Equilibria between Phases

To calculate the equilibrium condition between several phases, the condition that the chemical potential of each component $\mu_i$ is the same in each phase. The graphical construction for chemical potential can be used to obtain this condition--this condition will be called the ``common tangent'' condition.

Suppose another curve corresponding to another phase that can also form a solution of $A$- $B$ is considered. Below, another graphical construction will be demonstrated that will determine the properties of stable phases.

Consider two phases; to fix our ideas let one curve be the molar Gibbs free energy as a function of composition at constant pressure and temperature be for a solid solution. Another curve for the liquid solution will be added:

Figure 28-3: Molar Gibbs Free Energies for a liquid and solid solution at a particular pressure and temperature.

Questions:

1. What is the molar free energy charge for melting pure $A$? For melting pure $B$?

   
   

2. What is the molar free energy charge for melting a solid solution at $\ensuremath{{X}_\circ}$?

   
   

3. What is the free energy charge for forming a liquid solution from $\ensuremath{{X}_\circ}$ moles of pure $B$ and $1-\ensuremath{{X}_\circ}$ moles of pure $A$?

   
   

4. In the picture as it's drawn, rank the following with respect to stability from most stable to least stable at some fixed composition.

   $\ensuremath{{A}^{\mbox{solid}}}$-- $\ensuremath{{B}^{\mbox{solid}}}$

   A heterogeneous mixture of pure solid $A$ and pure solid $B$.

   $\ensuremath{{A}^{\mbox{solid}}}$-- $\ensuremath{{B}^{\mbox{liquid}}}$

   Heterogeneous mixture of pure solid $A$ and pure liquid $B$.

   $\ensuremath{{A}^{\mbox{liquid}}}$-- $\ensuremath{{B}^{\mbox{solid}}}$

   A heterogeneous mixture of pure liquid $A$ and pure solid $B$.

   $\ensuremath{{A}^{\mbox{liquid}}}$-- $\ensuremath{{B}^{\mbox{liquid}}}$

   A heterogeneous mixture of pure liquid $A$ and pure liquid $B$.

   $\ensuremath{{(AB)}^{\mbox{solid}}}$

   Homogeneous solid solution of $A$ and $B$.

   $\ensuremath{{(AB)}^{\mbox{liquid}}}$

   Homogeneous liquid solution of $A$ and $B$.

   
   

5. Considering that $\ensuremath{\overline{G}} = \ensuremath{\overline{H}} - T\ensuremath{\overline{S}}$, which curve will ``move'' the most as $T$ changes?

   
   

Consider the effect of lowering the temperature slightly.

Figure 28-4: Figure 28-3 drawn at a slightly lower temperature.

Figure 28-5: Figure 28-4 drawn at an even lower temperature than Figure 28-3.

Question: Which combination is the most stable in Figure 28-5?

Hint: Consider that at equilibrium $\ensuremath{{\mu_A}^{\mbox{liquid}}} = \ensuremath{{\mu_A}^{\mbox{solid}}}$ and $\ensuremath{{\mu_B}^{\mbox{liquid}}} = \ensuremath{{\mu_B}^{\mbox{liquid}}}$