For a reaction `A+B rarr` products, the rate of reaction was doubled when concentration of A was doubled. When concentration of A and B both was double, the rate was again doubled , order of reaction w.r.t. A and B are

Options

A. 1,1
B. 2,0
C. 1,0
D. 0,1

Answer

To determine the order of the reaction with respect to A and B, we can analyze the information given step by step.

### Step 1: Understand the Rate Law Expression
The rate of a reaction can be expressed as:
\[ \text{Rate} = k [A]^x [B]^y \]
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 First Condition
The first condition states that when the concentration of A is doubled (keeping B constant), the rate of reaction is also doubled.

This can be expressed mathematically as:
\[ 2 \times \text{Rate} = k (2[A])^x [B]^y \]
Simplifying this gives:
\[ 2 \times \text{Rate} = k \cdot 2^x [A]^x [B]^y \]

Since the original rate is:
\[ \text{Rate} = k [A]^x [B]^y \]

Setting the two equations equal to each other:
\[ 2 \cdot k [A]^x [B]^y = k \cdot 2^x [A]^x [B]^y \]

Dividing both sides by \( k [A]^x [B]^y \) (assuming \( [A] \) and \( [B] \) are not zero):
\[ 2 = 2^x \]

Taking logarithm base 2 on both sides:
\[ 1 = x \]

Thus, the order of reaction with respect to A is:
\[ x = 1 \]

### Step 3: Analyze the Second Condition
The second condition states that when both concentrations of A and B are doubled, the rate is again doubled.

This can be expressed as:
\[ 2 \times \text{Rate} = k (2[A])^x (2[B])^y \]
This simplifies to:
\[ 2 \times \text{Rate} = k \cdot 2^x \cdot 2^y [A]^x [B]^y \]
\[ 2 \times \text{Rate} = k \cdot 2^{x+y} [A]^x [B]^y \]

Setting this equal to the original rate:
\[ 2 \cdot k [A]^x [B]^y = k \cdot 2^{x+y} [A]^x [B]^y \]

Dividing both sides by \( k [A]^x [B]^y \):
\[ 2 = 2^{x+y} \]

Taking logarithm base 2 on both sides:
\[ 1 = x + y \]

### Step 4: Solve for y
We already found that \( x = 1 \). Substituting this into the equation:
\[ 1 = 1 + y \]
This simplifies to:
\[ y = 0 \]

### Conclusion
The order of the reaction with respect to A is 1 and with respect to B is 0. Therefore, the final answer is:
- Order with respect to A: 1
- Order with respect to B: 0

### Final Rate Law Expression
The rate law expression can be written as:
\[ \text{Rate} = k [A]^1 [B]^0 \]
or simply:
\[ \text{Rate} = k [A] \]