In the equilibrium `2SO_(2)(g)+O_(2)(g)hArr2SO_(3)(g)` , the partial pressure of `SO_(2),O_(2)` and `SO_(3)` are 0.662,0.10 and 0.331 atm respectively . What should be the partial pressure of Oxygen so that the equilibrium concentrations of `SO_(3)` are equal ?

Options

A. (a) `0.4atm`
B. (b) `1.0atm`
C. (c) `0.8atm`
D. (d) `0.25 atm`

Answer

To solve the problem step by step, we will follow the equilibrium expression and the provided data.

### Step 1: Write the equilibrium reaction
The equilibrium reaction is given as:
\[ 2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g) \]

### Step 2: Write the expression for the equilibrium constant \( K_p \)
The equilibrium constant \( K_p \) for the reaction can be expressed in terms of the partial pressures of the gases involved:
\[ K_p = \frac{(P_{SO_3})^2}{(P_{SO_2})^2 \cdot (P_{O_2})} \]

### Step 3: Substitute the known values into the \( K_p \) expression
Given the partial pressures:
- \( P_{SO_2} = 0.662 \, \text{atm} \)
- \( P_{O_2} = 0.10 \, \text{atm} \)
- \( P_{SO_3} = 0.331 \, \text{atm} \)

Substituting these values into the \( K_p \) expression:
\[ K_p = \frac{(0.331)^2}{(0.662)^2 \cdot (0.10)} \]

### Step 4: Calculate \( K_p \)
Calculating \( K_p \):
\[ K_p = \frac{0.109561}{0.438244 \cdot 0.10} = \frac{0.109561}{0.0438244} \approx 2.495 \]

### Step 5: Set up the new equilibrium condition
We need to find the partial pressure of \( O_2 \) such that the partial pressures of \( SO_3 \) and \( SO_2 \) are equal. Let:
- \( P_{SO_3} = P_{SO_2} = x \)

### Step 6: Write the new equilibrium expression
Substituting \( x \) into the \( K_p \) expression:
\[ K_p = \frac{x^2}{x^2 \cdot P_{O_2}} \]

### Step 7: Simplify the equation
Since \( x^2 \) cancels out:
\[ K_p = \frac{1}{P_{O_2}} \]

### Step 8: Solve for \( P_{O_2} \)
Rearranging gives:
\[ P_{O_2} = \frac{1}{K_p} \]
Substituting the value of \( K_p \):
\[ P_{O_2} = \frac{1}{2.495} \approx 0.401 \]

### Step 9: Final answer
Thus, the required partial pressure of \( O_2 \) is approximately:
\[ P_{O_2} \approx 0.4 \, \text{atm} \]