For a reaction `2A rarrB` Products , doubling the concentration of B alone doubles the rate and doubling the initial concentration of both the reactants increases the rate by a factor of 8 . The rate law for the reaction is

Options

A. Rate `= k [A] [B]^2`
B. Rate `= k [A]^2 [B]^2`
C. Rate `= k [A] [B]`
D. Rate `= k [A]^2 [B]`

Answer

To determine the rate law for the reaction \(2A + B \rightarrow \text{Products}\), we will analyze the information given in the problem step by step.

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

### Step 2: Analyze the Effect of Doubling [B]
According to the problem, doubling the concentration of \(B\) alone doubles the rate. This can be expressed mathematically:
\[ \text{If } [B] \text{ is doubled: } \text{Rate} = k [A]^x [2B]^y = k [A]^x [B]^y (2^y) \]
Setting the new rate equal to double the original rate:
\[ k [A]^x [2B]^y = 2 \cdot k [A]^x [B]^y \]
This simplifies to:
\[ 2^y = 2 \]
From this, we can deduce:
\[ y = 1 \]

### Step 3: Analyze the Effect of Doubling Both [A] and [B]
Next, we consider the second condition where doubling the initial concentrations of both reactants increases the rate by a factor of 8:
\[ \text{If } [A] \text{ and } [B] are both doubled: \text{Rate} = k [2A]^x [2B]^y = k (2^x [A]^x) (2^y [B]^y) = k [A]^x [B]^y (2^{x+y}) \]
Setting this equal to 8 times the original rate:
\[ k [2A]^x [2B]^y = 8 \cdot k [A]^x [B]^y \]
This simplifies to:
\[ 2^{x+y} = 8 \]
Since \(8 = 2^3\), we have:
\[ x + y = 3 \]

### Step 4: Solve for \(x\)
Now we have two equations:
1. \(y = 1\)
2. \(x + y = 3\)

Substituting \(y = 1\) into the second equation:
\[ x + 1 = 3 \implies x = 2 \]

### Conclusion: Write the Rate Law
Now that we have determined \(x\) and \(y\):
- \(x = 2\)
- \(y = 1\)

The rate law for the reaction is:
\[ \text{Rate} = k [A]^2 [B]^1 \]

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