Carbocation Stabilities and SN1/E1 Reaction Rates

Lecture outline A closer look at carbocation stabilities and SN1/E1 reaction rates We learned that an important factor in the SN1/E1 reaction pathway is the stability of the carbocation intermediate. The more stable the carbocation, the faster the reaction. But how do we know anything about the stabilities of carbocations? After all, they are very unstable, high-energy species that can't be observed directly under ordinary reaction conditions. In protic solvents, carbocations typically survive only for picoseconds (1 ps - 10–12s). Much of what we know (or assume we know) about carbocation stabilities comes from measurements of the rates of reactions that form them, like the ionization of an alkyl halide in a polar solvent. The key to connecting reaction rates (kinetics) with the stability of high-energy intermediates (thermodynamics) is the Hammond postulate. The Hammond postulate says that if two consecutive structures on a reaction coordinate are similar in energy, they are also similar in structure. For example, in the SN1/E1 reactions, we know that the carbocation intermediate is much higher in energy than the alkyl halide reactant, so the transition state for the ionization step must lie closer in energy to the carbocation than the alkyl halide. The Hammond postulate says that the transition state should therefore resemble the carbocation in structure. So factors that stabilize the carbocation also stabilize the transition state leading to it. Here are two examples of rxn coordinate diagrams for the ionization step that are reasonable and unreasonable according to the Hammond postulate. We use the Hammond postulate frequently in making assumptions about the nature of transition states and how reaction rates should be influenced by structural features in the reactants or products. For example, when a carbocation loses a proton to make a mixture of alkenes, how much should the ratio of alkenes depend on their relative stabilities? Does the transition state for that step of the E1 pathway resemble the alkene product or the carbocation? Since we're on the subject of kinetics, recall that reaction rates are expressed as the product of the concentration of one or more reactants and a rate constant, k. the rate constant depends on the activation energy, Ea, and the temperature. The form of the dependence was first recognized by Arrhenius — k = Ae–Ea/RT where R = 1.987 cal/molK (i.e., 1.987 cal mol–1 K–1), T is the absolute temperature (in Kelvin). "A" is given the trivial name "pre-exponential factor". This is related to the probability that reactants with sufficient energy will actually cross the barrier — according to transition state theory, this is related to the activation entropy. Typical A-values are 1011 - 1015 sec–1. We can determine the difference in Eas that corresponds to an observed difference in rate constants for two similar reactions by dividing the Arrhenius equations for the two. If the rxns are similar (e.g. ionization of a 3° RX and ionization of a 2° RX) the A values cancel, and we have RTln(k1/k2) = Ea(2) – Ea(1) Now plug in a rate ratio of 10 at room temp. What do you get for ΔEa? Plug in a rate ratio of 100, 1000, etc... see the pattern? (hint: yes, you do.) Now let's put that to use... The text (p 272, table 8.2) says that tBu–Cl ionizes 105 times as fast as isopropyl chloride (in what solvent?! that's not specified. Grrrrr.) What does this imply about the relative energies of tBu+ vs iPr+ ? (More correctly, this is difference in enthalpy changes for the two R–Cl —> R+ + Cl– ionizations.) Use the estimation method we developed above and the Hammond postulate... Of course, solvent plays a huge part in these ionizations, and solvent tends to attenuate the intrinsic differences in stabilities among cations. So let's get rid of the solvent and look at some gas phase ion data. On the following page are hydride affinities of some cations. These are the (negative) enthalpy changes for the process R+ + H:– —> R–H so the larger the number the less stable the cation. Gas phase hydride ion affinities (kcal/mol). H3C CH3CH2 !+ 312 !+ 273 !+ !+ 236 266 !+ !+ 225 265 !+ 256 225 !+ 265 !+ 246 !+ !+ !+ 247 248 225 !+ 231 212 !+ !+ 228 234 224 220 H2C CH 287 201 258 258 287 data from EVAnslyn, DADougherty, Modern Physical Organic Chemistry, 2006, p 88