The solvolysis of 2-bromooctane proceeds by an "SN1-like mechanism, but when a single enantiomer is used as substrate, the product mixture is found to have a significant amount of optical activity, indicating that the mechanism does not involve full racimization. If the specific rotation of (S)(+)-2-octanol is +89o and the reaction mixture after solvolysis has an apparent specific rotation of [a]apparent = +59o, what percentage of the reaction is proceeding through a mechanism involving stereochemical inversion?
[a] = a/(c x l)

If the fraction of the (+) enantiomer is x, then (1x) gives the fraction of the (-) enantiomer. For any mixture of the two, the apparent specific rotation will be given by:

x(+[a]) + (1x)(-[a]) = [a]apparent
Solution:

[a] = a/(c x l)

If the fraction of the (+) enantiomer is x, then (1x) gives the fraction of the (-) enantiomer. For any mixture of the two, the apparent specific rotation will be given by:

x(+[a]) + (1x)(-[a]) = [a]apparent

For this mixture:

x(+89o) + (1x)(-89o) = +59o
(+89x) - 89 (+89x) = +59
178x = 148
x = 0.83

Therefore, the mixture contains 83% of the (+) enantiomer and 17% of the (-) enantiomer.

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