The overall order of reaction between X & Y is 3. Which of the following rate equation must be correct, if on doubling the concentration of X, the rate of reaction gets doubled ?

Options

A. `r=K[X]^2[Y]^0`
B. `r=K[X]^1[Y]^2`
C. `r=K[X]^1[Y]^3`
D. `r=K[X]^2[Y]^1`

Answer

To solve the problem, we need to determine the correct rate equation for the reaction between X and Y given the overall order of the reaction and the effect of doubling the concentration of X on the rate of reaction.

### Step-by-Step Solution:

1. **Understanding the Overall Order of Reaction**:
- The overall order of the reaction is given as 3. This means that the sum of the orders with respect to each reactant (X and Y) is equal to 3.
- Let the order with respect to X be \( x \) and the order with respect to Y be \( y \). Therefore, we have:
\[ x + y = 3 \]

2. **Writing the Rate Equation**:
- The rate of the reaction can be expressed as:
\[ \text{Rate} = k [X]^x [Y]^y \]
- Here, \( k \) is the rate constant, and \( [X] \) and \( [Y] \) are the concentrations of reactants X and Y, respectively.

3. **Effect of Doubling the Concentration of X**:
- According to the problem, when the concentration of X is doubled (i.e., \( [X] \) becomes \( 2[X] \)), the rate of the reaction also doubles.
- This can be expressed mathematically as:
\[ 2 \times \text{Rate} = k (2[X])^x [Y]^y \]

4. **Setting Up the Equation**:
- We can set up the equation using the initial rate:
\[ 2 \times (k [X]^x [Y]^y) = k (2[X])^x [Y]^y \]
- Simplifying this gives:
\[ 2 = \frac{(2[X])^x}{[X]^x} \]

5. **Simplifying the Equation**:
- The left side simplifies to:
\[ 2 = 2^x \]
- This implies that:
\[ x = 1 \]

6. **Finding the Value of y**:
- Now that we have \( x = 1 \), we can substitute this back into the equation for the overall order:
\[ 1 + y = 3 \implies y = 2 \]

7. **Final Rate Equation**:
- Therefore, the rate equation can be written as:
\[ \text{Rate} = k [X]^1 [Y]^2 \]
- This simplifies to:
\[ \text{Rate} = k [X] [Y]^2 \]

8. **Conclusion**:
- The correct rate equation is \( \text{Rate} = k [X] [Y]^2 \).