For the reaction `A + B` products, it is observed that:
(1) on doubling the initial concentration of `A` only, the rate of reaction is also doubled and
(2) on doubling te initial concentration of both `A` and `B`, there is a charge by a factor of `8` in the rate of the reaction.
The rate of this reaction is given by

Options

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

Explanation

To determine the rate law for the reaction \( A + B \rightarrow \text{products} \), we will analyze the given observations step by step.

### Step 1: Write the general rate law expression
The rate of the reaction can be expressed as:
\[ R = k [A]^n [B]^m \]
where \( R \) is the rate of the reaction, \( k \) 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 observation
The first observation states that when the concentration of \( A \) is doubled, the rate of reaction is also doubled. Mathematically, this can be expressed as:
\[ 2R = k (2[A])^n [B]^m \]
Dividing both sides by the original rate equation \( R = k [A]^n [B]^m \):
\[ \frac{2R}{R} = \frac{k (2[A])^n [B]^m}{k [A]^n [B]^m} \]
This simplifies to:
\[ 2 = \frac{(2[A])^n}{[A]^n} = 2^n \]
From this, we can deduce:
\[ 2 = 2^n \implies n = 1 \]

### Step 3: Analyze the second observation
The second observation states that when the concentrations of both \( A \) and \( B \) are doubled, the rate increases by a factor of 8:
\[ 8R = k (2[A])^n (2[B])^m \]
Using \( n = 1 \) from the previous step, we can rewrite this as:
\[ 8R = k (2[A])^1 (2[B])^m = k (2[A][B]^m) \cdot 2^m \]
Dividing both sides by the original rate equation:
\[ \frac{8R}{R} = \frac{k (2[A])(2[B])^m}{k [A]^1 [B]^m} \]
This simplifies to:
\[ 8 = 2 \cdot 2^m \]
Thus, we have:
\[ 8 = 2^{1+m} \]
This implies:
\[ 8 = 2^3 \implies 1 + m = 3 \implies m = 2 \]

### Step 4: Write the final rate law
Now that we have determined the values of \( n \) and \( m \):
- \( n = 1 \)
- \( m = 2 \)

The rate law for the reaction is:
\[ R = k [A]^1 [B]^2 \]
or simply:
\[ R = k [A] [B]^2 \]

### Summary of the Solution
The rate of the reaction is given by:
\[ R = k [A] [B]^2 \]