= Solution
{c}
For $A,B>0$, differentiating the simplified <Köhler theory> curve gives
$$
\frac{dS_{\rm eq}}{dr}=-\frac A{r^2}+\frac{3B}{r^4},\qquad
\boxed{r_c=\sqrt{\frac{3B}{A}}.}
$$
The <derivative> is positive below $r_c$ and negative above it, so $r_c$ is the <Köhler activation radius>, the unique maximum. At that point $B/r_c^3=A/(3r_c)$, giving
$$
\boxed{S_c=1+\frac{2A}{3r_c}=1+\sqrt{\frac{4A^3}{27B}}.}
$$
The curve crosses $S=1$ at $r=\sqrt{B/A}=r_c/\sqrt3$, reaches $S_c$ at $r_c$, and approaches $1$ from above as $r\to\infty$. Its formal divergence to negative values as $r\to0$ is outside the dilute-<chemical solution> approximation; a finite dry solute core and composition-dependent properties must replace that extrapolation.
\Image[/past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2022-kohler-activation.png]
{title=Köhler equilibrium curve and growth directions at subcritical supersaturation}
Back to article page
