Lever Rule: Interactive Simulations

These simulations were prepared using Mathematica. Download the free CDF player, and then download the simulation CDF file (link given below or click on figure to download). Try to predict the behavior when a parameter changes before using a slider to change that parameter. Screencasts below explain how to use these simulations.

This simulation was prepared by Lisa M. Goss and is used under the Creative Commons license. It is published on the Wolfram Demonstration Project website (link). Open content licensed under CC BY-NC-SA.

A pressure-mole fraction (P-x-y) diagram shows the regions of phase stability for a benzene-toluene mixture at a fixed temperature. This mixture is modeled using Raoult’s law. The solid blue line is the bubble point line and the solid green line is the dew point line. For a point inside the two-phase region, the compositions of the two phases are determined by drawing horizontal lines to the blue liquid composition line and the green vapor composition line. 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 on the bar graph.

Try to answer these questions before determining the answer with the simulation. We suggest that you write down the reasons for your answers.

  1. Suppose liquid and vapor are in equilibrium for a binary mixture whose liquid is an ideal solution. When the overall mole fraction of the more volatile component increases at constant temperature and pressure, does the amount of vapor increase or decrease?
  2. Suppose liquid and vapor are in equilibrium. When the pressure increases at constant temperature, does the amount of vapor increase or decrease?

This simulation was prepared by Lisa M. Goss and is used under the Creative Commons license. It is published on the Wolfram Demonstration Project website (link). Open content licensed under CC BY-NC-SA.

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. The phases are (1) a liquid mixture of uranium and titanium of variable composition, (2) solid uranium U(s), (3) solid titanium Ti(s), and (4) a solid phase compound with the formula TiU2(s). 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 to the appropriate phase boundary. For a solid phase, a horizontal line is drawn 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 in the bar graph on the right. The composition of the liquid phase is indicated by the pink dotted line and also numerically at the top of the bar graph.

Try to answer these questions before determining the answer with the simulation. We suggest that you write down the reasons for your answers.

  1. Pure Ti(s) is in equilibrium with a Ti-U liquid mixture. How do the amounts of each phase change when the temperature increases (the overall composition does not change)?
  2. Are there places on the diagram where the lever rule does not apply?