### Problem 26

**Problem Statement:**
If the decomposition of \( AB \) into products is a first-order reaction with a rate constant \( k = 7.5 \times 10^{-3} \text{s}^{-1} \) at \( 10^\circ \text{C} \), how long will it take to reduce the initial concentration of \( AB \) from 0.54 M to 0.15 M?

**Solution:**

For a first-order reaction, the integrated rate law is given by:
\[
\ln \left( \frac{[AB]_0}{[AB]} \right) = kt
\]
where:
- \([AB]_0\) is the initial concentration of the reactant,
- \([AB]\) is the concentration of the reactant at time \( t \),
- \( k \) is the rate constant,
- \( t \) is the time.

Given:
\[
[AB]_0 = 0.54 \text{ M}
\]
\[
[AB] = 0.15 \text{ M}
\]
\[
k = 7.5 \times 10^{-3} \text{ s}^{-1}
\]

We need to find the time \( t \). Substituting the given values into the integrated rate law equation, we get:
\[
\ln \left( \frac{0.54}{0.15} \right) = 7.5 \times 10^{-3} \cdot t
\]

First, calculate the natural logarithm:
\[
\frac{0.54}{0.15} = 3.6
\]
\[
\ln (3.6) \approx 1.2809
\]

Now, solve for \( t \):
\[
1.2809 = 7.5 \times 10^{-3} \cdot t
\]
\[
t = \frac{1.2809}{7.5 \times 10^{-3}}
\]
\[
t \approx 170.79 \text{ seconds}
\]

Therefore, it will take approximately 170.79 seconds to reduce the initial concentration of \( AB \) from 0.54 M to 0.15 M.

---

Would you like more details on this solution or have any questions? Here are some relative questions to expand your understanding:

1. How does the half-life of a first-order reaction relate to its rate constant?
2. What is the significance of the rate constant \( k \) in first-order reactions?
3. How can you determine the order of a reaction from experimental data?
4. Why do first-order reactions have a constant half-life?
5. How would the time change if the reaction were zero-order instead of first-order?
6. What are some examples of first-order reactions in real life?
7. How do temperature and catalysts affect the rate constant \( k \)?
8. What is the difference between first-order and second-order reactions in terms of their rate laws?

**Tip:** Always remember that the units of the rate constant \( k \) can help determine the order of the reaction. For a first-order reaction, \( k \) has units of \( \text{time}^{-1} \).

Learn how to calculate the time required for a first-order reaction to reduce the concentration of AB from 0.54 M to 0.15 M. This problem involves chemical kinetics and the integrated rate law for first-order reactions. Suitable for advanced high school students and chemistry enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation