Ease of Decarboxylation in Substituted Cyclohexane Derivatives

To determine which compound undergoes decarboxylation most easily under mild conditions, we analyze the structural features and decarboxylation mechanisms of each given compound:

1. Option (A) is a $\beta$-dicarboxylic acid ($\text{HOOC-C-CH}_2\text{-COOH}$ derivative).
2. Option (B) is a $\beta$-keto acid ($2$-oxocyclohexanecarboxylic acid), where a carbonyl group ($\text{C=O}$) is located at the position $\beta$ to the carboxylic acid group ($-\text{COOH}$).
3. Option (C) is a $1,2$-dicarboxylic acid ($\alpha$-dicarboxylic acid system), which typically undergoes dehydration to form a cyclic anhydride upon heating rather than simple decarboxylation under mild conditions.
4. Option (D) is a $\gamma$-keto acid, where the carbonyl group is at the $\gamma$-position relative to the carboxylic acid group.

Decarboxylation Mechanism of $\beta$-Keto Acids:

$\beta$-keto acids undergo decarboxylation under very mild conditions (gentle heating or room temperature) through a concerted cyclic six-membered transition state:

1. Intramolecular Hydrogen Bonding: The carboxylic acid proton forms a stable six-membered ring via intramolecular hydrogen bonding with the carbonyl oxygen atom at the $\beta$-position.
2. Concerted Electron Shift: $\text{R-CO-CH}_2\text{-COOH} \xrightarrow{\Delta} \text{R-C(OH)=CH}_2 + \text{CO}_2 \uparrow$
3. Tautomerization: The enol intermediate rapidly tautomerizes to give the corresponding ketone (cyclohexanone).

Because the ketone carbonyl ($\text{C=O}$) is a significantly stronger electron-withdrawing group than the carboxylic acid group in $\beta$-dicarboxylic acids, the cyclic six-membered transition state is formed most readily in the $\beta$-keto acid (Option B), resulting in the lowest activation energy for decarboxylation.

Thus, Option B undergoes decarboxylation most readily under mild conditions.