text: Before any calculations can be made, a tie line is drawn on the phase diagram to determine the mass fraction of each element; on the phase diagram to the right it is line segment LS. This tie line is drawn horizontally at the compositions temperature from one phase to another (here the liquid to the solid). The mass fraction of element B at the liquidus is given by w (represented as w in this diagram) and the mass fraction of element B at the solidus is given by w (represented as w' in this diagram). The mass fraction of solid and liquid can then be calculated using the following lever rule equations:
  where w is the mass fraction of element B for the given composition (represented as w in this diagram).
  The numerator of each equation is the original composition that we are interested in is +/- the opposite lever arm. That is if you want the mass fraction of solid then take the difference between the liquid composition and the original composition. And then the denominator is the overall length of the arm so the difference between the solid and liquid compositions. If youre having difficulty realising why this is so, try visualising the composition when w approaches w'. Then the liquid concentration will start increasing.