The volumetric Gibbs energy used in the calculation of the

**critical radius**for growth of a nucleus changes as the temperature is increased (or decreased) above (or below) the equilibrium reaction temperature. Figure 2.06 will allow you to change the ΔG

_{rxn}value by changing the temperature beyond equilibrium |ΔT| (the overstepping value).

**Click on Figure 2.06 to open a larger version and move the |ΔT| slider**to see this effect. There are also sliders to change how the ΔG

_{rxn}value changes with temperature and to change the interfacial Gibbs energy (σ

_{A}). Because ΔG

_{rxn}for chemical reactions is commonly tabulated in molar units, the molar volume of the product phase is needed to convert the data to Gibbs energy per unit volume ΔG

_{V}. There is also a slider to change the molar volume.

As you increase the |ΔT| with its slider in Figure 2.06, you will see the ΔG

_{V}increase and the calculated critical radius decrease. Although the interfacial Gibbs energy does change with temperature, typically decreasing with increasing T and increasing with decreasing T, these changes are poorly known and believed to be small so they are not included in this model.

As an example, consider the kyanite = sillimanite reaction at a pressure of 0.6 GPa. You can see Gibbs energy data for this reaction here. The value of dΔG

_{rxn}/dΔT for this reaction is about -12 J/K/mole of Al

_{2}SiO

_{5}. The molar volume for sillimanite is about 50 cm

^{3}/mole of Al

_{2}SiO

_{5}. Using these values and an interfacial Gibbs energy of 0.015 J/m

^{2},

**what is the ΔT required to have a critical nucleus with a radius of 1 nm?**Use Figure 2.06 to answer the question, type your answer in the box.

Press "Enter" after you type in the number.

**Your answer is incorrect.**Open Figure 2.06 and use the sliders to set the dΔG

_{rxn}/dΔT to -12.0 J/mole/K, the molar volume to 50 cm

^{3}, and the interfacial energy to 0.015 J/m

^{2}. Then change the ΔT until the critical radius drops to 1 nm. Please try again.

Press "Enter" after you type in the number.

**No. You still have not given the correct answer.**According to the model, a ΔT of 120 to 130 °C beyond the equilibrium reaction temperature is required to have a (spherical) critical nucleus with a radius of 1 nm for the spontaneous growth of sillimanite from kyanite.

**Yes!**Using the specified parameters, the model yields a ΔT of 120 to 130 °C beyond the equilibrium reaction temperature to have a (spherical) critical nucleus with a radius of 1 nm for the spontaneous growth of sillimanite from kyanite.

Use the data of Figure 2.06 to help answer this question. Press "Enter" after you type in the number.

**Your answer is incorrect.**Use Figure 2.06 with the same parameters as for the last question. See how many forumla units there are in a 1 nm radius sphere of sillimanite and use the sillimanite formula to convert formula units to atoms. Please try again.

Press "Enter" after you type in the number.

**No. You still have not given the correct answer.**The total number of formula units of sillimanite in a 4.2 nm sphere is given in Figure 2.06 as about 50. There are 8 atoms per sillimanite formula (Al

_{2}SiO

_{5}). Therefore, there should be about 400 atoms in a 1 nm sphere of sillimanite.

**Yes.**The total number of formula units of sillimanite in a 4.2 nm sphere is given in Figure 2.06 as about 50. There are 8 atoms per sillimanite formula (Al

_{2}SiO

_{5}). Therefore, there should be about 400 atoms in a 1 nm sphere of sillimanite.

The thermodynamic (macroscopic) model of Figure 2.06 indicates that even if a large degree of reaction overstepping occurs so that the critical radius for spontaneous growth of a spherical sillimanite nucleus is reduced to 1 nm,

**400 atoms must be organized by the semi-random motions into the sillimanite structure to make the nucleus.**This seems like a large number atoms to organize. This number can be reduced further by a larger ΔT. It also can be reduced by nucleation on another phase with a lower interfacial energy.