Question

Which of the following pairs of gaseous contains the same number of molecules

• A. 11g of $CO_2$ and 7g of $N_2$
• B. 44g of $CO_2$ and 14g of $N_2$
• C. 22g of $CO_2$ and 28g of $N_2$
• D. All the above pairs of gases

Hint:

To quickly compare molecules, just find the moles of each gas ($n = \frac{m}{M}$). If the mole values are identical, then the number of molecules must be identical, regardless of the gas type.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks which pair of gas samples contains the same total number of molecules. According to Avogadro's law, equal moles of any two gases contain the same number of molecules. Thus, we need to calculate the number of moles for each gas in the given options and find the pair where the moles are equal.

Step 2: Key Formula or Approach:
The number of moles ($n$) is calculated by dividing the given mass ($m$) by the molar mass ($M$) of the substance: \[ n = \frac{m}{M} \] The number of molecules ($N$) is given by: \[ N = n \times N_A \] where $N_A$ is Avogadro's number ($6.022 \times 10^{23} \text{ molecules/mol}$). If two samples have equal moles, they have equal numbers of molecules.
The molar mass of carbon dioxide ($CO_2$) is: \[ M(CO_2) = 12 + (2 \times 16) = 44 \text{ g/mol} \] The molar mass of nitrogen gas ($N_2$) is: \[ M(N_2) = 2 \times 14 = 28 \text{ g/mol} \]

Step 3: Detailed Explanation:

• Let us analyze each option by calculating the number of moles:

• For Option (A):
Moles of $CO_2 = \frac{11 \text{ g}}{44 \text{ g/mol}} = 0.25 \text{ moles}$
Moles of $N_2 = \frac{7 \text{ g}}{28 \text{ g/mol}} = 0.25 \text{ moles}$
Since both gases have exactly $0.25$ moles, they contain the same number of molecules ($0.25 \times N_A$). This option is correct.

• For Option (B):
Moles of $CO_2 = \frac{44 \text{ g}}{44 \text{ g/mol}} = 1.0 \text{ mole}$
Moles of $N_2 = \frac{14 \text{ g}}{28 \text{ g/mol}} = 0.5 \text{ moles}$
The moles are unequal, so the number of molecules is different.

• For Option (C):
Moles of $CO_2 = \frac{22 \text{ g}}{44 \text{ g/mol}} = 0.5 \text{ moles}$
Moles of $N_2 = \frac{28 \text{ g}}{28 \text{ g/mol}} = 1.0 \text{ mole}$
The moles are unequal, so the number of molecules is different.

Step 4: Final Answer:
The pair containing $11 \text{ g}$ of $CO_2$ and $7 \text{ g}$ of $N_2$ has equal moles ($0.25 \text{ mol}$ each) and therefore contains the same number of molecules.