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Say, optically pure 2-bromobutane undergoes $\mathrm{S_N2}$ reaction with an alkali to form 2-butanol. The configuration about chiral carbon will now be inverted, but is it necessary that the sign of optical rotation will be opposite to that of substrate? i.e. if optical rotation of optically pure 2-bromobutane is $+x^\circ$, will optical rotation of the product be $- y^\circ$?

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if optical rotation of optically pure 2-bromobutane is some (+X) degrees, will optical rotation of the product be some (-Y) degrees?

No, there is no correlation between the configuration at a chiral carbon and the sign of optical rotation.

For example (from the Aldrich catalogue), the following chiral bromo compound, (R)-(+)-2-Bromo-3-methylbutyric acid,

enter image description here

has an optical rotation of [α]22/D +21°, c = 37 (c is the sample concentration in grams/100 ml solution) in benzene. The inverted hydroxyl analogue, (S)-(+)-2-Hydroxy-3-methylbutyric acid,

enter image description here

also rotates light in the (+) direction. Its optical rotation is [α]20/D +19°, c = 1 in chloroform.

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Optical activity determination is only by experiment(polarimetric) , it's not necessary that a dextro form should give laevo form after inversion of configuration, product may have dextro or leavo rotation.

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