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# Lever Rule Applied to the Benzene-Toluene Vapor Pressure Diagram

A vapor pressure diagram shows the regions of phase stability for a two-component mixture as a function of pressure and composition at a fixed temperature. Since benzene and toluene form a nearly ideal solution, the vapor pressure diagram for this mixture can be calculated from the vapor pressures of the two pure substances at a given temperature. When the pressure and composition of the mixture are at a point inside the region bounded by the red and blue lines, both phases can coexist. At all other combinations of pressure and composition, only one phase, either liquid or vapor, is stable. In the vapor pressure diagram for a benzene-toluene mixture at 20˚C shown above, the solid red line indicates the composition of the liquid and the solid blue line indicates the composition of the vapor. For a point inside the two-phase region, the compositions of the two phases are determined by drawing horizontal lines (here shown dashed) to the red liquid composition line and the blue vapor composition line. These are called "tie lines" or "levers". The lever rule is used to calculate the relative amounts of the two phases, which are shown on the bar graph on the right. The compositions of the two phases are indicated by dotted lines on the phase diagram and numerically at the top of the bar graph.

### DETAILS

Benzene and toluene form a nearly ideal solution obeying Raoult's law over the entire range of composition. Composition is usually expressed as a mole fraction but sometimes other units such as mole percent or mass percent are used. An "" usually indicates the mole fraction of the liquid phase, a "" is used for the vapor phase, and a "" for the total composition of all phases together. The amounts in the phases are calculated from the lever rule: . Here is the moles of liquid phase, is the moles of vapor phase, and and are the lengths of the levers indicated by dashed lines on the phase diagrams. The "relative amount" on the bar chart is calculated from and .
Snapshot 1: when the overall composition is 0.23 and the pressure is 5.0 kPa, only the liquid phase is stable and its composition is 0.23
Snapshot 2: when the overall composition is 0.83 and the pressure is 5.0 kPa, only the vapor phase is stable and its composition is 0.83
Snapshot 3: when the overall composition is 0.50 and the pressure is 5.0 kPa, both phases can coexist. The liquid phase has composition and the vapor phase has composition . The liquid lever is and the vapor lever is . The relative amounts of the phases are then . There are twice as many moles of vapor as moles of liquid as shown on the bar chart.

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