The Hammett correlation of the acid-catalyzed dehydration of...

The Hammett correlation of the acid-catalyzed dehydration of 1,2 -diaryl ethanols has been studied. The correlation resulting from substitution on both the 1 - and 2-aryl rings is: $\log k=-3.78\left(\sigma_{\mathrm{Ar}}^{+}+0.23 \sigma_{\mathrm{Ar}^{\prime}}\right)-3.18$. Rationalize the form of this correlation equation. What information does it give about the involvement of the $\mathrm{Ar}^{\prime}$ ring in the rate-determining step of the reaction?

Key Concepts

Hammett Equation

The Hammett equation is a linear free energy relationship that relates the logarithm of a reaction rate constant (or equilibrium constant) to the electronic nature of substituents on an aromatic ring. It uses sigma (?) constants, which quantify the electron-withdrawing or donating ability of substituents, to help predict how changes in substitution will affect reaction rates or equilibria under similar mechanistic circumstances.

Sigma Constants (? and ?+)

Sigma constants are numerical values that represent the electronic effect of a substituent relative to hydrogen. The ?+ constants, in particular, are used when positive charge is developing in the reaction intermediate or transition state, as in carbocation formation. These constants enable the quantification of how substituents stabilize or destabilize the developing charge via resonance or inductive effects.

Substituent Sensitivity and Reaction Mechanism

In a reaction where a carbocation-like transition state is generated, substituents that are able to donate electrons by resonance will stabilize the positive charge. The large coefficient for one aryl ring (using the ?+ value) indicates that substituents on that ring play a significant role in stabilizing the transition state. The relatively smaller coefficient for the second ring implies that its substituents have only a minor influence on the charge distribution during the rate-determining step.

Rate?Determining Step and Delocalization

The correlation shows that in the acid-catalyzed dehydration, the rate-determining step involves the formation of a carbocation where the positive charge is extensively delocalized onto the aryl ring with the larger coefficient. The small coefficient associated with the other aryl ring suggests that it is not significantly involved in stabilizing the charge in the transition state, meaning its electronic properties have only a modest effect on the reaction rate.

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