Question

127 ml of a certain gas diffuses at the same time as 100 ml of chlorine under the
same conditions, if the molecular weight of the gas is X, then the nearest integer value
of X is:

Answer 1

Hint: Graham's Law of Diffusion states that:
>The rate of diffusion of gases is inversely proportional to the square root of their densities.
\[Rat{{e}_{diffusion}}\propto \frac{1}{\sqrt{density}}\]
>Since volumes of different gases contain the same number of particles, the number of moles per litre at a given T and P is constant. Therefore, the density of a gas is directly proportional to its molar mass (M).
\[Rat{{e}_{diffusion}}\propto \frac{1}{\sqrt{Molar\,mass}}\]

Complete answer:
From above discussion we can finally write the equation for Graham's Law of Diffusion as:
\[\frac{{{V}_{1}}}{{{V}_{2}}}=\sqrt{\frac{{{d}_{2}}}{{{d}_{1}}}}=\sqrt{\frac{{{M}_{2}}}{{{M}_{1}}}}\]
     where, \[{{V}_{1}}\] and \[{{V}_{2}}\] is the volume of the two gases respectively.
          \[{{d}_{1}}\] and \[{{d}_{2}}\] is the density of the two gases respectively.
          \[{{M}_{1}}\] and \[{{M}_{2}}\] is the molar mass of the two gases respectively.

Given volume for Chlorine is 100ml and for unknown gas is 127ml.
We know the molecular weight of chlorine gas (\[C{{l}_{2}}\]) is 71amu and for the unknown gas, let’s take it X as given in the question.
Putting the values in the equation, we get:
\[\frac{127}{100}=\sqrt{\frac{71}{X}}\]
Taking squares on both the sides.
\[{{\left( \frac{127}{100} \right)}^{2}}={{\left( \sqrt{\frac{71}{X}} \right)}^{2}}\]
\[{{\left( 1.27 \right)}^{2}}=\frac{71}{X}\]
Now on rearranging the equation, we get:
\[X=\frac{71}{1.6129}=44.02\]
We can write it in the nearest integer value as 44.

Therefore, the molecular mass of the unknown gas is, X=44amu.

Note: Molar mass and molecular mass are two different concepts, though often interchangeable.
>The mass of one mole of a substance in grams is called molar mass of that substance.
>Molecular mass is calculated by taking the sum of all the atomic masses of the elements present in the molecule.
>In this case, we have calculated the molecular weight of Chlorine as 71amu, which is also similar to its molar mass of 71g/mol.