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 concentrations of both the reactants (A and B) are doubled, rate law for the reaction can be written as

Options

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

Answer

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

### Step-by-Step Solution:

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

2. **Analyzing the First Condition**:
The problem states that the rate is doubled when the concentration of B is doubled while keeping the concentration of A constant. This can be expressed mathematically as:
\[ 2 \cdot \text{Rate} = k [A]^x [2B]^y \]
Since the rate is doubled, we can set up the equation:
\[ 2 \cdot \text{Rate} = k [A]^x [B]^y \cdot 2^y \]
Dividing both sides by \( k [A]^x [B]^y \):
\[ 2 = 2^y \implies y = 1 \]
This means the order with respect to B is 1.

3. **Analyzing the Second Condition**:
The second condition states that when both concentrations of A and B are doubled, the rate increases by a factor of 8:
\[ 8 \cdot \text{Rate} = k [2A]^x [2B]^y \]
Substituting \( y = 1 \) into the equation:
\[ 8 \cdot \text{Rate} = k [2A]^x [2B]^1 \]
This simplifies to:
\[ 8 \cdot \text{Rate} = k [2^x A^x] [2B] \]
\[ 8 \cdot \text{Rate} = k [2^{x+1} A^x B] \]
Dividing both sides by \( k [A]^x [B] \):
\[ 8 = 2^{x+1} \]
Since \( 8 = 2^3 \), we can set the exponents equal:
\[ 3 = x + 1 \implies x = 2 \]
This means the order with respect to A is 2.

4. **Writing the Final Rate Law**:
Now that we have determined the orders with respect to both reactants:
- Order with respect to A (\( x \)) = 2
- Order with respect to B (\( y \)) = 1

The rate law can be written as:
\[ \text{Rate} = k [A]^2 [B]^1 \]

### Final Answer:
The rate law for the reaction is:
\[ \text{Rate} = k [A]^2 [B]^1 \]