Transition state theory — activated complex (intro)

Overview

Transition state theory (TST) explains why reactions have activation energies and how molecules transform from reactants to products. Instead of treating collisions as simple biliard balls, TST recognizes that molecules pass through a high-energy activated complex (or transition state) at the peak of the energy barrier.

Why this matters: Arrhenius told us reactions need activation energy $E_a$, but why? TST answers: molecules must climb an energy hill to rearrange bonds. The height of that hill determines the rate.

Core Concepts

Not the same as an intermediate: An intermediate is a stable (or metastable) species that sits in a local energy minimum. The transition state is at a maximum — a sadle point on the potential energy surface.

$E_{\text{activated complex}} = E_{\text{reactants}} + E_a$

where $E_a$ is the activation energy — the minimum kinetic energy reactants need to reach the transition state.

Derivation from first principles:

1. Potential energy surface: For a reaction $\text{A + B} \rightarrow \text{C}$, plot total energy vs. positions of all atoms. You get a multidimensional surface.
2. Reaction coordinate: The path of minimum energy from reactants to products traces a curve. The highest point along this curve is the transition state.
3. Why a maximum exists: Bonds must stretch/break before new ones form. Stretching costs energy (Coulombic repulsion, loss of bonding electron density). The system must "pay" this cost to rearrange.

$\Delta E_{\text{reaction}} = E_{\text{products}} - E_{\text{reactants}}$ $E_a^{\text{forward}} = E_{\text{TS}} - E_{\text{reactants}}$ $E_a^{\text{reverse}} = E_{\text{TS}} - E_{\text{products}}$

Connection to thermodynamics: $E_a^{\text{forward}} - E_a^{\text{reverse}} = \Delta E_{\text{reaction}}$

Step 1: Reactants approach. $\text{H-H}$ and $\text{I-I}$ bonds are intact.

Step 2: Activated complex forms. Imagine a square arrangement:

H---H
|   |
I---I

• The $\text{H-H}$ bond stretches from 74 pm to ~100 pm (weakened).
• The $\text{I-I}$ bond stretches from 267 pm to ~300 pm.
• New $\text{H-I}$ interactions begin (partial bonding).

Why this geometry? Electrons must delocalize from $\text{H-H}$ and $\text{I-I}$ to form $\text{H-I}$ bonds. At the transition state, all four atoms share electron density — maximum energy configuration.

Step 3: Products form. Two $\text{H-I}$ molecules separate (bond length161 pm).

Energy accounting:

• $E_a \approx 170 \text{ kJ/mol}$ (experimental).
• $\Delta H_{\text{rxn}} \approx +10 \text{ kJ/mol}$ (slightly endothermic).

Why this step? The activation energy pays for simultaneously breaking two bonds. Even though the reaction is only slightly endothermic, the transition state is much higher in energy because bonds break before they fully form.

Mechanism: Backside attack (Walden inversion).

Activated complex:

       δ-    δ-
    OH-----C-----Br
         / | \
        H

• The carbon is pentacoordinate (five bonds — three C-H, one partial C-O, one partial C-Br).
• Geometry is trigonal bipyramidal (the three H's form a triangle, OH and Br are axial).
• Both $\text{C-Br}$ (breaking) and $\text{C-O}$ (forming) are stretched beyond equilibrium.

Why pentacoordinate? Carbon normally allows 4 bonds (sp³). At the transition state, the incoming nucleophile (OH⁻) and leaving group (Br) share the carbon's bonding capacity. Electron density is delocalized — carbon is over-coordinated and unstable.

Energy: $E_a \approx 75 \text{ kJ/mol}$. The barrier comes from:

1. Electrostatic repulsion between OH⁻ and Br (both negatively charged).
2. Steric crowding (five groups around one carbon).
3. Weakening of the C-Br bond before-O strengthens.

Why this step? The $\text{S}_\text{N}\text{2}$ mechanism is concerted (one step), so bond-breaking and bond-making overlap. The transition state pays the cost of simultaneous changes.

Common Mistakes

Steel-man: Intermediates are real species with measurable lifetimes. It's natural to think the transition state is just a very short-lived intermediate.

The fix:

• Intermediate: Local minimum in energy. Exists long enough to potentially observe (nanoseconds to seconds). Example: carbocation in S_N1.
• Transition state: Local maximum. Exists for one bond vibration (~10⁻¹³ s, or 0.1 ps). It's a point of passage, not a resting place.

Analogy: Intermediate = a rest stop on a mountain trail. Transition state = the exact instant your foot leaves the ground mid-step.

Steel-man: For simple dissociations like $\text{H}_2 \rightarrow 2\text{ H}$, $E_a \approx D_{\text{H-H}}$. It's tempting to generalize.

The fix: In most reactions, bonds break and form simultaneously. $E_a$ is the energy to reach the transition state, which depends on:

1. Partial bond breaking.
2. Partial bond forming (which lowers the barrier).
3. Geometry changes (e.g., inversion, rotation).

Example: For $\text{H}_2 + \text{I}_2 \rightarrow 2\text{ HI}$, $E_a =170 \text{ kJ/mol}$. But $D_{\text{H-H}} = 436 \text{ kJ/mol}$ and $D_{\text{I-I}} = 151 \text{ kJ/mol}$. The activation energy is much less than breaking both bonds fully because new H-I bonds start forming in the transition state.

Steel-man: Products are more stable, so the system is "eager" to get there.

The fix: $\Delta H$ and $E_a$ are independent.

• $\Delta H$ compares reactants vs. products (thermodynamics).
• $E_a$ measures reactants vs. transition state (kinetics).

Example:

• $2 \text{H}_2 + \text{O}_2 \rightarrow 2 \text{H}_2\text{O}$ is highly exothermic ($\Delta H = -484 \text{ kJ/mol}$ for gaseous water), but $E_a \approx 200 \text{ kJ/mol}$ (needs a spark to ignite).
• $\text{H}_2 + \text{I}_2 \rightarrow 2\text{ HI}$ is endothermic ($\Delta H = +10 \text{ kJ/mol}$), but $E_a = 170 \text{ kJ/mol}$ (similar barrier).

Hammond's postulate (advanced): For exothermic reactions, the transition state resembles reactants (early barrier). For endothermic reactions, it resembles products (late barrier). But $\Delta H$ alone doesn't determine $E_a$.

Deep Dive: Why Does the Activated Complex Form?

Quantum mechanical view:

1. Reactants have occupied bonding orbitals (e.g., $\sigma_{\text{H-H}}$ for H₂).
2. To react, electrons must redistribute into new orbitals ($\sigma_{\text{H-I}}$ for HI).
3. Transition state: Electrons occupy antibonding or non-bonding orbitals temporarily. This raises energy.

Valence bond picture (for S_N2):

• The incoming nucleophile's lone pair overlaps with the C-Br $\sigma^*$ antibonding orbital.
• At the transition state, electron density is smeared across five atoms. This delocalization costs energy because electrons are less strongly bound.

Why a barrier at all?

• Pauli exclusion: Bringing two closed-shell molecules together causes electron-electron repulsion before bonding interactions dominate.
• Orbital overlap: New bonds require constructive overlap of orbitals, which happens only at specific geometries. Reaching that geometry costs energy.

Connections to Rate Laws

Eyring equation (preview): $k = \frac{k_B T}{h} e^{-\Delta G^\ddagger / RT}$

where $\Delta G^\ddagger$ is the Gibbs energy of activation (related to $E_a$ but includes entropy).

Key insight: The rate constant $k$ depends exponentially on the barrier height. A small change in $E_a$ causes a huge change in rate.

Example: Lowering $E_a$ by 10 kJ/mol at298 K increases the rate by a factor of ~55.

Active Recall

Summary Table

Connections

• Arrhenius equation — $E_a$ in $k = A e^{-E_a/RT}$ is the barrier to form the activated complex
• Collision theory — not all collisions succeed; only those with $KE \geq E_a$ reach the transition state
• Reaction coordinate diagrams — visual map of energy vs. progress; transition state at the peak
• Catalysis — catalysts lower $E_a$ by stabilizing the transition state
• Hammond's postulate — exothermic TS resembles reactants; endothermic TS resembles products
• Eyring equation — TST's quantitative prediction of rate constants from $\Delta G^\ddagger$
• Potential energy surfaces — multidimensional view; transition state is a saddle point
• SN1 vs SN2 mechanisms — different transition states explain different kinetics

#flashcards/chemistry

What is the activated complex? :: The highest-energy configuration along the reaction coordinate, where bonds are partially broken/formed. It's a fleeting maximum-energy state (10⁻¹³ s), not an isolatable intermediate.

Concept Map

Hinglish (regional understanding)