In a reaction `A+BhArrC+D` the rate constant of forward reaction & backward reaction is `K_(1)` and `K_(2)` then the equilibrium constant `(K)` for reaction is expressed as:

Options

A. `K_(C)=(K_(2))/(K_(1))`
B. `K_(C)=(K_(1))/(K_(2))`
C. `K_(C)=K_(1)xxK_(2)`
D. `K_(C)=K_(1)+K_(2)`

Answer

To find the equilibrium constant \( K \) for the reaction \( A + B \rightleftharpoons C + D \) with given rate constants \( K_1 \) for the forward reaction and \( K_2 \) for the backward reaction, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Reaction**:
The reaction given is:
\[ A + B \rightleftharpoons C + D \]
Here, \( A \) and \( B \) are reactants, while \( C \) and \( D \) are products.

2. **Understand Rate Constants**:
- The rate constant for the forward reaction (from reactants to products) is denoted as \( K_1 \).
- The rate constant for the backward reaction (from products back to reactants) is denoted as \( K_2 \).

3. **Define the Equilibrium Constant**:
The equilibrium constant \( K \) for a reaction at equilibrium is defined as the ratio of the rate of the forward reaction to the rate of the backward reaction:
\[ K = \frac{R_F}{R_B} \]
where \( R_F \) is the rate of the forward reaction and \( R_B \) is the rate of the backward reaction.

4. **Relate Rates to Rate Constants**:
At equilibrium, the rate of the forward reaction is equal to the rate of the backward reaction. Therefore, we can express the rates in terms of the rate constants:
\[ R_F = K_1 \cdot [A][B] \quad \text{(for forward reaction)} \]
\[ R_B = K_2 \cdot [C][D] \quad \text{(for backward reaction)} \]
However, at equilibrium, we can directly relate the rate constants to the equilibrium constant without needing to consider concentrations.

5. **Express the Equilibrium Constant**:
Since \( K \) is defined as the ratio of the forward rate constant to the backward rate constant, we can write:
\[ K = \frac{K_1}{K_2} \]

### Final Answer:
Thus, the equilibrium constant \( K \) for the reaction \( A + B \rightleftharpoons C + D \) is expressed as:
\[ K = \frac{K_1}{K_2} \]