text: Binary mixture VLE data at a certain overall pressure, such as 1 atm, showing mole fraction vapor and liquid concentrations when boiling at various temperatures can be shown as a two-dimensional graph called a boiling-point diagram. The mole fraction of component 1 in the mixture can be represented by the symbol .  The mole fraction of component 2, represented by , is related to  in a binary mixture as follows:
  In multi-component mixtures in general with n components, this becomes: 
  The preceding equilibrium equations are typically applied for each phase (liquid or vapor) individually, but the result can be plotted in a single diagram. In a binary boiling-point diagram, temperature () (or sometimes pressure) is graphed vs. .  At any given temperature (or pressure) where both phases are present, vapor with a certain mole fraction is in equilibrium with liquid with a certain mole fraction. The two mole fractions often differ.  These vapor and liquid mole fractions are represented by two points on the same horizontal isotherm (constant ) line.  When an entire range of temperatures vs. vapor and liquid mole fractions is graphed, two (usually curved) lines result.  The lower one, representing the mole fraction of the boiling liquid at various temperatures, is called the bubble point curve.  The upper one, representing the mole fraction of the vapor at various temperatures, is called the dew point curve.
  These two curves necessarily meet where the mixture becomes purely one component, namely where  (and , pure component 2) or  (and , pure component 1).  The temperatures at those two points correspond to the boiling points of each of the two pure components.
  For certain pairs of substances, the two curves also coincide at some point strictly between  and .  When they meet, they meet tangently; the dew-point temperature always lies above the boiling-point temperature for a given composition when they are not equal. The meeting point is called an azeotrope for that particular pair of substances.  It is characterized by an azeotrope temperature and an azeotropic composition, often expressed as a mole fraction.  There can be maximum-boiling azeotropes, where the azeotrope temperature is at a maximum in the boiling curves, or minimum-boiling azeotropes, where the azeotrope temperature is at a minimum in the boiling curves.
  If one wants to represent a VLE data for a three-component mixture as a boiling point "diagram", a three-dimensional graph can be used.  Two of the dimensions would be used to represent the composition mole fractions, and the third dimension would be the temperature.  Using two dimensions, the composition can be represented as an equilateral triangle in which each corner represents one of the pure components.  The edges of the triangle represent a mixture of the two components at each end of the edge. Any point inside the triangle represents the composition of a mixture of all three components. The mole fraction of each component would correspond to where a point lies along a line starting at that component's corner and perpendicular to the opposite edge. The bubble point and dew point data would become curved surfaces inside a triangular prism, which connect the three boiling points on the vertical temperature "axes". Each face of this triangular prism would represent a two-dimensional boiling-point diagram for the corresponding binary mixture. Due to their three-dimensional complexity, such boiling-point diagrams are rarely seen. Alternatively, the three-dimensional curved surfaces can be represented on a two-dimensional graph by the use of curved isotherm lines at graduated intervals, similar to iso-altitude lines on a map. Two sets of such isotherm lines are needed on such a two-dimensional graph: one set for the bubble point surface and another set for the dew point surface.