Deriving the Rate Constant of a First-Order Reaction and Calculating Remaining Substance After One Hour

Understanding the kinetics of chemical reactions is crucial for many applications, from pharmaceuticals to environmental chemistry. In a first-order reaction, the rate of decomposition is directly proportional to the concentration of the substance. This article will guide you through the process of determining the rate constant and predicting the remaining concentration of a substance after a specified time period, using a practical example.

Introduction to First-Order Reactions

A first-order reaction is characterized by a rate law equation of the form:

lnA??/A ?kt

ln([A?]/[A]): Natural logarithm of the ratio of the initial concentration to the concentration at time t. k: The rate constant, a characteristic of the reaction. t: Time, in seconds or minutes, depending on the units used for k.

Case Study: Substance Decomposition

In this example, we are given that 15% of a substance decomposes in the first 10 minutes. Our goal is to derive the rate constant and calculate the amount of substance remaining after one hour.

Step 1: Calculate the Rate Constant k

Given that 15% decomposes in the first 10 minutes, we express the remaining concentration as:

[A] [A?] - 0.15[A?] 0.85[A?]

Substitute this into the first-order equation:

ln([A?]/[0.85[A?]]) kt

This simplifies to:

ln(1/0.85) kt

By converting the time from minutes to seconds (10 minutes 600 seconds), we get:

ln(1.1765) k600

Calculating:

ln(1.1765) ≈ 0.1625

Substitute this into the equation:

0.1625 k600

Solving for k:

k ≈ 0.1625/600 ≈ 0.0002708 s?1

Step 2: Calculate Remaining Concentration After One Hour

To find out how much of the substance remains after one hour (60 minutes or 3600 seconds), we use the first-order decay equation:

ln([A?]/[A]) kt

Substitute t 3600 seconds:

ln([A?]/[A]) 0.0002708 * 3600

Calculating the right side:

0.0002708 * 3600 ≈ 0.97488

Now, we can calculate [A]:

[A]/[A?] e0.97488 ≈ 2.65

Thus, the fraction of the substance remaining is:

[A]/[A?] ≈ 1/2.65 ≈ 0.377

So, approximately 37.7% of the substance remains undecomposed after one hour.

Graphical Presentation

To visualize this, you can plot the decay of the substance over time using the first-order kinetics equation:

[A] [A?] e?kt

Time (X-axis): Time in seconds. Remaining concentration (Y-axis): Remaining concentration as a fraction of the initial concentration.

The graph will show an exponential decay curve, reflecting how the concentration decreases over time.

Summary

The rate constant k is approximately 0.0002708 s?1.

Approximately 37.7% of the substance will remain undecomposed after one hour.