Which of the following set contains same number of molecules?
(a) 1 gram `O_2`, 2 gram `SO_2`
(b) 1 gram `CO_2`, 1 gram `N_2O`
(c ) 1 gram CO, 1 gram `N_2`
(d) 1 gram `O_2`, 1 gram `O_3`

Options

A. (a), (b) and (c )
B. (a) and (b)
C. (c ) and (d)
D. (a), (b), (c ) and (d)

Answer

To determine which of the given sets contains the same number of molecules, we need to calculate the number of moles for each substance in the options provided. The number of moles can be calculated using the formula:

\[ \text{Number of moles} = \frac{\text{Given weight (g)}}{\text{Molar mass (g/mol)}} \]

Next, we will calculate the number of moles for each option.

### Option A: 1 gram of \( O_2 \) and 2 grams of \( SO_2 \)

1. **Molar mass of \( O_2 \)**:
- Oxygen (O) = 16 g/mol
- Molar mass of \( O_2 = 2 \times 16 = 32 \) g/mol

2. **Moles of \( O_2 \)**:
\[ \text{Moles of } O_2 = \frac{1 \text{ g}}{32 \text{ g/mol}} = 0.03125 \text{ moles} \]

3. **Molar mass of \( SO_2 \)**:
- Sulfur (S) = 32 g/mol
- Oxygen (O) = 16 g/mol
- Molar mass of \( SO_2 = 32 + 2 \times 16 = 64 \) g/mol

4. **Moles of \( SO_2 \)**:
\[ \text{Moles of } SO_2 = \frac{2 \text{ g}}{64 \text{ g/mol}} = 0.03125 \text{ moles} \]

**Conclusion for Option A**: Both \( O_2 \) and \( SO_2 \) have the same number of moles.

### Option B: 1 gram of \( CO_2 \) and 1 gram of \( N_2O \)

1. **Molar mass of \( CO_2 \)**:
- Carbon (C) = 12 g/mol
- Oxygen (O) = 16 g/mol
- Molar mass of \( CO_2 = 12 + 2 \times 16 = 44 \) g/mol

2. **Moles of \( CO_2 \)**:
\[ \text{Moles of } CO_2 = \frac{1 \text{ g}}{44 \text{ g/mol}} = 0.02273 \text{ moles} \]

3. **Molar mass of \( N_2O \)**:
- Nitrogen (N) = 14 g/mol
- Oxygen (O) = 16 g/mol
- Molar mass of \( N_2O = 2 \times 14 + 16 = 44 \) g/mol

4. **Moles of \( N_2O \)**:
\[ \text{Moles of } N_2O = \frac{1 \text{ g}}{44 \text{ g/mol}} = 0.02273 \text{ moles} \]

**Conclusion for Option B**: Both \( CO_2 \) and \( N_2O \) have the same number of moles.

### Option C: 1 gram of \( CO \) and 1 gram of \( N_2 \)

1. **Molar mass of \( CO \)**:
- Carbon (C) = 12 g/mol
- Oxygen (O) = 16 g/mol
- Molar mass of \( CO = 12 + 16 = 28 \) g/mol

2. **Moles of \( CO \)**:
\[ \text{Moles of } CO = \frac{1 \text{ g}}{28 \text{ g/mol}} = 0.03571 \text{ moles} \]

3. **Molar mass of \( N_2 \)**:
- Nitrogen (N) = 14 g/mol
- Molar mass of \( N_2 = 2 \times 14 = 28 \) g/mol

4. **Moles of \( N_2 \)**:
\[ \text{Moles of } N_2 = \frac{1 \text{ g}}{28 \text{ g/mol}} = 0.03571 \text{ moles} \]

**Conclusion for Option C**: Both \( CO \) and \( N_2 \) have the same number of moles.

### Option D: 1 gram of \( O_2 \) and 1 gram of \( O_3 \)

1. **Molar mass of \( O_2 \)**:
- As calculated before, \( O_2 = 32 \) g/mol.

2. **Moles of \( O_2 \)**:
\[ \text{Moles of } O_2 = \frac{1 \text{ g}}{32 \text{ g/mol}} = 0.03125 \text{ moles} \]

3. **Molar mass of \( O_3 \)**:
- Molar mass of \( O_3 = 3 \times 16 = 48 \) g/mol.

4. **Moles of \( O_3 \)**:
\[ \text{Moles of } O_3 = \frac{1 \text{ g}}{48 \text{ g/mol}} = 0.02083 \text{ moles} \]

**Conclusion for Option D**: \( O_2 \) and \( O_3 \) do not have the same number of moles.

### Final Answer:
The sets that contain the same number of molecules are:
- Option A: \( O_2 \) and \( SO_2 \)
- Option B: \( CO_2 \) and \( N_2O \)
- Option C: \( CO \) and \( N_2 \)