For a chemical reaction of the type `AhArrB, K=2.0 and BhArrC, K=0.01`. Equilibrium constant for the reaction `2ChArr 2A` is

Options

A. 25
B. 50
C. 2500
D. `4xx10^(-4)`

Answer

To find the equilibrium constant for the reaction \( 2C \rightleftharpoons 2A \), we will use the given equilibrium constants for the reactions \( A \rightleftharpoons B \) and \( B \rightleftharpoons C \).

### Step-by-Step Solution:

1. **Identify the Given Reactions and Their Equilibrium Constants:**
- For the reaction \( A \rightleftharpoons B \), the equilibrium constant \( K_1 = 2.0 \).
- For the reaction \( B \rightleftharpoons C \), the equilibrium constant \( K_2 = 0.01 \).

2. **Multiply the First Reaction by 2:**
- When we multiply the reaction \( A \rightleftharpoons B \) by 2, we get:
\[ 2A \rightleftharpoons 2B \]
- The new equilibrium constant \( K_1' \) becomes:
\[ K_1' = K_1^2 = (2.0)^2 = 4.0 \]

3. **Multiply the Second Reaction by 2:**
- For the reaction \( B \rightleftharpoons C \), multiplying by 2 gives:
\[ 2B \rightleftharpoons 2C \]
- The new equilibrium constant \( K_2' \) becomes:
\[ K_2' = K_2^2 = (0.01)^2 = 0.0001 = \frac{1}{10000} \]

4. **Reverse the Reactions:**
- Now, we need to reverse both reactions:
- From \( 2A \rightleftharpoons 2B \) to \( 2B \rightleftharpoons 2A \):
\[ K_{rev1} = \frac{1}{K_1'} = \frac{1}{4.0} = 0.25 \]
- From \( 2B \rightleftharpoons 2C \) to \( 2C \rightleftharpoons 2B \):
\[ K_{rev2} = \frac{1}{K_2'} = 10000 \]

5. **Add the Reversed Reactions:**
- Now we add the two reversed reactions:
\[ 2C \rightleftharpoons 2B \quad \text{and} \quad 2B \rightleftharpoons 2A \]
- The \( 2B \) cancels out, resulting in:
\[ 2C \rightleftharpoons 2A \]

6. **Calculate the Overall Equilibrium Constant:**
- The overall equilibrium constant \( K \) for the reaction \( 2C \rightleftharpoons 2A \) is the product of the equilibrium constants of the two reversed reactions:
\[ K = K_{rev1} \times K_{rev2} = 0.25 \times 10000 = 2500 \]

### Final Answer:
The equilibrium constant for the reaction \( 2C \rightleftharpoons 2A \) is \( K = 2500 \).

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