Concept: The acid strength of substituted benzoic acids is governed by the electronic effects (inductive and resonance) of the substituents on the benzene ring.
• Electron Withdrawing Groups (EWG): Groups like $-NO_2$ withdraw electron density from the ring, stabilizing the carboxylate anion and increasing acidity. The more EWGs present, the stronger the acid.
• Electron Donating Groups (EDG): Groups like $-OCH_3$ donate electron density into the ring, destabilizing the carboxylate anion and decreasing acidity.
• Order of Effects: Nitro group has strong $-I$ and $-M$ effects, while Methoxy group has a strong $+M$ effect that outweighs its $-I$ effect at the para position.
Step 1: Evaluate the effect of each substituent.
- (III) 3,4-dinitrobenzoic acid: Contains two strong electron-withdrawing nitro groups. This will be the strongest acid.
- (II) 4-nitrobenzoic acid: Contains one strong electron-withdrawing nitro group. It is stronger than benzoic acid but weaker than the dinitro derivative.
- (I) Benzoic acid: The reference standard with no substituents.
- (IV) 4-methoxybenzoic acid: The methoxy group ($+M$) increases electron density on the ring, making it the weakest acid.
Step 2: Determine the increasing order.
Combining these observations, the order of increasing acid strength is:
4-methoxybenzoic acid (IV) $<$ benzoic acid (I) $<$ 4-nitrobenzoic acid (II) $<$ 3,4-dinitrobenzoic acid (III).
This corresponds to Option (C).