 Solution

ID: past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2022/3/a/solution

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_{eq} (r) = \frac{e _{S}^{sol} ( r )}{e _{S} ( \infty )} = (1 - \frac{B}{r ^{3}}) exp (\frac{A}{r}) .

For a dilute chemical solution and a small curvature correction,

B / r^{3} ≪ 1

and

A / r ≪ 1

S_{eq} = 1 + \frac{A}{r} - \frac{B}{r ^{3}} + O ((A / r)^{2}, (A / r) (B / r^{3})) .

Thus the simplified Köhler theory relation is

S_{eq} (r) ≃ 1 + \frac{A}{r} - \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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