Reactivity Learn

Mayr's equation

Herbert Mayr's group at LMU Munich measured the rates of thousands of reactions between nucleophiles and electrophiles and found that one short equation describes them, at 20 °C:

log k = s_N (N + E)log k₂₀ °C, with k the second-order rate constant in M⁻¹ s⁻¹

N

the nucleophilicity of the nucleophile. It belongs to the nucleophile in one solvent: the same anion has different N values in water and in DMSO.

s_N

the sensitivity of that nucleophile, the slope of its line below. Most values lie between about 0.5 and 1.2.

E

the electrophilicity of the electrophile, treated as independent of the solvent.

Because log k = 0 when E = −N, a nucleophile's N is minus the E of an electrophile it attacks with k = 1 M⁻¹ s⁻¹. The parameters of a few hundred electrophiles and more than a thousand nucleophiles are collected in Mayr's Database of Reactivity Parameters, where every card in this app links to its entry.

How N, s_N and E are measured

Mayr and Patz fixed the scales in 1994 with two definitions: E = 0 for the bis(4-methoxyphenyl)methylium ion, and s_N = 1 for 2-methylpent-1-ene. Benzhydrylium ions Ar₂CH⁺ and quinone methides, whose reactivity can be tuned over a wide range by changing their substituents, became the reference electrophiles.

1. A new nucleophile is combined with several reference electrophiles of known E, and each rate constant k is measured, usually by following the coloured electrophile by UV–vis spectroscopy.
2. Plotting log k against E gives a straight line. Its slope is s_N, and the E where it crosses log k = 0 is −N.
3. A new electrophile is measured the other way round, against reference nucleophiles of known N and s_N: each gives E = log k / s_N − N, and the values are averaged.

In 2012 the whole benzhydrylium data set was refitted (Ammer, Nolte and Mayr); the database, and this app, use the current values.

Reading the numbers

• The −5 rule of thumb. When s_N is ignored, a pair reacts at room temperature if N + E > −5. That is the rule behind Reacting?; the feedback then works out log k with s_N.
• The fast end. Near k = 10⁹–10¹⁰ M⁻¹ s⁻¹ reactions become diffusion-controlled: every encounter reacts, the equation stops working (above about 10⁸) and selectivity is lost.
• One solvent at a time. N depends strongly on the solvent, especially for anions, which water and alcohols solvate by hydrogen bonding. The nitromethane anion has N = 20.71 in DMSO but 12.06 in water. Rank therefore only compares nucleophiles measured in the same solvent; E values can be compared across all sets.
• Crossing lines. Two nucleophiles with different s_N can swap order depending on the electrophile, as two lines in the chart above can cross. Ranking by N alone, as in Rank, is Mayr's own first approximation.
• Where the equation works. It was built for reactions in which one new bond forms and steric effects are small. Bulky partners (a trityl cation with a hindered amine) and SN2 reactions at alkyl halides are less well described.

Which atom reacts

Many nucleophiles have more than one nucleophilic atom (enolates, cyanide, thiocyanate, pyridones, amino alcohols), and many electrophiles more than one electrophilic atom (enones, molecules with two carbonyl groups). For ambident reactivity Mayr, Breugst and Ofial (2011) replaced the hard and soft acids and bases (HSAB) rule with three questions:

1. Is the reaction reversible? Then the more stable product wins (thermodynamic control): N-attack of thiocyanate, C-alkylation of enamines.
2. If not, which site is faster? Marcus theory splits the barrier into an intrinsic part and the reaction's driving force. Reactions with high intrinsic barriers, such as SN2 at carbon, follow the more stable product even under kinetic control: enolates are C-alkylated by alkyl halides.
3. Is it diffusion-controlled? Then every encounter reacts and there is no selectivity: very reactive carbocations attack cyanide at both ends.

For regioselectivity in addition, substitution and elimination reactions, the same two ideas, the stability of the cation, anion or radical formed and kinetic against thermodynamic control, explain most of the questions in the Regioselectivity tab. Each answer comes with a short reason.

Computed indices: ω and N from orbital energies

Mayr's parameters are measured. Computational chemists often estimate reactivity instead from a single calculation per molecule, with the conceptual DFT indices. With Koopmans' approximation, the ionisation energy is I ≈ −ε_HOMO and the electron affinity A ≈ −ε_LUMO, and:

μ = (ε_HOMO + ε_LUMO) / 2

electronic chemical potential: how strongly the molecule holds on to its electrons (Parr and Pearson 1983).

η = ε_LUMO − ε_HOMO

chemical hardness: resistance to charge transfer. Some authors include a factor ½; Domingo's scales below leave it out.

ω = μ² / (2η)

global electrophilicity index: the energy gained when the molecule takes up as much electron density from its surroundings as it can (Parr, von Szentpály and Liu 1999).

N = ε_HOMO(Nu) − ε_HOMO(TCE)

Domingo's nucleophilicity index, measured from the HOMO of tetracyanoethylene computed at the same level of theory (B3LYP/6-31G(d) in the published scales), so that it is positive for nearly all nucleophiles (Domingo, Chamorro and Pérez 2008).

The inverse of the electrophilicity, 1/ω, is sometimes used as a nucleophilicity index. It follows nucleophilicity only within limited series of molecules (it fails, for example, for captodative ethylenes), so the thresholds above are for Domingo's N.

Local versions: which atom

Condensed Fukui functions split a global index over the atoms. From the atomic charges q_k of the molecule with N, N+1 and N−1 electrons, all at the neutral geometry: f_k⁺ = q_k(N) − q_k(N+1) marks the atoms that nucleophiles attack, and f_k⁻ = q_k(N−1) − q_k(N) the atoms that attack electrophiles (Yang and Mortier 1986). Domingo's Parr functions use the atomic spin densities of the radical anion (P_k⁺) and radical cation (P_k⁻) instead, giving local indices ω_k = ω·P_k⁺ and N_k = N·P_k⁻.

How the two kinds of number relate

The conceptual DFT indices describe one isolated molecule from one calculation; Mayr's parameters are rate constants measured in solution, with solvation and steric effects included. They tend to agree within a family of similar molecules, but they are not on the same scale: Mayr's N and Domingo's N share a letter and nothing else.

References

1. H. Mayr, M. Patz, Scales of nucleophilicity and electrophilicity: a system for ordering polar organic and organometallic reactions. Angew. Chem. Int. Ed. Engl. 1994, 33, 938–957. doi:10.1002/anie.199409381
2. H. Mayr, T. Bug, M. F. Gotta, N. Hering, B. Irrgang, B. Janker, B. Kempf, R. Loos, A. R. Ofial, G. Remennikov, H. Schimmel, Reference scales for the characterization of cationic electrophiles and neutral nucleophiles. J. Am. Chem. Soc. 2001, 123, 9500–9512. doi:10.1021/ja010890y
3. H. Mayr, A. R. Ofial, Kinetics of electrophile–nucleophile combinations: a general approach to polar organic reactivity. Pure Appl. Chem. 2005, 77, 1807–1821. doi:10.1351/pac200577111807
4. J. Ammer, C. Nolte, H. Mayr, Free energy relationships for reactions of substituted benzhydrylium ions: from enthalpy over entropy to diffusion control. J. Am. Chem. Soc. 2012, 134, 13902–13911. doi:10.1021/ja306522b
5. H. Mayr, M. Breugst, A. R. Ofial, Farewell to the HSAB treatment of ambident reactivity. Angew. Chem. Int. Ed. 2011, 50, 6470–6505. doi:10.1002/anie.201007100
6. K. Wolff, A. R. Ofial, C. Proppe, Revisiting Mayr's reactivity database: expansion, sensitivity analysis, and uncertainty quantification. Org. Biomol. Chem. 2025, 23, 7188–7196. doi:10.1039/D5OB00686D
7. R. G. Parr, R. G. Pearson, Absolute hardness: companion parameter to absolute electronegativity. J. Am. Chem. Soc. 1983, 105, 7512–7516. doi:10.1021/ja00364a005
8. R. G. Parr, L. von Szentpály, S. Liu, Electrophilicity index. J. Am. Chem. Soc. 1999, 121, 1922–1924. doi:10.1021/ja983494x
9. R. G. Parr, W. Yang, Density functional approach to the frontier-electron theory of chemical reactivity. J. Am. Chem. Soc. 1984, 106, 4049–4050. doi:10.1021/ja00326a036
10. W. Yang, W. J. Mortier, The use of global and local molecular parameters for the analysis of the gas-phase basicity of amines. J. Am. Chem. Soc. 1986, 108, 5708–5711. doi:10.1021/ja00279a008
11. L. R. Domingo, E. Chamorro, P. Pérez, Understanding the reactivity of captodative ethylenes in polar cycloaddition reactions. A theoretical study. J. Org. Chem. 2008, 73, 4615–4624. doi:10.1021/jo800572a
12. L. R. Domingo, P. Pérez, J. A. Sáez, Understanding the local reactivity in polar organic reactions through electrophilic and nucleophilic Parr functions. RSC Adv. 2013, 3, 1486–1494. doi:10.1039/C2RA22886F
13. L. R. Domingo, M. Ríos-Gutiérrez, P. Pérez, Applications of the conceptual density functional theory indices to organic chemistry reactivity. Molecules 2016, 21, 748 (open access). doi:10.3390/molecules21060748

Each card's N, s_N or E comes from Mayr's Database of Reactivity Parameters; the card names its entry and the original paper, as the database asks.