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: Understand the Definitions
- \( K_p \) is the equilibrium constant in terms of partial pressures of the gases.
- \( K_c \) is the equilibrium constant in terms of molar concentrations of the gases.

### Step 2: Write the General Relation
The general relation between \( K_p \) and \( K_c \) is given by the formula:
\[ K_p = K_c \cdot R^T \cdot \Delta N \]
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 (moles of products - moles of reactants).

### Step 3: Calculate \( \Delta N \)
For the given reaction:
- Reactants: \( A(g) + B(g) \) → 2 moles of gas (1 mole of A + 1 mole of B)
- Products: \( C(g) + 2D(g) \) → 3 moles of gas (1 mole of C + 2 moles of D)

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

### Step 4: Substitute \( \Delta N \) into the Relation
Now substitute \( \Delta N \) into the relation:
\[ K_p = K_c \cdot R^T \cdot 1 \]
This simplifies to:
\[ K_p = K_c \cdot R^T \]

### Step 5: Conclusion
Thus, the relation between \( K_p \) and \( K_c \) for the reaction \( A(g) + B(g) \rightleftharpoons C(g) + 2D(g) \) is:
\[ K_p = K_c \cdot R^T \]