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 reactants A and B, we can analyze the information provided in the question step by step.

### Step 1: Understand the Rate Law
The rate of a reaction can be expressed using the rate law equation:
\[ R = k [A]^m [B]^n \]
where:
- \( R \) is the rate of the reaction,
- \( k \) is the rate constant,
- \( [A] \) and \( [B] \) are the concentrations of reactants A and B, respectively,
- \( m \) is the order of the reaction with respect to A,
- \( n \) is the order of the reaction with respect to B.

### Step 2: Analyze the First Condition
According to the first condition, when the concentration of A is doubled, the rate of reaction is also doubled:
- Let the initial concentration of A be \( [A] \) and the concentration of B be \( [B] \).
- When \( [A] \) is doubled, the new concentration is \( 2[A] \).

Substituting this into the rate law:
\[ R_{new} = k(2[A])^m[B]^n = 2^m k[A]^m[B]^n \]

Since the rate is doubled:
\[ 2R = 2^m R \]
This implies:
\[ 2 = 2^m \]
From this equation, we can conclude that:
\[ m = 1 \]
Thus, the order of reaction with respect to A is 1.

### Step 3: Analyze the Second Condition
Now, according to the second condition, when both concentrations of A and B are doubled, the rate is again doubled:
- The new concentrations are \( 2[A] \) and \( 2[B] \).

Substituting these into the rate law:
\[ R_{new} = k(2[A])^m(2[B])^n = k(2^m[A]^m)(2^n[B]^n) = 2^{m+n} k[A]^m[B]^n \]

Since the rate is doubled:
\[ 2R = 2^{m+n} R \]
This implies:
\[ 2 = 2^{m+n} \]
From this equation, we can conclude that:
\[ m+n = 1 \]

### Step 4: Solve for n
We already found that \( m = 1 \). Substituting this into the equation \( m+n = 1 \):
\[ 1 + n = 1 \]
This leads to:
\[ n = 0 \]
Thus, the order of reaction with respect to B is 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 Answer
The order of reaction with respect to A and B is:
**1 and 0, respectively.**