The Common Tangent Construction

The equilibrium condition, that the chemical potentials of components must have equal values in all phases, indicates that at equilibrium compositions that have the same tangent (i.e., a common tangent).

This result allows equilibrium to be determined by a geometrical construction: the common tangent construction.

Consider the region of lines that lies inside the common tangent point in Figure , a mixture of $\ensuremath{{f}^{\mbox{liquid}}} \ensuremath{{G^{sol}}^{\mb... ...math{{f}^{\mbox{solid}}} \ensuremath{{G^{sol}}^{\mbox{solid}}} (X_S,T,P)$ has lowest value of $\ensuremath{\overline{G}}$ where $\ensuremath{{f}^{\mbox{liquid}}}$ is the fraction of the system that is liquid and $\ensuremath{{f}^{\mbox{solid}}}$ fraction of system that is solid.

$$\ensuremath{{f}^{\mbox{solid}}}+ \ensuremath{{f}^{\mbox{liquid}}}= 1$$

This corresponds to a diagram that maps stable compositions of phase mixtures:

Figure: Construction of the equilibrium values of the compositions resulting from the lowest free energy in Figure .

Figure 29-2: Illustration of the physical composition of the states corresponding to the average compositions indicated in Figure 29-1.

There is a range of ``average compositions'' at $T < T^B_M$ in which the system has as its most stable form a mixture of liquid at composition $X_L$ and solid at composition $X_S$. The fractions of $\ensuremath{{f}^{\mbox{liquid}}}$ and $\ensuremath{{f}^{\mbox{solid}}}$ come from the requirements that the average composition is given by:

$$\input{equations/21-6A}$$ (29-1)

or, for the general case where the two phases in equilibrium are $\alpha$ and $\beta$:

$$\input{equations/phase-fracs}$$ (29-2)

Equation 29-2 is called the lever rule:

Figure 29-3: The lever rule as indicated by Equation 29-2--on the balance chubby kids get the short end of the stick.