Question

For a 1st order change R \(\rightarrow\) P, the concentration of Reactant R changes from 0.1 M to 0.025 M in 40 minutes. The rate of reaction when the concentration of R is 0.01 M is

• A. \( 1.73 \times 10^{-5} \text{ M min}^{-1} \)
• B. \( 3.47 \times 10^{-4} \text{ M min}^{-1} \)
• C. \( 3.47 \times 10^{-5} \text{ M min}^{-1} \)
• D. \( 1.73 \times 10^{-4} \text{ M min}^{-1} \)

Hint:

For first-order reactions, always check if the concentration change is a power of 2 (like \( 1/2, 1/4, 1/8 \)). If it is, you can skip the logarithmic formula and find the half-life directly by dividing the total time by the number of half-lives.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the rate of a first-order reaction at a specific concentration (\( 0.01 \text{ M} \)). To find this, we first need to determine the rate constant (\( k \)) using the time taken for a given change in concentration.
Step 2: Key Formula or Approach:
For a first-order reaction:
1. Integrated rate law: \( k = \frac{2.303}{t} \log\left(\frac{[R]_0}{[R]_t}\right) \)
2. Rate law: \( \text{Rate} = k[R] \)
3. Relation between \( k \) and half-life \( t_{1/2} \): \( k = \frac{0.693}{t_{1/2}} \)
Step 3: Detailed Explanation:
1. Finding the Rate Constant (\( k \)):
Initial concentration, \( [R]_0 = 0.1 \text{ M} \).
Final concentration after 40 min, \( [R]_t = 0.025 \text{ M} \).
Notice that \( \frac{[R]_t}{[R]_0} = \frac{0.025}{0.1} = \frac{1}{4} \).
Since the concentration becomes one-fourth, two half-lives have passed in 40 minutes (\( 0.1 \rightarrow 0.05 \rightarrow 0.025 \)).
Therefore, \( 2 \times t_{1/2} = 40 \text{ min} \implies t_{1/2} = 20 \text{ min} \).
Now, calculate the rate constant \( k \):
\[ k = \frac{0.693}{t_{1/2}} = \frac{0.693}{20 \text{ min}} = 0.03465 \text{ min}^{-1} \]
2. Calculating the Rate of Reaction:
We need the rate when \( [R] = 0.01 \text{ M} \).
\[ \text{Rate} = k \times [R] \]
\[ \text{Rate} = 0.03465 \text{ min}^{-1} \times 0.01 \text{ M} \]
\[ \text{Rate} = 0.0003465 \text{ M min}^{-1} = 3.465 \times 10^{-4} \text{ M min}^{-1} \]
Rounding to three significant figures, we get \( 3.47 \times 10^{-4} \text{ M min}^{-1} \).
Step 4: Final Answer:
The rate is \( 3.47 \times 10^{-4} \text{ M min}^{-1} \), which corresponds to option (B).