Unit 4: Free Energy Relationships

The Hammett Equation. The correlation of reaction equilibria and rates with changes in structure is a major goal of chemistry. In organic chemistry, the change in an equilibrium constant, K, or a rate constant, k, which results from the substitution of a specific group for hydrogen, the so-called substituent effect, is of special interest. Professor L.P. Hammett, of Columbia University, systematized much of the research in this area by defining a quantity s (the substituent constant) for any given substituent.as follows:

where K_o is the acid dissociation constant for the ionization of benzoic acid and K is the acid dissociation constant for the ionization of a substituted benzoic acid with a given substituent at a given position on the aromatic ring. Since log K_o is directly related to the standard free energy change accompanying the ionization of benzoic acid (via DG^o = -2.303RTlog K_o ), and log K is directly related to the standard free energy change accompanying the ionization of the substituted benzoic acid, the substituent constant is actually related to DDG^o, the difference in the free energy changes for the two ionization processes, i.e., a measure of the substituent effect expressed in terms of a free energy quantity. Since the K’s depend somewhat upon the temperature and critically upon the nature of the solvent, s is defined specifically for water at 25 ^oC. Also, since the magnitude of the subtituent effect depends upon the position of the subsituent upon the aromatic ring, there are different substituent constants for para, meta, and ortho substituents. Typically, these are distinguished as s_p , s_m , and s_o . If the ratio (K/ K_o) >1, i.e., the substituent has increased the acidity of the benzoic acid, s is positive. Such a substituent is considered to be an electron-withdrawing group (EWG), because electron density is increased at the reaction site in the product benzoate anion, and an EWG will favor this change by withdrawing electron density away away from the reaction site. Groups such as m-Cl, p-Cl, m-NO₂, p-NO₂, etc. which have relatively large dipole moments oriented with the positive end directed toward the reaction site are EWG’s. On the other hand, electron donating groups (groups which tend to increase the electron density near the reaction site) disfavor the ionization to a negatively charged ion and have K/K_o<1. These groups have negative sigma values. These include alkyl groups (at both the meta and para positions), para alkoxy groups, and p-amino groups. The Hammett substituent constants for many hundreds of substituent groups have been measured and tabulated. It should be noted that, by definition, Hammett substituent constants are relative to hydrogen as a basis of comparison.That is to say, s_H = 0.0.

Hammett further observed that if log (K/ K_o) for the ionization of other acids was plotted versus these sigma values, reasonably linear plots were obtained. For example, if the log (K/ K_o) data for the ionization of meta and para substituted phenylacetic acids in water at 25 ^oC was plotted versus his sigma values a linear plot of slope 0.489 was obtained. Similarly, if the data for ionization of substituted phenylpropionic acids was plotted vs sigma, a linear plot of slope 0.212 was obtained.

Hammett therefore proposed the equation now known by his name:

log (K/ K_o) = rs

where log (K/ K_o) represent the log of the relative equilibrium constants for any reaction, with the K for the substituted reactant in the numerator and the K_o for the unsubstituted version in the denominator. The value of the proportionality constant, rho, depends upon the specific reaction being studied, and was termed the reaction constant. Essentially, the Hammett equation proposes that the relative free energy changes occasioned by the changes in substituent are proportional the same changes in the benzoic acid series, or for that matter to any other series of equilibria. The magnitude of rho, in effect, determines how large the free energy changes are in a reaction series, that is, how large the substituent effects are. If K>1, the substituent effects are larger than in the standard benzoic acid ionization. If K<1, the substituent effects are smaller than in the standard reaction. We can easily see that the substituent effects are smaller for the arylpropionic acid series than for the arylacetic acid series, than for the standard reaction. Essentially this is because the change in electron density occasioned by the loss of a proton for the carboxyl group, although the same in each of these reaction series, is closest to the substituent and the aryl ring in the standard series and most remote in the arylpropionic acid series. The substituent effect is thus least in the latter series because of the remoteness of the reaction site from the substituent and the aryl ring.

In the ionization of phenols, the negative charge is generated at a benzylic-type position, i.e., it is generated on an atom (oxygen) which is directly attached to the ring. The rho value for the ionization of substituted phenols in water at 25 ^oC is, correspondingly greater than that for the standard benzoic acid ionization (2.1)

When the reaction site is actually in the aromatic ring, the substituent effects, as reflected by the rho value, are even larger than in the standard reaction series. This is seen in the ionization of substituted pyridinium ions in water at 25 ^oC, where r = +5.2. Incidentally, the second equation below indicates that the rho value for the reverse of a reaction has the opposite sign, but the same magnitude as for the forward reaction. In general reactions, like the standard reaction, in which electron density is increase at the reaction site, have positive rho values (posite rho may be taken as a positive increase in electron density). Reactions in which the electron density at the reaction center is decreased in going from reactant to product have negative rho values. The seoncd equation below is an example of the latter kind, in which a positive charge is generated at the reaction site. From the first equation below we should note that electron density increases do not require the development of a negative charge. The removal of a positive charge is equally as good at increasing electron density.

We have seen that rho values depend quantitatively upon the distance of the electron density change from the aryl ring. They also depend, of course, upon the magnitude of the electron density change. The development of partial negative charges is best seen in a context of the correlation of reaction rates with substituent constants (see below). However, a similar effect can be seen in the dependence of the magnitude of rho, for a given reaction, upon the nature of the solvent. As we have seen, the rho value (by definition) for the standard reaction series (in water) is 1.0. If the solvent is changed to methanol, the rho value increases by about 50%, to 1.54. In solvent ethanol it increases to almost twice the value in water (1.96). In acetonitrile, a further increase to 2.8 is observed. Clearly, solvation has a major effect on the magnitude of substituent effects, being largest when the carboxylate ion is not as highly solvated by hydrogen bonding. In effect, hydrogen bonding to the carboxylate ion partially quenches the unit negative charge on the carboxylate anion. Consequently, the rho value is least in water, the best hydrogen bonding solvent of all.

Correlation of Reaction Rates with Hammett s Values. In an entirely similar way, relative rate constants for various reaction series can be correlated with substituent constants. In this case, of course, the free energy changes referred to are between reactant and transition state. Once again, if electron density is increased in the TS relative to the reactant, positive rho values are expected, i.e., electron withdrawing groups will accelerate the reaction. If electron density is decreased, EDG’s will speed the reaction. The Hammet equation as applied to relative reaction rates would be expressed as follows:

log (k/ k_o) = rs

where, as before, k is the rate constant for the substituted version of the reactant and k_o is the rate constant for the unsubstituted version of the reactant (X = H). As before, larger rho values can correspond to electron density changes which are closer to the aryl ring, but in the case of transition states, also to larger changes in electron density. This is because transition states can have partial charges.

The Substituent Constants.