Part III: Chemical Kinetics | Chapter 10

Arrhenius & Collision Theory

A rigorous derivation of the Arrhenius equation from empirical kinetic data, collision theory from the kinetic molecular model, and the Boltzmann energy distribution that connects them

1. Introduction: Temperature Dependence of Reaction Rates

One of the most fundamental observations in chemical kinetics is that reaction rates increase dramatically with temperature. A rough rule of thumb holds that many reactions double their rate for every 10 K rise in temperature near room conditions. This seemingly simple observation conceals deep physics: the interplay between molecular motion, energy distributions, and the geometric requirements for reactive encounters.

The Central Question

If we measure the rate constant \(k\) for a reaction at several temperatures, we invariably find that \(\ln k\) plotted against \(1/T\) yields an approximately straight line with negative slope. Why? What molecular-level mechanism produces this universal behavior?

The answer lies in the Maxwell–Boltzmann energy distribution: only a fraction of molecular collisions possess sufficient energy to surmount the activation barrier, and that fraction grows exponentially with temperature.

Experimentally, the temperature dependence of the rate constant \(k(T)\) for an elementary reaction is captured by the Arrhenius equation:

\[ k(T) = A \, e^{-E_a / RT} \]

where \(A\) is the pre-exponential (frequency) factor, \(E_a\) is the activation energy, and \(R = 8.314 \;\text{J mol}^{-1}\text{K}^{-1}\) is the gas constant.

In this chapter, we derive the Arrhenius equation from its empirical origins, then provide a molecular justification through collision theory. We examine the Boltzmann factor, collision cross sections, steric factors, and the limits of the hard-sphere model. We conclude with computational explorations in Python and Fortran.

2. Derivation 1: The Arrhenius Equation

2.1 The Empirical Starting Point

In 1889, Svante Arrhenius noted that for a wide range of reactions, the temperature dependence of the rate constant could be described by the differential relation:

\[ \frac{d(\ln k)}{dT} = \frac{E_a}{RT^2} \]

Arrhenius's differential equation (1889). Here \(E_a\) is treated as a constant independent of temperature.

This relation was motivated by van't Hoff's equation for the temperature dependence of equilibrium constants. For an equilibrium \(A \rightleftharpoons B\) with forward rate constant \(k_f\) and reverse rate constant \(k_r\), the equilibrium constant \(K = k_f/k_r\) satisfies:

\[ \frac{d(\ln K)}{dT} = \frac{\Delta H^\circ}{RT^2} \quad \Longrightarrow \quad \frac{d(\ln k_f)}{dT} - \frac{d(\ln k_r)}{dT} = \frac{\Delta H^\circ}{RT^2} \]

Arrhenius proposed that each rate constant individually satisfies a relation of this form, with its own characteristic energy parameter \(E_a\).

2.2 Integration to the Arrhenius Form

If \(E_a\) is constant (independent of \(T\)), we can integrate directly. Separating variables:

\[ \int d(\ln k) = \int \frac{E_a}{RT^2} \, dT \]

The right-hand side integrates as:

\[ \ln k = -\frac{E_a}{RT} + C \]

where \(C\) is the integration constant. Exponentiating both sides and identifying \(A = e^C\):

\[ \boxed{k(T) = A \, e^{-E_a/(RT)}} \]

The Arrhenius equation

2.3 The Arrhenius Plot

Taking the natural logarithm of the Arrhenius equation gives the linearized form:

\[ \ln k = \ln A - \frac{E_a}{R} \cdot \frac{1}{T} \]

A plot of \(\ln k\) versus \(1/T\) (the Arrhenius plot) yields a straight line with slope \(-E_a/R\) and intercept \(\ln A\). This provides a powerful experimental method to determine both \(E_a\) and \(A\)from kinetic measurements at multiple temperatures.

Pre-exponential Factor A

The factor \(A\) has the same units as \(k\) and represents the rate constant in the hypothetical limit \(T \to \infty\). For bimolecular reactions, typical values are \(A \sim 10^{10}\text{–}10^{11} \; \text{L mol}^{-1}\text{s}^{-1}\).

Collision theory identifies \(A\) with the collision frequency (times the steric factor), providing a molecular interpretation.

Activation Energy E_a

The activation energy \(E_a\) represents the minimum kinetic energy (along the reaction coordinate) that reactant molecules must possess for a collision to be reactive. For typical reactions, \(E_a \sim 40\text{–}200 \; \text{kJ mol}^{-1}\).

Reactions with \(E_a < 40 \; \text{kJ mol}^{-1}\) are considered “fast” and those with \(E_a > 200 \; \text{kJ mol}^{-1}\) are negligibly slow at room temperature.

2.4 Two-Point Form

When data at only two temperatures are available, the Arrhenius equation gives:

\[ \ln\!\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right) \]

Two-point Arrhenius equation, obtained by subtracting \(\ln k_1 = \ln A - E_a/(RT_1)\) from\(\ln k_2 = \ln A - E_a/(RT_2)\).

3. Derivation 2: Collision Theory of Bimolecular Reactions

3.1 The Kinetic Theory Foundation

Collision theory provides a molecular-level derivation of the rate constant for bimolecular gas-phase reactions of the form \(A + B \to \text{products}\). The fundamental idea: for a reaction to occur, molecules A and B must (i) collide, and (ii) possess sufficient kinetic energy along the line of centers to overcome the activation barrier.

From kinetic molecular theory, two species A and B with number densities \(n_A\)and \(n_B\) collide at a rate per unit volume given by:

\[ Z_{AB} = n_A \, n_B \, \sigma_{AB} \, \langle v_{\text{rel}} \rangle \]

Collision frequency (collisions per unit volume per unit time)

where \(\sigma_{AB}\) is the collision cross section and \(\langle v_{\text{rel}} \rangle\) is the mean relative speed of the two species.

3.2 Mean Relative Speed

For a Maxwell–Boltzmann distribution, the mean relative speed of two species with masses\(m_A\) and \(m_B\) is:

\[ \langle v_{\text{rel}} \rangle = \sqrt{\frac{8 k_B T}{\pi \mu}} \]

where \(\mu = m_A m_B / (m_A + m_B)\) is the reduced mass and \(k_B\) is Boltzmann's constant.

This result follows from the fact that the relative velocity vector \(\mathbf{v}_{\text{rel}} = \mathbf{v}_A - \mathbf{v}_B\) is itself Maxwell–Boltzmann distributed with the reduced mass \(\mu\) replacing the single-particle mass.

3.3 Including the Energy Threshold

Not every collision leads to reaction. Only collisions where the kinetic energy along the line of centers exceeds \(E_a\) are reactive. The fraction of such collisions is the Boltzmann factor \(e^{-E_a/(k_B T)}\) (derived in Section 4). The reactive collision rate per unit volume is therefore:

\[ Z_{\text{reactive}} = n_A \, n_B \, \sigma_{AB} \, \sqrt{\frac{8 k_B T}{\pi \mu}} \, e^{-E_a/(k_B T)} \]

Since the macroscopic rate law gives \(\text{rate} = k \, [A][B]\) and \(n_i = N_A [i]\), we identify the bimolecular rate constant:

\[ \boxed{k = N_A \, \sigma_{AB} \, \sqrt{\frac{8 k_B T}{\pi \mu}} \, e^{-E_a/(RT)}} \]

Collision theory rate constant (note: \(E_a/(k_B T) = E_a/(RT)\) when \(E_a\) is expressed per mole)

3.4 The Steric Factor

Collision theory systematically overestimates rate constants, often by orders of magnitude. The reason: not all collisions with sufficient energy lead to reaction. Molecules must also collide with the correct relative orientation. We introduce the steric factor \(p\):

\[ k = p \, N_A \, \sigma_{AB} \, \sqrt{\frac{8 k_B T}{\pi \mu}} \, e^{-E_a/(RT)} \]

where \(0 < p \leq 1\) corrects for orientational requirements

For simple atom–atom reactions (e.g., \(\text{K} + \text{Br}_2\)), \(p \approx 1\). For complex molecular reactions (e.g., \(\text{C}_2\text{H}_4 + \text{HCl}\)), steric factors can be as small as \(p \sim 10^{-6}\). Transition state theory provides a more rigorous treatment of these geometric constraints through the entropy of activation.

4. Derivation 3: Energy Distribution and the Boltzmann Factor

4.1 The Maxwell–Boltzmann Energy Distribution

The Maxwell–Boltzmann speed distribution in three dimensions gives the probability that a molecule has translational kinetic energy between \(\varepsilon\) and \(\varepsilon + d\varepsilon\):

\[ f(\varepsilon) \, d\varepsilon = \frac{2\pi}{(\pi k_B T)^{3/2}} \, \varepsilon^{1/2} \, e^{-\varepsilon/(k_B T)} \, d\varepsilon \]

Energy distribution function for a 3D Maxwell–Boltzmann gas

However, for a bimolecular collision, what matters is the kinetic energy along the line of centers (the component of relative kinetic energy directed along the inter-molecular axis at the moment of collision). This is a one-dimensional problem.

4.2 Fraction of Collisions Above the Threshold

We need the fraction of collisions where the kinetic energy along the line of centers exceeds\(E_a\). Starting from the collision theory framework, the flux of molecules through a collision cross section with relative kinetic energy \(\varepsilon_r\) is weighted by the Boltzmann distribution. The fraction of collisions with energy\(\varepsilon_r \geq E_a\) is:

\[ f(\varepsilon_r \geq E_a) = \frac{\displaystyle \int_{E_a}^{\infty} \varepsilon_r \, e^{-\varepsilon_r/(k_B T)} \, d\varepsilon_r}{\displaystyle \int_{0}^{\infty} \varepsilon_r \, e^{-\varepsilon_r/(k_B T)} \, d\varepsilon_r} \]

The denominator evaluates to \((k_B T)^2\) (a standard Gamma function integral). For the numerator, we use integration by parts. Let \(u = \varepsilon_r/(k_B T)\):

\[ \int_{E_a}^{\infty} \varepsilon_r \, e^{-\varepsilon_r/(k_B T)} \, d\varepsilon_r = (k_B T)^2 \left(1 + \frac{E_a}{k_B T}\right) e^{-E_a/(k_B T)} \]

Therefore the fraction is:

\[ f(\varepsilon_r \geq E_a) = \left(1 + \frac{E_a}{k_B T}\right) e^{-E_a/(k_B T)} \]

4.3 The High-Barrier Approximation

For chemically relevant conditions where \(E_a \gg k_B T\) (which is almost always satisfied — at 300 K, \(k_B T \approx 2.5 \; \text{kJ mol}^{-1}\) while typical activation energies are 40–200 kJ/mol):

\[ \frac{E_a}{k_B T} \gg 1 \implies \left(1 + \frac{E_a}{k_B T}\right) \approx \frac{E_a}{k_B T} \]

But even this prefactor is absorbed into the pre-exponential factor (which already contains a\(\sqrt{T}\) dependence from collision theory). To leading order, the essential temperature dependence is dominated by the exponential:

\[ \boxed{f(\varepsilon_r \geq E_a) \approx e^{-E_a/(k_B T)} \quad \text{for } E_a \gg k_B T} \]

The Boltzmann factor: the fundamental origin of the Arrhenius exponential

This result is the bridge between statistical mechanics and chemical kinetics: the exponential sensitivity of reaction rates to temperature arises directly from the exponential tail of the Maxwell–Boltzmann distribution.

5. Derivation 4: Collision Cross Section and the Hard-Sphere Model

5.1 Hard-Sphere Collision Cross Section

In the simplest model, molecules A and B are treated as rigid spheres with radii \(r_A\)and \(r_B\). A collision occurs whenever the center-to-center distance equals\(d = r_A + r_B\). In the center-of-mass frame, the target molecule presents an effective area (the collision cross section):

\[ \boxed{\sigma_{AB} = \pi (r_A + r_B)^2 = \pi d^2} \]

Hard-sphere collision cross section

To derive this, consider molecule A moving through a gas of stationary B molecules. In time\(dt\), molecule A sweeps out a cylinder of length \(v_{\text{rel}} \, dt\)and cross-sectional area \(\pi d^2\). Any B molecule whose center lies within this cylinder will be struck. The number of collisions in time \(dt\) is:

\[ dN_{\text{coll}} = n_B \cdot \pi d^2 \cdot v_{\text{rel}} \, dt \]

Averaging over all A molecules and their velocity distribution yields the total collision rate\(Z_{AB} = n_A n_B \sigma_{AB} \langle v_{\text{rel}} \rangle\) as before.

5.2 Calculating Collision Rates for Real Reactions

For a concrete example, consider the gas-phase reaction \(\text{H}_2 + \text{I}_2 \to 2\text{HI}\)at \(T = 600 \; \text{K}\). Using kinetic radii\(r_{\text{H}_2} = 1.4 \; \text{\AA}\) and \(r_{\text{I}_2} = 2.15 \; \text{\AA}\):

Step 1: Cross section. \(\sigma = \pi(1.4 + 2.15)^2 \times 10^{-20} = 3.96 \times 10^{-19} \; \text{m}^2\)

Step 2: Reduced mass. \(\mu = \frac{2.016 \times 253.8}{2.016 + 253.8} \times \frac{10^{-3}}{6.022 \times 10^{23}} = 3.31 \times 10^{-27} \; \text{kg}\)

Step 3: Mean relative speed. \(\langle v_{\text{rel}} \rangle = \sqrt{\frac{8 \times 1.381 \times 10^{-23} \times 600}{\pi \times 3.31 \times 10^{-27}}} = 798 \; \text{m s}^{-1}\)

Step 4: Collision frequency. For 1 mol/L each (i.e., \(n = 6.022 \times 10^{26} \; \text{m}^{-3}\)):\(Z_{AB} = n^2 \sigma \langle v_{\text{rel}} \rangle \approx 1.15 \times 10^{35} \; \text{m}^{-3}\text{s}^{-1}\)

5.3 Why Steric Factors Are Small

For the \(\text{H}_2 + \text{I}_2\) reaction, collision theory with \(E_a = 171 \; \text{kJ mol}^{-1}\) predicts a rate constant of approximately\(k_{\text{CT}} \approx 1.6 \times 10^{6} \; \text{L mol}^{-1}\text{s}^{-1}\) at 600 K (before the steric factor), while the experimental value is\(k_{\text{exp}} \approx 8.7 \times 10^{-5} \; \text{L mol}^{-1}\text{s}^{-1}\). This implies \(p = k_{\text{exp}}/k_{\text{CT}} \sim 10^{-11}\).

Why is \(p\) so small? Several reasons conspire:

Orientational requirements: The reactive encounter requires specific relative orientations of the reactant molecules. For \(\text{H}_2 + \text{I}_2\), the approach must allow simultaneous H–I bond formation.
Energy localization: Not all of the relative kinetic energy is available along the reaction coordinate. Energy in rotational and non-reactive vibrational modes does not contribute to barrier crossing.
Quantum tunneling neglect: The hard-sphere model ignores quantum effects that can enhance or suppress reactivity depending on the potential energy surface.
Incorrect mechanism: For \(\text{H}_2 + \text{I}_2\)specifically, the reaction actually proceeds via a radical mechanism involving I atoms, not a simple bimolecular collision. This is a dramatic failure of the simple collision model.

The inadequacy of collision theory for reactions with very small steric factors motivated the development of transition state theory (Eyring, 1935), which replaces the crude geometric picture with a statistical mechanical treatment of the activated complex.

6. Applications

Industrial Catalysis

Catalysts work by lowering \(E_a\). The Haber process (\(\text{N}_2 + 3\text{H}_2 \to 2\text{NH}_3\)) has\(E_a \approx 335 \; \text{kJ mol}^{-1}\) uncatalyzed but only\(\sim 80 \; \text{kJ mol}^{-1}\) on an iron catalyst. This reduction increases the rate by a factor of \(\sim 10^{18}\) at 700 K.

The Arrhenius equation quantifies exactly how much a given reduction in \(E_a\)accelerates a reaction at a specified temperature.

Combustion Chemistry

Combustion involves hundreds of elementary reactions, each with its own Arrhenius parameters. The ignition delay time is controlled by the chain-branching reactions with the highest activation energies. For hydrogen combustion, the critical branching step \(\text{H} + \text{O}_2 \to \text{OH} + \text{O}\)has \(E_a \approx 70 \; \text{kJ mol}^{-1}\).

Detailed combustion mechanisms (e.g., GRI-Mech 3.0 for methane) contain Arrhenius parameters for hundreds of reactions, often with modified forms \(k = A T^n e^{-E_a/(RT)}\).

Food Science & Shelf Life

Food spoilage is governed by chemical and enzymatic reactions that follow Arrhenius kinetics. Refrigeration extends shelf life because lowering \(T\) exponentially decreases the rate of degradation reactions. The \(Q_{10}\) coefficient (factor by which rate increases per 10 K) is directly related to \(E_a\):

\(Q_{10} = \exp\!\left[\frac{E_a \cdot 10}{R T(T+10)}\right]\). For typical food degradation (\(E_a \approx 80 \; \text{kJ mol}^{-1}\)), \(Q_{10} \approx 2\text{–}3\)near room temperature.

Semiconductor Processing

Chemical vapor deposition (CVD), thermal oxidation, and dopant diffusion in semiconductor fabrication all follow Arrhenius kinetics. Silicon oxidation (the Deal–Grove model) has activation energies of \(\sim 120 \; \text{kJ mol}^{-1}\) for the linear rate constant and \(\sim 230 \; \text{kJ mol}^{-1}\) for the parabolic rate constant.

Precise temperature control in semiconductor processing is critical: a 10 K variation at 1000°C changes oxide growth rate by approximately 15%.

7. Historical Context

Svante Arrhenius (1889)

In his 1889 paper “Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren” (On the reaction velocity of the inversion of cane sugar by acids), Arrhenius proposed that molecules must acquire a minimum energy to react. He was studying the acid-catalyzed hydrolysis of sucrose and found that the temperature dependence of the rate constant followed the exponential form precisely. Arrhenius received the Nobel Prize in Chemistry in 1903, primarily for his work on electrolytic dissociation, but his kinetics equation proved equally enduring.

Arrhenius was also a pioneer of climate science, calculating in 1896 that doubling atmospheric CO\(_2\) would raise global temperatures by approximately 5°C — remarkably close to modern estimates.

Max Trautz (1916) and William C. McC. Lewis (1918)

The collision theory of chemical kinetics was developed independently by Max Trautz in Germany (1916) and William C. McC. Lewis in England (1918). Both applied the kinetic theory of gases — specifically, the Maxwell–Boltzmann velocity distribution — to calculate the frequency of molecular collisions with sufficient energy to react.

Trautz and Lewis showed that the collision theory pre-exponential factor was of the correct order of magnitude for simple reactions (within a factor of 10), but could be orders of magnitude too large for reactions involving complex molecules. This discrepancy — quantified by the steric factor \(p\) — motivated the search for more sophisticated theories.

The resolution came with Henry Eyring's transition state theory (1935), which replaced the collision picture with a statistical mechanical treatment of the activated complex, naturally accounting for orientational and vibrational effects through the entropy of activation.

8. Python Simulation: Arrhenius Analysis and Collision Theory

The following simulation generates Arrhenius plots for several reactions, extracts \(A\)and \(E_a\) from experimental data via linear regression, and compares collision theory predictions with measured rate constants.

Python

script.py147 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# =============================================================
# Arrhenius & Collision Theory Analysis
# =============================================================

# --- Physical constants ---
R = 8.314          # J/(mol K)
kB = 1.381e-23     # J/K
NA = 6.022e23      # 1/mol

# --- Reaction data: (name, Ea [kJ/mol], A [L/(mol s)], T_range, experimental_k_at_Ts) ---
reactions = {
    "H2 + I2 -> 2HI": {"Ea": 171.0, "A": 1.0e11, "T_range": (500, 900)},
    "2NOCl -> 2NO + Cl2": {"Ea": 100.0, "A": 9.4e9, "T_range": (300, 700)},
    "CH3Br + OH- -> CH3OH + Br-": {"Ea": 89.0, "A": 2.0e10, "T_range": (250, 450)},
    "2N2O5 -> 4NO2 + O2": {"Ea": 103.0, "A": 4.0e13, "T_range": (280, 350)},
}

# --- Part 1: Arrhenius plots ---
fig, axes = plt.subplots(2, 2, figsize=(12, 10))
axes = axes.flatten()

for idx, (name, params) in enumerate(reactions.items()):
    Ea = params["Ea"] * 1000  # Convert to J/mol
    A = params["A"]
    T_lo, T_hi = params["T_range"]
    T = np.linspace(T_lo, T_hi, 200)
    k = A * np.exp(-Ea / (R * T))
    inv_T = 1.0 / T
    ln_k = np.log(k)

ax = axes[idx]
    ax.plot(inv_T * 1000, ln_k, 'r-', linewidth=2)

# Linear fit to verify Ea extraction
    coeffs = np.polyfit(inv_T, ln_k, 1)
    Ea_fit = -coeffs[0] * R / 1000  # kJ/mol
    A_fit = np.exp(coeffs[1])

ax.set_xlabel('1000/T (1/K)')
    ax.set_ylabel('ln k')
    ax.set_title(name, fontsize=10)
    ax.grid(True, alpha=0.3)
    ax.text(0.05, 0.15, f'Ea(fit) = {Ea_fit:.1f} kJ/mol\nA(fit) = {A_fit:.2e}',
            transform=ax.transAxes, fontsize=8,
            bbox=dict(boxstyle='round', facecolor='lightyellow', alpha=0.8))

plt.suptitle('Arrhenius Plots for Several Reactions', fontsize=14, fontweight='bold')
plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight')
plt.close()

# --- Part 2: Extract Ea from synthetic experimental data ---
print("=" * 65)
print("  ARRHENIUS PARAMETER EXTRACTION FROM EXPERIMENTAL DATA")
print("=" * 65)

# Synthetic experimental data with noise for H2 + I2 reaction
np.random.seed(42)
T_exp = np.array([550, 600, 650, 700, 750, 800])
Ea_true = 171000.0  # J/mol
A_true = 1.0e11
k_true = A_true * np.exp(-Ea_true / (R * T_exp))
k_exp = k_true * np.exp(np.random.normal(0, 0.1, len(T_exp)))  # 10% noise

inv_T_exp = 1.0 / T_exp
ln_k_exp = np.log(k_exp)
coeffs_exp = np.polyfit(inv_T_exp, ln_k_exp, 1)
Ea_extracted = -coeffs_exp[0] * R / 1000
A_extracted = np.exp(coeffs_exp[1])

print(f"\nReaction: H2 + I2 -> 2HI")
print(f"{'T (K)':<10} {'k_exp (L/mol/s)':<22} {'k_fit (L/mol/s)':<22}")
print("-" * 54)
for i, T_val in enumerate(T_exp):
    k_fit_val = np.exp(coeffs_exp[1]) * np.exp(coeffs_exp[0] / T_val)
    print(f"{T_val:<10} {k_exp[i]:<22.6e} {k_fit_val:<22.6e}")

print(f"\nExtracted Ea = {Ea_extracted:.1f} kJ/mol (true: 171.0 kJ/mol)")
print(f"Extracted A  = {A_extracted:.3e} L/(mol s) (true: 1.000e+11)")

# --- Part 3: Collision theory predictions ---
print("\n" + "=" * 65)
print("  COLLISION THEORY vs EXPERIMENT")
print("=" * 65)

# Molecular data
mol_data = {
    "H2 + I2": {
        "r_A": 1.4e-10, "r_B": 2.15e-10,  # m (kinetic radii)
        "m_A": 2.016e-3/NA, "m_B": 253.8e-3/NA,  # kg
        "Ea": 171000.0, "k_exp_600K": 8.7e-5, "T": 600
    },
    "NO + O3": {
        "r_A": 1.17e-10, "r_B": 1.80e-10,
        "m_A": 30.01e-3/NA, "m_B": 48.00e-3/NA,
        "Ea": 10500.0, "k_exp_600K": 2.3e7, "T": 300
    },
    "K + Br2": {
        "r_A": 2.27e-10, "r_B": 1.98e-10,
        "m_A": 39.10e-3/NA, "m_B": 159.8e-3/NA,
        "Ea": 0.0, "k_exp_600K": 1.0e12, "T": 300
    },
}

print(f"\n{'Reaction':<15} {'T(K)':<7} {'sigma(m2)':<14} {'v_rel(m/s)':<12} "
      f"{'k_CT':<14} {'k_exp':<14} {'p=k/kCT':<12}")
print("-" * 100)

for name, d in mol_data.items():
    sigma = np.pi * (d["r_A"] + d["r_B"])**2
    mu = d["m_A"] * d["m_B"] / (d["m_A"] + d["m_B"])
    T = d["T"]
    v_rel = np.sqrt(8 * kB * T / (np.pi * mu))
    k_CT = NA * sigma * v_rel * np.exp(-d["Ea"] / (R * T))
    p = d["k_exp_600K"] / k_CT if k_CT > 0 else 0
    print(f"{name:<15} {T:<7} {sigma:<14.4e} {v_rel:<12.1f} "
          f"{k_CT:<14.4e} {d['k_exp_600K']:<14.4e} {p:<12.4e}")

# --- Part 4: Boltzmann fraction vs temperature ---
print("\n" + "=" * 65)
print("  BOLTZMANN FRACTION f(E >= Ea) vs TEMPERATURE")
print("=" * 65)

Ea_vals = [40, 80, 120, 171]  # kJ/mol
T_range = np.linspace(200, 1200, 50)

print(f"\n{'T(K)':<8}", end="")
for Ea_v in Ea_vals:
    print(f"{'Ea='+str(Ea_v)+'kJ':<16}", end="")
print()
print("-" * 72)

for T_val in T_range[::5]:
    print(f"{T_val:<8.0f}", end="")
    for Ea_v in Ea_vals:
        Ea_J = Ea_v * 1000
        frac = np.exp(-Ea_J / (R * T_val))
        print(f"{frac:<16.4e}", end="")
    print()

print("\n[Arrhenius plots saved to output.png]")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

9. Fortran Simulation: Collision Frequencies and Steric Factors

The following Fortran program computes collision frequencies, rate constants from collision theory, and steric factors for a set of gas-phase reactions. Fortran's native support for double-precision arithmetic makes it well-suited for these calculations where exponentials span many orders of magnitude.

Fortran

program.f90218 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 collision_theory
  implicit none
  integer, parameter :: dp = selected_real_kind(15, 307)

! Physical constants
  real(dp), parameter :: R_gas  = 8.314d0           ! J/(mol K)
  real(dp), parameter :: kB     = 1.381d-23         ! J/K
  real(dp), parameter :: NA     = 6.022d23          ! 1/mol
  real(dp), parameter :: pi_val = 3.14159265358979d0

! Reaction data arrays (up to 5 reactions)
  integer, parameter :: nrxn = 5
  character(len=30) :: rxn_name(nrxn)
  real(dp) :: r_A(nrxn), r_B(nrxn)     ! kinetic radii in m
  real(dp) :: mA(nrxn), mB(nrxn)       ! molecular masses in kg
  real(dp) :: Ea(nrxn)                   ! activation energy in J/mol
  real(dp) :: k_exp(nrxn)               ! experimental rate constant L/(mol s)
  real(dp) :: T_rxn(nrxn)               ! temperature in K

real(dp) :: sigma, mu_red, v_rel, k_CT, p_steric
  real(dp) :: Z_per_conc, boltz_frac
  integer  :: i, j
  integer  :: nT
  parameter(nT = 20)
  real(dp) :: T_arr(nT), k_arr(nT), frac_arr(nT)
  real(dp) :: T_lo, T_hi, dT, T_val

! --- Define reaction data ---
  rxn_name(1) = 'H2 + I2 -> 2HI'
  r_A(1) = 1.4d-10;   r_B(1) = 2.15d-10
  mA(1)  = 2.016d-3/NA; mB(1) = 253.8d-3/NA
  Ea(1)  = 171.0d3;   k_exp(1) = 8.7d-5;  T_rxn(1) = 600.0d0

rxn_name(2) = 'NO + O3 -> NO2 + O2'
  r_A(2) = 1.17d-10;  r_B(2) = 1.80d-10
  mA(2)  = 30.01d-3/NA; mB(2) = 48.00d-3/NA
  Ea(2)  = 10.5d3;    k_exp(2) = 2.3d7;   T_rxn(2) = 300.0d0

rxn_name(3) = 'K + Br2 -> KBr + Br'
  r_A(3) = 2.27d-10;  r_B(3) = 1.98d-10
  mA(3)  = 39.10d-3/NA; mB(3) = 159.8d-3/NA
  Ea(3)  = 0.0d0;     k_exp(3) = 1.0d12;  T_rxn(3) = 300.0d0

rxn_name(4) = '2NOCl -> 2NO + Cl2'
  r_A(4) = 1.90d-10;  r_B(4) = 1.90d-10
  mA(4)  = 65.46d-3/NA; mB(4) = 65.46d-3/NA
  Ea(4)  = 100.0d3;   k_exp(4) = 9.4d5;   T_rxn(4) = 500.0d0

rxn_name(5) = 'CH3Br + OH -> CH3OH+Br'
  r_A(5) = 2.00d-10;  r_B(5) = 1.40d-10
  mA(5)  = 94.94d-3/NA; mB(5) = 17.01d-3/NA
  Ea(5)  = 89.0d3;    k_exp(5) = 2.8d1;   T_rxn(5) = 350.0d0

! === Table 1: Collision theory summary ===
  write(*,'(A)') '================================================================='
  write(*,'(A)') '  COLLISION THEORY ANALYSIS (Fortran)'
  write(*,'(A)') '================================================================='
  write(*,'(A)') ''
  write(*,'(A)') 'TABLE 1: Rate constants and steric factors'
  write(*,'(A)') '-----------------------------------------------------------------'
  write(*,'(A6,A30,A14,A14,A14,A14)') &
       'Rxn', 'Name', 'sigma(m2)', 'v_rel(m/s)', 'k_CT', 'p_steric'
  write(*,'(A)') '-----------------------------------------------------------------'

do i = 1, nrxn
    sigma = pi_val * (r_A(i) + r_B(i))**2
    mu_red = mA(i) * mB(i) / (mA(i) + mB(i))
    v_rel = sqrt(8.0d0 * kB * T_rxn(i) / (pi_val * mu_red))
    boltz_frac = exp(-Ea(i) / (R_gas * T_rxn(i)))
    k_CT = NA * sigma * v_rel * boltz_frac

if (k_CT > 0.0d0) then
      p_steric = k_exp(i) / k_CT
    else
      p_steric = 0.0d0
    end if

write(*,'(I4,2X,A30,ES14.4,F14.1,ES14.4,ES14.4)') &
         i, rxn_name(i), sigma, v_rel, k_CT, p_steric
  end do

! === Table 2: Temperature dependence for H2 + I2 ===
  write(*,'(/,A)') '================================================================='
  write(*,'(A)') '  TEMPERATURE DEPENDENCE: H2 + I2 -> 2HI'
  write(*,'(A)') '================================================================='
  write(*,'(A)') ''

sigma = pi_val * (r_A(1) + r_B(1))**2
  mu_red = mA(1) * mB(1) / (mA(1) + mB(1))

T_lo = 400.0d0
  T_hi = 1000.0d0
  dT = (T_hi - T_lo) / dble(nT - 1)

write(*,'(A)') 'DATA_START'
  write(*,'(A)') 'T_K,k_CT,k_with_p,Boltzmann_frac'

do j = 1, nT
    T_val = T_lo + dble(j-1) * dT
    v_rel = sqrt(8.0d0 * kB * T_val / (pi_val * mu_red))
    boltz_frac = exp(-Ea(1) / (R_gas * T_val))
    k_CT = NA * sigma * v_rel * boltz_frac
    ! Use p = 5.4e-11 estimated from 600K data
    k_arr(j) = 5.4d-11 * k_CT
    frac_arr(j) = boltz_frac
    T_arr(j) = T_val
    write(*,'(F8.1,A,ES16.6,A,ES16.6,A,ES16.6)') &
         T_val, ',', k_CT, ',', k_arr(j), ',', boltz_frac
  end do
  write(*,'(A)') 'DATA_END'

! === Table 3: Collision frequencies at 1 mol/L ===
  write(*,'(/,A)') '================================================================='
  write(*,'(A)') '  COLLISION FREQUENCIES (n = 6.022e26 m^-3 = 1 mol/L)'
  write(*,'(A)') '================================================================='

do i = 1, nrxn
    sigma = pi_val * (r_A(i) + r_B(i))**2
    mu_red = mA(i) * mB(i) / (mA(i) + mB(i))
    v_rel = sqrt(8.0d0 * kB * T_rxn(i) / (pi_val * mu_red))
    Z_per_conc = NA * sigma * v_rel  ! This is Z_AB / (n_A * n_B) * NA^2

write(*,'(A,A30,A,ES12.4,A,F8.1,A)') &
         '  ', rxn_name(i), ': Z/n^2 = ', Z_per_conc, &
         ' m^3/(mol s) at T = ', T_rxn(i), ' K'
  end do

write(*,'(/,A)') 'Computation complete.'

end program collision_theory
"""

# Write, compile, and run
with open('collision_theory.f90', 'w') as f:
    f.write(fortran_source)

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

run_result = subprocess.run(
    ['./collision_theory'],
    capture_output=True, text=True, timeout=30
)
output = run_result.stdout
print(output)

# --- Parse DATA_START..DATA_END and plot ---
lines = output.split('\n')
data_lines = []
in_data = False
for line in lines:
    if 'DATA_START' in line:
        in_data = True
        continue
    if 'DATA_END' in line:
        in_data = False
        continue
    if in_data and line.strip() and not line.startswith('T_K'):
        parts = line.split(',')
        if len(parts) >= 4:
            try:
                vals = [float(p.strip()) for p in parts]
                data_lines.append(vals)
            except ValueError:
                pass

if data_lines:
    data = np.array(data_lines)
    T_vals = data[:, 0]
    k_CT_vals = data[:, 1]
    k_p_vals = data[:, 2]
    bf_vals = data[:, 3]

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))

# Arrhenius plot
    inv_T = 1000.0 / T_vals
    ax1.semilogy(inv_T, k_CT_vals, 'r-o', markersize=4, label='k_CT (collision theory)')
    ax1.semilogy(inv_T, k_p_vals, 'b-s', markersize=4, label='k_CT x p (corrected)')
    ax1.set_xlabel('1000/T (1/K)')
    ax1.set_ylabel('k (L/(mol s))')
    ax1.set_title('H2 + I2: Rate Constant vs Temperature')
    ax1.legend(fontsize=9)
    ax1.grid(True, alpha=0.3)

# Boltzmann fraction
    ax2.semilogy(T_vals, bf_vals, 'r-o', markersize=4)
    ax2.set_xlabel('Temperature (K)')
    ax2.set_ylabel('Boltzmann fraction exp(-Ea/RT)')
    ax2.set_title('Fraction of Collisions with E >= Ea')
    ax2.grid(True, alpha=0.3)

plt.tight_layout()
    plt.savefig('output.png', dpi=150, bbox_inches='tight')
    plt.close()
    print("\n[Fortran Arrhenius/Boltzmann plots saved to output.png]")
else:
    print("Warning: Could not parse DATA output for plotting.")

# Cleanup
for f in ['collision_theory.f90', 'collision_theory']:
    if os.path.exists(f):
        os.remove(f)

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Chapter Summary

The Arrhenius equation \(k = A e^{-E_a/(RT)}\) follows from integrating the empirical relation \(d(\ln k)/dT = E_a/(RT^2)\).
Collision theory derives the rate constant from kinetic theory:\(k = p N_A \sigma_{AB} \sqrt{8k_BT/(\pi\mu)} \, e^{-E_a/(RT)}\), identifying \(A\) with the collision frequency times the steric factor.
The Boltzmann factor \(e^{-E_a/(k_BT)}\) represents the fraction of collisions with energy exceeding \(E_a\), valid when \(E_a \gg k_BT\).
The hard-sphere cross section \(\sigma = \pi(r_A + r_B)^2\) provides a simple geometric model, but steric factors \(p \ll 1\) are needed for complex molecules.
Collision theory works best for simple atom–molecule reactions; for complex molecules,transition state theory provides a superior framework.

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