# Why is the surface area of 1000 water drops greater than one whole water drop?

Dec 11, 2017

The surface area to volume ratio of a sphere is inversely proportional to its radius. So small spheres have a higher surface area to volume ratio.

#### Explanation:

Surface Area To Volume Ratio: The surface area to volume ratio of a sphere is inversely related to its radius.

Considering a sphere of radius $R$,

$\frac{S}{V} = \setminus \frac{4 \setminus \pi {R}^{2}}{\frac{4}{3} \setminus \pi {R}^{3}} = \frac{3}{R}$

This quantity (surface area to volume ratio) is very important in some physics problems. For example, thermal energy dissipation, chemical absorption etc. depend on the surface area. In the thermal energy dissipation problem, dissipation happens through the surface. Spheres with high surface area to volume ratio, dissipate heat more efficiently and so cool fast.

At the time of formation, all planets were born molten and carried enormous quantities of thermal energy. Planets with a high surface area to volume ratio dissipate heat faster and so cool fast.

Geological activities like tectonic plate movement depends on the interior of the planet being in a molten state. This means planets with high heat dissipation rate will become geologically dead sooner. Since the surface are to volume ratio is inversely related to planetary radius, bigger planets like Earth and Venus are still geologically active while small objects such as Mercury, Mars and Moon are already geologically dead.

When a spherical blob of water of radius $R$ is atomized to 1000 droplets each of radius $r$, the total surface area increases. Let us calculate the total surface area for the 1000 blobs and compare it with a single blob.

Volume Conservation: $\setminus q \quad V = 1000. \left(\cancel{\frac{4}{3} \setminus \pi} {r}^{3}\right) = \cancel{\frac{4}{3} \setminus \pi} {R}^{3}$
$r = \frac{R}{10}$

Total surface area of 1000 small droplets : S_{"tot"}=1000.(4\pir^2);
Total surface area of a single big droplet : S_1 = 4\piR^2;

${S}_{\text{tot}} / {S}_{1} = \setminus \frac{1000. \left(\cancel{4 \setminus \pi} {r}^{2}\right)}{\cancel{4 \setminus \pi} {R}^{2}} = 1000 {\left(\frac{r}{R}\right)}^{2} = 10$

By atomizing a single blob of a liquid in to 1000 small droplets, the surface area increases 10 times for the same volume.