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Question: The gas which diffuses twice as quickly as \[S{O_2}\] is \[ A.\;C{H_4} \\\ B.\;{H_2} \\\ ...

The gas which diffuses twice as quickly as SO2S{O_2} is

A.  CH4 B.  H2 C.  O2 D.  He  A.\;C{H_4} \\\ B.\;{H_2} \\\ C.\;{O_2} \\\ D.\;He \\\
Explanation

Solution

The question is related to diffusion of gases. Diffusion is the process where a particle moves from higher concentration to lower concentration. It is basically the property that gases through liquids also diffuse to some extent. This movement is related to the weight and speed of the particle being diffused.

Complete answer:
As I have mentioned that it is related to weight and speed, the relation between rate of diffusion and its mass was given by a chemist Thomas Graham which is named as Graham’s law of diffusion. This law states that the rate of diffusion of any substance is inversely proportional to the square root of its molecular weight. The formula is as follows:

r1M r1r2=M2M1  \Rightarrow r \propto \dfrac{1}{{\sqrt M }} \\\ \Rightarrow \dfrac{{{r_1}}}{{{r_2}}} = \sqrt {\dfrac{{{M_2}}}{{{M_1}}}} \\\

Where ‘r’ is the rate of diffusion and ‘M’ is the molecular weight of the gas. In the question it is given that one of the gases diffusion rate is twice when compared to SO2S{O_2} . So, let us assume the other gas as ‘x’ . The diffusion rate of SO2S{O_2} and gas ‘x’ is in the ratio 1:2. Now by substituting the values in the formula ,we get

rSO2rx=MxMSO2 12=Mx64  \Rightarrow \dfrac{{{r_{S{O_2}}}}}{{{r_x}}} = \sqrt {\dfrac{{{M_x}}}{{{M_{S{O_2}}}}}} \\\ \Rightarrow \dfrac{1}{2} = \sqrt {\dfrac{{{M_x}}}{{64}}} \\\

Squaring on both sides, we get

14=Mx64 644=Mx Mx=16  \Rightarrow \dfrac{1}{4} = \dfrac{{{M_x}}}{{64}} \\\ \Rightarrow \dfrac{{64}}{4} = {M_x} \\\ \Rightarrow {M_x} = 16 \\\

The molecular weight of gas which we assumed as ‘x’ should be 16. Now, let us go through the options and find the molecular weights of given gases. The Molecular weight of CH4C{H_4} is 16, the molecular weight of H2{H_2} is 2, the molecular weight of O2{O_2} is 32 and the molecular weight of He is 4. So the correct option is A.

Note:
There is a theory called Kinetic molecular theory, which states that the gas particles are always in a constant motion moving in different directions and at random speed. Apart from the molar mass, gases movement is also affected by its size and this concept is referred to as effusion.