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. `0.4atm`
B. `1.0atm`
C. `0.8atm`
D. `0.25 atm`

Answer

To solve the problem, we need to determine the partial pressure of oxygen (O2) such that the equilibrium concentrations of SO2 and SO3 are equal in the reaction:

\[ 2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g) \]

Given the partial pressures:
- \( P_{SO_2} = 0.662 \, \text{atm} \)
- \( P_{O_2} = 0.10 \, \text{atm} \)
- \( P_{SO_3} = 0.331 \, \text{atm} \)

### Step 1: Write the expression for the equilibrium constant \( K_p \)

The equilibrium constant \( K_p \) for the reaction can be expressed as:

\[ K_p = \frac{(P_{SO_3})^2}{(P_{SO_2})^2 \cdot (P_{O_2})} \]

### Step 2: Substitute the known values into the \( K_p \) expression

Substituting the given partial pressures into the equation:

\[ K_p = \frac{(0.331)^2}{(0.662)^2 \cdot (0.10)} \]

### Step 3: Calculate \( K_p \)

Calculating the numerator:

\[ (0.331)^2 = 0.109561 \]

Calculating the denominator:

\[ (0.662)^2 = 0.438244 \quad \text{and} \quad 0.438244 \cdot 0.10 = 0.0438244 \]

Now substituting back to find \( K_p \):

\[ K_p = \frac{0.109561}{0.0438244} \approx 2.5 \]

### Step 4: Set up the condition for equal concentrations of \( SO_2 \) and \( SO_3 \)

If the equilibrium concentrations of \( SO_2 \) and \( SO_3 \) are equal, we can denote their partial pressures as \( x \). Thus, we have:

\[ P_{SO_2} = P_{SO_3} = x \]

### Step 5: Substitute into the \( K_p \) expression

Substituting \( x \) into the \( K_p \) expression gives:

\[ K_p = \frac{x^2}{x^2 \cdot P_{O_2}} = \frac{1}{P_{O_2}} \]

### Step 6: Solve for \( P_{O_2} \)

Equating this to the previously calculated \( K_p \):

\[ 2.5 = \frac{1}{P_{O_2}} \]

Rearranging gives:

\[ P_{O_2} = \frac{1}{2.5} = 0.4 \, \text{atm} \]

### Conclusion

The required partial pressure of oxygen (O2) to ensure that the equilibrium concentrations of SO2 and SO3 are equal is:

\[ \boxed{0.4 \, \text{atm}} \]