The lever rule is a simple, but extremely effective, mathematical tool used to find the amount of each phase in a two-phase region. This rule can be applied where the system follows equilibrium cooling and heating processes. Based on mass conservation (sum of the phases equals 100%) the lever rule draws its name from the analogy with a mechanical lever where, when the lever is balanced (equilibrium), the distance from the fulcrum is inversely proportional to the weight on each side.

Applying this concept in a phase diagram, where the distance refers to the chemical composition and the weights to the phase fraction: the relative lengths of lines (from the chosen chemical composition to the solubility limit line) are inversely proportional to the amounts of phases present. Let’s now report the simple equations of the lever rule explained with the mechanical analogy:

$$
\begin{cases}
F_1 · a = F_2 · b \\
F_1 + F_2 = 100\% \,of\, the\, load
\end{cases}
$$

The forces can be directly associated with the phase quantities; the total amount of phases is 100% (the sum of the two percentages). To guarantee equilibrium the amount of the left phase, which we call Q₁ (F₁), must be lower since the distance between the fulcrum and the weight is higher.

In simple terms, here is a clear explanation of how to proceed to apply the rule:

1. Decide the chemical composition (x-axis) and one specific temperature (y-axis), the intersection between the two pieces of information identifies the fulcrum of the lever;
2. The line of the lever, called the tie line, can be drawn at that temperature starting from the fulcrum and going left and right until the solubility limits are found;
3. Calculate the overall length of the tie line and the two arms based on the chemical composition of the abscissa;
4. Calculate the ratio between the left arm and the total length, it will be the fraction of the right-side phase. Similarly, you can get the values of the left-side phase. These are the formulas valid for this last step:
   $$
   \begin{cases}
   Q_1 =\frac{b}{(a + b)} · 100\%\\
   Q_2 =\frac{a}{(a + b)} · 100\%
   \end{cases}
   $$