Discussion
As a particle's diameter falls, its surface to volume ratio
Answer: C
Correct answer: rises in inverse proportion to the diameter
For a sphere, \[ \frac{S}{V} = \frac{4\pi r^2}{\frac{4}{3}\pi r^3} = \frac{3}{r} = \frac{6}{d}. \] Area falls as the square of the size and volume as the cube, so their ratio goes as the inverse of the first power. This single relation underlies nanoscale catalysis, the tendency to agglomerate, and the depression of the melting point.
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