$[20$ pts$,$ bonus$]$ Heterogeneous nucleation plays a crucial role in various industrial processes

and natural phenomena, such as crystallization, phase transformations, and cloud formation.

a$)$ Define heterogeneous nucleation and explain how it differs from homogeneous nucleation

in terms of the energetics and kinetics of the nucleation process.

b$)$ Describe the critical nucleus in the context of heterogeneous nucleation and discuss the

role of surface energy in determining the nucleation barrier.