91Ó°ÊÓ

A crystal with a lower Gibbs free energy is more stable at a given temperature and pressure. The Gibbs free energy per mole of two crystals is used to compare their stability.

(a) The Gibbs free energy of calcite and aragonite are

$$\begin{gathered} G_C=-1128·8\mathrm{kJ} \\ {\mathrm{G}}_{\mathrm{a}}=-1127·8\mathrm{kJ}\mathrm{S} \end{gathered}$$

Under typical conditions, the Gibbs free energy of 1 mole of calcite is 1-0KJ lower than that of a mole of aragonite, based on the above numbers. As a result, at room temperature and air pressure, or at the earth's surface, calcite is in a more stable phase.

The volume of the substance determines the dependency of the Gibbs free energy on pressure.

${\left(\frac{∂G}{∂P}\right)}_{T,N}=V$

Where,

G is Gibbs free energy

P is the atmospheric pressure

V is the volume of substance

The volumes of calcite and aragonite are

$$\begin{gathered} V_{\mathrm{c}}=36.93{\mathrm{cm}}^3 \\ {V}_{\mathrm{a}}=34.15{\mathrm{cm}}^3 \end{gathered}$$

Converting it into kJ/kbar

$$\begin{gathered} V_{\mathrm{c}}=36.93{\mathrm{cm}}^3\left(\frac{{10}^{-1}\mathrm{kJ}/\mathrm{kbar}}{1{\mathrm{cm}}^3}\right) \\ =3.693\mathrm{kJ}/\mathrm{kbar} \\ {V}_{\mathrm{a}}=34.15{\mathrm{cm}}^3\left(\frac{{10}^{-1}\mathrm{kJ}/\mathrm{kbar}}{1{\mathrm{cm}}^3}\right) \\ =3.415\mathrm{kJ}/\mathrm{kbar} \end{gathered}$$

Because the Aragonite has a smaller volume, it should be stable at high pressure.

Set $G_c=0$at $P=Ibar$for simplicity; then $G_a=1.0kJatP=1bar$

For calcite and aragonite, the equations relating Gibbs free energy, pressure, and volume are as follows:

$$\begin{gathered} G_C=V_{C}P \\ {G}_a=V_{a}P+1·0\mathrm{kJ} \end{gathered}$$

$G_a$should equal $G_c$, at the place where the graphs for Gibbs free energy and pressure for Calcite and Aragonite intersect.

$G_C=G_a$

Substituting the values of $G_{c}and{G}_a$

$V_{c}P=V_{a}P+1·0\mathrm{kJ}$

Substituting the values and solving for P

$$\begin{gathered} P=\frac{1·0\mathrm{kJ}}{V_C-V_a} \\ P=\frac{1.0\mathrm{kJ}}{3·693\mathrm{kJ}/\mathrm{kbar}-3.415\mathrm{kJ}/\mathrm{kbar}} \\ P=3.6\mathrm{kbar} \end{gathered}$$

The aragonite is more stable than calcite at pressure of 3.6 kbar