Which of the following has the smallest number of molecules?

The correct option is D

$0.1\mathrm{mole}$ of ${\mathrm{CO}}_2$ gas

Explanation for correct options

(D) $0.1\mathrm{mole}$ of ${\mathrm{CO}}_2$ gas

Step 1: Formula used to calculate the number of molecules

• $\mathrm{N}=\mathrm{n}\times {\mathrm{N}}_{\mathrm{A}}$
• $\mathrm{N}$ is the number of molecules.
• $\mathrm{n}$ is the number of moles.
• ${\mathrm{N}}_{\mathrm{A}}$ is the Avogadro's number.
• Number of moles is as follows:

$\mathrm{n}=\frac{\mathrm{m}}{\mathrm{M}}$

• $\mathrm{m}$ is the mass.
• $\mathrm{M}$ is the molecular mass.

Step 2: Determine the number of molecules in $0.1\mathrm{mole}$ of ${\mathrm{CO}}_2$ gas

• The number of moles of carbon dioxide, $\left({\mathrm{CO}}_2\right),\mathrm{m}=0.1\mathrm{mol}$
• Calculate the number of molecules is as follows:

$\begin{array}{rcl}\mathrm{N}& =& \mathrm{n}\times {\mathrm{N}}_{\mathrm{A}}\\ & =& 0.1\times 6.022\times {10}^{23}\mathrm{molecule}\\ & =& 6.022\times {10}^{22}\mathrm{molecules}\end{array}$

Explanation for incorrect options

(A) $22\mathrm{g}$ of ${\mathrm{CO}}_2$ gas

• Mass of carbon dioxide, $\left({\mathrm{CO}}_2\right),\mathrm{m}=22\mathrm{g}$
• Molecular mass of carbon dioxide, $\mathrm{M}=44\mathrm{g}{\mathrm{mol}}^{-1}$
• Calculate the number of molecules is as follows:

$\begin{array}{rcl}\mathrm{N}& =& \frac{\mathrm{m}}{\mathrm{M}}\times {\mathrm{N}}_{\mathrm{A}}\\ & =& \frac{22}{44}\times 6.022\times {10}^{23}\mathrm{molecules}\\ & =& 3.011\times {10}^{23}\mathrm{molecules}\end{array}$

(B) $11.2\mathrm{L}$ of ${\mathrm{CO}}_2$ gas

• Volume of carbon dioxide, $\left({\mathrm{CO}}_2\right),\mathrm{V}=11.2\mathrm{L}$
• $22.4\mathrm{L}$ of ${\mathrm{CO}}_2$ gas at standard temperature pressure (STP) contains $1\mathrm{mole}$ of ${\mathrm{CO}}_2$ gas.
• $11.2\mathrm{L}$ of ${\mathrm{CO}}_2$ gas at standard temperature pressure (STP) contains $0.5\mathrm{mole}$ of ${\mathrm{CO}}_2$ gas.
• Calculate the number of molecules is as follows:

$\begin{array}{rcl}\mathrm{N}& =& \mathrm{n}\times {\mathrm{N}}_{\mathrm{A}}\\ & =& 0.5\times 6.022\times {10}^{23}\mathrm{molecule}\\ & =& 3.011\times {10}^{22}\mathrm{molecules}\end{array}$

(C) $22.4\times {10}^3\mathrm{mL}$ of ${\mathrm{CO}}_2$ gas

• Volume of carbon dioxide, $\left({\mathrm{CO}}_2\right),\mathrm{V}=22.4\times {10}^3\mathrm{mL}$
• $22.4\times {10}^3\mathrm{mL}=22.4\mathrm{L}$
• $22.4\mathrm{L}$ of ${\mathrm{CO}}_2$ gas at standard temperature pressure (STP) contains $1\mathrm{mole}$ of ${\mathrm{CO}}_2$ gas.
• Calculate the number of molecules is as follows:

$\begin{array}{rcl}\mathrm{N}& =& \mathrm{n}\times {\mathrm{N}}_{\mathrm{A}}\\ & =& 1\times 6.022\times {10}^{23}\mathrm{molecule}\\ & =& 6.022\times {10}^{23}\mathrm{molecule}\end{array}$

Therefore, the correct option is(D) $0.1\mathrm{mole}$ of ${\mathrm{CO}}_2$ gas.