`{:(List - I("Reaction"), List -II((K_(p))/(K_(c)))),((A)A_(2)(g) + 3B_(2)(g) hArr 2AB_(3)(g),(P)(RT)^(-2)),((B) A_(2) (g)+ B_(2) (g) hArr 2 AB(g) ,(Q) (RT)^(@)),((C) A(s) + (3)/(2) B_(2) (g) hArr AB_(3)(g),(R) (RT)^(1//2)),((D) AB_(2) (g) hArr AB(g) +(1)/(2) B_(2)(g),(S) (RT)^(-1//2)):}`

Answer

To solve the question, we need to match each reaction with its corresponding \( \frac{K_p}{K_c} \) value based on the relationship between \( K_p \) and \( K_c \) for a given reaction. The relationship is given by:

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

where \( \Delta n \) is the change in the number of moles of gas, calculated as:

\[ \Delta n = \text{(moles of gaseous products)} - \text{(moles of gaseous reactants)} \]

Now, let's analyze each reaction step by step.

### Step 1: Analyze Reaction A
**Reaction:** \( A_2(g) + 3B_2(g) \rightleftharpoons 2AB_3(g) \)

- **Moles of Products:** 2 (from \( 2AB_3 \))
- **Moles of Reactants:** 1 (from \( A_2 \)) + 3 (from \( 3B_2 \)) = 4

\[ \Delta n = 2 - 4 = -2 \]

Thus,

\[ \frac{K_p}{K_c} = (RT)^{-2} \quad \text{(which corresponds to P)} \]

### Step 2: Analyze Reaction B
**Reaction:** \( A_2(g) + B_2(g) \rightleftharpoons 2AB(g) \)

- **Moles of Products:** 2 (from \( 2AB \))
- **Moles of Reactants:** 1 (from \( A_2 \)) + 1 (from \( B_2 \)) = 2

\[ \Delta n = 2 - 2 = 0 \]

Thus,

\[ \frac{K_p}{K_c} = (RT)^{0} \quad \text{(which corresponds to Q)} \]

### Step 3: Analyze Reaction C
**Reaction:** \( A(s) + \frac{3}{2}B_2(g) \rightleftharpoons AB_3(g) \)

- **Moles of Products:** 1 (from \( AB_3 \))
- **Moles of Reactants:** 0 (from solid \( A \)) + 1.5 (from \( \frac{3}{2}B_2 \)) = 1.5

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

Thus,

\[ \frac{K_p}{K_c} = (RT)^{-\frac{1}{2}} \quad \text{(which corresponds to S)} \]

### Step 4: Analyze Reaction D
**Reaction:** \( AB_2(g) \rightleftharpoons AB(g) + \frac{1}{2}B_2(g) \)

- **Moles of Products:** 1 (from \( AB \)) + 0.5 (from \( \frac{1}{2}B_2 \)) = 1.5
- **Moles of Reactants:** 1 (from \( AB_2 \))

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

Thus,

\[ \frac{K_p}{K_c} = (RT)^{\frac{1}{2}} \quad \text{(which corresponds to R)} \]

### Final Matching
Now we can summarize the matches:
- A corresponds to P
- B corresponds to Q
- C corresponds to S
- D corresponds to R

### Final Answer
- A - P
- B - Q
- C - S
- D - R