$$\rightleftharpoonup{xx}$$
$$\longleftharp{xx}$$,
$$\longrightharp{xx}$$,
Hofmann and Curtius rearrangements convert different carboxylic acid derivatives to primary amines.
Hofmann rearrangement begins with the base abstracting the N–H proton, followed by an α-substitution reaction with the halogen to form an N-haloamide.
Abstracting the second N–H proton provides a resonance-stabilized anion.
A further rearrangement via alkyl migration from the carbonyl carbon to the adjacent nitrogen with simultaneous loss of the halide ion produces an isocyanate intermediate.
Nucleophilic addition of water to the isocyanate forms carbamic acid, which spontaneously expels CO2, yielding an amine.
Curtius rearrangement involves a thermally induced concerted rearrangement of the azide to generate the isocyanate with the simultaneous loss of N2.
Hydration under acidic condition adds a molecule of water across the C=N bond, generating carbamic acid, which on spontaneous decarboxylation followed by neutralization gives the free amine.
Both rearrangements involve the formation of isocyanate via alkyl migration to the nitrogen atom, along with the loss of CO2 in the final step.
The leaving group, however, is different in each rearrangement.