In the process of cloud seeding, small "seed" particles are put into an under$-$cooled water vapor region in the sky in an attempt to nucleate water droplets, resulting in rain.

The spherical model can be used to model the nucleation process.

Assume the seed particle is a sphere of radius $r,$ which is less than the critical radius for homogeneous nucleation.

Also assume that the surface energy between the liquid and the solid seed particle is essentially zero.

a$)$ Write an equation for the change in the free energy due to the new volume of the liquid nucleus.

b$)$ Write an equation for the change in the free energy due to the new interfaces that have been created.

c$)$ Show that the total free energy change for this heterogeneous nucleation is the same as for homogeneous nucleation of a spherical nucleus of radius $R$ minus a constant that depends on the size of the seed particle.

d$)$ Will the critical radius for heterogeneous seed nucleation be LARGER or SMALLER than for homogeneous nucleation?