`2SO_(3)hArr2SO_(2) + O_(2)`. If `K_(c) = 100`, `alpha = 1`, half of the reaction is completed, the concentration of `SO_(3)` and `SO_(2)` are equal, the concentration of `O_(2)` is

Options

A. 0.001 M
B. 1/2 of `SO_(2)`
C. 2 times of `SO_(2)`
D. Data incomplete

Answer

To solve the problem, we need to analyze the given reaction and the information provided:

**Given Reaction:**
\[ 2SO_3 \rightleftharpoons 2SO_2 + O_2 \]

**Given Information:**
- \( K_c = 100 \)
- \( \alpha = 1 \) (degree of dissociation)
- At equilibrium, the concentration of \( SO_3 \) and \( SO_2 \) are equal.

### Step 1: Understanding the Reaction
The reaction shows that 2 moles of \( SO_3 \) produce 2 moles of \( SO_2 \) and 1 mole of \( O_2 \). This means that for every 2 moles of \( SO_3 \) that dissociate, 2 moles of \( SO_2 \) are produced along with 1 mole of \( O_2 \).

### Step 2: Setting Up Initial Concentrations
Let the initial concentration of \( SO_3 \) be \( [SO_3]_0 \). Since no specific values are given, we can assume \( [SO_3]_0 = x \).

### Step 3: Degree of Dissociation
Given \( \alpha = 1 \), this means that the reaction goes to completion. Therefore, all \( SO_3 \) will dissociate.

### Step 4: Equilibrium Concentrations
At equilibrium:
- The concentration of \( SO_3 \) will be \( 0 \) (since it fully dissociates).
- The concentration of \( SO_2 \) will be \( 2 \times \frac{x}{2} = x \) (for every 2 moles of \( SO_3 \), 2 moles of \( SO_2 \) are produced).
- The concentration of \( O_2 \) will be \( \frac{x}{2} \) (for every 2 moles of \( SO_3 \), 1 mole of \( O_2 \) is produced).

### Step 5: Equilibrium Expression
The equilibrium constant \( K_c \) is given by the expression:
\[ K_c = \frac{[SO_2]^2 [O_2]}{[SO_3]^2} \]

Substituting the equilibrium concentrations:
\[ K_c = \frac{(x)^2 \left(\frac{x}{2}\right)}{(0)^2} \]

However, since \( [SO_3] \) is 0 at completion, we cannot directly use this expression. Instead, we can analyze the relationship between the concentrations.

### Step 6: Concentration Relationship
Since it is stated that at equilibrium the concentrations of \( SO_3 \) and \( SO_2 \) are equal, we can set:
\[ [SO_2] = [SO_3] \]
This implies:
\[ x = 0 \] (which is not possible since \( SO_2 \) cannot have a concentration of zero).

### Step 7: Conclusion
Given that \( K_c \) is a large number (100), it indicates that the reaction favors the products. Therefore, the concentration of \( O_2 \) can be derived from the stoichiometry of the reaction, but without specific initial concentrations or volumes, we cannot calculate an exact value for \( [O_2] \).

### Final Answer
Since no specific values are provided, we cannot determine the exact concentration of \( O_2 \). Thus, the answer is that the data is incomplete, and we cannot find the concentration of \( O_2 \).

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