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]^(2)[B]`
B. Rate `= k[A][B]`
C. Rate `= k[A][B]^(2)`
D. Rate `= k[A]^(2)[B]^(2)`

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 varying concentrations of reactants.

### Step-by-Step Solution:

1. **Write the Rate Law Expression**:
The rate law for the reaction can be expressed as:
\[ \text{Rate} = k[A]^x[B]^y \]
where \( k \) is the rate constant, and \( x \) and \( y \) are the orders of the reaction with respect to reactants \( A \) and \( B \) respectively.

2. **Analyze the First Condition**:
According to the problem, when the concentration of \( B \) is doubled (from \( [B] \) to \( 2[B] \)), the rate doubles. This can be expressed mathematically as:
\[ 2R = k[A]^x[2B]^y \]
Simplifying this gives:
\[ 2R = k[A]^x \cdot 2^y[B]^y \]
Dividing both sides by the original rate \( R = k[A]^x[B]^y \):
\[ \frac{2R}{R} = \frac{k[A]^x \cdot 2^y[B]^y}{k[A]^x[B]^y} \]
This simplifies to:
\[ 2 = 2^y \]
From this, we can conclude that:
\[ y = 1 \]

3. **Analyze the Second Condition**:
The second condition states that when the concentrations of both \( A \) and \( B \) are doubled, the rate increases by a factor of 8. This can be expressed as:
\[ 8R = k[2A]^x[2B]^y \]
Simplifying this gives:
\[ 8R = k \cdot 2^x[A]^x \cdot 2^y[B]^y \]
Dividing both sides by \( R \):
\[ \frac{8R}{R} = \frac{k \cdot 2^x[A]^x \cdot 2^y[B]^y}{k[A]^x[B]^y} \]
This simplifies to:
\[ 8 = 2^{x+y} \]
Since \( 8 = 2^3 \), we can equate the exponents:
\[ x + y = 3 \]

4. **Substitute the Value of \( y \)**:
We already found that \( y = 1 \). Substituting this into the equation \( x + y = 3 \):
\[ x + 1 = 3 \]
Thus, we find:
\[ x = 2 \]

5. **Write the Final Rate Law**:
Now that we have determined the values of \( x \) and \( y \):
\[ x = 2, \quad y = 1 \]
The rate law for the reaction can be written as:
\[ \text{Rate} = k[A]^2[B]^1 \]
or simply:
\[ \text{Rate} = k[A]^2[B] \]

### Final Answer:
The rate law for the reaction is:
\[ \text{Rate} = k[A]^2[B] \]