For a reactions `A+Brarr` product ,it was found that rate of reaction increases four times if concentration of 'A' is doubled. But the rate of reaction remains unaffected, if concentration of 'B' is doubled . Hence , the rate law for the reaction is

Options

A. rate = k[A][B]
B. rate = `k[A]^2`
C. rate = `k[A]^2[B]^1`
D. rate = `k[A]^2[B]^2`

Answer

To determine the rate law for the reaction \( A + B \rightarrow \text{products} \), we can follow these steps:

### Step 1: Write the general 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, \( x \) is the order of the reaction with respect to \( A \), and \( y \) is the order of the reaction with respect to \( B \).

### Step 2: Analyze the effect of changing concentration of \( A \)
According to the problem, when the concentration of \( A \) is doubled, the rate of reaction increases four times. This can be expressed mathematically as:
\[ \text{If } [A] \text{ is doubled, } \text{Rate} \propto [A]^x \]
Let’s denote the initial concentration of \( A \) as \( [A] \) and the initial rate as \( r_1 \):
\[ r_1 = k [A]^x \]
When \( [A] \) is doubled, the new rate \( r_2 \) becomes:
\[ r_2 = k [2A]^x = k (2[A])^x = k \cdot 2^x \cdot [A]^x \]
Given that \( r_2 = 4r_1 \), we can set up the equation:
\[ k \cdot 2^x \cdot [A]^x = 4(k [A]^x) \]
Dividing both sides by \( k [A]^x \):
\[ 2^x = 4 \]
This implies:
\[ 2^x = 2^2 \implies x = 2 \]

### Step 3: Analyze the effect of changing concentration of \( B \)
The problem states that the rate of reaction remains unaffected when the concentration of \( B \) is doubled. This indicates that the rate does not depend on \( B \):
\[ \text{If } [B] \text{ is doubled, } \text{Rate} \propto [B]^y \]
Since the rate remains constant, we can conclude:
\[ y = 0 \]

### Step 4: Write the final rate law expression
Substituting the values of \( x \) and \( y \) into the rate law expression:
\[ \text{Rate} = k [A]^2 [B]^0 \]
Since \( [B]^0 = 1 \), we simplify this to:
\[ \text{Rate} = k [A]^2 \]

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