For the elementary reaction `MrarrN`, the rate of disappearance of `M` increases by a factor of `8` upon doubling the concentration of `M`. The order of the reaction will respect to `M` is

Options

A. `4`
B. `3`
C. `2`
D. `1`

Explanation

To determine the order of the reaction with respect to \( M \) for the elementary reaction \( M \rightarrow N \), we can follow these steps:

### Step 1: Write the Rate Law
The rate of the reaction can be expressed as:
\[ \text{Rate} = k [M]^n \]
where \( k \) is the rate constant, \( [M] \) is the concentration of \( M \), and \( n \) is the order of the reaction with respect to \( M \).

### Step 2: Analyze the Change in Concentration
According to the problem, when the concentration of \( M \) is doubled (from \( [M] \) to \( 2[M] \)), the rate of disappearance of \( M \) increases by a factor of 8. This can be expressed as:
\[ \text{Rate}_2 = 8 \times \text{Rate}_1 \]

### Step 3: Write the Rates for Both Concentrations
Using the rate law for both conditions:
- For the initial concentration:
\[ \text{Rate}_1 = k [M]^n \]
- For the doubled concentration:
\[ \text{Rate}_2 = k (2[M])^n = k \cdot 2^n \cdot [M]^n \]

### Step 4: Set Up the Equation
From the relationship established in Step 2, we can set up the equation:
\[ k \cdot 2^n \cdot [M]^n = 8 \cdot (k [M]^n) \]

### Step 5: Simplify the Equation
We can cancel \( k [M]^n \) from both sides (assuming \( k \) and \( [M] \) are not zero):
\[ 2^n = 8 \]

### Step 6: Solve for \( n \)
We know that \( 8 \) can be expressed as \( 2^3 \):
\[ 2^n = 2^3 \]
Thus, by equating the exponents, we find:
\[ n = 3 \]

### Conclusion
The order of the reaction with respect to \( M \) is \( n = 3 \).