Cloud Droplet Coalescence Calculation

The problem asks for the number of smaller cloud droplets that combine (coalesce) to form a single larger rain droplet. This process conserves the total volume of water.

Volume Conservation Principle

• Assume both cloud and rain droplets are perfect spheres.
• The volume of a sphere is given by the formula: $ V = \frac{4}{3} \pi r^3 $.
• Let $N$ be the number of cloud droplets.
• Let $r_c$ be the radius of a cloud droplet and $r_r$ be the radius of a rain droplet.
• The total volume of $N$ cloud droplets must equal the volume of one rain droplet: $ N \times V_c = V_r $.
• Therefore, $ N = \frac{V_r}{V_c} = \frac{\frac{4}{3} \pi (r_r)^3}{\frac{4}{3} \pi (r_c)^3} = \left(\frac{r_r}{r_c}\right)^3 $.

Applying Radii and Calculating

• Given cloud droplet radius, $r_c = 10 \text{ \mu m}$.
• Given rain droplet radius, $r_r = 1 \text{ mm}$.
• First, convert radii to the same units. Since $1 \text{ mm} = 1000 \text{ \mu m}$, we have $r_r = 1000 \text{ \mu m}$.
• Now, calculate the ratio of the radii: $ \frac{r_r}{r_c} = \frac{1000 \text{ \mu m}}{10 \text{ \mu m}} = 100 $.
• Finally, calculate the number of droplets $N$: $ N = \left(\frac{r_r}{r_c}\right)^3 = (100)^3 = (10^2)^3 = 10^6 $.

Thus, $10^6$ cloud droplets are needed to form one rain droplet.