How did Schoenberg adapt sonata form for 12-tone music?

Arnold Schoenberg adapted sonata form for his 12-tone compositions by utilizing the form's traditional structural outlines while replacing its foundational principle of key contrast with a reliance on contrasting themes.1

Because 12-tone music is atonal—deliberately avoiding the establishment of a central key or tonal center—Schoenberg could not rely on the traditional Classical mechanism of modulating between different keys to create tension and resolution. Instead, he organized his compositions using a "row," or series, of the 12 tones of the chromatic scale, which served as the melodic and harmonic basis for the work.12

While Schoenberg successfully applied this method in many contexts, some critics have noted that forcing the 12-tone system into the rigid outlines of the traditional sonata form could result in a diminished effectiveness of the musical structure.3