The reaction for which `K_(P)=K_(C)` is satisfied

Options

A. `A(g)+2B(g)hArr3C(g)`
B. `A(s)hArrB(g)`
C. `2A(g)hArrB(g)+C(g)`
D. `A(s)+B(g)toC(s)+2D(g)`

Answer

To determine the reaction for which \( K_p = K_c \), we need to understand the relationship between these two equilibrium constants. The formula that relates \( K_p \) and \( K_c \) is:

\[ K_p = K_c \cdot R T^{\Delta n} \]

Where:
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin,
- \( \Delta n \) is the change in the number of moles of gas, calculated as \( n_P - n_R \) (number of moles of gaseous products minus number of moles of gaseous reactants).

For \( K_p \) to equal \( K_c \), the term \( R T^{\Delta n} \) must equal 1, which occurs when \( \Delta n = 0 \).

### Step-by-Step Solution:

1. **Identify the reactions and their states**: We need to analyze the given reactions and identify the states of the reactants and products (gaseous, liquid, solid).

2. **Calculate \( \Delta n \) for each reaction**:
- For each reaction, count the number of moles of gaseous products (\( n_P \)) and the number of moles of gaseous reactants (\( n_R \)).
- Use the formula \( \Delta n = n_P - n_R \).

3. **Check if \( \Delta n = 0 \)**:
- If \( \Delta n = 0 \), then \( K_p = K_c \).

4. **Evaluate the options**:
- For each reaction provided in the options, perform the calculations and determine if \( K_p = K_c \).

### Example Reactions:

1. **First Reaction**: \( A + 2B \rightleftharpoons 3C \)
- \( n_P = 3 \) (3 moles of C)
- \( n_R = 1 + 2 = 3 \) (1 mole of A and 2 moles of B)
- \( \Delta n = 3 - 3 = 0 \) → \( K_p = K_c \)

2. **Second Reaction**: \( C (s) \rightleftharpoons D (g) \)
- \( n_P = 1 \) (1 mole of D)
- \( n_R = 0 \) (C is solid, not counted)
- \( \Delta n = 1 - 0 = 1 \) → \( K_p \neq K_c \)

3. **Third Reaction**: \( 2A (g) \rightleftharpoons B (g) + C (g) \)
- \( n_P = 2 \) (1 mole of B and 1 mole of C)
- \( n_R = 2 \) (2 moles of A)
- \( \Delta n = 2 - 2 = 0 \) → \( K_p = K_c \)

4. **Fourth Reaction**: \( 2D (g) \rightleftharpoons E (s) \)
- \( n_P = 0 \) (E is solid, not counted)
- \( n_R = 2 \) (2 moles of D)
- \( \Delta n = 0 - 2 = -2 \) → \( K_p \neq K_c \)

### Conclusion:
The reactions for which \( K_p = K_c \) are the first and third reactions.