= Solution
{c}
The <dilute Köhler curve> follows because the <Kelvin equation> multiplies the planar-solution <saturation pressure> by the curvature correction. Combining it with <Raoult's law> therefore gives
$$
S_{\rm eq}(r)=\frac{e_S^{\rm sol}(r)}{e_S(\infty)}
=\left(1-\frac B{r^3}\right)\exp\left(\frac Ar\right).
$$
For a dilute <chemical solution> and a small curvature correction, $B/r^3\ll1$ and $A/r\ll1$. A first-order <Taylor expansion> gives
$$
S_{\rm eq}=1+\frac Ar-\frac B{r^3}
+O\left((A/r)^2,(A/r)(B/r^3)\right).
$$
Thus the simplified <Köhler theory> relation is
$$
\boxed{S_{\rm eq}(r)\simeq1+\frac Ar-\frac B{r^3}.}
$$
The curvature term raises the required ambient <vapor pressure>; the solute term lowers it. These laws combine multiplicatively before linearization, so the displayed simplified relation is an approximation, not an exact identity.
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