Two gaseous equilibria `SO_(2)(g)+(1)/(2)O_(2)(g)hArr SO_(3)(g) and 2SO_(3)(g) hArr 2SO_(2)(g)+O_(2)(g)` have equilibrium constants `K_(1) and K_(2)` respectively at 298 K. Which of the following relationships is correct?

Options

A. `K_(1)=K_(2)`
B. `K_(1)= sqrt(K_(2))`
C. `K_(1)= sqrt((1)/(K_(2)))`
D. `K_(2)=(1)/(K_(1))`

Answer

To solve the problem, we need to analyze the two given equilibria and their corresponding equilibrium constants.

### Step 1: Write the two equilibrium reactions and their constants
The two reactions are:
1. \( SO_2(g) + \frac{1}{2} O_2(g) \rightleftharpoons SO_3(g) \) with equilibrium constant \( K_1 \)
2. \( 2SO_3(g) \rightleftharpoons 2SO_2(g) + O_2(g) \) with equilibrium constant \( K_2 \)

### Step 2: Invert the second reaction to relate it to the first
To relate \( K_2 \) to \( K_1 \), we first need to invert the second reaction. When we invert a reaction, the equilibrium constant becomes the reciprocal of the original constant. Thus, inverting the second reaction gives:
\[ SO_3(g) \rightleftharpoons SO_2(g) + \frac{1}{2} O_2(g) \]
The new equilibrium constant for this reaction, which we will denote as \( K' \), is:
\[ K' = \frac{1}{K_2} \]

### Step 3: Multiply the inverted reaction by 2
Next, we need to multiply the inverted reaction by 2 to match the stoichiometry of the original second reaction:
\[ 2SO_3(g) \rightleftharpoons 2SO_2(g) + O_2(g) \]
When we multiply a reaction by a factor, the equilibrium constant is raised to the power of that factor. Therefore, the new equilibrium constant \( K'' \) for this multiplied reaction is:
\[ K'' = (K')^2 = \left(\frac{1}{K_2}\right)^2 = \frac{1}{K_2^2} \]

### Step 4: Relate \( K_1 \) and \( K_2 \)
Now, we can relate \( K_1 \) to \( K_2 \) using the fact that the equilibrium constant for the first reaction is equal to the equilibrium constant of the multiplied inverted reaction:
\[ K_1 = \frac{1}{K_2^2} \]

### Step 5: Rearranging the equation
From the equation \( K_1 = \frac{1}{K_2^2} \), we can rearrange it to find a relationship between \( K_1 \) and \( K_2 \):
\[ K_2^2 = \frac{1}{K_1} \]
Taking the square root of both sides gives:
\[ K_2 = \frac{1}{\sqrt{K_1}} \]

### Final Relationship
Thus, the relationship between \( K_1 \) and \( K_2 \) can be expressed as:
\[ K_1 = \frac{1}{K_2^{1/2}} \]
or equivalently:
\[ K_1 = \frac{1}{\sqrt{K_2}} \]

### Conclusion
The correct relationship is:
\[ K_1 = \frac{1}{\sqrt{K_2}} \]