Organic Chemistry · Part 1 of 9 · Free
Foundations — How Mechanisms Work — formula sheet
Every key expression and definition from Reaction Mechanisms, Part 1, on one page. Free to read, no sign-in.
Key expressions
second-order perturbation stabilisation from a donor–acceptor interaction
∆E_stab ≈ 2 × |⟨ψ_donor|H|ψ_acceptor⟩|² ÷ (ε_donor − ε_acceptor)
∆E_stab ≈ 2 × |⟨ψ_donor|H|ψ_acceptor⟩|² ÷ (ε_donor − ε_acceptor)
the two contributions to the interaction energy: electrostatic and orbital
∆E = q_Nuq_E / εR + 2 Σ_occΣ_unocc (c_Nuc_Eβ)² / (ε_HOMO − ε_LUMO) [Klopman–Salem, simplified]
∆E = q_Nuq_E / εR + 2 Σ_occΣ_unocc (c_Nuc_Eβ)² / (ε_HOMO − ε_LUMO) [Klopman–Salem, simplified]
nucleophilicity parameter n and substrate sensitivity s
log(k / k_0) = s · n [Swain–Scott]
log(k / k_0) = s · n [Swain–Scott]
the dual-substituent-parameter separation of inductive and resonance effects
σ_p = σ_I + σ_R σ_m ≈ σ_I + ⅓σ_R
σ_p = σ_I + σ_R σ_m ≈ σ_I + ⅓σ_R
the Arrhenius equation
k = A e^−E_a/RT ln k = ln A − E_a/RT
k = A e^−E_a/RT ln k = ln A − E_a/RT
the Eyring equation from transition state theory
k = (k_BT / h) e^−∆G‡/RT = (k_BT / h) e^∆S‡/R e^−∆H‡/RT
k = (k_BT / h) e^−∆G‡/RT = (k_BT / h) e^∆S‡/R e^−∆H‡/RT
the Eyring equation with a transmission coefficient
k = κ (k_BT / h) e^−∆G‡/RT
k = κ (k_BT / h) e^−∆G‡/RT
the Bell–Evans–Polanyi relation
E_a = E_0 + α∆H (0 < α < 1)
E_a = E_0 + α∆H (0 < α < 1)
the Marcus equation
∆G‡ = ∆G‡_0 (1 + ∆G°/4∆G‡_0)²
∆G‡ = ∆G‡_0 (1 + ∆G°/4∆G‡_0)²
the Curtin–Hammett product ratio — transition states only
[P_1] / [P_2] = e^−(∆G‡_1 − ∆G‡_2)/RT
[P_1] / [P_2] = e^−(∆G‡_1 − ∆G‡_2)/RT
product ratio under kinetic control
kinetic limit: [P_K]/[P_T] = k_K/k_T = e^−∆∆G‡/RT
kinetic limit: [P_K]/[P_T] = k_K/k_T = e^−∆∆G‡/RT
product ratio under thermodynamic control
thermodynamic limit: [P_T]/[P_K] = K = e^−∆∆G°/RT
thermodynamic limit: [P_T]/[P_K] = K = e^−∆∆G°/RT
converting a pK_a difference into a free energy
∆G° = 2.303 RT × ∆pK_a ≈ 1.36 × ∆pK_a kcal mol^−1 at 298 K
∆G° = 2.303 RT × ∆pK_a ≈ 1.36 × ∆pK_a kcal mol^−1 at 298 K
the equilibrium constant for any acid–base reaction
K_eq = 10^[pK_a(conjugate acid of the base) − pK_a(acid)]
K_eq = 10^[pK_a(conjugate acid of the base) − pK_a(acid)]
absolute hardness η and electronic chemical potential μ
η = (I − A)/2 ≈ (ε_LUMO − ε_HOMO)/2 μ = −(I + A)/2 ≈ (ε_HOMO + ε_LUMO)/2
η = (I − A)/2 ≈ (ε_LUMO − ε_HOMO)/2 μ = −(I + A)/2 ≈ (ε_HOMO + ε_LUMO)/2
Definitions worth memorising
Mechanism: a description of a chemical reaction at the level of individual bonds — which bonds break, which form, in what order, through what intermediates, over what energy barriers, and with what stereochemical consequence. A mechanism is a hypothesis about an unobservable sequence, supported by kinetic, stereochemical, isotopic and spectroscopic evidence. It is never ‘proved’; it is consistent with the evidence, and its rivals are not.
Curly arrow (full-headed, also called double-barbed): the movement of one pair of electrons. Its tail is placed on the pair that moves — a lone pair, or a bond. Its head is placed where that pair ends up — between two atoms if it becomes a bond, or on an atom if it becomes a lone pair.
Fishhook arrow (half-headed, also called single-barbed): the movement of a single electron. Used only for radical and single-electron-transfer mechanisms, which are Part 3 and parts of Part 8. Mixing the two notations in one scheme is a guaranteed lost mark. Note that ‘double-headed arrow’ is reserved for something else entirely — the straight, two-ended resonance arrow ↔ — so do not use it for the curly arrow.
Nucleophile: a species that donates an electron pair to form a covalent bond. Its reactive site carries a high-energy filled orbital — a lone pair, a π bond, or occasionally a polarised σ bond. ‘Nucleus-loving’: it seeks positive charge.
Electrophile: a species that accepts an electron pair. Its reactive site carries a low-energy empty orbital — a vacant p orbital, a π*, or a σ* of a bond to a good leaving group. ‘Electron-loving’.
Basicity is a thermodynamic property: the position of the equilibrium B + H⁺ ⇌ BH⁺, measured by pK_a of the conjugate acid. Nucleophilicity is a kinetic property: the rate constant for attack on a defined electrophilic carbon, measured relative to a standard.
Inductive effect: the polarisation of a σ bond caused by the difference in electronegativity between the atoms it joins, transmitted along the σ framework (and through space) to nearby atoms, attenuating strongly with distance. Written −I for withdrawal and +I for donation, both defined relative to hydrogen.
Resonance effect: delocalisation of electron density through a π system, by overlap of a p-type orbital on the substituent with the π system of the substrate. Written +M (or +R) for donation into the system and −M (−R) for withdrawal from it.
Hyperconjugation: delocalisation involving a σ bond as donor (usually σ(C–H) or σ(C–C)) into an adjacent empty or partially empty p or π* orbital. It requires alignment: the donor bond must be roughly parallel to the acceptor orbital, and the interaction falls off as cos²θ.
Steric effect: the energetic cost of forcing atoms closer than the sum of their van der Waals radii. It is a repulsive, short-range effect and, crucially, it acts on transition states as well as on ground states — which is why it changes rates and not only stabilities.
Reaction coordinate: a composite progress variable describing the lowest-energy path from reactants to products. It is not time, not distance, and not any single bond length — at different points along it, different geometric changes dominate.
Transition state (‡): the point of highest free energy along that lowest-energy path. It is a maximum, so it has no lifetime, cannot be isolated, and has no concentration in the ordinary sense.
∆G‡ (free energy of activation): the height of the transition state above the reactants. It fixes the rate. ∆G°: the free-energy difference between reactants and products. It fixes the equilibrium constant. These are independent quantities.
The lifetime criterion: a proposed intermediate is a real chemical species only if it lives longer than a bond vibration, roughly 10^−13 s. If the calculated lifetime is shorter than that, the species does not exist as a distinct entity and the mechanism must be concerted — not because concertedness is preferred, but because the stepwise alternative is not available.
Rate-determining (rate-limiting) step: the step whose transition state is the highest point on the free-energy profile measured from the resting state of the system. It is decided by the absolute energies of the transition states, not by the heights of the individual climbs.
Hammond postulate: if two states occur consecutively along a reaction coordinate and are close in energy, their interconversion involves only a small reorganisation of structure — so species close in energy are close in structure.
Curtin–Hammett principle: if two conformers (or other rapidly interconverting isomers) react to give different products, and their interconversion is fast compared with either reaction, then the product ratio is determined solely by the difference in the free energies of the two transition states — and is independent of the equilibrium populations of the two conformers.
Kinetic control: the product ratio is determined by the relative rates of the competing product-forming steps — that is, by the relative heights of the competing transition states. The kinetic product is the one formed over the lower barrier. Requires that product formation be effectively irreversible under the conditions used.
Thermodynamic control: the product ratio is determined by the relative stabilities of the products, because they are in equilibrium with each other (usually via the starting material). The thermodynamic product is the more stable one. Requires that product formation be reversible under the conditions used.
pK_a: for the equilibrium HA ⇌ H⁺ + A⁻, pK_a = −log K_a. A lower pK_a means a stronger acid. Each unit is a factor of ten in K_a and, at 298 K, 1.36 kcal mol^−1 of free energy.
Hard: small, high charge density, weakly polarisable, with frontier orbitals far apart in energy. Soft: large, low charge density, highly polarisable, with frontier orbitals close together. The principle: hard acids prefer hard bases and soft acids prefer soft bases — not because opposites repel, but because a hard–hard pair maximises the electrostatic term and a soft–soft pair maximises the orbital term, whereas a mismatched pair maximises neither.
Where these come from
This sheet is distilled from Reaction Mechanisms, Part 1 — 10 sections that derive every one of these results and show you how to use them.
Read Part 1 All formula sheets