If ice homogeneously nucleates at \(-40^{\circ} \mathrm{C}\) calculate the critical radius given values of \(-3.1 \times 10^{8} \mathrm{J} / \mathrm{m}^{3}\) and \(25 \times 10^{-3} \mathrm{J} / \mathrm{m}^{2},\) respec tively, for the latent heat of fusion and the surface free energy.

The Gibbs-Thomson equation is pivotal in materials science and plays a crucial role when studying phase changes, such as the nucleation of crystals from a liquid or vapor phase. It describes the relationship between the curvature of a surface and the change in chemical potentials caused by this curvature, which in turn affects the melting or freezing point of the material. Specifically, it predicts how small particles, due to their curved surfaces, have different melting points compared to the bulk material.

The equation is given by:
\[T_m - T = \frac{2 \gamma V_m}{r R T_m}\]In this formula, \(T_m\) stands for the melting temperature of the bulk material, \(T\) is the temperature at which the material is currently, \(\gamma\) is the surface free energy, \(V_m\) is the molar volume, \(r\) is the radius of the particle, and \(R\) is the universal gas constant. It allows us to calculate the critical radius, which is the size below which a particle will melt at a lower temperature than its standard melting point. This has far-reaching implications in understanding the stability of nanoparticles, the formation of precipitates in alloys, and even the formation of frost at subfreezing temperatures.

The latent heat of fusion, represented by \(L\), is the amount of energy required to convert a substance from solid to liquid state without changing its temperature. This concept is fundamental in understanding phase transitions. For water, this value is positive, which means energy is absorbed during melting. However, for ice nucleation, the process being studied is freezing, thus the energy is considered to be releasing, leading to a negative sign for \(L\) in the context of the exercise.

The latent heat of fusion is part of the equation to determine the molar volume, which is essential in applying the Gibbs-Thomson equation. Understanding this property provides an insight into how much energy is involved in phase changes and how that energy contributes to the stability of small particles or droplets as they change phase.

Surface free energy, denoted by \(\gamma\), represents the excess energy at the surface of a material compared to its bulk counterpart. It's the energetic cost of creating a new surface and is a measure of the system's tendency to minimize its surface area. Materials with higher surface free energy tend to form smaller droplets or particles, as they 'pull' themselves together to minimize the surface area exposed.

In the context of the exercise, surface free energy plays a central role in the Gibbs-Thomson equation. It is directly proportional to the temperature difference required for a phase change to occur for small particles. A higher value of \(\gamma\) means that smaller particles will require a lower temperature to freeze than larger ones. This is why nanoparticles can remain liquid at temperatures where larger particles have solidified, a principle utilized in various technological applications and in the study of atmospheric phenomena like frost formation.