Question

A gaseous mixture contains ${\rm{C}}{{\rm{O}}_{\rm{2}}}$ (g) and ${{\rm{N}}_{\rm{2}}}{\rm{O}}\left( g \right)$ in 2:5 ratio by mass. The ratio of the number of molecules is:
A. 5:2
B. 2:5
C. 1:2
D. 5:4

Answer 1

Here, first we have to calculate the moles of ${\rm{C}}{{\rm{O}}_{\rm{2}}}$ (g) and ${{\rm{N}}_{\rm{2}}}{\rm{O}}\left( g \right)$ and then to find the number of molecules of both the gases we have to use the formula,
Number of moles = $\dfrac{{{\rm{Number}}\,{\rm{of}}\,{\rm{molecules}}}}{{{\rm{Avogadro's}}\,{\rm{number}}}}$

Complete step by step answer:
The ratio by mass of ${\rm{C}}{{\rm{O}}_{\rm{2}}}$ (g) and ${{\rm{N}}_{\rm{2}}}{\rm{O}}\left( g \right)$ is 2:5, that means mass of ${\rm{C}}{{\rm{O}}_{\rm{2}}}$ is 2x and mass of ${{\rm{N}}_{\rm{2}}}{\rm{O}}\left( g \right)$ is 5x.
Now, we have to calculate the number of moles of ${\rm{C}}{{\rm{O}}_{\rm{2}}}$ and ${{\rm{N}}_{\rm{2}}}{\rm{O}}\left( g \right)$.
The formula to calculate mole number is,
Number of moles = $\dfrac{{{\rm{Mass}}}}{{{\rm{Molar}}\,{\rm{Mass}}}}$
For ${\rm{C}}{{\rm{O}}_{\rm{2}}}$, mass is 2x and molar mass of ${\rm{C}}{{\rm{O}}_{\rm{2}}}$
is $44\,{\rm{g}}\,{\rm{mo}}{{\rm{l}}^{ - 1}}$.
Number of moles of ${\rm{C}}{{\rm{O}}_{\rm{2}}}$ = $\dfrac{{2x}}{{44}}$
For ${{\rm{N}}_{\rm{2}}}{\rm{O}}\left( g \right)$, mass is 5x and molar mass of ${{\rm{N}}_{\rm{2}}}{\rm{O}}\left( g \right)$ is $44\,{\rm{g}}\,{\rm{mo}}{{\rm{l}}^{ - 1}}$.
Number of moles of ${{\rm{N}}_{\rm{2}}}{\rm{O}}\left( g \right)$ = $\dfrac{{5x}}{{44}}$
Now, we have to calculate the number of molecules of each gas.
Number of molecules = Number of moles$\times$
Avogadro’s number
We know the Avogadro’s number is $6.022 \times {10^{23}}$.
For ${\rm{C}}{{\rm{O}}_{\rm{2}}}$ number of molecules = $\dfrac{{2x}}{{44}} \times 6.022 \times {10^{23}}$
For ${{\rm{N}}_{\rm{2}}}{\rm{O}}$ number of molecules = $\dfrac{{5x}}{{44}} \times 6.022 \times {10^{23}}$
So, the ratio of number of molecules of ${\rm{C}}{{\rm{O}}_{\rm{2}}}$ to number of molecules of ${{\rm{N}}_{\rm{2}}}{\rm{O}}$ is,
Ratio = $\dfrac{{\dfrac{{2x}}{{44}} \times 6.022 \times {{10}^{23}}}}{{\dfrac{{5x}}{{44}} \times 6.022 \times {{10}^{23}}}} = \dfrac{2}{5}$
Therefore, the ratio is the same as the ratio by mass given that is. 2:5

So, the correct answer is Option B.

Note: It is to be noted that the number $6.022 \times {10^{23}}$ is named in honor of the Italian physicist Amedeo Avogadro. The Avogadro's number aids in counting very small particles. Different kinds of particles, such as molecules, atoms, ions, electrons are representative particles. One mole of anything consists of $6.022 \times {10^{23}}$ representative particles. For example, one mole of oxygen consists of $6.022 \times {10^{23}}$ molecules of oxygen. Hence, the relation between mole and Avogadro’s number is 1 mol = $6.022 \times {10^{23}}$