Part III, Chapter 8

Reaction Mechanisms

Elementary steps, steady-state, pre-equilibrium, Lindemann, and chain reactions

Historical Context

The idea that overall reactions proceed through a series of elementary steps emerged in the early 20th century. Max Bodenstein's work on the hydrogen-bromine reaction (1906) revealed complex kinetics that could only be explained by a chain mechanism. Frederick Lindemann (1922) proposed the first mechanism for unimolecular reactions, explaining how molecules acquire enough energy to react even without bimolecular collisions.

The steady-state approximation, developed by Bodenstein and later formalized by others, became the primary tool for deriving rate laws from complex mechanisms. Cyril Hinshelwood and Nikolay Semenov shared the 1956 Nobel Prize for their work on chain reaction mechanisms.

Key contributors: Max Bodenstein (1906), Frederick Lindemann (1922), Cyril Hinshelwood (Nobel 1956), Nikolay Semenov (Nobel 1956).

Derivation 1: Elementary Steps and Molecularity

An elementary step is a single molecular event. Its rate law can be written directly from its stoichiometry (unlike overall reactions). The molecularity classifies elementary steps:

Unimolecular (molecularity = 1)

A → products. Rate = k[A]. Examples: isomerization, decomposition, radioactive decay.

Bimolecular (molecularity = 2)

A + B → products. Rate = k[A][B]. The most common type. Requires collision between two molecules.

Termolecular (molecularity = 3)

A + B + C → products. Rate = k[A][B][C]. Very rare because simultaneous three-body collisions are improbable. Usually indicates the mechanism is more complex.

Key distinction: Molecularity is a property of an elementary step (always a positive integer). Reaction order is an empirical quantity that can be fractional, zero, or negative. They coincide only for elementary steps.

Derivation 2: Steady-State Approximation

Consider consecutive reactions A → B → C with rate constants $k_1$ and $k_2$. The exact differential equations are:

$$\frac{d[\text{A}]}{dt} = -k_1[\text{A}], \quad \frac{d[\text{B}]}{dt} = k_1[\text{A}] - k_2[\text{B}], \quad \frac{d[\text{C}]}{dt} = k_2[\text{B}]$$

When $k_2 \gg k_1$, the intermediate B is consumed as fast as it is formed. Setting $d[\text{B}]/dt \approx 0$:

Steady-State Result

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

The rate-determining step (the slowest step) controls the overall rate. When$k_2 \gg k_1$, the first step is rate-determining and the overall rate equals $k_1[\text{A}]$. The SSA is valid when the intermediate concentration is small and changes slowly compared to reactants and products.

Derivation 3: Pre-Equilibrium Approximation

Consider a mechanism where a fast equilibrium precedes a slow product-forming step:

$$\text{A} + \text{B} \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} \text{I} \xrightarrow{k_2} \text{P}$$

If $k_1, k_{-1} \gg k_2$, the first step reaches equilibrium before significant product forms. The equilibrium constant is $K = k_1/k_{-1}$:

$$K = \frac{[\text{I}]}{[\text{A}][\text{B}]} = \frac{k_1}{k_{-1}} \implies [\text{I}] = K[\text{A}][\text{B}]$$

The rate of product formation:

$$\text{rate} = k_2[\text{I}] = k_2 K [\text{A}][\text{B}] = \frac{k_1 k_2}{k_{-1}}[\text{A}][\text{B}]$$

Note that the SSA applied to [I] gives the same result: setting$d[\text{I}]/dt = k_1[\text{A}][\text{B}] - k_{-1}[\text{I}] - k_2[\text{I}] = 0$yields $[\text{I}] = k_1[\text{A}][\text{B}]/(k_{-1} + k_2) \approx K[\text{A}][\text{B}]$when $k_{-1} \gg k_2$.

Derivation 4: Lindemann Mechanism

The Lindemann mechanism explains unimolecular reactions by proposing that molecules must first be energized by collisions:

$$\text{A} + \text{M} \xrightarrow{k_1} \text{A}^* + \text{M} \quad \text{(activation)}$$

$$\text{A}^* + \text{M} \xrightarrow{k_{-1}} \text{A} + \text{M} \quad \text{(deactivation)}$$

$$\text{A}^* \xrightarrow{k_2} \text{products} \quad \text{(reaction)}$$

Applying the SSA to [A*]:

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

Lindemann Rate Law

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

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

$$\text{rate} = \frac{k_1 k_2}{k_{-1}}[\text{A}] \quad \text{(first order)}$$

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

$$\text{rate} = k_1[\text{A}][\text{M}] \quad \text{(second order)}$$

The transition from first to second order with decreasing pressure is called the "falloff" region, and its observation was a key early success for the Lindemann theory.

Derivation 5: Chain Reactions (H2-Br2)

The hydrogen-bromine reaction produces a complex rate law that can only be explained by a chain mechanism. The elementary steps are:

Initiation: $\text{Br}_2 \xrightarrow{k_i} 2\text{Br}\cdot$

Propagation: $\text{Br}\cdot + \text{H}_2 \xrightarrow{k_p} \text{HBr} + \text{H}\cdot$

Propagation: $\text{H}\cdot + \text{Br}_2 \xrightarrow{k_p'} \text{HBr} + \text{Br}\cdot$

Inhibition: $\text{H}\cdot + \text{HBr} \xrightarrow{k_{inh}} \text{H}_2 + \text{Br}\cdot$

Termination: $2\text{Br}\cdot \xrightarrow{k_t} \text{Br}_2$

Applying the SSA to both radical intermediates [Br] and [H]:

Overall Rate Law

$$\frac{d[\text{HBr}]}{dt} = \frac{k[\text{H}_2][\text{Br}_2]^{1/2}}{1 + k'[\text{HBr}]/[\text{Br}_2]}$$

The fractional order ($1/2$ in [Br2]) and inhibition by product [HBr] are characteristic of chain mechanisms and cannot arise from a single elementary step. The chain length (number of propagation cycles per initiation event) determines the efficiency of the chain.

Applications

Combustion

Hydrocarbon combustion involves hundreds of elementary steps and radical intermediates. Chain branching (one radical producing two) leads to explosive behavior when the branching rate exceeds the termination rate.

Polymerization

Free radical polymerization follows a chain mechanism: initiation produces radicals, propagation grows the chain, and termination (combination or disproportionation) stops growth. The kinetic chain length determines molecular weight.

Ozone Chemistry

Stratospheric ozone depletion by CFCs involves a catalytic chain mechanism. A single Cl atom can destroy thousands of O3 molecules before being removed, demonstrating the amplifying power of chain reactions.

Enzyme Catalysis

The Michaelis-Menten mechanism (E + S ⇌ ES → E + P) is a pre-equilibrium (or steady-state) mechanism. Understanding the intermediate ES complex is essential for drug design and enzyme engineering.

Simulation: Mechanism Analysis

This simulation compares exact numerical solutions with analytical approximations for four mechanism types: consecutive reactions (SSA), pre-equilibrium, Lindemann falloff, and chain reaction kinetics. Observe when the approximations break down.

Reaction Mechanisms: SSA, Pre-Equilibrium, Lindemann, Chain

Python

script.py159 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
from scipy.integrate import odeint

# ============================================================
# Reaction Mechanisms: Steady-State, Pre-Equilibrium, Lindemann
# ============================================================

fig, axes = plt.subplots(2, 2, figsize=(14, 12))

# ---- Panel 1: Consecutive reactions A -> B -> C ----
# Steady-state approximation on intermediate B
ax1 = axes[0, 0]
k1_val = 0.5
k2_val = 2.0
t = np.linspace(0, 10, 500)
A0 = 1.0

def consecutive(y, t_val, k1, k2):
    A, B, C = y
    dAdt = -k1 * A
    dBdt = k1 * A - k2 * B
    dCdt = k2 * B
    return [dAdt, dBdt, dCdt]

sol = odeint(consecutive, [A0, 0, 0], t, args=(k1_val, k2_val))
A_t, B_t, C_t = sol[:, 0], sol[:, 1], sol[:, 2]

# Steady-state approximation: d[B]/dt = 0 => [B]_ss = k1[A]/k2
B_ss = k1_val * A_t / k2_val

ax1.plot(t, A_t, 'r-', lw=2.5, label='[A] (exact)')
ax1.plot(t, B_t, 'y-', lw=2.5, label='[B] (exact)')
ax1.plot(t, C_t, 'lime', lw=2.5, label='[C] (exact)')
ax1.plot(t, B_ss, 'y--', lw=2, alpha=0.7, label='[B]_ss (SSA)')
ax1.set_xlabel('Time', fontsize=11, color='white')
ax1.set_ylabel('Concentration', fontsize=11, color='white')
ax1.set_title('A -> B -> C  (k1=' + str(k1_val) + ', k2=' + str(k2_val) + ')', fontsize=13, fontweight='bold', color='white')
ax1.legend(fontsize=9, facecolor='black', edgecolor='gray', labelcolor='white')
ax1.set_facecolor('#1a1a2e')
ax1.tick_params(colors='white')
ax1.grid(True, alpha=0.3)

# ---- Panel 2: Pre-equilibrium mechanism ----
# A + B <=> I (fast) -> P (slow)
ax2 = axes[0, 1]
kf = 10.0   # forward (fast)
kr = 5.0    # reverse (fast)
kp = 0.2    # product formation (slow)
K_eq = kf / kr

def pre_eq(y, t_val, kf_v, kr_v, kp_v):
    A, B, I, P = y
    dAdt = -kf_v * A * B + kr_v * I
    dBdt = -kf_v * A * B + kr_v * I
    dIdt = kf_v * A * B - kr_v * I - kp_v * I
    dPdt = kp_v * I
    return [dAdt, dBdt, dIdt, dPdt]

t2 = np.linspace(0, 15, 500)
sol2 = odeint(pre_eq, [1.0, 1.0, 0, 0], t2, args=(kf, kr, kp))

ax2.plot(t2, sol2[:, 0], 'r-', lw=2.5, label='[A]')
ax2.plot(t2, sol2[:, 2], 'y-', lw=2.5, label='[I] (intermediate)')
ax2.plot(t2, sol2[:, 3], 'lime', lw=2.5, label='[P] (product)')

# Pre-equilibrium approx: [I] = K[A][B], rate = kp*K*[A][B]
I_preq = K_eq * sol2[:, 0] * sol2[:, 1]
ax2.plot(t2, I_preq, 'y--', lw=2, alpha=0.7, label='[I]_preq = K[A][B]')

ax2.set_xlabel('Time', fontsize=11, color='white')
ax2.set_ylabel('Concentration', fontsize=11, color='white')
ax2.set_title('Pre-Equilibrium: A+B <=> I -> P', fontsize=13, fontweight='bold', color='white')
ax2.legend(fontsize=9, facecolor='black', edgecolor='gray', labelcolor='white')
ax2.set_facecolor('#1a1a2e')
ax2.tick_params(colors='white')
ax2.grid(True, alpha=0.3)

# ---- Panel 3: Lindemann mechanism ----
ax3 = axes[1, 0]
# A + M -> A* + M  (k1)
# A* + M -> A + M  (k-1)
# A* -> P          (k2)
# Rate = k1*k2*[A]*[M] / (k_neg1*[M] + k2)

k1_L = 1.0
k_neg1 = 2.0
k2_L = 0.5

M_range = np.logspace(-2, 3, 500)

# Effective first-order rate constant
k_uni = k1_L * k2_L * M_range / (k_neg1 * M_range + k2_L)

# High and low pressure limits
k_high = k1_L * k2_L / k_neg1  # high [M] limit
k_low_slope = k1_L  # low [M] limit: k_uni ~ k1 * [M]

ax3.loglog(M_range, k_uni, 'r-', lw=2.5, label='k_uni (Lindemann)')
ax3.axhline(y=k_high, color='yellow', linestyle='--', alpha=0.7, label='High-P limit: k1*k2/k_-1')
ax3.loglog(M_range, k1_L * M_range, 'c--', lw=1.5, alpha=0.5, label='Low-P limit: k1*[M]')
ax3.set_xlabel('[M] (concentration)', fontsize=11, color='white')
ax3.set_ylabel('k_uni (effective rate constant)', fontsize=11, color='white')
ax3.set_title('Lindemann Falloff', fontsize=13, fontweight='bold', color='white')
ax3.legend(fontsize=9, facecolor='black', edgecolor='gray', labelcolor='white')
ax3.set_facecolor('#1a1a2e')
ax3.tick_params(colors='white')
ax3.grid(True, alpha=0.3)

# ---- Panel 4: Chain reaction (H2-Br2) ----
ax4 = axes[1, 1]
# Rate = k * [H2] * [Br2]^0.5 / (1 + k'*[HBr]/[Br2])
# Simplified: plot rate vs [H2] and [Br2]

H2_range = np.linspace(0.01, 2, 200)
Br2_vals = [0.1, 0.5, 1.0, 2.0]
colors_br = ['#ff6b6b', '#ffa502', '#2ed573', '#1e90ff']
k_chain = 1.0

for Br2, clr in zip(Br2_vals, colors_br):
    rate = k_chain * H2_range * np.sqrt(Br2)
    ax4.plot(H2_range, rate, color=clr, lw=2, label='[Br2] = ' + str(Br2))

ax4.set_xlabel('[H2]', fontsize=11, color='white')
ax4.set_ylabel('Rate', fontsize=11, color='white')
ax4.set_title('H2 + Br2 Chain Reaction Rate', fontsize=13, fontweight='bold', color='white')
ax4.legend(fontsize=10, facecolor='black', edgecolor='gray', labelcolor='white')
ax4.set_facecolor('#1a1a2e')
ax4.tick_params(colors='white')
ax4.grid(True, alpha=0.3)

for ax in axes.flat:
    ax.spines['bottom'].set_color('gray')
    ax.spines['left'].set_color('gray')
    ax.spines['top'].set_color('gray')
    ax.spines['right'].set_color('gray')

fig.patch.set_facecolor('#0a0a1a')
plt.tight_layout()
plt.savefig('output.png', dpi=100, bbox_inches='tight', facecolor='#0a0a1a')
plt.close()

print("Reaction Mechanisms Analysis")
print("=" * 55)
print()
print("Consecutive A -> B -> C (k1=" + str(k1_val) + ", k2=" + str(k2_val) + "):")
print("  SSA valid when k2 >> k1 (intermediate consumed fast)")
print("  [B]_ss = k1[A]/k2 = " + str(round(k1_val/k2_val, 3)) + " * [A]")
print()
print("Pre-equilibrium A+B <=> I -> P:")
print("  K_eq = kf/kr = " + str(K_eq))
print("  rate = kp * K * [A][B] = " + str(round(kp * K_eq, 3)) + " * [A][B]")
print()
print("Lindemann mechanism:")
print("  High-P limit: k_inf = k1*k2/k_-1 = " + str(round(k_high, 4)))
print("  Falloff pressure: [M]_1/2 = k2/k_-1 = " + str(round(k2_L/k_neg1, 4)))

Click Run to execute the Python code

Code will be executed with Python 3 on the server

The Rate-Determining Step

In a multi-step mechanism, the rate-determining step (RDS) is the slowest step that limits the overall rate. Identifying the RDS simplifies kinetic analysis enormously.

Criterion for Rate-Determining Step

A step is rate-determining if its rate constant is much smaller than those of all subsequent steps. All steps before the RDS are in (pre-)equilibrium; the RDS controls the flux through the mechanism.

Example: SN1 Mechanism

Step 1 (slow, RDS): R-X → R+ + X- (unimolecular ionization)

Step 2 (fast): R+ + Nu- → R-Nu (nucleophilic attack)

Overall rate = k1[RX] (first order; independent of [Nu-])

Example: SN2 Mechanism

Single concerted step (simultaneously RDS): Nu- + R-X → Nu-R + X-

Overall rate = k[Nu-][RX] (second order; depends on both species)

Caution: The concept of a single rate-determining step is an approximation. In reality, multiple steps may contribute comparably to the overall rate. The full kinetic analysis (solving all differential equations simultaneously) is always more accurate.

Chain Branching and Explosion Limits

Some chain reactions can undergo explosive behavior when chain branching (one radical producing two or more radicals) outpaces termination. The hydrogen-oxygen reaction famously exhibits three explosion limits.

First limit (low P): Below this pressure, radicals diffuse to the walls and are destroyed faster than they can branch. No explosion.

Second limit (intermediate P): Above this pressure, three-body termination reactions (which require collisions with a third body M) become fast enough to quench branching. No explosion between limits 2 and 3.

Third limit (high P): At very high pressures, the thermal energy released by the exothermic chain reaction cannot be dissipated fast enough. Thermal runaway leads to explosion.

The mathematical condition for explosion is that the net branching factor exceeds unity:

$$\phi = \frac{k_b}{k_t + k_w} > 1 \quad \text{(explosion)}$$

where $k_b$ is the branching rate, $k_t$ is the gas-phase termination rate, and $k_w$ is the wall termination rate. Understanding these limits is essential for safe handling of flammable gases.

Catalytic Mechanisms

A catalyst participates in the mechanism but is regenerated at the end of the catalytic cycle. It lowers the activation energy without changing the thermodynamic equilibrium position.

Homogeneous Catalysis

Catalyst is in the same phase as reactants. Example: acid-catalyzed ester hydrolysis. The catalyst enters the rate law: $r = k[\text{ester}][\text{H}^+]$.

Heterogeneous Catalysis

Catalyst is a solid surface. Follows adsorption (Langmuir), surface reaction, and desorption steps. The Langmuir-Hinshelwood mechanism is the most common model for surface kinetics.

Enzyme Catalysis

The Michaelis-Menten mechanism: E + S ⇌ ES → E + P. Applying the SSA to [ES] gives $v = V_{\max}[\text{S}]/(K_M + [\text{S}])$ where$K_M = (k_{-1} + k_2)/k_1$.

Autocatalysis

The product catalyzes its own formation: A + B → 2B. Rate = k[A][B]. The concentration of B shows sigmoidal growth (slow start, rapid acceleration, then deceleration as A is consumed).

Chapter Summary

• Elementary steps: Rate law equals stoichiometry. Molecularity = 1, 2, or 3.

• Steady-state approximation: Set $d[\text{I}]/dt = 0$ for reactive intermediates.

• Pre-equilibrium: Fast equilibrium precedes rate-determining step; $[\text{I}] = K[\text{A}][\text{B}]$.

• Lindemann: Explains unimolecular reactions; first order at high P, second order at low P.

• Chain reactions: Initiation, propagation, termination; can produce fractional orders and complex rate laws.

← Rate Laws Next: Transition State Theory →