The major product (P) in the following reaction is :
The given reaction involves an \( \alpha \)-keto aldehyde (glyoxal derivative) reacting with KOH under heating. This is a Cannizzaro reaction, which is a disproportionation reaction of aldehydes lacking an \( \alpha \)-hydrogen in the presence of a strong base to give a primary alcohol and a carboxylate salt. The starting material is Ph-CO-CHO. This molecule lacks an \( \alpha \)-hydrogen. In the presence of KOH, the following steps occur:
1. Hydroxide ion attacks the carbonyl carbon of the aldehyde group:
2. Hydride transfer: The alkoxide ion formed acts as a hydride donor to another molecule of the aldehyde. The hydride shifts to the carbonyl carbon of the ketone group in the other molecule.
3. Proton transfer: The resulting species undergo proton transfer to give an alcoholate and a carboxylic acid. In the presence of KOH, the carboxylic acid is deprotonated to form a carboxylate salt. The products of the Cannizzaro reaction of Ph-CO-CHO are Ph-CH(OH)-COO\( ^- \)K\( ^+ \) (a \( \alpha \)-hydroxy carboxylate salt) and Ph-CO-CH\( _2 \)OH (an \( \alpha \)-hydroxy ketone, formed by reduction of the ketone and oxidation of the aldehyde if we consider the other pathway of hydride transfer). However, in this specific molecule, the aldehyde group is more reactive towards nucleophilic attack by KOH compared to the ketone group. Therefore, the Cannizzaro reaction primarily occurs at the aldehyde center, leading to the oxidation of the aldehyde to a carboxylate and the reduction of the aldehyde to an alcohol. Considering the mechanism shown in the solution, the hydride transfer occurs from the carbon that was originally the aldehyde group to the carbon that was originally the ketone group. This leads to the formation of the \( \alpha \)-hydroxy carboxylate salt.
Therefore, the major product (P) is Ph-CH(OH)-COO\( ^- \)K\( ^+ \).
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: