= Solution
{c}
A rising parcel cools, lowering the <saturation pressure>. Once its <water vapor> supply exceeds that value, <atmospheric supersaturation> develops. <Homogeneous nucleation> of pure <water> is strongly inhibited at the small <atmospheric supersaturations> typical of <clouds>; existing <aerosol> particles provide <cloud condensation nuclei>. Their dry sizes and composition, together with the parcel's ascent rate, determine how many <cloud droplets> activate.
<Köhler theory> combines the <Kelvin equation>, which raises equilibrium <vapor pressure> over a curved surface, with <Raoult's law>, which lowers it for a <chemical solution>. In a dilute diameter convention, with dry diameter $D_d$ and wet diameter $D$,
$$
s_{\rm eq}(D)\simeq\frac{A_K}D-\frac{\kappa_hD_d^3}{D^3},\qquad
A_K=\frac{4\sigma_wM_w}{RT\rho_w},\qquad
s_c(D_d,\kappa_h)=\left(\frac{4A_K^3}{27\kappa_hD_d^3}\right)^{1/2}.
$$
Here $s=e_v/E(T)-1$ is <atmospheric supersaturation>, $\sigma_w$ is <surface tension>, and $\kappa_h$ measures <aerosol hygroscopicity>, not the dry-<adiabatic process> exponent. The maximum of the <Köhler curve> is the <critical supersaturation>. Larger or more hygroscopic particles generally have lower thresholds. <Cloud-droplet activation> requires overcoming that maximum; total <aerosol> number is not the activated number.
Given an observed <aerosol> <number size distribution> $n_j(D_d)=dN_j/d\ln D_d$ for composition classes $j$, first assign each class a hygroscopicity and compute its <critical supersaturation>. If the parcel reaches maximum <atmospheric supersaturation> $s_{\max}$, the equilibrium-threshold estimate is
$$
D_{d,\min,j}=\left(\frac{4A_K^3}{27\kappa_{h,j}s_{\max}^2}\right)^{1/3},\qquad
\boxed{N_{\rm act}=\sum_j\int_{D_{d,\min,j}}^\infty n_j(D_d)\,d\ln D_d.}
$$
For each fitted <lognormal distribution>, this is its upper-tail <integral>; it can be evaluated analytically with an <error function> or numerically from measured bins.
The value $s_{\max}$ must be calculated, not guessed from the <aerosol> distribution alone. Prescribe parcel upward <velocity> $w$, initial <pressure>, <temperature> and <humidity>, then integrate expansion cooling, <latent heat> release, conservation of total <water>, and diffusional <cloud droplet> growth. Schematically,
$$
\frac{ds}{dt}=a_s w-b_s\frac{dq_l}{dt},\qquad
\frac{d r_i^2}{dt}=2G\,[s-s_{{\rm eq},i}(r_i)],
$$
where $a_s,b_s>0$, $G$ includes <water vapor> <diffusion> and <thermal conduction>, and $q_l$ is liquid <mass fraction>. Rising motion produces <atmospheric supersaturation>; growing particles consume it. Their competition selects the peak and which particles cross their activation barriers. Assume a well-mixed closed parcel, liquid <water>, soluble spherical particles, specified mixing state, and negligible <collision-coalescence> and fallout during activation. Slow kinetic growth, weakly soluble particles or <entrainment> require a more detailed treatment than the threshold <integral>.
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