In a reaction , `A + B rarr` Product, rate is doubled when the concentration of `B` is doubled, and rate increases by a factor of `8` when the concentration of both the reactants (A and B) are doubled, rate law for the reaction can be written as

Options

A. `rate =K[A][b]^(2)`
B. `rate = k [A]^(2)[B]^(2)`
C. `rate =K [A][B]`
D. `rate=k [A]^(2)[B]`

Explanation

To solve the problem, we need to determine the rate law for the reaction \( A + B \rightarrow \text{Product} \) based on the information provided about how the rate changes with the concentrations of the reactants.

### Step-by-Step Solution:

1. **Assume a Rate Law Expression:**
We start by assuming a general rate law for the reaction:
\[ \text{Rate} = k [A]^a [B]^b \]
where \( k \) is the rate constant, \( a \) is the order of the reaction with respect to \( A \), and \( b \) is the order of the reaction with respect to \( B \).

2. **Initial Rate Expression:**
Let the initial rate of the reaction be denoted as \( r_1 \):
\[ r_1 = k [A]^a [B]^b \]

3. **Doubling Concentration of B:**
When the concentration of \( B \) is doubled (keeping \( A \) constant), the new rate \( r_2 \) becomes:
\[ r_2 = k [A]^a [2B]^b = k [A]^a (2^b [B]^b) = 2^b r_1 \]
According to the problem, \( r_2 = 2r_1 \). Thus, we have:
\[ 2^b r_1 = 2r_1 \]
Dividing both sides by \( r_1 \) (assuming \( r_1 \neq 0 \)):
\[ 2^b = 2 \implies b = 1 \]

4. **Doubling Concentrations of A and B:**
Now, when both \( A \) and \( B \) are doubled, the new rate \( r_3 \) is given by:
\[ r_3 = k [2A]^a [2B]^b = k (2^a [A]^a) (2^b [B]^b) = 2^{a+b} r_1 \]
We know from the problem that \( r_3 = 8r_1 \). Thus, we have:
\[ 2^{a+b} r_1 = 8r_1 \]
Dividing both sides by \( r_1 \):
\[ 2^{a+b} = 8 \]
Since \( 8 = 2^3 \), we can equate the exponents:
\[ a + b = 3 \]

5. **Substituting the Value of b:**
We already found that \( b = 1 \). Now substituting this into the equation \( a + b = 3 \):
\[ a + 1 = 3 \implies a = 2 \]

6. **Final Rate Law Expression:**
Now that we have determined \( a \) and \( b \), we can write the rate law for the reaction:
\[ \text{Rate} = k [A]^2 [B]^1 \]
or simply:
\[ \text{Rate} = k [A]^2 [B] \]

### Conclusion:
The rate law for the reaction \( A + B \rightarrow \text{Product} \) is:
\[ \text{Rate} = k [A]^2 [B] \]