Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Interdisciplinary paper IDP1 / 3 / d / i / Solution

Hold liquid-water content and cloud thickness fixed, as required for a partial sensitivity to droplet number. Then $\tau =K{N}_d^{1/3}$, so $d\tau /d{N}_d=\tau /(3N_d)$. Differentiating the albedo relation gives $dA/d\tau =7/(\tau +7{)}^2$, henceSince $A=\tau /(\tau +7)$ and $1-A=7/(\tau +7)$,The factor $A(1-A)$ peaks at $A=1/2$, so the albedo response to a fixed fractional droplet-number change is greatest at intermediate albedo and weaker near either saturation limit.