Lever Rule for the Uranium-Titanium Solid-Liquid Phase Diagram

A solid-liquid phase diagram for the uranium-titanium system shows the regions of phase stability as a function of the temperature and overall composition of the system. The phases are (1) a liquid mixture of uranium and titanium of variable composition, (2) solid uranium , (3) solid titanium , and (4) a solid phase compound with the formula . In this phase diagram, the phase boundaries are shown as solid black lines. The phase boundaries involving liquids are assumed to be a linear function of composition. When the temperature and composition are at a point above all the phase boundaries, only the liquid mixture is stable. At any point inside the phase boundaries, two phases are stable. The phases present, as well as their relative amounts, are shown in the bar graph on the right. When the liquid phase is present, its composition is found by drawing a horizontal line (here shown as a dashed red line) to the appropriate phase boundary. For a solid phase, a horizontal line is drawn (here shown dashed) to the vertical phase boundary corresponding to the pure solid. These horizontal lines 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 composition of the liquid phase is indicated by the red dotted line and numerically at the top of the bar graph.


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"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 represents the moles of one of the phases, represents the moles of the other 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 of the system is and the temperature is 675˚C, two phases are stable: a liquid mixture of uranium and titanium of composition , and the solid compound ; there are approximately equal amounts of these two phases
Snapshot 2: when the overall composition of the system is and the temperature is 625˚C, two phases are stable: solid titanium and solid ; there is approximately twice as much titanium as there is of the compound
Snapshot 3: when the overall composition of the system is and the temperature is 750˚C, only one phase is stable: a liquid mixture of uranium and titanium
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