The rate of reaction between two `A` and `B` decreases by factor `4` if the concentration of reactant `B` is doubled. The order of this reaction with respect to `B` is

Options

A. `-1`
B. `-2`
C. 1
D. 2

Explanation

To determine the order of the reaction with respect to reactant B, let's analyze the information given in the question step by step.

### Step 1: Understand the Rate Law
The rate of a reaction can be expressed by the rate law:
\[ \text{Rate} = k [A]^x [B]^y \]
where:
- \( k \) is the rate constant,
- \( [A] \) is the concentration of reactant A,
- \( [B] \) is the concentration of reactant B,
- \( x \) is the order of the reaction with respect to A,
- \( y \) is the order of the reaction with respect to B.

### Step 2: Analyze the Given Information
According to the problem, when the concentration of B is doubled, the rate of reaction decreases by a factor of 4. This can be expressed mathematically as:
\[ \text{New Rate} = \frac{\text{Initial Rate}}{4} \]

### Step 3: Set Up the Equation
Let the initial rate be \( I \). When the concentration of B is doubled, the new concentration of B becomes \( 2[B] \). Thus, we can write:
\[ \frac{I}{4} = k [A]^x (2[B])^y \]

### Step 4: Write the Initial Rate Expression
The initial rate can be expressed as:
\[ I = k [A]^x [B]^y \]

### Step 5: Substitute and Simplify
Now, substituting the expression for the initial rate into the equation for the new rate:
\[ \frac{I}{4} = k [A]^x (2[B])^y \]
Substituting \( I \):
\[ \frac{k [A]^x [B]^y}{4} = k [A]^x (2[B])^y \]

### Step 6: Cancel Common Terms
We can cancel \( k [A]^x \) from both sides (assuming \( k \) and \( [A] \) are not zero):
\[ \frac{[B]^y}{4} = (2[B])^y \]

### Step 7: Rearranging the Equation
This simplifies to:
\[ \frac{[B]^y}{4} = 2^y [B]^y \]

### Step 8: Solve for \( y \)
Now, we can rearrange this equation:
\[ 1 = 4 \cdot 2^y \]
This implies:
\[ 4 = 2^y \]
Taking logarithm base 2:
\[ 2^2 = 2^y \]
Thus, we find:
\[ y = -2 \]

### Conclusion
The order of the reaction with respect to B is \( -2 \).

### Final Answer
The order of the reaction with respect to B is **-2** (Option B).

---