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 Concept:
This problem is fundamentally about the Mole Concept and Avogadro's Hypothesis.
Avogadro's law states that equal volumes of all gases, under the same conditions of temperature and pressure, contain the same number of molecules.
A more direct way to express this is: one mole of any substance contains exactly the same number of particles (Avogadro's number, \(N_A \approx 6.022 \times 10^{23}\)).
Therefore, to find which pair of gas samples contains the same number of molecules, we simply need to find the pair that contains the same number of moles.
Even if the gases are different (like Carbon Dioxide and Nitrogen), if the mole count is identical, the molecule count is identical.
Key Formula or Approach:
The formula for calculating the number of moles (\(n\)) from a given mass is:
\[ n = \frac{\text{Given Mass (m)}}{\text{Molar Mass (M)}} \] First, we must calculate the molar masses of the gases involved:
1. Molar mass of \(CO_2\): Carbon (12) + 2 \(\times\) Oxygen (16) = \(12 + 32 = 44 \text{ g/mol}\).
2. Molar mass of \(N_2\): 2 \(\times\) Nitrogen (14) = \(28 \text{ g/mol}\).
Step 2: Detailed Explanation:
Let's evaluate each option by calculating the moles for both gases:
For Option (A):
Moles of \(CO_2 = \frac{11 \text{ g}}{44 \text{ g/mol}} = 0.25 \text{ mol}\)
Moles of \(N_2 = \frac{7 \text{ g}}{28 \text{ g/mol}} = 0.25 \text{ mol}\)
Since \(0.25 = 0.25\), both samples contain exactly the same number of molecules (\(0.25 \times N_A\)). This option is correct.
For Option (B):
Moles of \(CO_2 = \frac{44 \text{ g}}{44 \text{ g/mol}} = 1.0 \text{ mol}\)
Moles of \(N_2 = \frac{14 \text{ g}}{28 \text{ g/mol}} = 0.5 \text{ mol}\)
The mole counts are different, so the molecule counts are different.
For Option (C):
Moles of \(CO_2 = \frac{22 \text{ g}}{44 \text{ g/mol}} = 0.5 \text{ mol}\)
Moles of \(N_2 = \frac{28 \text{ g}}{28 \text{ g/mol}} = 1.0 \text{ mol}\)
The mole counts are different, so the molecule counts are different.
Since only Option (A) is correct, Option (D) "All the above" is logically eliminated.
The physics/chemistry principle here is that the number of molecules depends solely on the number of moles, regardless of the complexity or weight of the individual molecules.
Step 3: Final Answer:
By calculating the moles, we find that 11g of \(CO_2\) and 7g of \(N_2\) both represent 0.25 moles of gas.
Thus, they contain the same number of molecules. Option (A) is the correct answer.