Given that equilibrium constant for the reaction `2SO_(2)(g) + O_(2)(g)hArr2SO_(3)(g)` has a value of 278 at a particular temperature. What is the value of the equilibrium constant for the following reaction at the same temperature ? `SO_(3)(g)hArrSO_(2)(g) + (1)/(2)O_(2)(g)`

Options

A. `1.8xx10^(-3)`
B. `3.6xx10^(-3)`
C. `6.0xx10^(-2)`
D. `1.3xx10^(-5)`

Explanation

To find the equilibrium constant for the reaction \( SO_3(g) \rightleftharpoons SO_2(g) + \frac{1}{2} O_2(g) \), we will use the given equilibrium constant for the reaction \( 2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g) \) which is \( K_c = 278 \).

### Step-by-Step Solution:

1. **Write the given reaction and its equilibrium constant**:
The reaction we have is:
\[ 2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g) \]
with \( K_c = 278 \).

2. **Reverse the reaction**:
To find the equilibrium constant for the reaction \( SO_3(g) \rightleftharpoons SO_2(g) + \frac{1}{2} O_2(g) \), we first need to reverse the original reaction. When we reverse a reaction, the equilibrium constant becomes the reciprocal:
\[ 2SO_3(g) \rightleftharpoons 2SO_2(g) + O_2(g) \]
Therefore, the equilibrium constant for this reversed reaction is:
\[ K' = \frac{1}{K_c} = \frac{1}{278} \]

3. **Divide the reversed reaction by 2**:
Now, we need to divide the entire equation by 2 to match our target reaction:
\[ SO_3(g) \rightleftharpoons SO_2(g) + \frac{1}{2} O_2(g) \]
When we divide the reaction by 2, the equilibrium constant is raised to the power of \( \frac{1}{2} \):
\[ K'' = \sqrt{K'} = \sqrt{\frac{1}{278}} \]

4. **Calculate the equilibrium constant**:
Now we can compute \( K'' \):
\[ K'' = \frac{1}{\sqrt{278}} \]

5. **Estimate the value**:
To estimate \( \sqrt{278} \), we know that \( 16^2 = 256 \) and \( 17^2 = 289 \). Therefore, \( \sqrt{278} \) is between 16 and 17. We can approximate:
\[ \sqrt{278} \approx 16.7 \]
Thus,
\[ K'' \approx \frac{1}{16.7} \approx 0.06 \]

### Final Answer:
The equilibrium constant for the reaction \( SO_3(g) \rightleftharpoons SO_2(g) + \frac{1}{2} O_2(g) \) is approximately \( 0.06 \).

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