4by12

Fermi Questions

Just found this nice list of Fermi questions. According to this Science Olympics site, A “Fermi question” is a question in physics which seeks a fast, rough estimate of quantity which is either difficult or impossible to measure directly.

Here’s a nice question from the list:

If the mass of one teaspoon of water could be converted entirely into energy in the form of heat, what volume of water, initially at room temperature, could it bring to a boil? (litres).

Let’s see now. First, how much energy would we get by converting a teaspoon of water into heat? By far the largest contributor would be the rest mass of the water. The mass/energy conversion ratio is given by . A teaspoon contains a few grams of water, say 4 grams. So the amount of heat we get from it is:

Where J stands for Joule. 1 Joule is about 0.25 kilo-calories. Next, how much energy do we need to invest to boil 1 litre of water starting at room temperature? First, we need to heat the water from about 20 degress celsius to 100 degrees. Raising the temperature of 1 gram of water by 1 degree celsius takes about 4 Joules. So to heat 1 litre:

(K means Kelvin, which is for our purposes the same as degrees celsius)

But that’s just half the story. Once we’re at 100 degrees, transforming the water from liquid to gas requires an additional investment of energy, known as latent heat. This has nothing to do with temperature — the temperature doesn’t change at all during the process of vaporization itself. For water at 100 deg celsius, the latent heat for vaporization is about 2 kJ/gr. So to vaporize 1 litre of water we need:

Clearly the latent heat is the important factor here. So, the energy required to boil a litre of water is . Hence, the energy in our teaspoon can be used to boil this many litres:

Or about 150 million litres. To put that number in perspective, let’s measure the water volume in olympic swimming pools. i.e. how many olympic pools can a single teaspoon of water evaporate? According to this site, an olympic pool holds

of water. Therefore, a teaspoon-worth of water can vaporize about 60 olympic swimming pools!