A ternary system refers to a mixture composed of three components. Its phase diagram requires at least three dimensions, typically representing either temperature or pressure
, along with two independent mole fractions.
Question
Show that the height of an equilateral triangle is
, where
is the side length, and hence,
.
Answer
Using Pythagorean theorem, , which rearranges to
. The area of the triangle is
. Since
, we have
, which rearranges to
.
Since , the perpendicular distances DE, DF and DG becomes
,
and
respectively if we set
. This normalisation allows us to represent the ternary system in an equilateral triangle, assuming both
and
are held constant (see diagram above). For instance, the red dot in the diagram corresponds to mole fractions
,
and
. In fact, once two independent mole fractions are specified, the third is automatically determined due to the constraint
. This effectively reduces the degrees of freedom to two, allowing the system to be represented on a two-dimensional diagram.
The ternary phase diagram below represents the acetone-water-diethyl ether system at 30oC and 1 atm. Under these conditions, water and diethyl ether are only partially miscible, while the other two binary pairs are fully miscible. The region under the curve consists of two liquid phases in equilibrium, whereas the region above the curve represents a single-phase liquid.
Since temperature and pressure are constant, tie lines must remain in the plane of the diagram, but they are not required to be parallel. Their orientations are determined experimentally. The compositions of the two coexisting phases at a given point under the curve are found at the ends of the tie line that intersects the point. For example, point f lies within the two-phase region and corresponds to a water-rich, ether-poor phase () of composition e, and a water-poor, ether-rich phase (
) of composition g. Moreover, point e is richer in acetone than point g. Finally, the lever rule also applies here: the relative amounts of the two phases are inversely proportional to the lengths of the tie line segments, i.e.
.