An S_N2 reaction at an asymmetric carbon of a compound always gives

a product with opposite optical rotation

The correct answer is option 2. a product with opposite optical rotation.

An \(S_N2\) (bimolecular nucleophilic substitution) reaction at an asymmetric carbon involves a nucleophile attacking the carbon center from the side opposite to the leaving group. This results in a single-step mechanism where the bond formation and bond breaking occur simultaneously.

Here is a more detailed explanation:

1. Backside Attack: In an \(S_N2\) reaction, the nucleophile approaches the electrophilic carbon atom from the opposite side of the leaving group. This is called a backside attack.

2. Transition State: During the reaction, a transition state is formed where the carbon is partially bonded to both the nucleophile and the leaving group. This state is highly unstable and exists momentarily.

3. Inversion of Configuration: As the nucleophile forms a new bond with the carbon atom, the leaving group departs. This simultaneous bond formation and breaking cause the carbon center to invert its configuration. This is often referred to as the "umbrella flip."

4. Chirality and Optical Rotation: If the carbon center is asymmetric (chiral), the inversion results in the formation of the enantiomer of the original molecule. Since enantiomers have identical physical properties except for the direction in which they rotate plane-polarized light, the product will have the opposite optical rotation to the original substrate.

For example, if the original molecule rotates plane-polarized light to the right (dextrorotatory), the product will rotate it to the left (levorotatory), and vice versa.

Therefore, the correct answer is a product with opposite optical rotation. This inversion of configuration and consequent change in optical rotation is a hallmark of the SN2 reaction mechanism at a chiral center.