The equilibrium constant for the given reaction:
`SO_(3(g))hArrSO_(2(g))+1//2O_(2(g))`, `(K_(c)=4.9xx10^(-2))`
The value of `K_(c)` for the reaction:
`2SO_(2(g))+O_(2(g))hArr2SO_(3(g))`, will be :

Options

A. `416`
B. `2.40xx10^(-3)`
C. `9.8xx10^(-2)`
D. `4.9xx10^(-2)`

Explanation

To determine the equilibrium constant \( K_c \) for the reaction

\[ 2 \text{SO}_2(g) + \text{O}_2(g) \rightleftharpoons 2 \text{SO}_3(g) \]

given the equilibrium constant for the reaction

\[ \text{SO}_3(g) \rightleftharpoons \text{SO}_2(g) + \frac{1}{2} \text{O}_2(g) \]

is \( K_c = 4.9 \times 10^{-2} \), we can follow these steps:

### Step 1: Write the expression for the first reaction
The equilibrium constant \( K_c \) for the first reaction is given by:

\[ K_c = \frac{[\text{SO}_2][\text{O}_2]^{1/2}}{[\text{SO}_3]} \]

### Step 2: Reverse the first reaction
When we reverse a reaction, the equilibrium constant for the reverse reaction is the reciprocal of the original equilibrium constant. Therefore, for the reaction:

\[ \text{SO}_2(g) + \frac{1}{2} \text{O}_2(g) \rightleftharpoons \text{SO}_3(g) \]

the equilibrium constant \( K_c' \) is:

\[ K_c' = \frac{1}{K_c} = \frac{1}{4.9 \times 10^{-2}} \]

### Step 3: Square the equilibrium constant
The second reaction involves doubling the coefficients of the products and reactants:

\[ 2 \text{SO}_2(g) + \text{O}_2(g) \rightleftharpoons 2 \text{SO}_3(g) \]

When we multiply the coefficients of a balanced equation by a factor \( n \), the equilibrium constant for the new reaction is given by:

\[ K_c'' = (K_c')^n \]

In this case, \( n = 2 \):

\[ K_c'' = (K_c')^2 = \left(\frac{1}{4.9 \times 10^{-2}}\right)^2 \]

### Step 4: Calculate \( K_c'' \)
Now we calculate \( K_c'' \):

\[ K_c'' = \left(\frac{1}{4.9 \times 10^{-2}}\right)^2 = \frac{1}{(4.9 \times 10^{-2})^2} \]

Calculating \( (4.9 \times 10^{-2})^2 \):

\[ (4.9 \times 10^{-2})^2 = 24.01 \times 10^{-4} = 2.401 \times 10^{-3} \]

Thus,

\[ K_c'' = \frac{1}{2.401 \times 10^{-3}} \approx 416.49 \]

### Final Answer
Therefore, the value of \( K_c \) for the reaction

\[ 2 \text{SO}_2(g) + \text{O}_2(g) \rightleftharpoons 2 \text{SO}_3(g) \]

is approximately:

\[ K_c \approx 416.5 \]

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