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# Lever Rule Applied to Phase Diagram for Partially Miscible Liquids

The miscibility of a pair of liquids refers to the degree to which they mix spontaneously. A pair of partially miscible liquids mixes under some conditions but not at others. The phase diagram for a pair of partially miscible liquids and illustrates their behavior as a function of temperature and overall composition. For conditions corresponding to a point inside the curve, two phases are present (i.e., two layers are observed in the container). One phase is mostly substance with some dissolved in it. The other phase is mostly substance with some dissolved in it. Inside the two phase regions, the compositions of the two phases are determined by drawing horizontal lines (here shown dashed) to the red line (for the composition of the "-rich" phase) and the blue line (for the "-rich" phase). These are called "tie lines" or "levers". The lever rule is used to calculate the relative amounts of the two phases, which are represented 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. Outside the curve, and mix spontaneously to form one layer. The highest temperature at which two phases form is called the upper consolute temperature and is indicated by a black dot on the phase diagram.

### DETAILS

Composition is usually expressed as a mole fraction but sometimes other units such as mole percent or mass percent are used. The amounts of the phases are calculated from the lever rule: . Here and are the number of moles in the two liquid phases, while and are the lengths of the levers, as 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 0.6 and the temperature is 110 ˚C, there are two phases; one of the phases is mostly ) and the other phase is mostly ); there is twice as much of the phase with as there is of the phase with .
Snapshot 2: when the overall composition is and the temperature is 115 ˚C, the substances are miscible and form a single phase
Snapshot 3: when the overall composition () is 0.3 and the temperature is 105 ˚C, there are two phases; one of the phases is mostly ) and the other phase is mostly ). There is about three times as much of the phase with as there is of the phase with .

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