For the reaction : A+2B ? `AB_(2)`, the rate of forward reaction is `(dx)/(dt) =1xx10^(5) [A] [B]^(2)-1xx10^(4) [AB_(2)]`. The rate constants for forward and backward reactions are :

Options

A. `1xx10^(-5) L^(2) m^(-2) s^(-1), 1xx10^(4) sec`
B. `1xx10^(5) sec^(-1), 1xx10^(4) L^(2) m^(-2) s^(-1)`
C. `1xx10^(5) L^(2) m^(-2) s^(-1), 1xx10^(4) sec^(-1)`
D. `1xx10^(-5) L m^(-1) s^(-1), 1xx10^(-4) sec`

Answer

To solve the problem, we need to determine the rate constants for the forward and backward reactions from the given rate equation.

### Step-by-Step Solution:

1. **Understand the Reaction**: The reaction is given as:
\[ A + 2B \rightleftharpoons AB_2 \]

2. **Given Rate Equation**: The rate of the forward reaction is given by:
\[ \frac{dx}{dt} = 1 \times 10^5 [A][B]^2 - 1 \times 10^4 [AB_2] \]

3. **Identify the Rate of Forward Reaction**: The forward reaction rate can be expressed as:
\[ \text{Rate}_{\text{forward}} = k_f [A][B]^2 \]
where \( k_f \) is the rate constant for the forward reaction.

4. **Identify the Rate of Backward Reaction**: The backward reaction rate can be expressed as:
\[ \text{Rate}_{\text{backward}} = k_b [AB_2] \]
where \( k_b \) is the rate constant for the backward reaction.

5. **Set Up the Equation**: From the given rate equation, we can equate the forward and backward rates:
\[ k_f [A][B]^2 - k_b [AB_2] = 0 \]

6. **Comparing Coefficients**: From the rate equation:
\[ \frac{dx}{dt} = 1 \times 10^5 [A][B]^2 - 1 \times 10^4 [AB_2] \]
We can identify:
- \( k_f = 1 \times 10^5 \)
- \( k_b = 1 \times 10^4 \)

7. **Final Values**: Therefore, the rate constants for the forward and backward reactions are:
\[ k_f = 1 \times 10^5 \, \text{mol}^{-1} \, \text{L}^2 \, \text{s}^{-1} \]
\[ k_b = 1 \times 10^4 \, \text{mol}^{-1} \, \text{L}^2 \, \text{s}^{-1} \]

### Summary of the Solution:
- The rate constant for the forward reaction \( k_f \) is \( 1 \times 10^5 \).
- The rate constant for the backward reaction \( k_b \) is \( 1 \times 10^4 \).

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