Modeling First-Order Chemical Reactions

Let’s explore a problem in Chemistry related to chemical kinetics, specifically the rate of a first-order reaction.

A first-order reaction is one where the rate of reaction depends linearly on the concentration of a single reactant.

Problem Statement

We will model the concentration of a reactant over time for a first-order reaction and plot the concentration as a function of time.
The rate law for a first-order reaction is given by:
$$
\frac{d[A]}{dt} = -k[A]
$$

  • $ [A] $ = concentration of the reactant
  • $ k $ = rate constant
  • $ t $ = time

The integrated rate law for a first-order reaction is:
$$
[A] = [A]_0 e^{-kt}
$$

  • $ [A]_0 $ = initial concentration of the reactant

Python Code

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import numpy as np
import matplotlib.pyplot as plt

# Parameters
A0 = 1.0  # Initial concentration in mol/L
k = 0.1   # Rate constant in s^-1
time = np.linspace(0, 50, 500)  # Time range from 0 to 50 seconds

# Calculate concentration over time
A = A0 * np.exp(-k * time)

# Plot concentration vs time
plt.figure(figsize=(10, 6))
plt.plot(time, A, color='blue', linewidth=2, label='[A]')
plt.title('First-Order Reaction: Concentration vs Time')
plt.xlabel('Time (s)')
plt.ylabel('Concentration [A] (mol/L)')
plt.grid(True)

# Highlight half-life
half_life = np.log(2) / k
plt.axvline(x=half_life, color='red', linestyle='--', label=f'Half-life = {half_life:.2f} s')
plt.legend()
plt.show()

Explanation

  1. Rate Law:

    • The rate of a first-order reaction is proportional to the concentration of the reactant:
      $$
      \frac{d[A]}{dt} = -k[A]
      $$

  2. Integrated Rate Law:

    • The integrated rate law for a first-order reaction is:
      $$
      [A] = [A]_0 e^{-kt}
      $$
    • This equation describes how the concentration of the reactant decreases exponentially over time.

  3. Parameters:

    • $ [A]_0 = 1.0 \text{mol/L} $: Initial concentration of the reactant.
    • $ k = 0.1 \text{s}^{-1} $: Rate constant.

  4. Time Range:

    • We define a time range from $0$ to $50$ seconds using np.linspace.

  5. Plotting:

    • The concentration $ [A] $ is plotted as a function of time.
    • A vertical dashed line is added to highlight the half-life of the reaction.

Result

The graph shows the concentration of the reactant $ [A] $ as a function of time:

  • At $ t = 0 $, the concentration is $ [A]_0 = 1.0 , \text{mol/L} $.
  • As time increases, the concentration decreases exponentially.
  • The half-life $( t_{1/2} )$ is the time required for the concentration to reduce to half of its initial value.
    For this reaction, the half-life is:
    $$
    t_{1/2} = \frac{\ln(2)}{k} \approx 6.93 \text{s}
    $$

Interpretation

  • Exponential Decay: The concentration of the reactant decreases exponentially over time, which is characteristic of first-order kinetics.
  • Half-Life: The half-life is constant for a first-order reaction and is independent of the initial concentration.
    This is a key feature of first-order reactions.

This example demonstrates how $Python$ can be used to model and visualize chemical kinetics, providing insights into the behavior of chemical reactions.