= Solution
In a sufficiently supersaturated <vapor>, making <liquid> lowers bulk <Gibbs free energy>, but making a small droplet creates a costly liquid-vapor interface. The volume contribution scales as $r^3$ and the interfacial cost as $r^2$. Identifying a positive bulk driving <force> density $|\Delta g_v|$ and <interfacial tension> $\gamma$ gives $A=4|\Delta g_v|/3$ and $B=4\gamma$. In the supplied notation,
$$
\frac{d\Delta G}{dr}=\pi r(-3Ar+2B),\qquad \boxed{r_* =\frac{2B}{3A}},\qquad \Delta G_* =\frac{4\pi B^3}{27A^2}.
$$
At $r_*$, $d^2\Delta G/dr^2=-2\pi B<0$: this is the barrier summit. Subcritical droplets tend to shrink, whereas supercritical droplets can grow by reducing <Gibbs free energy>. Fluctuations must cross the barrier before bulk condensation proceeds, especially in a clean <vapor> without effective sites for <heterogeneous nucleation>. A <temperature> just below 100 °C implies a supersaturated <vapor> only with a suitable <pressure>, such as a <vapor> maintained near one atmosphere; <temperature> alone does not specify the driving <force>. \b[The <nucleation> barrier can delay condensation even when bulk <liquid> is thermodynamically favored.]
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2017/paper-1-nucleation.png]
{title=A <classical nucleation theory> barrier and a schematic <La Mer diagram>}
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