Final answer:
The best reagent for converting (s)-3-tosyloxy-2-methylpentane to (r)-3-bromo-2-methylpentane is bromine, which allows for an S_N2 substitution promoting inversion of stereochemistry.
Step-by-step explanation:
To convert (s)-3-tosyloxy-2-methylpentane to (r)-3-bromo-2-methylpentane, you need to choose reagents that will enable an inversion of stereochemistry at the tertiary carbon atom where substitution is to take place. The conversion requires a backside attack, meaning that a nucleophilic substitution reaction is necessary. The most appropriate reagent among the given options for this conversion would be bromine because its use can lead to a bromide ion which can perform a nucleophilic substitution, specifically a bimolecular nucleophilic substitution (SN2) reaction, resulting in an inversion of the stereochemistry at the carbon atom. Neither sodium hydroxide nor hydrochloric acid would directly provide a bromide ion for the reaction, and a palladium catalyst is generally used for cross-coupling reactions, not for such substitutions.