1. The problem asks to explain the temperature-composition (T-x-y) diagram using examples of a minimum-boiling azeotrope and a partially miscible liquid system. 2. A T-x-y diagram shows the relationship between temperature (T), liquid composition (x), and vapor composition (y) in a binary mixture at equilibrium. 3. For a minimum-boiling azeotrope, the mixture boils at a lower temperature than either pure component, forming a point where liquid and vapor compositions are equal. 4. The T-x-y diagram for a minimum-boiling azeotrope has a characteristic minimum boiling point where the liquid and vapor curves meet. 5. For a partially miscible liquid system, the T-x-y diagram shows regions where two liquid phases coexist, with compositions that do not fully mix. 6. The diagram includes a miscibility gap where the liquid phase splits into two layers, each with different compositions. 7. These diagrams help understand phase behavior, boiling points, and separation processes in mixtures. Final answer: The T-x-y diagram illustrates phase equilibrium in mixtures, showing minimum boiling azeotropes as points of equal liquid and vapor composition at the lowest boiling temperature, and partially miscible systems as regions with two liquid phases of distinct compositions.