The equilibrium constant of a reaction is `20.0`. At equilibrium, the reate constant of forward reaction is `10.0`. The rate constant for backward reaction is :

Options

A. 0.5
B. `2.0`
C. `10.0`
D. `200.0`

Answer

To find the rate constant for the backward reaction, we can use the relationship between the equilibrium constant (K) and the rate constants of the forward (k_f) and backward (k_b) reactions.

### Step-by-Step Solution:

1. **Understanding the Equilibrium Constant**:
The equilibrium constant (K) for a reaction at equilibrium is defined as the ratio of the rate constant of the forward reaction (k_f) to the rate constant of the backward reaction (k_b). This can be expressed mathematically as:
\[ K = \frac{k_f}{k_b} \]

2. **Substituting Known Values**:
We are given:
- Equilibrium constant, \( K = 20.0 \)
- Rate constant of the forward reaction, \( k_f = 10.0 \)

We can substitute these values into the equation:
\[ 20.0 = \frac{10.0}{k_b} \]

3. **Rearranging the Equation**:
To find \( k_b \), we can rearrange the equation:
\[ k_b = \frac{10.0}{20.0} \]

4. **Calculating \( k_b \)**:
Now, we perform the calculation:
\[ k_b = \frac{10.0}{20.0} = 0.5 \]

5. **Final Answer**:
The rate constant for the backward reaction is:
\[ k_b = 0.5 \]

### Summary of the Solution:
The rate constant for the backward reaction is \( 0.5 \).

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