For the reaction. `SO_(2(g)) + (1)/(2) O_(2(g)) hArr SO_(3(g)). If K_(p) = K_(c)(RT)^(x)`. when the symbols have usual meaning. the value of x is (assuming ideality)

Options

A. `-1`
B. `- (1)/(2)`
C. `+ (1)/(2)`
D. `+1`

Answer

To solve the problem, we need to find the value of \( x \) in the equation \( K_p = K_c (RT)^x \) for the reaction:

\[ SO_2(g) + \frac{1}{2} O_2(g) \rightleftharpoons SO_3(g) \]

### Step 1: Identify the reaction and the change in moles
The balanced chemical equation shows that 1 mole of \( SO_2 \) and \( \frac{1}{2} \) mole of \( O_2 \) react to form 1 mole of \( SO_3 \).

### Step 2: Calculate the change in moles (\( \Delta n \))
To find \( \Delta n \), we use the formula:

\[ \Delta n = n_{\text{products}} - n_{\text{reactants}} \]

- Moles of products: \( 1 \) (from \( SO_3 \))
- Moles of reactants: \( 1 + \frac{1}{2} = 1.5 \) (from \( SO_2 \) and \( O_2 \))

Thus,

\[ \Delta n = 1 - 1.5 = -0.5 \]

### Step 3: Relate \( \Delta n \) to \( x \)
From the relationship given in the question:

\[ K_p = K_c (RT)^{\Delta n} \]

We can substitute \( \Delta n \) into this equation:

\[ K_p = K_c (RT)^{-0.5} \]

### Step 4: Identify \( x \)
From the equation \( K_p = K_c (RT)^x \), we can see that:

\[ x = -0.5 \]

### Conclusion
Thus, the value of \( x \) is:

\[ \boxed{-\frac{1}{2}} \]