\(\alpha\)- and \(\beta\)-glucose differ in the orientation of –OH group around

\(C_1\)

The answer is (1) \(C_1\).

\(\alpha\) and \(\beta\) glucose are two different anomers of glucose. Anomers are stereoisomers of cyclic forms of sugars that differ in the configuration at the anomeric carbon, which is the carbon that bears the aldehyde or ketone functional group in the sugar's open-chain form.

In the case of glucose, the anomeric carbon is carbon-1. The only difference between alpha-glucose and beta-glucose is the configuration of the hydroxyl group on carbon-1. In alpha-glucose, the hydroxyl group is on the same side of the ring as the aldehyde group. In beta-glucose, the hydroxyl group is on the opposite side of the ring as the aldehyde group.

Therefore, the only option that is the anomeric carbon of glucose is \(C_1\).

The other options are incorrect. Option (2) \(C_2\), is incorrect because the hydroxyl group on carbon-2 is the same in both alpha-glucose and beta-glucose. Option (3) \(C_3\), is incorrect because the hydroxyl group on carbon-3 is the same in both alpha-glucose and beta-glucose. Option (4) \(C_4\), is incorrect because the hydroxyl group on carbon-4 is not part of the anomeric carbon.