If the ration of the masses of `SO_(3)` and `O_(2)` gases confined in a vessel is `1 : 1` , then the ratio of their partial pressure would be

Options

A. `1 : 2`
B. `2 : 5`
C. `2 : 1`
D. `1 : 2`

Explanation

To solve the problem of finding the ratio of the partial pressures of SO₃ and O₂ gases confined in a vessel with a mass ratio of 1:1, we can follow these steps:

### Step 1: Define the Masses
Let the mass of SO₃ be \( m_{SO_3} = x \) grams and the mass of O₂ be \( m_{O_2} = x \) grams, since the ratio of their masses is given as 1:1.

### Step 2: Calculate Moles of Each Gas
Using the molar masses:
- Molar mass of SO₃ = 80 g/mol
- Molar mass of O₂ = 32 g/mol

Calculate the number of moles for each gas:
- Moles of SO₃:
\[ n_{SO_3} = \frac{m_{SO_3}}{M_{SO_3}} = \frac{x}{80} \]
- Moles of O₂:
\[ n_{O_2} = \frac{m_{O_2}}{M_{O_2}} = \frac{x}{32} \]

### Step 3: Calculate Total Moles
Now, we can find the total number of moles in the vessel:
\[ n_{total} = n_{SO_3} + n_{O_2} = \frac{x}{80} + \frac{x}{32} \]

To simplify this, we need a common denominator:
\[ n_{total} = \frac{x}{80} + \frac{2.5x}{80} = \frac{3.5x}{80} = \frac{7x}{160} \]

### Step 4: Calculate Mole Fractions
Now, we can calculate the mole fractions of each gas:
- Mole fraction of SO₃:
\[ \chi_{SO_3} = \frac{n_{SO_3}}{n_{total}} = \frac{\frac{x}{80}}{\frac{7x}{160}} = \frac{160}{560} = \frac{2}{7} \]
- Mole fraction of O₂:
\[ \chi_{O_2} = \frac{n_{O_2}}{n_{total}} = \frac{\frac{x}{32}}{\frac{7x}{160}} = \frac{160}{224} = \frac{5}{7} \]

### Step 5: Calculate Partial Pressures
Using Dalton's Law of Partial Pressures, the partial pressures can be calculated as:
- Partial pressure of SO₃:
\[ P_{SO_3} = \chi_{SO_3} \cdot P_{total} = \frac{2}{7} P \]
- Partial pressure of O₂:
\[ P_{O_2} = \chi_{O_2} \cdot P_{total} = \frac{5}{7} P \]

### Step 6: Find the Ratio of Partial Pressures
Now, we can find the ratio of the partial pressures:
\[ \frac{P_{SO_3}}{P_{O_2}} = \frac{\frac{2}{7} P}{\frac{5}{7} P} = \frac{2}{5} \]

### Final Answer
Thus, the ratio of the partial pressures of SO₃ to O₂ is:
\[ \frac{P_{SO_3}}{P_{O_2}} = 2:5 \]