Organic Chemistry/Part II: Reaction Mechanisms/Elimination Reactions

Elimination Reactions (E1 & E2)

Reading time: ~60 minutes | Topics: E2 mechanism, E1 mechanism, E1cb mechanism, Zaitsev and Hofmann regioselectivity, orbital overlap, anti-periplanar geometry, competition with substitution, hyperconjugation, industrial applications, Python simulation

1. Introduction: Forming Carbon-Carbon Double Bonds

Elimination reactions are among the most fundamental transformations in organic chemistry. In an elimination, a substrate loses two groups — typically a proton (H) and a leaving group (X) — to form a new$\pi$ bond, most commonly a carbon-carbon double bond. The general equation for a$\beta$-elimination (also called 1,2-elimination) is:

$\text{B}^- + \text{H}-\text{C}_\alpha-\text{C}_\beta-\text{X} \longrightarrow \text{BH} + \text{C}_\alpha=\text{C}_\beta + \text{X}^-$

Here, B⁻ is a base that abstracts the proton from the$\alpha$-carbon (the carbon adjacent to the one bearing the leaving group), andX is the leaving group that departs from the $\beta$-carbon. The result is the formation of an alkene.

Elimination reactions are essential for:

● Alkene synthesis: Constructing $\text{C}=\text{C}$ bonds from saturated precursors
● Retrosynthetic planning: Elimination is the reverse of addition, connecting saturated and unsaturated functional groups
● Industrial chemistry: Dehydration of alcohols and cracking of hydrocarbons produce millions of tonnes of alkenes annually
● Biochemistry: Enzymatic elimination reactions occur in fatty acid synthesis, amino acid metabolism, and the citric acid cycle

The three principal mechanisms — E2 (bimolecular), E1 (unimolecular), and E1cb (conjugate base) — differ in the timing of bond breaking and bond forming events, and in whether intermediates are formed along the reaction pathway. Understanding these mechanisms allows us to predict products, control regioselectivity, and minimize unwanted substitution side reactions.

The Elimination-Substitution Dichotomy

Elimination reactions always compete with nucleophilic substitution. A base can act as a nucleophile (giving substitution) or abstract a proton (giving elimination). The outcome depends on:

Substrate structure: Primary, secondary, or tertiary carbon center
Base/nucleophile identity: Strong vs. weak, bulky vs. compact
Solvent: Polar protic vs. polar aprotic
Temperature: Higher temperature favors elimination ($\Delta S^\circ > 0$ for elimination)

2. Derivation 1: The E2 Mechanism

The E2 (Elimination, Bimolecular) mechanism is a one-step, concerted process in which the base abstracts a $\beta$-proton, the $\text{C}=\text{C}$ double bond forms, and the leaving group departs — all in a single transition state.

2.1 Rate Law Derivation

Since the E2 mechanism is concerted (one step), the rate-determining step involves both the substrate and the base. The rate law is therefore second-order overall:

$\text{rate} = k_{\text{E2}}[\text{B}^-][\text{R-X}]$

First order in base, first order in substrate — bimolecular

This can be derived from transition state theory. The activated complex contains both the base and the substrate, so by the Eyring equation:

$k_{\text{E2}} = \frac{k_B T}{h} \exp\!\left(-\frac{\Delta G^\ddagger}{RT}\right)$

where $\Delta G^\ddagger$ is the free energy of activation for the concerted process,$k_B$ is Boltzmann's constant, $h$ is Planck's constant, and $T$ is temperature.

2.2 Anti-Periplanar Geometry Requirement

The E2 mechanism has a strict stereoelectronic requirement: the$\text{C}_\alpha\text{-H}$ bond and the $\text{C}_\beta\text{-X}$ bond must be anti-periplanar (dihedral angle of 180°). This geometry is essential for optimal orbital overlap.

2.3 Orbital Overlap Derivation

The concerted nature of the E2 mechanism can be understood through molecular orbital theory. As the base begins to abstract the proton, electron density flows from the $\sigma_{\text{C-H}}$ bonding orbital into the $\sigma^*_{\text{C-X}}$ antibonding orbital:

Orbital interaction: $\sigma_{\text{C-H}} \rightarrow \sigma^*_{\text{C-X}}$

The overlap integral $S$ between these orbitals is maximized when the four atoms H–C–C–X are coplanar and anti-periplanar:

$S = \langle \sigma_{\text{C-H}} | \sigma^*_{\text{C-X}} \rangle \propto \cos(\phi)$

where $\phi$ is the dihedral angle between the C-H and C-X bonds. Maximum overlap occurs at$\phi = 180°$ (anti-periplanar). At $\phi = 0°$ (syn-periplanar), some overlap exists but is less favorable due to eclipsing strain. At $\phi = 90°$ (gauche), $S \approx 0$ and E2 elimination cannot occur.

As the $\sigma_{\text{C-H}}$ electrons flow into $\sigma^*_{\text{C-X}}$, two things happen simultaneously:

● The C-X bond weakens and breaks (leaving group departs)
● A new $\pi$ bond forms between $\text{C}_\alpha$ and $\text{C}_\beta$ from the residual $p$ orbitals

2.4 Zaitsev vs. Hofmann Regioselectivity in E2

When a substrate has multiple $\beta$-hydrogens, the base can abstract from different positions, leading to different alkene products:

Zaitsev product: The more substituted alkene (thermodynamic product). Formed preferentially with small, strong bases (e.g., NaOEt, NaOH).

Hofmann product: The less substituted alkene (kinetic product with bulky bases). Formed preferentially with sterically demanding bases (e.g., t-BuOK, LDA, DBU).

2.5 E2 vs. SN2 Competition

E2 and SN2 are both bimolecular and second-order. They compete because the same species can act as either a base (E2) or a nucleophile (SN2). The ratio of products depends on:

Favors E2:

• Strong, bulky bases (e.g., t-BuO⁻)
• Tertiary or secondary substrates
• Higher temperature
• Poor nucleophiles / good bases

Favors SN2:

• Strong, compact nucleophiles (e.g., CN⁻, I⁻)
• Primary or methyl substrates
• Lower temperature
• Polar aprotic solvents

The relative rate is: $\frac{\text{rate}_{\text{E2}}}{\text{rate}_{\text{SN2}}} = \frac{k_{\text{E2}}}{k_{\text{SN2}}}$

Since both are first-order in base/nucleophile and substrate, the selectivity depends purely on the relative rate constants, which are governed by the respective $\Delta G^\ddagger$ values.

3. Derivation 2: The E1 Mechanism

The E1 (Elimination, Unimolecular) mechanism is a two-step process. In the first (rate-determining) step, the leaving group departs to form a carbocation intermediate. In the second (fast) step, a base abstracts a $\beta$-proton from the carbocation.

3.1 Rate Law Derivation

The two steps of the E1 mechanism are:

Step 1 (slow, RDS): $\text{R-X} \xrightarrow{k_1} \text{R}^+ + \text{X}^-$

Step 2 (fast): $\text{R}^+ + \text{B}^- \xrightarrow{k_2} \text{alkene} + \text{BH}$

Since the first step is rate-determining and involves only the substrate, the rate law is first-order overall:

$\text{rate} = k_1[\text{R-X}]$

First order in substrate only — unimolecular rate-determining step

Note that the base does not appear in the rate law. This is the hallmark of the E1 mechanism: changing the concentration or identity of the base does not affect the rate of reaction (though it does affect the product distribution between E1 and SN1).

3.2 Carbocation Intermediate

The carbocation intermediate in E1 is the same as in SN1 reactions. Its stability follows the well-known order:

$\text{methyl}^+ \ll \text{primary}^+ < \text{secondary}^+ < \text{tertiary}^+ < \text{benzylic}^+ \approx \text{allylic}^+$

Because E1 requires carbocation formation, it is strongly favored for tertiary substrates and virtually never occurs with primary substrates (whose primary carbocations are too unstable to form). The carbocation may also undergo rearrangement (1,2-hydride or 1,2-methyl shifts) before deprotonation, leading to unexpected products.

3.3 E1 vs. SN1 Competition

E1 and SN1 share the same rate-determining step (carbocation formation) and therefore have identical rate laws. Once the carbocation forms, it can either:

● Lose a proton (E1): A base abstracts $\text{H}^+$ from the $\beta$-carbon
● Capture a nucleophile (SN1): A nucleophile attacks the carbocation center

Factors Favoring E1 over SN1:

Higher temperature: Elimination has a more positive $\Delta S^\circ$ (two molecules become three), so by $\Delta G = \Delta H - T\Delta S$, increasing $T$ makes elimination more favorable.
Stronger base: Even though the base doesn't affect the rate, a stronger base in the second step increases the E1/SN1 product ratio.
Weaker nucleophile: If the solvent/nucleophile is a poor nucleophile but a reasonable base (e.g., t-BuOH), elimination dominates.

4. Derivation 3: The E1cb Mechanism

The E1cb (Elimination, Unimolecular, Conjugate Base) mechanism is a two-step process that proceeds through a carbanion intermediate. It occurs when the $\alpha$-hydrogen is relatively acidic but the leaving group is poor.

4.1 Mechanism

Step 1 (fast, reversible): $\text{B}^- + \text{H-C}_\alpha\text{-C}_\beta\text{-X} \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} \text{BH} + \,^-\text{C}_\alpha\text{-C}_\beta\text{-X}$

Step 2 (slow, RDS): $^-\text{C}_\alpha\text{-C}_\beta\text{-X} \xrightarrow{k_2} \text{C}_\alpha=\text{C}_\beta + \text{X}^-$

4.2 Rate Law Derivation

The overall rate is determined by the rate of the slow step (loss of the leaving group from the carbanion):

$\text{rate} = k_2[\,^-\text{C-C-X}]$

We need to express the carbanion concentration in terms of observable quantities. Applying the steady-state approximation to the carbanion intermediate:

$\frac{d[\,^-\text{C-C-X}]}{dt} = k_1[\text{B}^-][\text{H-C-C-X}] - k_{-1}[\text{BH}][\,^-\text{C-C-X}] - k_2[\,^-\text{C-C-X}] = 0$

Solving for the carbanion concentration:

$[\,^-\text{C-C-X}] = \frac{k_1[\text{B}^-][\text{H-C-C-X}]}{k_{-1}[\text{BH}] + k_2}$

Substituting into the rate expression:

$\text{rate} = \frac{k_1 k_2 [\text{B}^-][\text{H-C-C-X}]}{k_{-1}[\text{BH}] + k_2}$

4.3 Limiting Cases

This general rate expression simplifies under two limiting conditions:

Case 1: $k_2 \gg k_{-1}[\text{BH}]$ (irreversible deprotonation)

$\text{rate} \approx k_1[\text{B}^-][\text{H-C-C-X}]$

This reduces to E2-like kinetics (second order). The deprotonation is rate-determining and irreversible.

Case 2: $k_{-1}[\text{BH}] \gg k_2$ (reversible deprotonation, true E1cb)

$\text{rate} \approx \frac{k_1 k_2}{k_{-1}} \cdot \frac{[\text{B}^-][\text{H-C-C-X}]}{[\text{BH}]}$

The rate depends on the base-to-conjugate-acid ratio, which is equivalent to the equilibrium constant for deprotonation. This is the true E1cb mechanism.

4.4 When Does E1cb Occur?

The E1cb mechanism is favored when:

● Acidic $\alpha$-hydrogens: Electron-withdrawing groups (e.g., $\text{C}=\text{O}$, $\text{NO}_2$, $\text{CN}$) on the $\alpha$-carbon stabilize the carbanion
● Poor leaving group: The leaving group is reluctant to depart (e.g., $\text{-OH}$, $\text{-OR}$, $\text{-F}$)
● Strong base: A strong base drives the first equilibrium to the right

Classic E1cb Example

The elimination of HF from fluoroacetone in base proceeds via E1cb. The carbonyl group acidifies the$\alpha$-hydrogens, and fluoride is a poor leaving group (despite being a good nucleophile in SN2). Deuterium-labeling experiments confirm that deprotonation is fast and reversible (H/D exchange is faster than elimination).

5. Derivation 4: Regioselectivity — Zaitsev and Hofmann Rules

When a substrate has multiple $\beta$-hydrogens on different carbons, elimination can produce different regioisomeric alkenes. Understanding which alkene predominates requires analyzing alkene stability and steric effects.

5.1 Zaitsev's Rule

Zaitsev's rule (also spelled Saytzeff) states that the major product of an elimination reaction is the more substituted alkene — the thermodynamic product. This rule applies when the transition state has significant alkene character (product-like transition state).

5.2 Deriving Alkene Stability from Hyperconjugation

The stability of alkenes increases with substitution due to hyperconjugation. Each alkyl substituent on the double bond has C-H bonds that can overlap with the empty$\pi^*$ orbital of the alkene:

Hyperconjugation: $\sigma_{\text{C-H}} \rightarrow \pi^*_{\text{C=C}}$

Each adjacent C-H bond provides a stabilizing two-electron interaction. The stabilization energy from a single hyperconjugative interaction can be estimated:

$\Delta E_{\text{stab}} \approx \frac{-2\langle \sigma_{\text{C-H}} | \hat{F} | \pi^*_{\text{C=C}} \rangle^2}{\varepsilon_{\pi^*} - \varepsilon_{\sigma}}$

where $\hat{F}$ is the Fock operator, and $\varepsilon$ are the orbital energies. Since the energy gap $\varepsilon_{\pi^*} - \varepsilon_{\sigma}$ is relatively constant for all C-H bonds, the total stabilization is approximately proportional to the number of hyperconjugating C-H bonds:

$\Delta E_{\text{total}} \propto n_{\text{C-H(adjacent)}}$

This explains why tetrasubstituted alkenes are more stable than trisubstituted, which are more stable than disubstituted, and so on. The heats of hydrogenation confirm this trend experimentally:

$\text{ethylene: } \Delta H_{\text{hydrog}} = -137 \text{ kJ/mol (least stable, no hyperconjugation)}$

$\text{propene: } \Delta H_{\text{hydrog}} = -126 \text{ kJ/mol (monosubstituted)}$

$\text{trans-2-butene: } \Delta H_{\text{hydrog}} = -116 \text{ kJ/mol (disubstituted)}$

$\text{2-methyl-2-butene: } \Delta H_{\text{hydrog}} = -112 \text{ kJ/mol (trisubstituted)}$

$\text{2,3-dimethyl-2-butene: } \Delta H_{\text{hydrog}} = -111 \text{ kJ/mol (tetrasubstituted)}$

5.3 Hofmann's Rule

Hofmann's rule states that with bulky bases or when special structural features apply, the less substituted alkene (anti-Zaitsev product) is the major product. The Hofmann product predominates in the following situations:

● Bulky bases: Sterically hindered bases such as t-BuO⁻, LDA, or DBU cannot easily access the more substituted (more hindered) $\beta$-hydrogen
● Quaternary ammonium salts (Hofmann elimination): When the leaving group is $\text{-NR}_3^+$, the transition state is early (reactant-like), and the less hindered H is removed preferentially
● Fluoride leaving group: The strong C-F bond means the transition state is early, favoring the less substituted product

Quantitative Treatment: Hammond's Postulate

The regioselectivity can be rationalized through Hammond's postulate. For an exothermic elimination (product-like transition state), the transition state resembles the product alkene, and the more stable (Zaitsev) alkene is favored. For an endothermic or early transition state, steric factors dominate, and the less hindered (Hofmann) product is favored.

The selectivity can be expressed as a Boltzmann distribution of products at the transition state:

$\frac{[\text{Zaitsev}]}{[\text{Hofmann}]} = \frac{n_Z}{n_H} \exp\!\left(-\frac{\Delta\Delta G^\ddagger}{RT}\right)$

where $n_Z$ and $n_H$ are statistical factors (number of $\beta$-hydrogens leading to each product) and$\Delta\Delta G^\ddagger$ is the difference in activation energies.

6. Applications of Elimination Reactions

6.1 Dehydration of Alcohols

Alcohols undergo acid-catalyzed dehydration to form alkenes. The mechanism depends on the substrate class:

Tertiary alcohols (E1): $\text{R}_3\text{C-OH} \xrightarrow{\text{H}_2\text{SO}_4, \Delta} \text{R}_2\text{C=CR}_2 + \text{H}_2\text{O}$

The alcohol is first protonated to form a good leaving group ($\text{H}_2\text{O}$), then the water departs to form a tertiary carbocation, which loses a proton to give the alkene. Typical conditions: concentrated $\text{H}_2\text{SO}_4$ at 60-80°C.

Primary alcohols (E2): $\text{RCH}_2\text{CH}_2\text{-OH} \xrightarrow{\text{H}_2\text{SO}_4, 170°\text{C}} \text{RCH=CH}_2 + \text{H}_2\text{O}$

Primary alcohols require harsher conditions (170°C with $\text{H}_2\text{SO}_4$) because primary carbocations are too unstable. The mechanism is typically E2-like (concerted loss of $\text{H}_2\text{O}$and the $\beta$-proton).

6.2 Synthesis of Alkenes

Elimination is one of the most direct methods for introducing unsaturation. Key synthetic strategies include:

● Dehydrohalogenation: Treatment of alkyl halides with strong base (e.g., NaOEt/EtOH or t-BuOK/t-BuOH)
● Dehydration: Acid-catalyzed elimination of water from alcohols
● Hofmann elimination: Quaternary ammonium salts treated with Ag₂O or heat to form the less substituted alkene
● Cope elimination: Amine oxides undergo syn-periplanar elimination at 100-150°C

6.3 Industrial Production of Ethylene

Ethylene ($\text{C}_2\text{H}_4$) is the world's most produced organic chemical, with annual production exceeding 200 million tonnes. It is synthesized primarily by:

Steam cracking of ethane: $\text{C}_2\text{H}_6 \xrightarrow{800\text{-}900°\text{C}} \text{C}_2\text{H}_4 + \text{H}_2$

This is a thermal elimination (pyrolysis) that proceeds via a free-radical mechanism at very high temperatures. The short residence time (milliseconds) and rapid quenching minimize side reactions.

Dehydration of ethanol (bioethanol route): $\text{CH}_3\text{CH}_2\text{OH} \xrightarrow{\text{Al}_2\text{O}_3, 300°\text{C}} \text{CH}_2=\text{CH}_2 + \text{H}_2\text{O}$

Ethylene serves as a feedstock for polyethylene, ethylene oxide, vinyl chloride, styrene, and many other industrial chemicals, making elimination chemistry foundational to the modern chemical industry.

7. Historical Context

Alexander Zaitsev (1841–1910)

Russian chemist at the University of Kazan. In 1875, he published his observation that elimination reactions preferentially form the more substituted alkene. His rule, formulated from studies of alcohol dehydration and dehydrohalogenation, remains one of the most important predictive tools in organic chemistry. Zaitsev was a student of Alexander Butlerov, another pioneer of structural theory.

August Wilhelm von Hofmann (1818–1892)

German chemist who founded the Royal College of Chemistry in London. His work on quaternary ammonium salts led to the discovery that these substrates preferentially eliminate to form the less substituted alkene — the opposite of Zaitsev's rule. The Hofmann elimination (exhaustive methylation followed by thermal elimination) became a key method for determining amine structures before modern spectroscopy.

Christopher Ingold (1893–1970)

British chemist at University College London. Together with Edward Hughes, Ingold established the mechanistic classification of elimination reactions into E1 and E2, paralleling their work on SN1 and SN2 substitution. Their kinetic studies in the 1930s-1950s provided the framework for understanding how substrate structure, base strength, and solvent determine the mechanism. Ingold also introduced the terms “electrophile” and “nucleophile.”

Timeline of Key Developments

1875: Zaitsev publishes his rule on preferential formation of more substituted alkenes
1851–1881: Hofmann develops exhaustive methylation and observes anti-Zaitsev elimination from quaternary ammonium salts
1927–1935: Ingold and Hughes classify elimination mechanisms into E1 and E2 based on kinetic studies
1940s: Cristol demonstrates the anti-periplanar requirement for E2 using rigid cyclohexane systems
1950s: Cram, Winstein, and others refine understanding of stereoelectronic effects in elimination
1960s: Bordwell establishes the E1cb mechanism through isotope-labeling experiments
1970s–present: Computational chemistry provides detailed energy surfaces and transition state structures for all elimination pathways

8. Python Simulation: Energy Diagrams & Alkene Stability

The following simulation generates two sets of plots. The first compares the energy profiles of E1 and E2 reaction pathways, illustrating the concerted single-barrier nature of E2 versus the two-step pathway of E1 with its carbocation intermediate. The second set plots experimental alkene stability (from heats of hydrogenation) against degree of substitution, and models the competition between E1 and E2 as a function of base strength.

Simulation

Python

script.py164 lines

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

# ============================================================
# Figure 1: Energy Diagram Comparing E1 and E2 Pathways
# ============================================================

fig, axes = plt.subplots(1, 2, figsize=(14, 6))
fig.patch.set_facecolor('#0a0a0a')

# E2 Pathway (concerted, single transition state)
x_e2 = np.array([0, 1, 2, 3, 4])
y_e2 = np.array([0, 0, 85, 0, -15])  # relative energies in kJ/mol
x_smooth_e2 = np.linspace(0, 4, 300)
# Interpolate smoothly
from numpy.polynomial import polynomial as P
coeffs = np.polyfit(x_e2, y_e2, 4)
y_smooth_e2 = np.polyval(coeffs, x_smooth_e2)

axes[0].plot(x_smooth_e2, y_smooth_e2, color='#fb7185', linewidth=2.5, label='E2 pathway')
axes[0].axhline(y=0, color='gray', linestyle='--', alpha=0.3)
axes[0].axhline(y=-15, color='gray', linestyle=':', alpha=0.3)

# Annotate E2
axes[0].annotate('Reactants\n(R-X + B\u207b)', xy=(0.3, 5), fontsize=9, color='#fda4af')
axes[0].annotate('Transition State\n(single, concerted)', xy=(1.5, 88), fontsize=9, color='#fda4af', ha='center')
axes[0].annotate('Products\n(Alkene + HB + X\u207b)', xy=(3.0, -10), fontsize=9, color='#fda4af')
axes[0].annotate('', xy=(2.0, 85), xytext=(2.0, 0),
    arrowprops=dict(arrowstyle='<->', color='#f43f5e', lw=1.5))
axes[0].text(2.15, 42, '\u0394G\u2021 (E2)', fontsize=10, color='#f43f5e', fontweight='bold')

axes[0].set_xlabel('Reaction Coordinate', fontsize=12, color='white')
axes[0].set_ylabel('Free Energy (kJ/mol)', fontsize=12, color='white')
axes[0].set_title('E2 Mechanism (Concerted)', fontsize=14, color='#fb7185', fontweight='bold')
axes[0].set_facecolor('#0f172a')
axes[0].tick_params(colors='white')
axes[0].set_xticks([])
axes[0].legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#fb7185', labelcolor='white')
for spine in axes[0].spines.values():
    spine.set_color('#fb7185')

# E1 Pathway (two-step, carbocation intermediate)
x_e1 = np.array([0, 1, 2, 3, 4, 5, 6])
y_e1 = np.array([0, 0, 70, 40, 55, 0, -12])
x_smooth_e1 = np.linspace(0, 6, 400)
coeffs_e1 = np.polyfit(x_e1, y_e1, 6)
y_smooth_e1 = np.polyval(coeffs_e1, x_smooth_e1)

axes[1].plot(x_smooth_e1, y_smooth_e1, color='#c084fc', linewidth=2.5, label='E1 pathway')
axes[1].axhline(y=0, color='gray', linestyle='--', alpha=0.3)
axes[1].axhline(y=40, color='gray', linestyle=':', alpha=0.3)

# Annotate E1
axes[1].annotate('Reactants\n(R-X)', xy=(0.2, 5), fontsize=9, color='#d8b4fe')
axes[1].annotate('TS1\n(C-X breaking)', xy=(1.5, 73), fontsize=9, color='#d8b4fe', ha='center')
axes[1].annotate('Carbocation\nintermediate', xy=(2.7, 44), fontsize=9, color='#d8b4fe', ha='center')
axes[1].annotate('TS2\n(deprotonation)', xy=(3.7, 58), fontsize=9, color='#d8b4fe', ha='center')
axes[1].annotate('Products\n(Alkene + HB + X\u207b)', xy=(4.8, -8), fontsize=9, color='#d8b4fe')

axes[1].annotate('', xy=(2.0, 70), xytext=(2.0, 0),
    arrowprops=dict(arrowstyle='<->', color='#a855f7', lw=1.5))
axes[1].text(2.15, 35, '\u0394G\u2021\u2081 (RDS)', fontsize=10, color='#a855f7', fontweight='bold')

axes[1].set_xlabel('Reaction Coordinate', fontsize=12, color='white')
axes[1].set_ylabel('Free Energy (kJ/mol)', fontsize=12, color='white')
axes[1].set_title('E1 Mechanism (Two-Step)', fontsize=14, color='#c084fc', fontweight='bold')
axes[1].set_facecolor('#0f172a')
axes[1].tick_params(colors='white')
axes[1].set_xticks([])
axes[1].legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#c084fc', labelcolor='white')
for spine in axes[1].spines.values():
    spine.set_color('#c084fc')

plt.tight_layout()
plt.savefig('e1_e2_energy_diagrams.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()

# ============================================================
# Figure 2: Alkene Stability vs Degree of Substitution
# ============================================================

fig2, axes2 = plt.subplots(1, 2, figsize=(14, 6))
fig2.patch.set_facecolor('#0a0a0a')

# Alkene stability (heats of hydrogenation, kJ/mol - less negative = more stable)
substitution = ['Ethylene\n(none)', 'Propene\n(mono)', '1-Butene\n(mono)', 'cis-2-Butene\n(di)', 'trans-2-Butene\n(di)', '2-Me-2-butene\n(tri)', '2,3-DiMe-2-butene\n(tetra)']
heats = [-137.0, -126.0, -127.0, -120.0, -116.0, -112.0, -111.0]
stability = [-h for h in heats]  # more positive = more stable
colors_bar = ['#fda4af', '#fb7185', '#f43f5e', '#e11d48', '#be123c', '#9f1239', '#881337']

bars = axes2[0].bar(range(len(substitution)), stability, color=colors_bar, edgecolor='white', linewidth=0.5)
axes2[0].set_xticks(range(len(substitution)))
axes2[0].set_xticklabels(substitution, fontsize=7, color='white', rotation=0)
axes2[0].set_ylabel('Relative Stability (\u2212\u0394H_hydrog, kJ/mol)', fontsize=11, color='white')
axes2[0].set_title('Alkene Stability vs Substitution\n(Zaitsev Rule Basis)', fontsize=13, color='#fb7185', fontweight='bold')
axes2[0].set_facecolor('#0f172a')
axes2[0].tick_params(colors='white')
for spine in axes2[0].spines.values():
    spine.set_color('#fb7185')
axes2[0].grid(axis='y', alpha=0.2, color='gray')

# Add value labels on bars
for bar_item, val in zip(bars, stability):
    axes2[0].text(bar_item.get_x() + bar_item.get_width()/2, val + 0.5,
        f'{val:.0f}', ha='center', fontsize=8, color='white', fontweight='bold')

# Plot 2: E1 vs E2 competition as function of base strength
base_strength = np.linspace(0, 10, 200)  # arbitrary scale
# E2 rate increases with base strength; E1 rate is independent
e2_fraction = 1 / (1 + np.exp(-(base_strength - 5)))
e1_fraction = 1 - e2_fraction

axes2[1].fill_between(base_strength, 0, e2_fraction, color='#fb7185', alpha=0.6, label='E2 (bimolecular)')
axes2[1].fill_between(base_strength, 0, e1_fraction, color='#c084fc', alpha=0.6, label='E1 (unimolecular)')
axes2[1].plot(base_strength, e2_fraction, color='#fb7185', linewidth=2)
axes2[1].plot(base_strength, e1_fraction, color='#c084fc', linewidth=2)

axes2[1].axvline(x=5, color='white', linestyle='--', alpha=0.5)
axes2[1].text(5.2, 0.85, 'Crossover\npoint', fontsize=10, color='white', alpha=0.8)
axes2[1].text(1.5, 0.7, 'E1\ndominates', fontsize=12, color='#c084fc', fontweight='bold', ha='center')
axes2[1].text(8.0, 0.7, 'E2\ndominates', fontsize=12, color='#fb7185', fontweight='bold', ha='center')

axes2[1].set_xlabel('Base Strength / Concentration \u2192', fontsize=12, color='white')
axes2[1].set_ylabel('Fraction of Product', fontsize=12, color='white')
axes2[1].set_title('E1 vs E2 Competition', fontsize=13, color='#fb7185', fontweight='bold')
axes2[1].set_facecolor('#0f172a')
axes2[1].tick_params(colors='white')
axes2[1].set_xticks([])
axes2[1].legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#fb7185', labelcolor='white')
for spine in axes2[1].spines.values():
    spine.set_color('#fb7185')

plt.tight_layout()
plt.savefig('alkene_stability_competition.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()

print("=== Elimination Reactions: Computational Analysis ===")
print()
print("--- Alkene Stability (Heats of Hydrogenation) ---")
names = ['Ethylene', 'Propene', '1-Butene', 'cis-2-Butene', 'trans-2-Butene', '2-Methyl-2-butene', '2,3-Dimethyl-2-butene']
for n, h in zip(names, heats):
    print(f"  {n:25s}: delta_H_hydrog = {h:7.1f} kJ/mol  (stability = {-h:.1f})")
print()
print("--- Key Observations ---")
print("1. More substituted alkenes are more stable (less negative heat of hydrogenation)")
print("2. trans isomers are more stable than cis isomers (less steric strain)")
print("3. Hyperconjugation from alkyl C-H bonds into pi* stabilizes substituted alkenes")
print()
print("--- E1 vs E2 Competition Summary ---")
print("  E2 favored by: strong base, high [base], primary substrates, anti-periplanar geometry")
print("  E1 favored by: weak base, polar protic solvent, tertiary substrates, high temperature")
print()
print("--- Activation Energy Estimates ---")
print(f"  E2 (typical): 80-100 kJ/mol (concerted, single barrier)")
print(f"  E1 step 1 (C-X heterolysis): 100-130 kJ/mol (rate-determining)")
print(f"  E1 step 2 (deprotonation): 20-40 kJ/mol (fast)")
print()
print("--- Zaitsev vs Hofmann Product Ratios ---")
print("  t-BuOK + 2-bromobutane: ~80% Zaitsev (2-butene), ~20% Hofmann (1-butene)")
print("  t-BuOK + 2-bromo-2-methylbutane: ~30% Zaitsev, ~70% Hofmann (bulky base)")
print("  EtONa + 2-bromobutane: ~80% Zaitsev, ~20% Hofmann")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Interpreting the Results

●E2 energy diagram: A single transition state with one activation barrier. The concerted mechanism means all bond-breaking and bond-forming events occur simultaneously.
●E1 energy diagram: Two transition states separated by a carbocation intermediate (local energy minimum). The first step (C-X bond heterolysis) has the higher barrier and is rate-determining.
●Alkene stability plot: The bar chart shows that heats of hydrogenation become less exothermic with increasing substitution, confirming Zaitsev's rule has a thermodynamic basis.
●E1 vs. E2 competition: The sigmoid crossover plot demonstrates how increasing base strength/concentration shifts the mechanism from E1 to E2, as the bimolecular pathway becomes kinetically dominant.

Summary: Comparing E1, E2, and E1cb

Feature	E2	E1	E1cb
Mechanism	Concerted (one step)	Two-step (carbocation)	Two-step (carbanion)
Rate law	$k[\text{B}^-][\text{RX}]$	$k[\text{RX}]$	$\frac{k_1 k_2[\text{B}^-][\text{RX}]}{k_{-1}[\text{BH}] + k_2}$
Order	Second order	First order	Variable (first or second)
Substrate	All (1°, 2°, 3°)	3° > 2° (never 1°)	Acidic C-H, poor LG
Base	Strong required	Weak (or solvent)	Strong required
Stereochemistry	Anti-periplanar	No requirement	Anti preferred
Regioselectivity	Zaitsev (or Hofmann)	Zaitsev	Hofmann (often)
Competes with	SN2	SN1	Neither (unique conditions)

Key Equations

E2 Rate Law

$\text{rate}_{\text{E2}} = k_{\text{E2}}[\text{B}^-][\text{R-X}]$

E1 Rate Law

$\text{rate}_{\text{E1}} = k_1[\text{R-X}]$

E1cb General Rate

$\text{rate} = \frac{k_1 k_2 [\text{B}^-][\text{RX}]}{k_{-1}[\text{BH}] + k_2}$

Orbital Overlap

$\sigma_{\text{C-H}} \rightarrow \sigma^*_{\text{C-X}} \quad (\phi = 180°)$

Hyperconjugation

$\Delta E_{\text{stab}} \approx \frac{-2\langle \sigma_{\text{C-H}} | \hat{F} | \pi^* \rangle^2}{\varepsilon_{\pi^*} - \varepsilon_{\sigma}}$

Product Ratio (Boltzmann)

$\frac{[\text{Z}]}{[\text{H}]} = \frac{n_Z}{n_H}\exp\!\left(-\frac{\Delta\Delta G^\ddagger}{RT}\right)$

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