Question

The vapour density of a mixture containing ${\text{N}}{{\text{O}}_{\text{2}}}$ and ${{\text{N}}_{\text{2}}}{{\text{O}}_{\text{4}}}$ is $38.3$ at $300{\text{ K}}$. The number of moles of ${\text{N}}{{\text{O}}_{\text{2}}}$ in $100{\text{ g}}$ of the mixture is approximate:
A) 0.44
B) 4.4
C) 33.4
D) 3.34

Answer 1

We are given that the total number of moles is 100. First calculate the molar mass of the mixture using the relation between vapour density and molar mass. Calculate the number of moles of each ${\text{N}}{{\text{O}}_{\text{2}}}$, ${{\text{N}}_{\text{2}}}{{\text{O}}_{\text{4}}}$ and the mixture of ${\text{N}}{{\text{O}}_{\text{2}}}$ and ${{\text{N}}_{\text{2}}}{{\text{O}}_{\text{4}}}$. Then calculate the mass of ${\text{N}}{{\text{O}}_{\text{2}}}$ from the number of moles.

Complete solution:
First we will calculate the molecular mass using the equation as follows:
${\text{Molecular mass}} = 2 \times {\text{Vapour density}}$
We are given that the vapour density of a mixture containing ${\text{N}}{{\text{O}}_{\text{2}}}$ and ${{\text{N}}_{\text{2}}}{{\text{O}}_{\text{4}}}$ is $38.3$. Thus,
${\text{Molecular mass}} = 2 \times {\text{38}}{\text{.3}}$
${\text{Molecular mass}} = 76.6$
Thus, the molecular mass of a mixture containing ${\text{N}}{{\text{O}}_{\text{2}}}$ and ${{\text{N}}_{\text{2}}}{{\text{O}}_{\text{4}}}$ is $76.6$.
We are given that the mass of a mixture containing ${\text{N}}{{\text{O}}_{\text{2}}}$ and ${{\text{N}}_{\text{2}}}{{\text{O}}_{\text{4}}}$ is $100{\text{ g}}$. Let the mass of ${\text{N}}{{\text{O}}_{\text{2}}}$ in the mixture be $x{\text{ g}}$. Thus,
${\text{Mass of }}{{\text{N}}_{\text{2}}}{{\text{O}}_{\text{4}}} = 100 - x$
Now, calculate the number of moles of ${\text{N}}{{\text{O}}_{\text{2}}}$, ${{\text{N}}_{\text{2}}}{{\text{O}}_{\text{4}}}$ and the mixture of ${\text{N}}{{\text{O}}_{\text{2}}}$ and ${{\text{N}}_{\text{2}}}{{\text{O}}_{\text{4}}}$ using the equation as follows:
${\text{Number of moles}} = \dfrac{{{\text{Mass}}}}{{{\text{Molar mass}}}}$
The molar mass of ${\text{N}}{{\text{O}}_{\text{2}}}$ is $46{\text{ g/mol}}$. Thus,
${\text{Number of moles of N}}{{\text{O}}_2} = \dfrac{x}{{{\text{46}}}}$
The molar mass of ${{\text{N}}_{\text{2}}}{{\text{O}}_{\text{4}}}$ is $92{\text{ g/mol}}$. Thus,
${\text{Number of moles of }}{{\text{N}}_{\text{2}}}{{\text{O}}_{\text{4}}} = \dfrac{{{\text{100}} - x}}{{{\text{92}}}}$
The molar mass of a mixture containing ${\text{N}}{{\text{O}}_{\text{2}}}$ and ${{\text{N}}_{\text{2}}}{{\text{O}}_{\text{4}}}$ is $76.6$. Thus,
${\text{Number of moles of mixture of N}}{{\text{O}}_2}{\text{ and }}{{\text{N}}_{\text{2}}}{{\text{O}}_{\text{4}}} = \dfrac{{100}}{{{\text{76}}}}$
Thus,
$\dfrac{x}{{{\text{46}}}} + \dfrac{{{\text{100}} - x}}{{{\text{92}}}} = \dfrac{{100}}{{{\text{76}}}}$
$x = 20.10{\text{ g}}$
Thus, the mass of ${\text{N}}{{\text{O}}_{\text{2}}}$ in the mixture is $20.10{\text{ g}}$.
Calculate the number of moles of ${\text{N}}{{\text{O}}_{\text{2}}}$ as follows:
${\text{Number of moles of N}}{{\text{O}}_2} = \dfrac{{20.10}}{{{\text{46}}}} = 0.44{\text{ mol}}$
Thus, the number of moles of ${\text{N}}{{\text{O}}_{\text{2}}}$ in $100{\text{ g}}$ of the mixture is 0.44.

Thus, the correct option is (A) 0.44.

Note: Vapour density is the density of vapours in relation to the density of hydrogen. Vapour density is mass of a certain volume of a substance divided by mass of the same volume of hydrogen. The relation between molar mass and density is a direct proportion

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