The relation between `K_(P) and K_(C)` for the reaction `A(g)+B(g) hArr C(g)+2D(g)` is -

Options

A. `K_(P)=K_(C)[RT]^(-1)`
B. `K_(P).K_(C^(-1))=RT`
C. `K_(C)K_(P^(-1))=RT`
D. `K_(P)=K_(C)[RT]^(3)`

Answer

To find the relation between \( K_P \) and \( K_C \) for the reaction \( A(g) + B(g) \rightleftharpoons C(g) + 2D(g) \), we can follow these steps:

### Step 1: Identify the Reaction
The given reaction is:
\[ A(g) + B(g) \rightleftharpoons C(g) + 2D(g) \]

### Step 2: Determine the Change in Moles of Gas (ΔN_g)
To find \( \Delta N_g \), we need to calculate the difference between the total moles of gaseous products and the total moles of gaseous reactants.

- **Reactants**:
- \( A(g) \): 1 mole
- \( B(g) \): 1 mole
- Total moles of reactants = \( 1 + 1 = 2 \)

- **Products**:
- \( C(g) \): 1 mole
- \( D(g) \): 2 moles
- Total moles of products = \( 1 + 2 = 3 \)

Now, we can calculate \( \Delta N_g \):
\[ \Delta N_g = \text{Total moles of products} - \text{Total moles of reactants} = 3 - 2 = 1 \]

### Step 3: Use the Relation Between \( K_P \) and \( K_C \)
The relation between \( K_P \) and \( K_C \) is given by the formula:
\[ K_P = K_C (RT)^{\Delta N_g} \]
where:
- \( R \) is the universal gas constant
- \( T \) is the temperature in Kelvin

### Step 4: Substitute \( \Delta N_g \) into the Formula
Since we found \( \Delta N_g = 1 \), we can substitute this value into the equation:
\[ K_P = K_C (RT)^{1} \]
This simplifies to:
\[ K_P = K_C \cdot RT \]

### Conclusion
The relation between \( K_P \) and \( K_C \) for the given reaction is:
\[ K_P = K_C \cdot RT \]

### Final Answer
Thus, the answer is \( K_P = K_C RT \).

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