# Taft equation
The Taft equation is a linear free-energy relationship in physical organic chemistry that separates the effect of an aliphatic or ortho substituent on a reaction rate or equilibrium into an additive polar term and a steric term. In the form codified by IUPAC, it is written
\[ \log_{10} k = \log_{10} k_{0} + \rho^{*} \sigma^{*} + \delta E_{\mathrm{s}} \]
where \( \sigma^{*} \) is the polar substituent constant, \( E_{\mathrm{s}} \) is the steric substituent constant, and \( \rho^{*} \) and \( \delta \) measure the sensitivity of the reaction series to polar and steric effects, respectively[1](https://goldbook.iupac.org/terms/view/T06247/html). The term most often designates the family of equations that emerged from Robert W. Taft's analysis of the reactivities of aliphatic esters in the early 1950s; one-parameter forms apply when either the polar or the steric term can be neglected[1](https://goldbook.iupac.org/terms/view/T06247/html).
| Key fact | Detail |
|---|---|
| Full form | \( \log_{10} k = \log_{10} k_{0} + \rho^{*} \sigma^{*} + \delta E_{\mathrm{s}} \), with one-parameter forms when either term is negligible[1](https://goldbook.iupac.org/terms/view/T06247/html) |
| Origin papers | Taft, *J. Am. Chem. Soc.* **74**, 3120–3128 (1952), polar and steric constants from esterification and hydrolysis rates[2](https://cir.nii.ac.jp/crid/1360855569761999744); *J. Am. Chem. Soc.* **75**, 4231–4238 (1953)[3](https://pubs.acs.org/doi/10.1021/ja01113a027) |
| Polar constant | \( \sigma^{*} = \frac{1}{2.48}\{\log (k_{\mathrm{X}}/k_{\mathrm{H}})_{\text{base}} - \log (k_{\mathrm{X}}/k_{\mathrm{H}})_{\text{acid}}\} \) from acid- and base-catalyzed ester hydrolysis[4](https://www.sciencedirect.com/topics/chemistry/reaction-rate-constant) |
| Steric constant | \( E_{\mathrm{s}} \) defined from acid-catalyzed hydrolysis of aliphatic esters with methyl as the reference; groups smaller than methyl (H, F) give positive \( E_{\mathrm{s}} \)[5](https://www.sips.org.in/wp-content/uploads/2021/06/QSAR.pdf) |
| Modern usage | \( \sigma^{*} \) is now usually replaced by the related inductive constant \( \sigma_{I} \)[1](https://goldbook.iupac.org/terms/view/T06247/html) |
| Known weaknesses | Taft averaged four reactions to define \( E_{\mathrm{s}} \), later judged an important source of error; some values for highly hindered alkyl groups appeared contradictory[6](https://www.chem.tsukuba.ac.jp/junji/domestic7_utility/PDFfiles/revised_Taft.pdf) |
## Origin and the Taft–Ingold hypothesis
The [Hammett equation](https://www.edgepedia.org/hammett-equation), the standard linear free-energy relationship for meta- and para-substituted aromatic systems, does not apply cleanly to aliphatic compounds or ortho-substituted benzene derivatives, because the substituent sits close enough to the reaction center for steric interference, and Hammett σ values concern groups attached to a resonating aromatic system[7](https://www.orientjchem.org/vol33no5/role-of-taft-equation-in-selecting-the-site-of-attack-in-the-reactions-of-aliphatic-amines-and-tliii/). Taft built on an idea Christopher Ingold had suggested: ester hydrolysis proceeds by acid- and base-catalyzed mechanisms with identical steric demands but opposite sensitivity to polar effects[8](https://unseel.com/chemistry/taft-equation). The basic idea of evaluating the polar effect from the difference between matched acid- and base-catalyzed series was explicitly credited to Ingold in Taft's 1952 paper[9](https://electronicsandbooks.com/edt/manual/Magazine/J/Journal%20of%20the%20American%20Chemical%20Society%20US/1952%20%20%28vol%20074%29/12%20%20%282953-3206%29/3116-3120.pdf).
In that paper Taft defined the polar constant through the equation \( \log k/k_{0} = E_{\sigma}(\rho_{B} - \rho_{A}) \), where the subscripts A and B refer to identical acid- and base-catalyzed reaction series, and obtained the steric constant from \( \log k/k_{0} = E_{\mathrm{s}} + E_{\sigma} \)[9](https://electronicsandbooks.com/edt/manual/Magazine/J/Journal%20of%20the%20American%20Chemical%20Society%20US/1952%20%20%28vol%20074%29/12%20%20%282953-3206%29/3116-3120.pdf). The polar constants were presented as nearly quantitative measures of the polarities of a large number of aliphatic and ortho-benzoate substituents, derived from rates of esterification and hydrolysis of esters[9](https://electronicsandbooks.com/edt/manual/Magazine/J/Journal%20of%20the%20American%20Chemical%20Society%20US/1952%20%20%28vol%20074%29/12%20%20%282953-3206%29/3116-3120.pdf). A companion 1953 paper extended the treatment to the general proportionality of polar effects of substituent groups[3](https://pubs.acs.org/doi/10.1021/ja01113a027).
## The two constants
**Polar constant σ*.** The operational definition subtracts the acid-catalyzed from the base-catalyzed relative rate of ester hydrolysis and divides by 2.48:
\[ \sigma^{*} = \frac{1}{2.48}\left\{ \log \left( \frac{k_{\mathrm{X}}}{k_{\mathrm{H}}} \right)_{\text{base}} - \log \left( \frac{k_{\mathrm{X}}}{k_{\mathrm{H}}} \right)_{\text{acid}} \right\} \]
Because the two mechanisms have the same steric requirements, the difference isolates the polar effect[4](https://www.sciencedirect.com/topics/chemistry/reaction-rate-constant). The constant measures the electron-attracting ability of the substituent transmitted through the aliphatic chain, a purely inductive effect[7](https://www.orientjchem.org/vol33no5/role-of-taft-equation-in-selecting-the-site-of-attack-in-the-reactions-of-aliphatic-amines-and-tliii/). Taft concluded that σ* values are universal parameters characteristic of the substituent alone, independent of the rest of the molecule or of the property under study[10](https://www.russchemrev.org/RCR2997pdf). Later work connected the scale to independent measures: a 1957 study discussed σ* in relation to Mulliken's concept of electronegativity[11](https://cir.nii.ac.jp/crid/1360292618963564288), and for 34 groups σ* and σI correlate well (r = 0.962 and 0.948) with Huheey's group electronegativity[12](https://onlinelibrary.wiley.com/doi/abs/10.1002/poc.610040205).
**Steric constant Es.** The steric constant is defined from acid-catalyzed hydrolysis of aliphatic esters, with the methyl group as reference: \( E_{\mathrm{s}} = \log (k_{\mathrm{X}}/k_{0}) \), where \( k_{\mathrm{X}} \) is the rate for the ester bearing substituent X and \( k_{0} \) the rate for the reference ester[5](https://www.sips.org.in/wp-content/uploads/2021/06/QSAR.pdf). Substituents smaller than methyl, such as H and F, hydrolyze faster than the reference, so \( k_{\mathrm{X}} > k_{0} \) and \( E_{\mathrm{s}} \) is positive; bulkier groups give negative values[5](https://www.sips.org.in/wp-content/uploads/2021/06/QSAR.pdf). In the original papers Taft used four closely related acid-catalyzed reactions to define the basic set: hydrolysis of ethyl esters in 70% v/v acetone at 25 °C, esterification of carboxylic acids in methanol at 25 °C, esterification in ethanol at 25 °C, and hydrolysis of ethyl esters in 60% v/v aqueous acetone, assuming all four had the same steric sensitivity and averaging the results[6](https://www.chem.tsukuba.ac.jp/junji/domestic7_utility/PDFfiles/revised_Taft.pdf).
## Criticisms and revised scales
Later workers identified specific defects in the original \( E_{\mathrm{s}} \) scale. The averaging of four reactions was called gratuitous and an important source of error; the correct procedure is to choose a single reference reaction[6](https://www.chem.tsukuba.ac.jp/junji/domestic7_utility/PDFfiles/revised_Taft.pdf). Taft also mixed in results from Newman for esterification in methanol at 40 °C without correcting for the temperature dependence of the steric sensitivity \( \delta \)[6](https://www.chem.tsukuba.ac.jp/junji/domestic7_utility/PDFfiles/revised_Taft.pdf). Some values for highly hindered alkyl groups appeared contradictory: the reported values for t-BuMe₂C– and t-Bu(t-BuCH₂)MeC– were −3.9 and −4.0, implying that replacing a methyl by a neopentyl group scarcely changes the steric effect, which one of the values must misstate[6](https://www.chem.tsukuba.ac.jp/junji/domestic7_utility/PDFfiles/revised_Taft.pdf).
A reexamination chose acid-catalyzed esterification of carboxylic acids in methanol at 40 °C as a single reference reaction, for which \( \delta = 1 \) by definition, and tabulated a revised, broader, and homogeneous scale \( E_{\mathrm{s}}' \) covering 94 groups, including highly congested alkyl groups[6](https://www.chem.tsukuba.ac.jp/junji/domestic7_utility/PDFfiles/revised_Taft.pdf). Other authors proposed corrected steric constants, for example removing a hyperconjugation contribution that Hancock identified in the original values[6](https://www.chem.tsukuba.ac.jp/junji/domestic7_utility/PDFfiles/revised_Taft.pdf). A separate caution concerns interpretation: for acidic and alkaline ester hydrolyses the Taft–Ingold assumption itself seems correct, while Taft's averaging procedure remains questionable[13](https://www.sciencedirect.com/science/article/pii/S004040200192424X).
## Behavior and limits of the constants
The two constants respond differently to structural changes. A study of acid and alkaline hydrolysis of esters bearing unsaturated substituents in aqueous acetone found that when the unsaturation in R¹ is conjugated with the ester group in the initial state, \( E_{\mathrm{s}} \) is considerably more negative than normal steric interactions would predict, while with no conjugation \( E_{\mathrm{s}} \) matches ordinary steric behavior. The polar constant σ*, by contrast, has fairly large positive values regardless of conjugation, attributed to the electron-withdrawing power of unsaturated substituents, which falls off as the seat of unsaturation moves away from the ester group[14](https://pubs.rsc.org/en/content/articlelanding/1971/j2/j29710001502).
The linear additive model also has scope limits. Curvature, changing mechanism, solvation, resonance, or an unsuitable substituent scale can defeat it, and the equation supports interpolation within a calibrated reaction family more strongly than extrapolation to a new mechanism or substituent class; legitimate use requires a coherent series compared under stable conditions of solvent, temperature, and catalysis.
## Applications
The equation became a standard tool of physical organic chemistry and a component of quantitative structure–activity relationships (QSAR), where steric parameters in the Taft tradition stand alongside electronic and hydrophobic descriptors[15](https://pubmed.ncbi.nlm.nih.gov/12470285/). In enzyme–ligand work, the success rate of structure–activity relationships built on the Hammett equation or the Taft equation alone is low, but adding a hydrophobic term yields statistically significant correlations for enzymatic rate or equilibrium data when structural changes are not too great[16](https://www.sciencedirect.com/science/chapter/bookseries/pii/0076687991020266).
Taft analysis remains in active use for dissecting substituent effects in specific reactions. In a study of bromination of alkyl- and alkoxybenzenes in aqueous solution, the polar sensitivity \( \rho^{*} \) and steric sensitivity \( \delta \) were fitted from \( \log (k_{\mathrm{S}}/k_{\mathrm{H}}) = \rho^{*}\sigma^{*} + \delta E_{\mathrm{s}} \); \( \rho^{*} \) was negative for all subgroups, indicating buildup of positive charge in the intermediate, and \( \delta \) was 2.4 ± 0.4 for para and 2.6 ± 0.2 for ortho bromination, with electronic effects outweighing steric effects overall but both contributing appreciably[17](https://pubs.rsc.org/en/content/getauthorversionpdf/D1CP02422A). Modified Taft equations have also been applied where steric accessibility is the dominant variable, for example in describing the catalytic activity of aliphatic tertiary amines through constants \( E_{N} \) equal to the Taft steric constants of isosteric hydrocarbon substituents[18](https://www.russchemrev.org/RCR2130pdf).
## Relation to the Hammett equation
The Taft equation is not the Hammett equation with renamed symbols. Hammett's relation concerns meta- and para-substituted aromatic systems with a single substituent constant, \( \log_{10}(k/k_{0}) = \rho\sigma \)[19](https://old.goldbook.iupac.org/plain/H02732-plain.html), while the Taft family was designed for aliphatic substitution and explicitly separates polar and steric effects[8](https://unseel.com/chemistry/taft-equation). The polar term plays the role of Hammett's inductive component transplanted to sp³ centers, and \( E_{\mathrm{s}} \) was the first quantitative, transferable steric parameter in physical organic chemistry[8](https://unseel.com/chemistry/taft-equation). In modern usage the polar constant σ* has largely been replaced by the related inductive constant σI[1](https://goldbook.iupac.org/terms/view/T06247/html).
## References
1. [Taft equation, IUPAC Compendium of Chemical Terminology (Gold Book), T06247](https://goldbook.iupac.org/terms/view/T06247/html)
2. [R. W. Taft, Jr. (1952). Polar and Steric Substituent Constants for Aliphatic and o-Benzoate Groups from Rates of Esterification and Hydrolysis of Esters. *J. Am. Chem. Soc.* 74 (12), 3120–3128, CiNii Research record](https://cir.nii.ac.jp/crid/1360855569761999744)
3. [R. W. Taft, Jr. (1953). The General Nature of the Proportionality of Polar Effects of Substituent Groups in Organic Chemistry. *J. Am. Chem. Soc.* 75 (17), 4231–4238](https://pubs.acs.org/doi/10.1021/ja01113a027)
4. [Reaction Rate Constant, ScienceDirect Topics (Taft σ* definition)](https://www.sciencedirect.com/topics/chemistry/reaction-rate-constant)
5. [Taft's Steric Factor (Es), QSAR course notes, SIPS](https://www.sips.org.in/wp-content/uploads/2021/06/QSAR.pdf)
6. [A Critical Examination of the Taft Steric Parameter E_s; Definition of a Revised, Broader and Homogeneous Scale (Tetrahedron, PII 0040-4020(78)88431-2)](https://www.chem.tsukuba.ac.jp/junji/domestic7_utility/PDFfiles/revised_Taft.pdf)
7. [Taft Equation – A Convenient Tool to Decide the Position of Attack in the Reactions of Aliphatic Amines and Tl(III), Oriental Journal of Chemistry](https://www.orientjchem.org/vol33no5/role-of-taft-equation-in-selecting-the-site-of-attack-in-the-reactions-of-aliphatic-amines-and-tliii/)
8. [The Taft Equation – Separating Steric and Electronic Effects on Reactivity, Unseel](https://unseel.com/chemistry/taft-equation)
9. [R. W. Taft, Jr., *J. Am. Chem. Soc.* 74 (1952), pp. 3116–3120, full-text PDF](https://electronicsandbooks.com/edt/manual/Magazine/J/Journal%20of%20the%20American%20Chemical%20Society%20US/1952%20%20%28vol%20074%29/12%20%20%282953-3206%29/3116-3120.pdf)
10. [Russian Chemical Reviews 30 (9), 1961, on Taft σ* values](https://www.russchemrev.org/RCR2997pdf)
11. [Concerning the Electron-Withdrawing Power and the Electronegativity of Groups, *J. Chem. Phys.* 26, 93 (1957), CiNii record](https://cir.nii.ac.jp/crid/1360292618963564288)
12. [Datta (1991). Taft's substituent constants, σ* and σI, and Huheey's group electronegativity. *J. Phys. Org. Chem.*](https://onlinelibrary.wiley.com/doi/abs/10.1002/poc.610040205)
13. [Cautions regarding the physical interpretation of statistically based separation of the ortho substituent effect, III, Tetrahedron](https://www.sciencedirect.com/science/article/pii/S004040200192424X)
14. [C. G. Evans and J. D. R. Thomas (1971). Effect of unsaturated substituents on the hydrolysis of esters. *J. Chem. Soc. B*, 1502](https://pubs.rsc.org/en/content/articlelanding/1971/j2/j29710001502)
15. [QSAR: then and now, PubMed](https://pubmed.ncbi.nlm.nih.gov/12470285/)
16. [Quantitative structure-activity relationships and molecular graphics in evaluation of enzyme-ligand interactions, Methods in Enzymology 202 (1991)](https://www.sciencedirect.com/science/chapter/bookseries/pii/0076687991020266)
17. [Schammel et al., Structural effects on the bromination rate and selectivity of alkylbenzenes and alkoxybenzenes in aqueous solution, *Phys. Chem. Chem. Phys.*](https://pubs.rsc.org/en/content/getauthorversionpdf/D1CP02422A)
18. [Russian Chemical Reviews 46 (3), 1977, p. 251, modified Taft equation for tertiary amines](https://www.russchemrev.org/RCR2130pdf)
19. [Hammett equation (Hammett relation), IUPAC Gold Book, H02732](https://old.goldbook.iupac.org/plain/H02732-plain.html)
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*Topic: Encyclopedia › Physical world and mathematics › Chemistry › Organic substances › Organic reactions, structure and reference › Organic reactions and synthetic methods › Physical organic chemistry and reaction mechanisms › Linear free-energy relationships and kinetics › Taft equation: polar and steric parameters*
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