An elementary reaction A and B is second order reaction. Which of the following rate equation must be correct?

Options

A. `r=k[A]^(2)[B]^(0)`
B. `r=k[A]^(3//2)[B]^(1//2)`
C. `r=k[A]^(0)[B]^(2)`
D. `r=k[A][B]`

Answer

To determine the correct rate equation for a second-order elementary reaction involving reactants A and B, we can follow these steps:

### Step 1: Understand the Concept of Elementary Reactions
An elementary reaction is a reaction that occurs in a single step. The order of an elementary reaction is equal to its molecularity, which is the number of reactant molecules involved in the reaction.

### Step 2: Identify the Molecularity
For the reaction A + B → Products, the molecularity is 2 because there are two reactant molecules (A and B) involved in the reaction.

### Step 3: Relate Order and Molecularity
Since the reaction is second order, the sum of the powers of the concentration terms in the rate equation must equal 2. Therefore, if we denote the rate equation as:
\[ \text{Rate} = k[A]^x[B]^y \]
then we have:
\[ x + y = 2 \]

### Step 4: Determine Constraints on x and y
- Both \( x \) and \( y \) must be greater than 0 because they represent the number of molecules reacting.
- Both \( x \) and \( y \) must be integers, as fractional orders would imply a complex reaction rather than an elementary one.

### Step 5: Analyze the Given Options
Now, we need to evaluate the provided options based on the above criteria:

1. **Option 1**: \( \text{Rate} = k[A]^2[B]^0 \)
- Here, \( x = 2 \) and \( y = 0 \). This option is invalid because \( y \) cannot be zero.

2. **Option 2**: \( \text{Rate} = k[A]^{3/2}[B]^{1/2} \)
- Here, \( x = 3/2 \) and \( y = 1/2 \). This option is invalid because both \( x \) and \( y \) are fractional.

3. **Option 3**: \( \text{Rate} = k[A]^0[B]^2 \)
- Here, \( x = 0 \) and \( y = 2 \). This option is invalid because \( x \) cannot be zero.

4. **Option 4**: \( \text{Rate} = k[A]^1[B]^1 \)
- Here, \( x = 1 \) and \( y = 1 \). This option is valid because \( x + y = 2 \), both \( x \) and \( y \) are greater than zero, and they are integers.

### Conclusion
The correct rate equation for the second-order elementary reaction A + B is:
\[ \text{Rate} = k[A]^1[B]^1 \]