An Excursion into Phase Tetrahedra-Where Physical Chemistry and Geometry Meet

Phase diagrams are effective means of graphically representing the phase behavior of single and multicomponent systems. In many undergraduate chemistry curricula students will be introduced to mainly single- and two-component systems in the form of x-y graphs. Although treated in some standard texts, three- and four-component systems are less commonly encountered by students. Moreover, the procedure described here is more detailed than procedures encountered in standard textbooks.In this article we show how phase behavior of ternary (three-component) and quaternary (four-component) systems can be represented as phase triangles and tetrahedra. In particular, we discuss a quaternary system composed of oil, water, surface-active agent (surfactant), and cosurfactant. It is shown that although the behavior can be fully described by a phase tetrahedron, it is more convenient to display two-dimensional representations given by slices cut through the tetrahedron. Of the infinite number of the resulting triangular phase diagrams, there will obviously be some that share common features. We have exhibited two such representations of pseudo-ternary phase diagrams for the quaternary model system. We show that all compositions in which there is one fixed ratio between three of the components lie on a line that is common to the two pseudo-ternary phase triangles. This chemically based conclusion may also be elegantly reached by using purely geometrical considerations, thus illustrating another connection between purely abstract mathematical laws and chemistry.