Question

10.2 (a) Rewrite the expression for the total free energy change for nucleation (Equation 10.1) for the case of a cubic nucleus of edge lengt.h (instead of a sphere of radius r). Now dif ferentiate this expression with respect to a (per Equation 10.2) and solve for both the critical cube edge length, a*, and ΔG*. (b) Is AG* greater for a cube or a sphere? Why'?

Answer 1

To solve for the free energy change ΔG* of nucleation for a cubic nucleus, we must consider the sum of surface and volume free energies, differentiate with respect to cube edge length a, and find the equilibrium condition. We then determine a* and calculate ΔG*. A comparison of ΔG* between a cube and a sphere involves analyzing the surface area-to-volume ratio.

Step-by-step explanation:

The question asks to rewrite the expression for the total free energy change for nucleation in the case of a cubic nucleus and to solve for the critical cube edge length a* and ΔG*.

Additionally, it requests a comparison of ΔG* values between a cube and a sphere.

In the scenario with a cubic nucleus, you must consider both the surface free energy, which is proportional to the surface area of the cube, and the volume free energy, which is a function of the volume of the cube.

The total free energy change ΔG is the sum of these two contributions.

After setting up the total free energy equation with these parameters, you take the derivative concerning the cube edge length a, set it to zero to find conditions for equilibrium, and solve for a*.

This value of a* represents the critical cube edge length at which nucleation becomes favorable.

Next, you would insert a* back into the total free energy equation to obtain ΔG*, the energy barrier for nucleation.

To determine whether ΔG* is greater for a cube or a sphere, you need to consider the ratio of surface area to volume for each shape, since the surface contributes positively to free energy while the volume contributes negatively.

Answer 2

To find the critical edge length a* and critical free energy change ΔG* for a cubic nucleus, one would set up the free energy equation considering the volume and surface area of the cube, differentiate with respect to edge length a, and solve. The values of a* and ΔG* can then be determined. The ΔG* for a cube and a sphere will differ due to their different surface-to-volume ratios.

Step-by-step explanation:

To rewrite the expression for the total free energy change, ΔG, for nucleation in the case of a cubic nucleus with an edge length a, we would consider both the volume and the surface area of the cube. The free energy change due to the formation of a new phase nucleus includes a term proportional to the volume of the nucleus that accounts for the decrease in free energy (negative term) and a term proportional to the surface area of the nucleus that represents the energy cost of creating a new interface (positive term).

ΔG for a cubic nucleus can be written as: ΔG = -αa^3 + βa^2, where α is the volume free energy gain and β is the surface free energy cost per unit area. To find the critical cube edge length a* and critical free energy change ΔG*, we differentiate the expression with respect to a and set the derivative equal to zero:

dΔG/da = -3αa^2 + 2βa = 0

From this, a* = 2β / 3α is the critical edge length, and ΔG* is obtained by plugging a* back into the expression for ΔG.

Comparison between ΔG* for a cube and a sphere

The value of ΔG* will vary depending on the shape of the nucleus because the surface-to-volume ratio is different for a cube than for a sphere. Typically, ΔG* would be different for a cube and sphere, with the exact relationship depending on the surface energy and the volume energy density.