In a reaction A + B `to` Products
(i) If the initial concentration of A is doubled and B is kept constant, the rate of the reaction is doubled
(ii) If the initial concentration both A and B are doubled the rate of reaction becomes eight times
The rate law of the reaction is :

Options

A. `k[A]^2`
B. `k[A]^2[B]^2`
C. `k[A][B]^2`
D. None of these

Answer

To determine the rate law of the reaction \( A + B \to \text{Products} \), we will analyze the information given in the question step by step.

### Step 1: Establish the Rate Law Expression
The rate law for the reaction can be expressed as:
\[ \text{Rate} = k [A]^n [B]^m \]
where \( k \) is the rate constant, \( n \) is the order of the reaction with respect to \( A \), and \( m \) is the order of the reaction with respect to \( B \).

### Step 2: Analyze the First Condition
According to the first condition:
- If the initial concentration of \( A \) is doubled (i.e., \( [A] \to 2[A] \)) and \( [B] \) is kept constant, the rate of the reaction doubles.

This can be expressed mathematically as:
\[ 2R = k (2[A])^n [B]^m \]
Since \( [B] \) is constant, we can simplify this to:
\[ 2R = k \cdot 2^n [A]^n [B]^m \]
Dividing both sides by \( R \):
\[ 2 = 2^n \]
This implies:
\[ n = 1 \]

### Step 3: Analyze the Second Condition
According to the second condition:
- If the initial concentrations of both \( A \) and \( B \) are doubled (i.e., \( [A] \to 2[A] \) and \( [B] \to 2[B] \)), the rate of the reaction becomes eight times the original rate.

This can be expressed as:
\[ 8R = k (2[A])^n (2[B])^m \]
Simplifying this gives:
\[ 8R = k \cdot 2^n [A]^n \cdot 2^m [B]^m \]
Dividing both sides by \( R \):
\[ 8 = 2^n \cdot 2^m \]
This can be rewritten as:
\[ 8 = 2^{n+m} \]
Since \( 8 = 2^3 \), we have:
\[ n + m = 3 \]

### Step 4: Solve for \( m \)
From Step 2, we found \( n = 1 \). Substituting this into the equation \( n + m = 3 \):
\[ 1 + m = 3 \]
Thus:
\[ m = 2 \]

### Step 5: Write the Final Rate Law
Now that we have determined the values of \( n \) and \( m \):
- \( n = 1 \)
- \( m = 2 \)

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

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

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