Hammett equation

The Hammett equation establishes a quantitative relationship between the structure of chemical reactants and their reactivity . It is a linear free energy relationship ( Linear Free Energy Relationships , LFERs). It applies in general, but is parameterized differently for different reactions or reactants . The equation falls within the sub-area of physical organic chemistry .

General

The American chemist Louis Plack Hammett developed this relationship for substitution reactions on bi-substituted benzenes. If, in the case of the alkaline hydrolysis of substituted benzoic acid esters , one considers the relative rate constants based on the unsubstituted ester , as well as the relative pKa value of the ester based on the correspondingly substituted benzoic acid, the graph of a linear function is obtained with logarithmic representation . Ortho -substituted benzoic acids are an exception , since entropic effects play a role in reactivity due to the proximity of the ester group and the second substituent.

The general form of the equation is:

${\displaystyle \log(k_{\rm {rel}})\propto \log(K_{\rm {rel}})=\rho \sigma }$

With

k : rate constant

K : equilibrium constant

The influence of the substituent can be described by the difference in the free Gibb's enthalpy of the different reactions. x denotes an undefined substituent, H stands for the reference substituent hydrogen:

${\displaystyle \Delta G_{x}-\Delta G_{H}=\log \!\left({\frac {K_{x}}{K_{H}}}\right)=\sigma _{x}}$

In addition:

${\displaystyle {\frac {K_{x}}{K_{H}}}=K_{\rm {rel}}}$  and   ${\displaystyle {}{\frac {k_{x}}{k_{H}}}=k_{\rm {rel}}}$

You can see that it can also be applied against . In addition, one adds a proportionality factor so that one obtains the general form of Hammett's equation mentioned above. ${\displaystyle \log(k_{\rm {rel}})}$ ${\displaystyle \sigma }$ ${\displaystyle \rho }$

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A kinetic variable is correlated with a thermodynamic one in order to conclude that there is a correlation between reactivity and structure. The connection between reactivity and kinetics, as well as between structure and thermodynamics, is used to establish a quantitative connection between reactivity and structure via the third connection between kinetics and thermodynamics in the Hammett relationship.

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For the side chain reactions of ortho- and para-substituted benzene derivatives, the following two forms of the Hammett equation apply to the reaction rates or equilibria.

${\displaystyle \log(k)=\log(k_{o})+\rho \cdot \sigma }$

${\displaystyle \log(K)=\log(K_{o})+\rho \cdot \sigma }$

log ( k _o ) and log ( K _o ) are the intercepts when the log ( k ) or log ( K ) plots against the substitution constant σ with a constant reaction constant ρ.

Selected substitution constants of substituted benzoic acids:

Substituent	σ (meta)	σ (para)
CH 3	−0.07	−0.16
C 6 H 5	0.06	−0.01
COOH	0.37	0.45
OMe	0.10	−0.27
OH	0.13	−0.38
Cl	0.37	0.22
NH 2	0.00	−0.57
NMe 2	−0.21	−0.83
NO 2	0.71	0.78
SH	0.25	0.15
SO 2 Me	0.64	0.73

parameter

${\displaystyle \sigma }$is called the substituent parameter, it is postulated that it is independent of the reaction parameter . This is only an approximation because a reaction with different substituents never takes exactly the same route. However, this assumption is permissible for substituents in para or meta position. Qualitatively, it can be stated that the reaction parameter represents the sensitivity of a reaction to substituent effects. ${\displaystyle \rho }$

Substituent parameter σ ⁰

Since the size of the substituent parameter also depends on other reaction conditions, such as e.g. B. depends on the solvent , standardized substituent parameters are generally used, which have been averaged over many reactions. The magnitude of this quantity characterizes the ability of a substituent to affect the electron distribution in the transition state. The values ​​for are tabulated and different for each reaction, as well as the position and type of substituent. ${\displaystyle \sigma ^{0}}$ ${\displaystyle \sigma }$

A distinction is made between two different substituent effects that lead to a substituent parameter.

• Resonance or mesomerism effect (R or M)
• Inductive effect (I)

+ I effects can only be caused by alkyl residues - via hyperconjugation - or substituents that are more electropositive than carbon, e.g. B. silicon or boron - are triggered. If inductive and mesomeric effects are opposing, the mesomeric effect generally dominates. There are essentially four types of substituents:

• Alkyl groups, -SiR ₃ , -BR ₂ + I

• ${\displaystyle \pi }$-Acceptor groups -R, -I

z. B. carbonyl, nitro, nitrile or sulfate groups

• Groups with unbound electron pairs + R, -I

z. B. sec. Amine, ether, thioether, halide groups

• Cationic groups -I

z. B. -NR ₃ ⁺ or -PR ₃ ⁺

Non-linearity

Mechanistic Effects

Mechanistic effects are the cause of a change in the reaction parameter . Under certain circumstances, two different reaction mechanisms can compete with one another, which have a comparable activation energy but a very different electron requirement. This can lead to a non-linear curve shape of the Hammett graph. If the reaction mechanism changes, the graph can be broken down into linear sub-areas. There are therefore linear sections with different slopes and thus different reaction parameters . ${\displaystyle \rho }$ ${\displaystyle \rho }$

Furthermore, mechanisms with intermediate steps can lead to non-linear behavior. In general, the slowest reaction step in a sequence is responsible for the overall speed. By changing the substituents, the rate-determining step of a reaction sequence can change, resulting in non-linear curves. Possibly. the function also no longer consists of linear sub-areas if the change takes place slowly and continuously. In this case, graphs with a curvature are obtained.

Conjugation effects

Conjugation effects lead to a variation in the substituent parameter . If a substituent with a second substituent is in the para position, a conjugated system can occur in the transition state which includes both substituents. This effect must be differentiated from the normal mesomerism effect and also affects reactivity. This leads to a non-linear curve. To counteract this, further substituent constants and were developed . The former for donors, the latter for acceptors. A reference reaction is established for both cases in order to obtain the relative values. ${\displaystyle \sigma }$ ${\displaystyle \sigma ^{+}}$ ${\displaystyle \sigma ^{-}}$ ${\displaystyle \pi }$ ${\displaystyle \pi }$

This fact can help to elucidate the mechanism of a reaction. If there is a correlation with or parameters , then there is probably a through-conjugated transition state. ${\displaystyle \sigma ^{+}}$ ${\displaystyle \sigma ^{-}}$ ${\displaystyle \sigma ^{0}}$

The hydrolysis of substituted cumyl chloride is used as the reference reaction for values . For values, the dissociation of substituted aniline is used . ${\displaystyle \sigma ^{+}}$ ${\displaystyle \sigma ^{-}}$

Despite these three possible parameter systems, non-linear behavior can occur. In this case, an additional parameter r is introduced , which serves as a weighting factor for a sum of and values. If the value is one, the curve shape only correlates with , if it is zero only with . This means that reactions that lie between these extremes can also be mapped. ${\displaystyle \sigma ^{+}}$ ${\displaystyle \sigma ^{0}}$ ${\displaystyle \sigma ^{+}}$ ${\displaystyle \sigma ^{0}}$

Extension to Taft

In aliphatic systems, the substituent and reaction center are usually closer to one another, so that steric effects must also be taken into account. Furthermore, the parameters and their components are separated according to the I and R effect. For example, consider the hydrolysis of an aliphatic ester and make the following assumptions: ${\displaystyle \sigma }$ ${\displaystyle \rho }$

1. Electronic effects are weak in acid hydrolysis
2. Electronic effects are strong in alkaline hydrolysis
3. No resonance effects along a saturated C chain

In general, the linear free enthalpy relationship can be established for this case: With the assumptions (1) and (3) it follows: (*) In addition, the assumptions (2) and (3) give: With the last two equations a set of new parameters can be found Define so that the steric parameter just cancels out: If you take a value from the Hammett parameters, the new parameter can be calculated. The Taft equation can thus be set up: The steric parameter can be calculated from (*), the methyl substituent is chosen as the reference for the rate constant. The Taft equation thus provides a formalism to quantify systems in which steric effects play a role in the course of the reaction.
${\displaystyle \log \!\left({\frac {k_{x}}{k_{H}}}\right)=\rho _{I}\sigma _{I}+\rho _{R}\sigma _{R}+E_{s}}$

${\displaystyle \log \!\left({\frac {k_{x}^{A}}{k_{H}^{A}}}\right)=E_{s}}$

${\displaystyle \log \!\left({\frac {k_{x}^{B}}{k_{H}^{B}}}\right)=\rho _{I}\sigma _{I}+E_{s}}$
${\displaystyle E_{s}}$
${\displaystyle \log \!\left({\frac {k_{x}^{A}}{k_{H}^{A}}}\right)-\log \!\left({\frac {k_{x}^{B}}{k_{H}^{B}}}\right)=\rho _{I}\sigma _{I}=\rho ^{*}\sigma ^{*}}$
${\displaystyle \rho ^{*}}$ ${\displaystyle \sigma ^{*}}$
${\displaystyle \log \!\left({\frac {k_{x}}{k_{M}e}}\right)=\rho ^{*}\sigma ^{*}(+\rho _{s}E_{s})}$
${\displaystyle E_{s}}$

literature

• John Shorter: The Hammett Equation - and what became of it in fifty years . In: Chemistry in Our Time . tape 19 , no. 6 , 1985, pp. 197-208 , doi : 10.1002 / ciuz.19850190604 .

Individual evidence

1. ↑ Ulrich Lüning: Organic reactions - An introduction to the reaction pathways and mechanisms . 2nd Edition. Spektrum, Munich 2007, ISBN 978-3-8274-1834-0 , pp. 21 .