Evaporation

When I was a lad, I remember looking a drops of rain that had plopped on the sidewalk. It was a light rain so I could make out the individual drops. They gradually evaporated. I noticed that if I used my finger and spread the raindrops, out they evaporated faster. At my tender age I had no idea why. Still not 100% certain, but my guess now is that if you have a drop of water it is losing molecules, i.e., evaporating, from its surface. If you take a drop of water, it is evaporating at a certain rate. If you separate that drop into two drops, it will evaporate faster because there is much more surface area for which it can use to evaporate.

So, let's examine this as a math application. Suppose a drop of water is spherical and has a volume of 10 whatevers. Then suppose we separate that drop into two drops of volume 5 whatevers each. Then we look at their total surface areas. Will they come out the same?

We start with the fact that it has a volume of 10:  (4/3)πr³ = 10
If we solve for r we get a value of r = 1.3365
We can then find its surface area: 4π(1.3365)² = 22.446

Now what if we now have two spheres of 5 each.
Their radii would be:  (4/3)πr³ = 5, so r = 1.0608
The surface area of one drop is 14.140. There are two drops, though, so the total surface area of them would be 28.28

Comparing the two situations, we can in fact state that, since 28.28/22.466 = 1.26, there is 26% more surface area, and so I will postulate a 26% faster drying time for the two drops over the one drop.

That was to be the end of the story, but I thought what if you separate one spherical drop into two? Is it always going to be a 26% greater surface area.

Unfortunately this is beyond my skill level to show all this. You might recall that I only recently found how to write integer exponents. This process involves having the cube root of a fraction all taken to a power of two and other assorted difficulties. So let me map this out leaving a few gaps for you or students to work through. It really is a great problem, though, with the opportunity to review simplifying - some major simplifying.

• Let us say that  (4/3)πr³ = V
• Solve for r
• Substitute this expression into 4πr²
• Now, find the radius for a sphere that his half the original volume:    (4/3)πr³ = (v/2)
• Substitute this r into 4πr²
• Make a ratio of the two radii and simplify 
• You end up with cube root of 16 divided by 2, which is 1.26
• Ta-da. An increase of 26%

Satisfying