Tutorial

In this tutorial, you will use Numbat to calculate how many bananas you would need to power a house. This is based on an article in the great what if? series by the author of the xkcd comics.

Bananas contain potassium. In its natural form, potassium contains a tiny fraction (0.0117%) of the isotope ⁴⁰K, which is radioactive. The idea is to use the radioactive decay energy as a power source. Open an interactive Numbat session by typing numbat in your favorite terminal emulator. We start by entering a few facts about potassium-40:

let halflife = 1.25 billion years
let occurrence = 0.0117%
let molar_mass = 40 g / mol

New constants are introduced with the let keyword. We define these physical quantities with their respective physical units (years, percent, g / mol) in order to profit from Numbat’s unit-safety and unit-conversion features later on.

Our first goal is to compute the radioactivity of natural potassium. Instead of dealing with the half-life, we want to know the decay rate. When entering the following computation, you can try Numbat’s auto-completion functionality. Instead of typing out halflife, just type half and press Tab.

let decay_rate = ln(2) / halflife

As you can see, we can use typical mathematical functions such as the natural logarithm ln. Next, we are interested how much radioactivity comes from a certain mass of potassium:

let radioactivity =
    N_A * occurrence * decay_rate / molar_mass -> Bq / g

The -> Bq / g part at the end converts the expression to Becquerel per gram. If you type in radioactivity, you should see a result of roughly 31 Bq / g, i.e. 31 radioactive decays per second, per gram of potassium.

The unit conversion also serves another purpose. If anything would be wrong with our calculation at the units-level, Numbat would detect that and show an error. Unit safety is a powerful concept not just because you can eliminate an entire category of errors, but also because it makes your computations more readable.

We are interested in the radioactivity of bananas, so we first introduce a new (base) unit:

unit banana

This lets us write readable code like

let potassium_per_banana = 451 mg / banana

let radioactivity_banana = potassium_per_banana * radioactivity -> Bq / banana

and should give you a result of roughly 14 Bq / banana. Adding unit conversions at the end of unit definitions is one way to enforce unit safety. An even more powerful way to do this is to add type annotations: For example, to define the decay energy for a single potassium-40 atom, you can optionally add a : Energy annotation that will be enforced by Numbat:

let energy_per_decay: Energy = 11% × 1.5 MeV + 89% × 1.3 MeV

This also works with custom units since Numbat adds new physical dimensions (types) implicitly:

let power_per_banana: Power / Banana = radioactivity_banana * energy_per_decay

You’ll also notice that types can be combined via mathematical operators such as / in this example.

How many bananas we need to power a household is going to depend on the average power consumption of that household. So we are defining a simple function

fn household_power(annual_consumption: Energy) -> Power = annual_consumption / year

This allows us to finally answer the original question (for a typical US household in 2021)

household_power(10000 kWh) / power_per_banana

This should give you a result of roughly 4×10¹⁴ bananas¹.

Attribution

The images in this tutorial are from https://what-if.xkcd.com/158/. They are licensed under the Creative Commons Attribution-NonCommercial 2.5 License. Details and usage notes can be found at https://xkcd.com/license.html.