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 based on the information given. Here's the step-by-step solution:

### Step 1: Understand the overall order of the reaction
The overall order of the reaction between X and Y is given as 3. This means that the sum of the powers of the concentrations of reactants X and Y in the rate equation must equal 3.

### Step 2: Set up the rate equation
The general form of the rate equation can be expressed as:
\[ \text{Rate} = k [X]^m [Y]^n \]
where \( m \) is the order with respect to X and \( n \) is the order with respect to Y. According to the information provided:
\[ m + n = 3 \]

### Step 3: Analyze the effect of doubling the concentration of X
It is stated that when the concentration of X is doubled, the rate of the reaction also doubles. This indicates that the rate is directly proportional to the concentration of X. Therefore, we can conclude that:
\[ \text{Rate} \propto [X]^m \]
If we double the concentration of X, the new rate becomes:
\[ \text{New Rate} = k (2[X])^m [Y]^n = k \cdot 2^m [X]^m [Y]^n \]
Since the rate doubles, we can set up the equation:
\[ 2 \cdot \text{Rate} = 2 \cdot k [X]^m [Y]^n \]
This implies:
\[ k \cdot 2^m [X]^m [Y]^n = 2 \cdot k [X]^m [Y]^n \]
From this, we can deduce that:
\[ 2^m = 2 \]
Thus, \( m = 1 \).

### Step 4: Determine the order with respect to Y
Now that we know \( m = 1 \), we can find \( n \) using the overall order equation:
\[ m + n = 3 \]
Substituting \( m = 1 \):
\[ 1 + n = 3 \]
This gives us:
\[ n = 2 \]

### Step 5: Write the final rate equation
Now we can write the rate equation based on the values of \( m \) and \( n \):
\[ \text{Rate} = k [X]^1 [Y]^2 \]
or simply:
\[ \text{Rate} = k [X] [Y]^2 \]

### Conclusion
The correct rate equation that satisfies all the conditions given in the problem is:
\[ \text{Rate} = k [X] [Y]^2 \]