4-Bromobicyclo \([2,2,2]\) octane-1-carboxylic acid is more acidic than 5 -bromopentanoic acid. Explain this observation by considering the transmission of the electrical effect of the C-Br dipole.

First, let's draw the molecular structures of 4-bromobicyclo[2,2,2]octane-1-carboxylic acid, 5-bromopentanoic acid, and their corresponding anions, which form after losing a proton from the carboxylic acid group. This will help us better understand their geometries and the orientations of the C-Br dipoles. 4-Bromobicyclo[2,2,2]octane-1-carboxylic acid: Structure: \[\begin{array}{c} \text{COOH}\\ | \\ \text{Br-C}_\text{3}\text{H}_\text{6}-\text{C}_\text{3}\text{H}_\text{6}-\text{C}_\text{3}\text{H}_\text{6} \end{array} \] Anion: \[\begin{array}{c} \text{COO}^-\\ | \\ \text{Br-C}_\text{3}\text{H}_\text{6}-\text{C}_\text{3}\text{H}_\text{6}-\text{C}_\text{3}\text{H}_\text{6} \end{array} \] 5-Bromopentanoic acid: Structure: \[\text{CH}_\text{3}-\text{CH}_\text{2}-\text{CH}_\text{2}-\text{CH}(Br)-\text{CH}_\text{2}-\text{COOH}\] Anion: \[\text{CH}_\text{3}-\text{CH}_\text{2}-\text{CH}_\text{2}-\text{CH}(Br)-\text{CH}_\text{2}-\text{COO}^-\]

Now that we have the structures, we can analyze the electrical effect of the C-Br dipole, remembering that the bromine atom is more electronegative than carbon, so the C-Br bond is polarized with a partial positive charge on carbon and a partial negative charge on bromine. In 4-bromobicyclo[2,2,2]octane-1-carboxylic acid, the C-Br dipole is oriented in such a way that it can transmit its electrical effect directly onto the carboxylic acid group through the ring system. The presence of bromine atom helps to stabilize the negative charge on the carboxylate anion when deprotonation occurs, making the compound more acidic. On the other hand, in 5-bromopentanoic acid, the C-Br dipole is located farther away from the carboxylic acid group and the transmission of its electrical effect is less efficient due to the presence of an alkyl chain. This makes the carboxylate anion less stable when deprotonation occurs, resulting in a weaker acid.

Based on our analysis, we can conclude that 4-bromobicyclo[2,2,2]octane-1-carboxylic acid is more acidic than 5-bromopentanoic acid, because the C-Br dipole in the former is better at transmitting its electrical effect to the carboxylic acid group via the ring system, thus stabilizing the carboxylate anion formed upon deprotonation and increasing acidity.