A non-reacting gaseous mixture contains SO_2 and SO_3 in the mass ratio of 1 : 5. Find the ratio of the number of molecules

Options

A. 1 : 1
B. 4 : 5
C. 1 : 4
D. 1 : 5

Answer

To solve the problem of finding the ratio of the number of molecules of SO₂ and SO₃ in a gaseous mixture with a mass ratio of 1:5, we can follow these steps:

### Step 1: Define the Masses
Let the mass of SO₂ be \( x \) grams. According to the given mass ratio of SO₂ to SO₃ (1:5), the mass of SO₃ will be \( 5x \) grams.

### Step 2: Calculate Molar Masses
- The molar mass of SO₂:
- Sulfur (S) = 32 g/mol
- Oxygen (O) = 16 g/mol, and there are 2 oxygen atoms in SO₂.
- Molar mass of SO₂ = \( 32 + (16 \times 2) = 32 + 32 = 64 \) g/mol.

- The molar mass of SO₃:
- Sulfur (S) = 32 g/mol
- Oxygen (O) = 16 g/mol, and there are 3 oxygen atoms in SO₃.
- Molar mass of SO₃ = \( 32 + (16 \times 3) = 32 + 48 = 80 \) g/mol.

### Step 3: Calculate the Number of Moles
- Number of moles of SO₂:
\[ \text{Number of moles of SO₂} = \frac{\text{mass of SO₂}}{\text{molar mass of SO₂}} = \frac{x}{64} \]

- Number of moles of SO₃:
\[ \text{Number of moles of SO₃} = \frac{\text{mass of SO₃}}{\text{molar mass of SO₃}} = \frac{5x}{80} \]

### Step 4: Calculate the Number of Molecules
Using Avogadro's number (\( N_A = 6.022 \times 10^{23} \)):
- Number of molecules of SO₂:
\[ N_{SO₂} = \left(\frac{x}{64}\right) \times N_A \]

- Number of molecules of SO₃:
\[ N_{SO₃} = \left(\frac{5x}{80}\right) \times N_A \]

### Step 5: Find the Ratio of Number of Molecules
Now, we want to find the ratio \( \frac{N_{SO₂}}{N_{SO₃}} \):
\[ \frac{N_{SO₂}}{N_{SO₃}} = \frac{\left(\frac{x}{64}\right) \times N_A}{\left(\frac{5x}{80}\right) \times N_A} \]

The \( N_A \) and \( x \) cancel out:
\[ \frac{N_{SO₂}}{N_{SO₃}} = \frac{\frac{1}{64}}{\frac{5}{80}} = \frac{80}{320} = \frac{1}{4} \]

### Conclusion
Thus, the ratio of the number of molecules of SO₂ to SO₃ is \( 1:4 \).