Question

For chemical reaction M \(\rightarrow\) N, the rate becomes 8 times when the concentration of M doubles, find the order of reaction with respect to M?

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

You can solve these by simple inspection:
If concentration \(\times 2 \implies\) rate \(\times 2\) (\(2^1\)), order = 1
If concentration \(\times 2 \implies\) rate \(\times 4\) (\(2^2\)), order = 2
If concentration \(\times 2 \implies\) rate \(\times 8\) (\(2^3\)), order = 3

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The order of a reaction dictates how the rate is affected by changes in reactant concentration. It is expressed mathematically as an exponent in the experimental rate law equation.
Step 2: Key Formula or Approach:
The general rate law for a single-reactant reaction \(\text{M} \rightarrow \text{N}\) is expressed as:
\[ \text{Rate} = k[\text{M}]^n \]
Where:
- \(k\) is the rate constant.
- \([\text{M}]\) is the concentration of reactant M.
- \(n\) is the order of the reaction with respect to M.
We can set up a ratio of rates under two different conditions to solve for the exponent \(n\).
Step 3: Detailed Explanation:
Let the initial concentration be \([\text{M}]_1\).
The initial rate is: \(\text{Rate}_1 = k([\text{M}]_1)^n\)
The problem states that when the concentration is doubled, \([\text{M}]_2 = 2[\text{M}]_1\).
Under these new conditions, the rate becomes 8 times the initial rate: \(\text{Rate}_2 = 8 \times \text{Rate}_1\).
Write the rate law for the second condition:
\[ \text{Rate}_2 = k([\text{M}]_2)^n \]
Substitute the known relations into this equation:
\[ 8 \times \text{Rate}_1 = k(2[\text{M}]_1)^n \]
\[ 8 \times \text{Rate}_1 = k \cdot 2^n \cdot ([\text{M}]_1)^n \]
We know that \(\text{Rate}_1 = k([\text{M}]_1)^n\). Substitute this into the right side:
\[ 8 \times \text{Rate}_1 = 2^n \times (\text{Rate}_1) \]
Divide both sides by \(\text{Rate}_1\) (assuming \(\text{Rate}_1 \neq 0\)):
\[ 8 = 2^n \]
Since \(8\) can be written as \(2^3\):
\[ 2^3 = 2^n \]
Therefore, by equating the exponents, we find:
\[ n = 3 \]
Step 4: Final Answer:
The order of the reaction is 3.