Part III: Chemical Kinetics | Chapter 1

Reaction Rate Laws

A rigorous treatment of rate laws, integrated rate expressions, mechanistic approximations, and the Lindemann–Hinshelwood theory of unimolecular reactions

1. Introduction: Reaction Rates and Their Measurement

Chemical kinetics is the study of how fast reactions occur andwhat mechanisms govern their progress. While thermodynamics tells us whether a reaction is favorable ($\Delta G < 0$), kinetics tells us how quickly equilibrium is approached.

The rate of reaction for a general reaction$\text{a}A + \text{b}B \rightarrow \text{c}C + \text{d}D$ is defined as:

$$r = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt}$$

The stoichiometric coefficients ensure a unique rate regardless of which species is monitored.

The rate law is an experimentally determined expression relating the rate to the concentrations of reactants:

$$\boxed{r = k[A]^m[B]^n}$$

Here $k$ is the rate constant, and $m$, $n$ are the orders with respect to $A$ and $B$. The overall order is $m + n$. Orders need not be integers and must be determined experimentally.

Experimental Methods

Spectroscopic Methods

UV-Vis absorption (Beer–Lambert law), IR spectroscopy, fluorescence. Concentration tracked in real time via absorbance changes.

Pressure Measurements

For gas-phase reactions where the total number of moles changes, pressure changes can be related to extent of reaction.

Conductimetry

Ionic reactions in solution monitored by electrical conductivity changes as ions are produced or consumed.

Stopped-Flow & Flash Photolysis

Fast mixing or pulsed initiation allows measurement of reactions with half-lives as short as microseconds.

2. Derivation 1: Integrated Rate Laws

For a reaction $A \rightarrow \text{products}$, the differential rate law takes the form $-d[A]/dt = k[A]^n$. We now integrate this for $n = 0, 1, 2$.

2.1 Zeroth-Order Kinetics ($n = 0$)

When the rate is independent of concentration (common for surface-catalyzed reactions at saturation):

$$-\frac{d[A]}{dt} = k$$

Separate variables and integrate from $t = 0$ to $t$:

$$\int_{[A]_0}^{[A]} d[A]' = -k\int_0^t dt'$$

$$\boxed{[A](t) = [A]_0 - kt}$$

A plot of $[A]$ vs $t$ is linear with slope $-k$. The reaction proceeds at constant rate until $[A]$ reaches zero.

The half-life is the time for $[A]$ to decrease to $[A]_0/2$:

$$t_{1/2} = \frac{[A]_0}{2k}$$

The half-life depends on the initial concentration — a signature of zeroth-order kinetics.

2.2 First-Order Kinetics ($n = 1$)

First-order reactions are ubiquitous: radioactive decay, isomerization, unimolecular decomposition.

$$-\frac{d[A]}{dt} = k[A]$$

Separate variables:

$$\int_{[A]_0}^{[A]} \frac{d[A]'}{[A]'} = -k\int_0^t dt'$$

$$\ln\!\left(\frac{[A]}{[A]_0}\right) = -kt$$

$$\boxed{[A](t) = [A]_0\,e^{-kt}}$$

Exponential decay. A plot of $\ln[A]$ vs $t$ gives a straight line with slope $-k$.

$$t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k}$$

The half-life is independent of initial concentration — the hallmark of first-order kinetics.

2.3 Second-Order Kinetics ($n = 2$)

Second-order in a single reactant is common for dimerization and bimolecular recombination.

$$-\frac{d[A]}{dt} = k[A]^2$$

Separate and integrate:

$$\int_{[A]_0}^{[A]} \frac{d[A]'}{[A]'^2} = -k\int_0^t dt'$$

$$\boxed{\frac{1}{[A](t)} = \frac{1}{[A]_0} + kt}$$

A plot of $1/[A]$ vs $t$ is linear with slope $k$.

$$t_{1/2} = \frac{1}{k[A]_0}$$

The half-life is inversely proportional to initial concentration.

2.4 Determining Reaction Order from Data

Order	Integrated Form	Linear Plot	Half-Life	Units of k
0	$[A] = [A]_0 - kt$	$[A]$ vs $t$	$[A]_0/2k$	mol/L/s
1	$\ln[A] = \ln[A]_0 - kt$	$\ln[A]$ vs $t$	$\ln 2/k$	1/s
2	$1/[A] = 1/[A]_0 + kt$	$1/[A]$ vs $t$	$1/(k[A]_0)$	L/mol/s

Practical Procedure

Plot the experimental data as $[A]$ vs $t$,$\ln[A]$ vs $t$, and $1/[A]$ vs $t$. The plot that yields a straight line determines the order. The slope gives $k$(with appropriate sign). Alternatively, examine how $t_{1/2}$ varies with initial concentration: constant means first order, proportional to $[A]_0$ means zeroth order, inversely proportional means second order.

3. Derivation 2: Method of Initial Rates

The method of initial rates is a systematic approach to determining the rate law when multiple reactants are involved. For the rate law $r = k[A]^m[B]^n$, we measure the initial rate $r_0$ for several experiments with different initial concentrations.

3.1 Isolating Individual Orders

The key principle: vary one concentration at a time while keeping all others constant. Consider two experiments where only $[A]_0$ changes:

Experiment 1: $r_1 = k[A]_1^m[B]_0^n$

Experiment 2: $r_2 = k[A]_2^m[B]_0^n$

Take the ratio:

$$\frac{r_2}{r_1} = \left(\frac{[A]_2}{[A]_1}\right)^m$$

Solving for $m$:

$$\boxed{m = \frac{\ln(r_2/r_1)}{\ln([A]_2/[A]_1)}}$$

The same procedure is repeated for $[B]$ (holding $[A]$ constant) to determine $n$. Once all orders are known, $k$ is found by substituting any experiment's data into the rate law.

3.2 Worked Example

Reaction: $2\text{NO}(g) + \text{Cl}_2(g) \rightarrow 2\text{NOCl}(g)$

Exp.	[NO]₀ (M)	[Cl₂]₀ (M)	r₀ (M/s)
1	0.10	0.10	$1.0 \times 10^{-3}$
2	0.20	0.10	$4.0 \times 10^{-3}$
3	0.10	0.20	$2.0 \times 10^{-3}$

Order in NO: Comparing Exp. 1 and 2 ($[Cl_2]$ constant): $m = \ln(4.0/1.0)/\ln(0.20/0.10) = \ln 4/\ln 2 = 2$

Order in Cl₂: Comparing Exp. 1 and 3 ($[NO]$ constant): $n = \ln(2.0/1.0)/\ln(0.20/0.10) = \ln 2/\ln 2 = 1$

Rate law: $r = k[\text{NO}]^2[\text{Cl}_2]$ (overall third order)

Rate constant: $k = r_1/([\text{NO}]_1^2[\text{Cl}_2]_1) = 1.0 \times 10^{-3}/(0.10^2 \times 0.10) = 1.0\;\text{L}^2\text{mol}^{-2}\text{s}^{-1}$

4. Derivation 3: Complex Reaction Mechanisms — Steady-State Approximation

Most reactions proceed through a series of elementary steps involving reactive intermediates. The steady-state approximation (SSA) is a powerful method for treating these mechanisms analytically.

4.1 Consecutive First-Order Reactions

Consider the sequential mechanism:

$$A \xrightarrow{k_1} B \xrightarrow{k_2} C$$

The exact kinetic equations are:

$$\frac{d[A]}{dt} = -k_1[A]$$

$$\frac{d[B]}{dt} = k_1[A] - k_2[B]$$

$$\frac{d[C]}{dt} = k_2[B]$$

The exact solution for $[A](t) = [A]_0 e^{-k_1 t}$ is straightforward. Substituting into the equation for $[B]$ and solving the first-order linear ODE:

$$[B](t) = \frac{k_1[A]_0}{k_2 - k_1}\left(e^{-k_1 t} - e^{-k_2 t}\right)$$

$$[C](t) = [A]_0 - [A](t) - [B](t)$$

4.2 Applying the Steady-State Approximation

When the intermediate $B$ is highly reactive ($k_2 \gg k_1$), it is consumed nearly as fast as it is formed. After a brief induction period, its concentration remains approximately constant. We set:

$$\frac{d[B]}{dt} \approx 0$$

Therefore:

$$k_1[A] - k_2[B]_{ss} = 0$$

$$\boxed{[B]_{ss} = \frac{k_1}{k_2}[A]}$$

The steady-state concentration of $B$ is proportional to $[A]$, with the ratio $k_1/k_2$ determining the (small) fraction of intermediate present.

The rate of product formation becomes:

$$\frac{d[C]}{dt} = k_2[B]_{ss} = k_2 \cdot \frac{k_1}{k_2}[A] = k_1[A]$$

The overall rate is governed by the slow step ($k_1$), the rate-determining step.

4.3 Validity of the SSA

The steady-state approximation is valid when: (1) the intermediate is highly reactive ($k_2 \gg k_1$), so its concentration is always small compared to reactant and product; (2) sufficient time has elapsed after the initial induction period ($t \gg 1/k_2$); (3) the intermediate does not accumulate ($[B]_{ss} \ll [A]$). The exact condition can be written as $k_1/k_2 \ll 1$. When $k_1 \approx k_2$, the SSA fails and the full analytical or numerical solution must be used.

5. Derivation 4: Pre-Equilibrium Approximation

When the first step is a fast, reversible equilibrium followed by a slow product-forming step, we use the pre-equilibrium approximation.

$$A \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} B \xrightarrow{k_2} C$$

The kinetic equations become:

$$\frac{d[A]}{dt} = -k_1[A] + k_{-1}[B]$$

$$\frac{d[B]}{dt} = k_1[A] - k_{-1}[B] - k_2[B]$$

$$\frac{d[C]}{dt} = k_2[B]$$

5.1 Deriving the Rate Law

The pre-equilibrium assumption states that the first step reaches equilibrium much faster than the second step proceeds ($k_1, k_{-1} \gg k_2$). Therefore:

Fast equilibrium established:

$$K_{eq} = \frac{k_1}{k_{-1}} = \frac{[B]_{eq}}{[A]_{eq}}$$

So the intermediate concentration is:

$$[B] = \frac{k_1}{k_{-1}}[A]$$

Substituting into the rate of product formation:

$$\boxed{r = \frac{d[C]}{dt} = k_2[B] = \frac{k_1 k_2}{k_{-1}}[A] = K_{eq}\,k_2\,[A]}$$

The effective rate constant is $k_{\text{eff}} = k_1 k_2/k_{-1}$.

5.2 Comparison with Steady-State Result

Applying the SSA to the same mechanism ($d[B]/dt = 0$):

$$k_1[A] - k_{-1}[B] - k_2[B] = 0$$

$$[B]_{ss} = \frac{k_1[A]}{k_{-1} + k_2}$$

$$r_{ss} = k_2[B]_{ss} = \frac{k_1 k_2}{k_{-1} + k_2}[A]$$

In the pre-equilibrium limit ($k_{-1} \gg k_2$), the denominator $k_{-1} + k_2 \approx k_{-1}$, recovering the pre-equilibrium result $r = k_1 k_2[A]/k_{-1}$. The steady-state result is more general.

6. Derivation 5: Lindemann–Hinshelwood Mechanism

The Lindemann mechanism (1922) explains how apparently unimolecular reactions can arise from bimolecular collisions. This was one of the great puzzles of early kinetics: how can a molecule “decide” to decompose on its own?

6.1 The Mechanism

Step 1 (Activation): $A + M \xrightarrow{k_1} A^* + M$

Step 2 (Deactivation): $A^* + M \xrightarrow{k_{-1}} A + M$

Step 3 (Reaction): $A^* \xrightarrow{k_2} \text{Products}$

$M$ is any collision partner (bath gas molecule), and $A^*$ is the energized molecule with sufficient internal energy to react.

6.2 Derivation of the Rate Law

Apply the steady-state approximation to the energized intermediate $A^*$:

$$\frac{d[A^*]}{dt} = k_1[A][M] - k_{-1}[A^*][M] - k_2[A^*] = 0$$

Solve for $[A^*]_{ss}$:

$$[A^*]_{ss} = \frac{k_1[A][M]}{k_{-1}[M] + k_2}$$

The overall rate is:

$$\boxed{r = k_2[A^*]_{ss} = \frac{k_1 k_2 [A][M]}{k_{-1}[M] + k_2}}$$

6.3 Limiting Behaviors

High Pressure ($k_{-1}[M] \gg k_2$)

$$r \approx \frac{k_1 k_2}{k_{-1}}[A] = k_\infty [A]$$

First-order kinetics. Collisions are frequent, so deactivation dominates over reaction. The rate is independent of $[M]$, and the effective rate constant is $k_\infty = k_1 k_2/k_{-1}$.

Low Pressure ($k_2 \gg k_{-1}[M]$)

$$r \approx k_1[A][M]$$

Second-order kinetics. Collisions are rare. Every energized molecule reacts before it can be deactivated. The activation step is rate-determining.

We can define the effective unimolecular rate constant:

$$k_{\text{uni}} = \frac{k_1 k_2 [M]}{k_{-1}[M] + k_2}$$

such that $r = k_{\text{uni}}[A]$. At high pressure,$k_{\text{uni}} \to k_\infty = k_1 k_2/k_{-1}$ (constant). At low pressure,$k_{\text{uni}} \to k_1[M]$ (decreases linearly with pressure).

The Lindemann Plot

Taking the reciprocal: $1/k_{\text{uni}} = 1/k_\infty + k_2/(k_1 k_2 [M]) = 1/k_\infty + 1/(k_1[M])$. A plot of $1/k_{\text{uni}}$ vs $1/[M]$ is linear with intercept $1/k_\infty$ and slope $1/k_1$. While the Lindemann mechanism captures the qualitative fall-off behavior, Hinshelwood improved it by accounting for the energy distribution of molecules using statistical mechanics.

7. Applications

Atmospheric Chemistry: Ozone Depletion

The Chapman cycle for stratospheric ozone involves consecutive and competing reactions. Chlorine radicals from CFCs act as catalysts in the chain mechanism:

$\text{Cl} + \text{O}_3 \rightarrow \text{ClO} + \text{O}_2$

$\text{ClO} + \text{O} \rightarrow \text{Cl} + \text{O}_2$

The steady-state approximation applied to ClO gives the overall catalytic rate. A single Cl atom can destroy ~100,000 ozone molecules.

Pharmacokinetics

Drug elimination typically follows first-order kinetics. The plasma concentration of a drug after intravenous administration:

$C(t) = C_0\,e^{-k_e t}$

where $k_e$ is the elimination rate constant. The half-life $t_{1/2} = \ln 2/k_e$ determines dosing intervals. Aspirin: $t_{1/2} \approx 3.5$ hours. Warfarin: $t_{1/2} \approx 40$ hours.

Radioactive Decay

Nuclear decay is the prototypical first-order process. The decay constant $\lambda$ is independent of temperature, pressure, and chemical environment:

$N(t) = N_0\,e^{-\lambda t}$

Carbon-14 dating uses $t_{1/2} = 5730$ years. Uranium-238 has $t_{1/2} = 4.47 \times 10^9$ years, useful for geological dating.

Catalysis

Heterogeneous catalysis on surfaces often exhibits zeroth-order kinetics at high reactant pressures (surface saturation), transitioning to first-order at low pressures (Langmuir kinetics):

$r = \frac{k K [A]}{1 + K[A]}$

The Langmuir–Hinshelwood mechanism combines adsorption equilibria with surface reaction kinetics, widely used in industrial process design.

8. Historical Context

Jacobus van't Hoff (1884) — PublishedÉtudes de Dynamique Chimique, establishing the mathematical foundations of chemical kinetics. Introduced the concepts of reaction order and the temperature dependence of rate constants (later formalized by Arrhenius). Awarded the first Nobel Prize in Chemistry (1901).

Max Bodenstein (1871–1942) — Pioneer of gas-phase kinetics. Studied the H₂ + Br₂ reaction in detail and proposed the chain mechanism concept. His meticulous experimental work on photochemical reactions laid the groundwork for the steady-state approximation.

Frederick Lindemann (1922) — Proposed the collisional activation mechanism for unimolecular reactions at the Faraday Society meeting. His simple two-step model explained how molecules acquire the energy needed for decomposition through bimolecular collisions. Later became Viscount Cherwell and served as Churchill's chief scientific advisor.

Cyril Hinshelwood (1897–1967) — Extended Lindemann's theory by incorporating statistical mechanics to describe the energy distribution of activated molecules. Shared the 1956 Nobel Prize in Chemistry with Semenov for research into the mechanism of chemical reactions, particularly chain reactions.

9. Python Simulation: Reaction Kinetics

The following simulation solves the kinetic ODEs numerically using the Euler method (no external ODE solvers), and plots concentration profiles for first-order, second-order, and consecutive reactions.

Reaction Kinetics: Integrated Rate Laws & Consecutive Reactions

Python

Solves and plots concentration vs time for zeroth, first, second order, and consecutive A->B->C reactions using Euler method

reaction_kinetics.py175 lines

#!/usr/bin/env python3
"""
reaction_kinetics.py
Numerical simulation of reaction rate laws and concentration profiles.
Uses Euler method for ODE integration (no scipy).
"""
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ============================================================
# Parameters
# ============================================================
t_max = 10.0       # Total time (arbitrary units)
dt = 0.001         # Time step for Euler integration
nsteps = int(t_max / dt)
t = np.linspace(0, t_max, nsteps + 1)

A0 = 1.0           # Initial concentration of A
k1_first = 0.5     # First-order rate constant
k2_second = 0.8    # Second-order rate constant

# Consecutive reaction parameters: A --(k_a)--> B --(k_b)--> C
k_a = 0.6
k_b = 0.15

# ============================================================
# 1. Analytical solutions for comparison
# ============================================================
# First order: [A] = A0 * exp(-k*t)
A_first_exact = A0 * np.exp(-k1_first * t)

# Second order: 1/[A] = 1/A0 + k*t  =>  [A] = A0 / (1 + k*A0*t)
A_second_exact = A0 / (1.0 + k2_second * A0 * t)

# Consecutive: exact solutions
A_consec_exact = A0 * np.exp(-k_a * t)
B_consec_exact = A0 * k_a / (k_b - k_a) * (np.exp(-k_a * t) - np.exp(-k_b * t))
C_consec_exact = A0 - A_consec_exact - B_consec_exact

# ============================================================
# 2. Euler method numerical solutions
# ============================================================
# First order
A_first_num = np.zeros(nsteps + 1)
A_first_num[0] = A0
for i in range(nsteps):
    A_first_num[i + 1] = A_first_num[i] + dt * (-k1_first * A_first_num[i])

# Second order
A_second_num = np.zeros(nsteps + 1)
A_second_num[0] = A0
for i in range(nsteps):
    A_second_num[i + 1] = A_second_num[i] + dt * (-k2_second * A_second_num[i]**2)

# Consecutive reactions
A_con = np.zeros(nsteps + 1)
B_con = np.zeros(nsteps + 1)
C_con = np.zeros(nsteps + 1)
A_con[0] = A0
B_con[0] = 0.0
C_con[0] = 0.0
for i in range(nsteps):
    dA = -k_a * A_con[i]
    dB = k_a * A_con[i] - k_b * B_con[i]
    dC = k_b * B_con[i]
    A_con[i + 1] = A_con[i] + dt * dA
    B_con[i + 1] = B_con[i] + dt * dB
    C_con[i + 1] = C_con[i] + dt * dC

# ============================================================
# 3. Half-life calculations
# ============================================================
# First order
t_half_first = np.log(2) / k1_first

# Second order
t_half_second = 1.0 / (k2_second * A0)

# Find numerical half-life for consecutive (time when [A] = A0/2)
idx_half = np.argmin(np.abs(A_con - A0 / 2))
t_half_consec_A = t[idx_half]

# Find time of max [B] in consecutive
idx_max_B = np.argmax(B_con)
t_max_B = t[idx_max_B]

print("=" * 60)
print("    REACTION RATE LAWS: NUMERICAL RESULTS")
print("=" * 60)
print(f"\nFirst-order reaction (k = {k1_first} /s):")
print(f"  Half-life (analytical): t_1/2 = ln(2)/k = {t_half_first:.4f} s")
print(f"  [A] at t=5s (exact):    {A0 * np.exp(-k1_first * 5):.6f} M")
print(f"  [A] at t=5s (Euler):    {A_first_num[int(5/dt)]:.6f} M")
print(f"\nSecond-order reaction (k = {k2_second} L/mol/s, [A]_0 = {A0} M):")
print(f"  Half-life (analytical): t_1/2 = 1/(k*[A]_0) = {t_half_second:.4f} s")
print(f"  [A] at t=5s (exact):    {A0 / (1 + k2_second * A0 * 5):.6f} M")
print(f"  [A] at t=5s (Euler):    {A_second_num[int(5/dt)]:.6f} M")
print(f"\nConsecutive reactions A -> B -> C:")
print(f"  k_a = {k_a} /s,  k_b = {k_b} /s")
print(f"  Half-life of A: {t_half_consec_A:.4f} s")
print(f"  Time of max [B]: {t_max_B:.4f} s")
print(f"  Max [B]: {B_con[idx_max_B]:.6f} M")
print(f"  [A]+[B]+[C] at t=10s: {A_con[-1]+B_con[-1]+C_con[-1]:.6f} M (conservation check)")

# ============================================================
# 4. Plotting
# ============================================================
fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor('#0f172a')
fig.suptitle('Reaction Rate Laws: Concentration Profiles',
             color='#e2e8f0', fontsize=15, fontweight='bold', y=0.98)

for ax in axes.flat:
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    for spine in ax.spines.values():
        spine.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4)

# Panel 1: First-order decay
ax = axes[0, 0]
ax.plot(t, A_first_exact, color='#f87171', linewidth=2, label='Exact')
ax.plot(t[::500], A_first_num[::500], 'o', color='#fbbf24', markersize=4, label='Euler')
ax.axhline(y=A0/2, color='#94a3b8', linestyle='--', alpha=0.5)
ax.axvline(x=t_half_first, color='#94a3b8', linestyle='--', alpha=0.5, label=f't_1/2 = {t_half_first:.2f}')
ax.set_xlabel('Time (s)', color='#cbd5e1')
ax.set_ylabel('[A] (M)', color='#cbd5e1')
ax.set_title('First-Order Decay', color='#e2e8f0', fontweight='bold')
ax.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=8)

# Panel 2: Second-order decay
ax = axes[0, 1]
ax.plot(t, A_second_exact, color='#60a5fa', linewidth=2, label='Exact')
ax.plot(t[::500], A_second_num[::500], 's', color='#34d399', markersize=4, label='Euler')
ax.axhline(y=A0/2, color='#94a3b8', linestyle='--', alpha=0.5)
ax.axvline(x=t_half_second, color='#94a3b8', linestyle='--', alpha=0.5, label=f't_1/2 = {t_half_second:.2f}')
ax.set_xlabel('Time (s)', color='#cbd5e1')
ax.set_ylabel('[A] (M)', color='#cbd5e1')
ax.set_title('Second-Order Decay', color='#e2e8f0', fontweight='bold')
ax.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=8)

# Panel 3: Consecutive reactions
ax = axes[1, 0]
ax.plot(t, A_con, color='#f87171', linewidth=2, label='[A] (Euler)')
ax.plot(t, B_con, color='#fbbf24', linewidth=2, label='[B] (Euler)')
ax.plot(t, C_con, color='#34d399', linewidth=2, label='[C] (Euler)')
ax.plot(t[::500], A_consec_exact[::500], 'x', color='#f87171', markersize=5, alpha=0.6)
ax.plot(t[::500], B_consec_exact[::500], 'x', color='#fbbf24', markersize=5, alpha=0.6)
ax.plot(t[::500], C_consec_exact[::500], 'x', color='#34d399', markersize=5, alpha=0.6)
ax.axvline(x=t_max_B, color='#94a3b8', linestyle=':', alpha=0.5, label=f't_max(B) = {t_max_B:.2f}')
ax.set_xlabel('Time (s)', color='#cbd5e1')
ax.set_ylabel('Concentration (M)', color='#cbd5e1')
ax.set_title('Consecutive: A -> B -> C', color='#e2e8f0', fontweight='bold')
ax.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=8)

# Panel 4: Comparison of orders (log scale)
ax = axes[1, 1]
# Zeroth order
k_zero = A0 / (2 * t_half_first)  # same half-life as first order for comparison
A_zero = np.maximum(A0 - k_zero * t, 0)
ax.plot(t, A_zero, color='#c084fc', linewidth=2, label='Zeroth order')
ax.plot(t, A_first_exact, color='#f87171', linewidth=2, label='First order')
ax.plot(t, A_second_exact, color='#60a5fa', linewidth=2, label='Second order')
ax.set_xlabel('Time (s)', color='#cbd5e1')
ax.set_ylabel('[A] (M)', color='#cbd5e1')
ax.set_title('Comparison of Reaction Orders', color='#e2e8f0', fontweight='bold')
ax.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=8)
ax.set_ylim(-0.05, 1.1)

plt.tight_layout(rect=[0, 0, 1, 0.95])
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())
print("\nPlot saved to output.png")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

10. Fortran Simulation: Half-Lives and Rate Constants

The Fortran code computes half-lives, rate constants, and concentration profiles for various reaction orders with high numerical precision.

Fortran + Python: Half-Lives and Rate Constants

Python

Compiles and runs a Fortran program computing half-lives, rate constants, and concentration profiles for zeroth, first, and second order reactions

reaction_rates_fortran.py219 lines

import subprocess, sys, os, csv, io
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import numpy as np

# --- Embedded Fortran source ---
fortran_source = r"""
program reaction_rates
  implicit none
  integer, parameter :: dp = selected_real_kind(15, 307)

! Parameters
  real(dp), parameter :: A0 = 1.0d0      ! Initial concentration (M)
  real(dp), parameter :: t_max = 10.0d0   ! Max time (s)
  integer, parameter  :: nsteps = 100000
  real(dp), parameter :: dt = t_max / real(nsteps, dp)

! Rate constants
  real(dp), parameter :: k_zero  = 0.08d0   ! Zeroth order (M/s)
  real(dp), parameter :: k_first = 0.5d0    ! First order (1/s)
  real(dp), parameter :: k_second = 0.8d0   ! Second order (L/mol/s)

! Variables
  real(dp) :: t_half_zero, t_half_first, t_half_second
  real(dp) :: A_z, A_f, A_s, t_curr
  integer :: i

! Half-life calculations (analytical)
  t_half_zero   = A0 / (2.0d0 * k_zero)
  t_half_first  = log(2.0d0) / k_first
  t_half_second = 1.0d0 / (k_second * A0)

write(*,'(A)') '============================================================'
  write(*,'(A)') '  REACTION RATE LAWS: FORTRAN HIGH-PRECISION COMPUTATION'
  write(*,'(A)') '============================================================'
  write(*,'(A)') ''
  write(*,'(A)')          '  Reaction Order Analysis'
  write(*,'(A)')          '  -----------------------'
  write(*,'(A,F12.6,A)')  '  Zeroth order  (k = 0.08 M/s):    t_1/2 = ', t_half_zero, ' s'
  write(*,'(A,F12.6,A)')  '  First order   (k = 0.50 1/s):    t_1/2 = ', t_half_first, ' s'
  write(*,'(A,F12.6,A)')  '  Second order  (k = 0.80 L/mol/s):t_1/2 = ', t_half_second, ' s'
  write(*,'(A)') ''
  write(*,'(A)')          '  Verification: [A] at t = t_1/2'
  write(*,'(A,F12.8,A)')  '  Zeroth:  [A] = ', A0 - k_zero * t_half_zero, ' M (expect 0.5)'
  write(*,'(A,F12.8,A)')  '  First:   [A] = ', A0 * exp(-k_first * t_half_first), ' M (expect 0.5)'
  write(*,'(A,F12.8,A)')  '  Second:  [A] = ', A0 / (1.0d0 + k_second * A0 * t_half_second), ' M (expect 0.5)'
  write(*,'(A)') ''

! Numerical integration: successive half-lives for each order
  write(*,'(A)') '  Successive Half-Lives (numerical, Euler):'
  write(*,'(A)') '  ----------------------------------------'

! First order: should have constant half-life
  write(*,'(A)') '  First-order successive half-lives:'
  A_f = A0
  t_curr = 0.0d0
  do i = 1, 5
    call find_half_life_first(A_f, k_first, dt, t_curr, A_f, t_curr)
    write(*,'(A,I1,A,F10.6,A,F10.6,A)') '    t_1/2(', i, ') = ', t_curr, &
      ' s,  [A] = ', A_f, ' M'
  end do

! Second order: half-life should double each time
  write(*,'(A)') '  Second-order successive half-lives:'
  A_s = A0
  t_curr = 0.0d0
  do i = 1, 5
    call find_half_life_second(A_s, k_second, dt, t_curr, A_s, t_curr)
    write(*,'(A,I1,A,F10.6,A,F10.6,A)') '    t_1/2(', i, ') = ', t_curr, &
      ' s,  [A] = ', A_s, ' M'
  end do

write(*,'(A)') ''

! Output DATA for plotting
  write(*,'(A)') 'DATA_START'
  write(*,'(A)') 'time,A_zero,A_first,A_second'
  do i = 0, nsteps, 100
    t_curr = real(i, dp) * dt
    A_z = max(A0 - k_zero * t_curr, 0.0d0)
    A_f = A0 * exp(-k_first * t_curr)
    A_s = A0 / (1.0d0 + k_second * A0 * t_curr)
    write(*,'(F8.4,A,F10.6,A,F10.6,A,F10.6)') t_curr, ',', A_z, ',', A_f, ',', A_s
  end do
  write(*,'(A)') 'DATA_END'

write(*,'(A)') ''
  write(*,'(A)') '============================================================'

contains

subroutine find_half_life_first(A_start, k, h, t_start_in, A_end, t_end)
    real(dp), intent(in)  :: A_start, k, h, t_start_in
    real(dp), intent(out) :: A_end, t_end
    real(dp) :: A_curr, t_curr, target
    target = A_start / 2.0d0
    A_curr = A_start
    t_curr = 0.0d0
    do while (A_curr > target .and. t_curr < 100.0d0)
      A_curr = A_curr - h * k * A_curr
      t_curr = t_curr + h
    end do
    A_end = A_curr
    t_end = t_curr
  end subroutine

subroutine find_half_life_second(A_start, k, h, t_start_in, A_end, t_end)
    real(dp), intent(in)  :: A_start, k, h, t_start_in
    real(dp), intent(out) :: A_end, t_end
    real(dp) :: A_curr, t_curr, target
    target = A_start / 2.0d0
    A_curr = A_start
    t_curr = 0.0d0
    do while (A_curr > target .and. t_curr < 1000.0d0)
      A_curr = A_curr - h * k * A_curr**2
      t_curr = t_curr + h
    end do
    A_end = A_curr
    t_end = t_curr
  end subroutine

end program reaction_rates
"""

# Write, compile, and run the Fortran code
with open('reaction_rates.f90', 'w') as f:
    f.write(fortran_source)

comp = subprocess.run(['gfortran', '-O2', '-o', './reaction_rates', 'reaction_rates.f90'],
                      capture_output=True, text=True)
if comp.returncode != 0:
    print("Compilation error:")
    print(comp.stderr)
    sys.exit(1)

result = subprocess.run(['./reaction_rates'], capture_output=True, text=True, timeout=30)
stdout = result.stdout

# Print text output (outside DATA markers)
in_data = False
data_lines = []
for line in stdout.splitlines():
    if line.strip() == 'DATA_START':
        in_data = True
        continue
    elif line.strip() == 'DATA_END':
        in_data = False
        continue
    if in_data:
        data_lines.append(line)
    else:
        print(line)

# Parse CSV data
reader = csv.reader(data_lines)
header = next(reader)
rows = []
for row in reader:
    try:
        rows.append([float(x) for x in row])
    except ValueError:
        continue

data = np.array(rows)
t_arr     = data[:, 0]
A_zero    = data[:, 1]
A_first   = data[:, 2]
A_second  = data[:, 3]

# --- Plot with dark theme ---
fig, axes = plt.subplots(1, 2, figsize=(14, 5))
fig.patch.set_facecolor('#0f172a')
fig.suptitle('Reaction Rate Laws: Fortran Computation',
             color='#e2e8f0', fontsize=15, fontweight='bold', y=1.02)

# Panel 1: Concentration vs time
ax = axes[0]
ax.plot(t_arr, A_zero, color='#c084fc', linewidth=2, label='Zeroth order')
ax.plot(t_arr, A_first, color='#f87171', linewidth=2, label='First order')
ax.plot(t_arr, A_second, color='#60a5fa', linewidth=2, label='Second order')
ax.axhline(y=0.5, color='#94a3b8', linestyle='--', alpha=0.5, label='[A] = A0/2')
ax.set_xlabel('Time (s)', color='#cbd5e1')
ax.set_ylabel('[A] (M)', color='#cbd5e1')
ax.set_title('Concentration Profiles (Fortran)', color='#e2e8f0', fontweight='bold')
ax.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)
ax.set_ylim(-0.05, 1.1)

# Panel 2: Linear plots for order determination
ax = axes[1]
# First order: ln[A] vs t should be linear
ln_A_first = np.log(np.maximum(A_first, 1e-15))
ax.plot(t_arr, ln_A_first, color='#f87171', linewidth=2, label='ln[A] (1st order)')
# Second order: 1/[A] vs t should be linear
inv_A_second = 1.0 / np.maximum(A_second, 1e-15)
ax2 = ax.twinx()
ax2.plot(t_arr, inv_A_second, color='#60a5fa', linewidth=2, linestyle='--', label='1/[A] (2nd order)')
ax.set_xlabel('Time (s)', color='#cbd5e1')
ax.set_ylabel('ln[A]', color='#f87171')
ax2.set_ylabel('1/[A] (L/mol)', color='#60a5fa')
ax2.tick_params(colors='#60a5fa')
ax.tick_params(axis='y', colors='#f87171')
ax.set_title('Linear Plots for Order Determination', color='#e2e8f0', fontweight='bold')
lines1, labels1 = ax.get_legend_handles_labels()
lines2, labels2 = ax2.get_legend_handles_labels()
ax.legend(lines1 + lines2, labels1 + labels2,
          facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())
print("\nPlot saved to output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Compilation & Output

$ gfortran -O2 -o reaction_rates reaction_rates.f90

$ ./reaction_rates

# Computes analytical half-lives and successive half-lives via Euler method

# Verifies that first-order half-life is constant, second-order doubles

The Fortran implementation achieves double-precision accuracy throughout, and the successive half-life computation demonstrates the characteristic behavior of each reaction order.

Summary of Key Results

Integrated Rate Laws

Zeroth: $[A] = [A]_0 - kt$ | First: $[A] = [A]_0\,e^{-kt}$ | Second: $1/[A] = 1/[A]_0 + kt$

Method of Initial Rates

$$m = \frac{\ln(r_2/r_1)}{\ln([A]_2/[A]_1)}$$

Steady-State Approximation

$$\text{For } A \to B \to C: \quad [B]_{ss} = \frac{k_1}{k_2}[A], \quad r = k_1[A]$$

Pre-Equilibrium Approximation

$$\text{For } A \rightleftharpoons B \to C: \quad r = \frac{k_1 k_2}{k_{-1}}[A]$$

Lindemann–Hinshelwood Mechanism

$$k_{\text{uni}} = \frac{k_1 k_2 [M]}{k_{-1}[M] + k_2} \quad \xrightarrow{\text{high }P}\quad \frac{k_1 k_2}{k_{-1}} \quad \xrightarrow{\text{low }P}\quad k_1[M]$$

Bibliography

Textbooks & Monographs

Atkins, P. & de Paula, J. (2023). Atkins' Physical Chemistry, 12th ed. Oxford University Press. — Standard reference for integrated rate laws and reaction mechanisms.
Steinfeld, J.I., Francisco, J.S. & Hase, W.L. (1999). Chemical Kinetics and Dynamics, 2nd ed. Prentice Hall. — Comprehensive treatment of kinetic theory including Lindemann–Hinshelwood theory.
Houston, P.L. (2006). Chemical Kinetics and Reaction Dynamics. Dover. — Excellent graduate-level treatment of steady-state and pre-equilibrium approximations.
Laidler, K.J. (1987). Chemical Kinetics, 3rd ed. Harper & Row. — Classic textbook by a pioneer of modern kinetics, with detailed historical accounts.

Key Papers

van't Hoff, J.H. (1884). Études de Dynamique Chimique. Frederik Muller. — Founding work establishing mathematical rate laws.
Lindemann, F.A. (1922). “Discussion on the radiation theory of chemical action,” Trans. Faraday Soc. 17, 598–606. — Original proposal of the collisional activation mechanism.
Hinshelwood, C.N. (1926). “On the theory of unimolecular reactions,” Proc. R. Soc. Lond. A 113, 230–233. — Extension of Lindemann theory using statistical mechanics.

← Part III Overview Back to Chemical Kinetics →

Order	Integrated Form	Linear Plot	Half-Life	Units of k
0	\([A] = [A]_0 - kt\)	\([A]\) vs \(t\)	\([A]_0/2k\)	mol/L/s
1	\(\ln[A] = \ln[A]_0 - kt\)	\(\ln[A]\) vs \(t\)	\(\ln 2/k\)	1/s
2	\(1/[A] = 1/[A]_0 + kt\)	\(1/[A]\) vs \(t\)	\(1/(k[A]_0)\)	L/mol/s