Most liquids contain heterogeneous nucleating particles, and so it is often difficult

to observe homogenous nucleation of crystals in liquids. In a classic experiment

$($with over $800$ citations!$),$ David Turnbull $(1915-2007)$ tracked the critical

undercooling for the nucleation of $Hg$ crystals versus the droplet size. Turnbull

chose $Hg$ for his experiment since it can be readily purified by distillation. The

results of his experiment are presented schematically below.

$\text{'}$ observed

s$.$ droplet size

a$.$ Provided that nucleation does not occur at the surface nor at the substrate

on which the droplet is sitting, explain in words the trend in the figure.

Hint: consider the supplemental reading on Canvas, "Phase Changes"

$($D$.$ Tumbull, Solid State Phys$.3,1956)$ and particularly pages $281-282.$

b$.$ If the dotted line occurs for droplet volumes of around ${10}^{-7}c{m}^3,$ estimate

the number of heterogeneous nuclei per $c{m}^3$ of liquid. How does this

number compare to the density of micrometer$-$sized particles in the air?

Note that this is sometimes referred to as the "atmospheric aerosol." You

may need to do some research; please cite your sources accordingly.