The thermodynamics of polymer phase separation constitutes a crucial topic in polymer physics

and the associated parameters pertaining to the critical point of the phase diagram

namely

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and

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present a challenge and a pivotal aspect in this area of study. This paper aims to address the inadequacies and insufficiencies of conventional derivative methods employed in teaching

based on the mathematical properties of the free energy-composition curve itself. We present a new approach to derive

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and

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using two fundamental mathematical tools

derivatives

and elementary inequalities

which not only simplifies the process but also offers a fresh perspective

thereby enhancing students’ comprehension and knowledge of the theory associated with the critical point of phase separation.