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Conditions for the Appearance of a New Phase

Last time, it was observed that a a soluble species (i.e, salt) cannot continue to be added to a phase (i.e. salty water) and continue to lower the freezing point.

What happens when too much salt is added to a solution?

Clearly, a solubility limit must exist and a new phase will appear.

This can be qualitatively understood by considering the behavior of the molar Gibbs free energy of forming a solution as a function of the amount of solute $\bgroup\color{blue}$ X_B$\egroup$ :

To quantify the conditions for the appearance of new phases, consider the thermodynamics of binary (i.e. two component) alloys¹

The Gibbs-Duhem equation of a system consisting of two components $\bgroup\color{blue}$ A$\egroup$ and $\bgroup\color{blue}$ B$\egroup$ is:

$\bgroup\color{blue}$\displaystyle 0 = SdT - VdP + \sum N_i d\mu_i = SdT - VdP + N_A d\mu_A + N_B d\mu_B$\egroup$

Consider a closed system consisting of one mole of molecules: $\bgroup\color{blue}$ N_A + N_B = 1$\egroup$ mole:

$\displaystyle \input{equations/20-3A}$

(28-1)

The state of the system (per mole) should be representable by three independent parameters, $\bgroup\color{blue}$ (T, P, X_B )$\egroup$ or $\bgroup\color{blue}$ (T, P, X_A)$\egroup$ . Therefore, using $\bgroup\color{blue}$ \mu_A = \mu_A(P,T,X_B)$\egroup$ and $\bgroup\color{blue}$ \mu_B = \mu_B(P,T,X_B)$\egroup$ in the Gibbs-Duhem equation:

$\displaystyle \input{equations/20-4A}$

(28-2)

where

$\bgroup\color{blue}$\displaystyle d\mu_A = \ensuremath{\frac{\partial{\mu_A}}{\p... ...\partial{T}}}dP + \ensuremath{\frac{\partial{\mu_A}}{\partial{X_B}}}dX_B$\egroup$

Note that $\bgroup\color{blue}$ \ensuremath{\frac{\partial{\mu_A}}{\partial{T}}} = -\ensuremath{\overline{S_A}}$\egroup$ and $\bgroup\color{blue}$ \ensuremath{\frac{\partial{\mu_B}}{\partial{T}}} = \ensuremath{\overline{V_B}}$\egroup$ , thus,

$\displaystyle \input{equations/20-4B}$

(28-3)

The first and third terms cancel and the second and fourth terms cancel. Therefore,

$\displaystyle \input{equations/20-4C}$

(28-4)

This is a general result for binary solution. It is the form of the Gibbs-Duhem equation for solutions. Equation 28-4 gives a relation between the derivatives of the chemical potentials but not a relation between the chemical potentials themselves.

Next: Graphical Constructions for the Up: Lecture_28_web Previous: Lecture_28_web

W. Craig Carter 2002-11-27