= Solution
A convex water surface has higher saturation vapour pressure than a flat one. Equality of chemical potentials with the Laplace pressure $2\sigma/r$ gives the <Kelvin equation>
$$
\boxed{e_s(r)=e_s(\infty)\exp\!\left(\frac{2\sigma M_w}{\rho_wRT r}\right).}
$$
Small pure droplets therefore need large supersaturation; many subcritical embryos evaporate, making homogeneous droplet formation inefficient under ordinary cloud conditions. <cloud condensation nucleus>[Cloud condensation nuclei] provide a surface and, when soluble, lower water activity. <Köhler theory> combines the curvature increase with this solute decrease, giving a critical supersaturation and size for activation.
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