Kinetic Resolution - Theory of Kinetic Resolution

Theory of Kinetic Resolution

Kinetic resolution is a possible method for irreversibly differentiating a pair of enantiomers due to (potentially) different activation energies. While both enantiomers are at the same Gibbs free energy level by definition, and the products of the reaction with both enantiomers are also at equal levels, the, or transition state energy, can differ. In the image below, the R enantiomer has a lower and would thus react faster than the S enantiomer.

The ideal kinetic resolution is that in which only one enantiomer reacts, i.e. k_R>>k_S. The selectivity (s) of a kinetic resolution is related to the rate constants of the reaction of the R and S enantiomers, k_R and k_S respectively, by s=k_R/k_S, for k_R>k_S. This selectivity can also be referred to as the relative rates of reaction. This can be written in terms of the free energy difference between the high- and low-energy transitions states, .

The selectivity can also be expressed in terms of ee of the product and conversion (c), if first-order kinetics are assumed. If it is assumed that the S enantiomer of the starting material racemate will be recovered in excess, it is possible to express the concentrations (mole fractions) of the S and R enantiomers as

where ee is the ee of the product. Note that for c=0, which signifies the beginning of the reaction, where these signify the initial concentrations of the enantiomers. Then, for stoichiometric chiral resolving agent B*,

Note that, if the resolving agent is stoichiometric and achiral, with a chiral catalyst, the term does not appear. Regardless, with a similar expression for R, we can express s as

If we wish to express this in terms of the enantiomeric excess of the recovered starting material, ee″, we must make use of the fact that, for products R' and S' from R and S, respectively

From here, we see that

which gives us

which, when we plug into our expression for s derived above, yield

Additionally, the expressions for c and ee can be parametrized to give explicit expressions for C and ee in terms of t. First, solving explicitly for and as functions of t yields

which, plugged in to expressions for ee and c, gives

Without loss of generality, we can allow k_S=1, which gives k_R=s, simplifying the above expressions. Similarly, an expression for ee″ as a function of t can be derived

Thus, plots of ee and ee″ vs. c can be generated with t as the parameter and different values of s generating different curves, as shown below.

As can be seen, high enantiomeric excesses are much readily attainable for the unreacted starting material. In contrast, in order to get good ee's and yield of the product, very high selectivities are necessary. With a selectivity of just 10, 99% ee is possible with approximately 70% conversion, resulting in about 30% yield of 99% ee starting material. In contrast, with a selectivity of 10, ee″ above approximately 80% is unattainable, and significantly lower ee″ values are realized for more realistic conversions. A selectivity in excess of 50 is required for highly enantioenriched product. Note that there is a tradeoff between ee and conversion, and thus yield. The higher conversion a kinetic resolution is carried to, the lower the yield of unreacted starting material, but the higher ee of the recovered substrate.

It must be noted that this is a simplified version of the true kinetics of kinetic resolution. The assumption that the reaction is first order is limiting, and it is likely that the dependence on substrate depends on conversion, resulting in a much more complicated picture. As a result, a common approach is to measure and report only yields and ee's, as the formula for k_rel only applies to an idealized kinetic resolution. It is simple to consider an initial substrate-catalyst complex forming, which would negate the first-order kinetics. However, the general conclusions drawn are still helpful to understand the effect of selectivity and conversion on ee.