Taking the reaction, `A +2B rarr` Products to be of second order, which of the following is/are the correct rate law expresisons `(s)`?

Options

A. `(dx)/(dt) = k[A]^(2)`
B. `(dx)/(st) = k[A][B]^(2)`
C. `(dx)/(dt) = k[A][B]`
D. `(dx)/(dt) = k_(1)[A]+k_(2)[B]^(2)`

Explanation

To determine the correct rate law expressions for the reaction \( A + 2B \rightarrow \text{Products} \) given that it is of second order, we will analyze the possible rate laws based on the stoichiometry of the reaction and the overall order.

### Step-by-Step Solution:

1. **Understanding Reaction Order**:
- The overall order of the reaction is given as 2. This means that the sum of the powers of the concentration terms in the rate law must equal 2.

2. **Possible Rate Law Expressions**:
- The rate law can be expressed in different forms depending on how the concentrations of reactants are included. The general form of the rate law can be:
\[ \text{Rate} = k [A]^m [B]^n \]
- Where \( m \) and \( n \) are the orders with respect to reactants A and B, respectively.

3. **Analyzing Different Combinations**:
- Since the overall order is 2, we can have the following combinations:
- **Case 1**: \( m = 2 \) and \( n = 0 \) (Rate = \( k [A]^2 [B]^0 \))
- This gives us a rate law of:
\[ \text{Rate} = k [A]^2 \]
- The overall order is \( 2 + 0 = 2 \) (Valid).

- **Case 2**: \( m = 1 \) and \( n = 1 \) (Rate = \( k [A]^1 [B]^1 \))
- This gives us a rate law of:
\[ \text{Rate} = k [A][B] \]
- The overall order is \( 1 + 1 = 2 \) (Valid).

- **Case 3**: \( m = 0 \) and \( n = 2 \) (Rate = \( k [A]^0 [B]^2 \))
- This gives us a rate law of:
\[ \text{Rate} = k [B]^2 \]
- The overall order is \( 0 + 2 = 2 \) (Valid).

4. **Evaluating Given Options**:
- Now, we will evaluate the provided options to see which ones match the valid rate laws derived:
- **Option 1**: \( \text{Rate} = k [A]^2 \) (Valid, order = 2)
- **Option 2**: \( \text{Rate} = k [A][B]^2 \) (Invalid, order = 3)
- **Option 3**: \( \text{Rate} = k [A][B] \) (Valid, order = 2)
- **Option 4**: \( \text{Rate} = k_1 [A] + k_2 [B]^2 \) (Invalid, order = 3)

5. **Conclusion**:
- The correct rate law expressions for the reaction \( A + 2B \rightarrow \text{Products} \) that are of second order are:
- Option 1: \( \text{Rate} = k [A]^2 \)
- Option 3: \( \text{Rate} = k [A][B] \)