For the reaction, `SO_(2)(g) + (1)/(2)O_(2)(g)hArrSO_(3) (g)`, If `K_(p) = K_(c) (RT)^(x)` where the symbols have usual meaning then, 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 determine the value of \( x \) in the equation \( K_p = K_c (RT)^x \) for the given reaction:

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

### Step-by-Step Solution:

1. **Identify the Reaction and Stoichiometry**:
- The balanced chemical equation is:
\[ SO_2(g) + \frac{1}{2}O_2(g) \rightleftharpoons SO_3(g) \]
- Here, the stoichiometric coefficients are:
- For \( SO_2 \): 1
- For \( O_2 \): \( \frac{1}{2} \)
- For \( SO_3 \): 1

2. **Calculate \( \Delta N_g \)**:
- \( \Delta N_g \) is defined as the change in the number of moles of gaseous products minus the change in the number of moles of gaseous reactants.
- For this reaction:
\[ \Delta N_g = \text{(moles of products)} - \text{(moles of reactants)} = 1 - \left(1 + \frac{1}{2}\right) \]
- Simplifying this gives:
\[ \Delta N_g = 1 - \left(1 + 0.5\right) = 1 - 1.5 = -0.5 \]

3. **Relate \( K_p \) and \( K_c \)**:
- The relationship between \( K_p \) and \( K_c \) is given by the formula:
\[ K_p = K_c (RT)^{\Delta N_g} \]
- Substituting \( \Delta N_g \) into this equation:
\[ K_p = K_c (RT)^{-0.5} \]

4. **Identify \( x \)**:
- From the equation \( K_p = K_c (RT)^x \), we can see that:
\[ x = \Delta N_g = -0.5 \]

5. **Final Answer**:
- Therefore, the value of \( x \) is:
\[ \boxed{-\frac{1}{2}} \]