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ELI5 Yahaaa

An argument about very large numbers

Every Breath You Take Has Been Breathed Before

One breath holds more molecules than the whole atmosphere holds breaths. That single fact does all the work.

Molecules in one breath Around 10 thousandmillion millionmillionBreaths in the wholeatmosphere Around a hundredtimes fewer
Because the first number is bigger, the argument works.

  1. Step 1 of 5

    Start with how much is in one breath.

    Each mark: athousand millionmillion moleculesOne breath: tenthousand of thesemarksAnd there isnothing unusualabout that breath

    A normal breath is about half a litre of air. That sounds like nothing, but a litre of air holds something like twenty-five thousand million million million molecules. Written out, the number of molecules you just inhaled has twenty-two digits.

  2. Step 2 of 5

    Now count how many breaths the atmosphere is.

    Molecules in one breathAbout 1.2 × 10²²Half-litre breaths in the atmosphereAbout 1 × 10²⁰
    One breath contains roughly a hundred molecules for every breath the sky can be divided into.

    The whole atmosphere weighs about five thousand million million tonnes. Divide it into half-litre portions and you get a huge number. It is still smaller than the number of molecules in a single breath, by a factor of about a hundred.

  3. Step 3 of 5

    So give the air two thousand years to mix.

    Winds carry air around the globe High-altitudecurrents mixit vertically The whole atmosphere turns over in a year or two Givencenturies, itis everywhere

    Air does not sit still. Winds circulate it, weather stirs it vertically, and the whole atmosphere turns over on a timescale of a year or two. Given centuries, any particular breath is spread thin and evenly across the entire planet.

  4. Step 4 of 5

    Then the arithmetic does the rest.

    1 An oldbreath goesoutItsmoleculesdisperse2 Centuries ofmixingEvenlyspreadworldwide3 You inhale abit of thewhole skyAbout 10²²molecules ofit4 Some ofthem, onaverageAround ahundred

    One old breath is now spread evenly through the sky. Your breath holds about a hundred molecules for every breath-sized portion of that sky. So on average it holds a hundred or so molecules from that old breath. Not certainly. On average.

  5. Step 5 of 5

    It is a statement about averages, and it comes with caveats.

    The claim, statedcarefullyNitrogen: inert, well mixed, the argument holdsOxygen and CO₂: consumed and dissolved
    The reason it works is that most of the air is nitrogen and nothing much happens to it.

    Not every molecule stays in circulation: oxygen gets used up and carbon dioxide dissolves into the sea. Nitrogen, which is most of the air, is far more inert and does stay. So the honest version of this claim is about nitrogen, is a probability rather than a certainty, and is still remarkable.

    The same reasoning is why you cannot claim it about a glass of water. Water leaves the mixed pool constantly — it freezes into ice caps, sinks into deep ocean and locks into rock.

The short version

A single breath holds about a hundred times more molecules than the atmosphere holds breaths, so after centuries of mixing, any old breath is on average represented in yours — for nitrogen at least.

Try it yourself

Put a drop of food colouring in a full bath and stir. It vanishes, but every cupful you scoop out has some in it. That is the atmosphere, with a much bigger bath and a much smaller drop.

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