Chapter 10: Problem 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 length \(a\) (instead of a sphere of radius \(r\) ). Now differentiate this expression with respect to \(a\) (per Equation 10.2) and solve for both the critical cube edge length, \(a^{}\), and \(\Delta G^{}\). (b) Is \(\Delta G^{*}\) greater for a cube or a sphere? Why?

Short Answer

Expert verified

Answer: The free energy change (ΔG^*) is greater for a spherical nucleus than for a cubic nucleus. This occurs because the sphere has a lower surface area-to-volume ratio, which leads to a lower energy state for the system during nucleation. Consequently, the critical nucleus size and free energy change are larger for a spherical nucleus compared to a cubic nucleus.

Step by step solution

Rewrite the expression for the total free energy change for nucleation for a cubic nucleus

The expression for the total free energy change for nucleation of a spherical nucleus of radius r is given by Equation 10.1: ΔG = 4/3 π r^3 Δg_v + 4 π r^2 γ For a cubic nucleus of edge length a, we need to modify this expression. The volume of the cube is given by V = a^3 and the surface area of the cube is given by A = 6a^2. Substitute these expressions for volume and surface area into the total free energy change: ΔG = a^3 Δg_v + 6 a^2 γ

Differentiate the expression with respect to the edge length a

To find the critical cube edge length a^* and critical free energy change ΔG^*, we need to find the minimum of the free energy change with respect to the edge length a. This can be done by taking the derivative of ΔG with respect to a and setting it equal to zero: d(ΔG)/da = 3a^2 Δg_v + 12a γ = 0

Solve for the critical cube edge length, a^*

To find the value of a^*, divide both sides by 3a: a Δg_v + 4γ = 0 Now, solve for a^*: a^* = -4γ / Δg_v

Solve for the critical free energy change, ΔG^*

Substitute the critical cube edge length a^* back into the expression for ΔG: ΔG^* = (-4γ / Δg_v)^3 Δg_v + 6 (-4γ / Δg_v)^2 γ Simplifying the expression gives: ΔG^* = 96γ^3 / Δg_v^2

Compare ΔG^* of a cube and a sphere

The free energy change ΔG^* for a spherical nucleus is given by: ΔG^* = 16πγ^3 / (3Δg_v^2) Now to determine which is greater, we compare the two expressions: cube: ΔG^* = 96γ^3 / Δg_v^2 sphere: ΔG^* = 16πγ^3 / (3Δg_v^2) Since π ≈ 3.14, 16π > 96, so the spherical ΔG^* is greater: ΔG^*_sphere > ΔG^*_cube Free energy change ΔG^* is greater for a sphere than for a cube. This happens because the sphere has a lower surface area-to-volume ratio, which leads to a lower energy state for the system during nucleation. As a result, the critical nucleus size and free energy change are larger for a spherical nucleus in comparison to a cubic nucleus.

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Short Answer

Step by step solution

Rewrite the expression for the total free energy change for nucleation for a cubic nucleus

Differentiate the expression with respect to the edge length a

Solve for the critical cube edge length, a^*

Solve for the critical free energy change, ΔG^*

Compare ΔG^* of a cube and a sphere

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