{gmo}}{{\text{l}}^{{\text{ - 1}}}}} }} $
On finding their ratio:
 $ \dfrac{{{\text{rat}}{{\text{e}}_{{\text{C}}{{\text{O}}_{\text{2}}}}}}}{{{\text{rat}}{{\text{e}}_{{{\text{H}}_{\text{2}}}}}}} = \sqrt {\dfrac{{2.016}}{{44.01}}} = 0.214 $
As predicted, hydrogen gas will effuse at a faster rate than carbon dioxide.
Therefore, if it takes 32 s for carbon dioxide to effuse, and hydrogen effuses $ 4.67 $ times faster:
 $ {{\text{t}}_{{H_2}}} = \dfrac{{32}}{{4.67}} = 6.85{\text{s}} $
So, the time taken by hydrogen is $ 6.85 $ seconds.

Additional information:
We should know that the diffusion is faster at higher temperatures because the gas molecules have greater kinetic energy. Effusion refers to the movement of gas particles through a small hole. Graham's Law states that the effusion rate of a gas is inversely proportional to the square root of the mass of its particles.

Note:
We already know that the gases with different densities can be separated using Graham's law. It is also helpful in determining the molar mass of unknown gases by comparing the rate of diffusion of unknown gas to known gas. We can separate the isotopes of an element using Graham's law. A common example is enriching uranium from its isotope.