A View from the Back of the Envelope top

Only a breath away
aka "Caesar's last breath"
Deep understanding

As you watch a sunset, you breath some atoms exhaled by each human who has ever watched a sunset.

As you read this page, you will breath atoms breathed by Franklin when he read the final draft of the Declaration of Independence for the first time.

As you head out the door, you share breath with a fur-clad traveler leaving home in Europe 10 thousand years ago. As you drive to work, with his struggles across the Alps. As you return home, with his/her return.

A few deep breaths get you atoms from your mother giving birth, from yourself taking your first breaths, and from... a Ceasar's last breath (the common name of this observation).

Your breath is about a liter of air, and thus about 1022 atoms. The atmosphere is about 1044 atoms. The breath's atoms will be well mixed back into the atmosphere after something like 10 years. Then, any randomly selected atmospheric atom has about a 10-22 chance (1022/1044) of being from the your breath. Suppose you take another breath, again of 1022 atoms. You can expect something like one of them, on average, to be from your original breath. The exact number will vary. It is not a direct hop from one breath to the next - the atom may have been breathed by others (or by yourself), in between the two breaths.


Which atoms to use? - Nitrogen vs Argon
Which atoms to use? There are two factors - how common, and how complex. The atmosphere's most common components as N2, O2, Ar, and CO2, at roughly 0.8, 0.2, 0.01, 0.0004 (by number of molecules, though at this roughness, `by mass' is about the same). Ar is even worse if you consider atoms, for with a foo2 molecule you get two foo atoms. So N atoms are something like 5000 times more common than Ar. (Averaged over the atmosphere - Ar is slightly heavier than N2, so it will tend to hang out near the ground, but I dont expect this to make even a × 10 difference.) However, while Ar basically just hangs out in the atmosphere, N2, O2, and CO2 are part of biological cycles. An N atoms spends some of its time in the atmosphere, but also some time in seawater, and buried as dead ocean critters. Argon is thus the choice of simplicity - you don't have to consider the bio part. But Nitrogen is so much more common, it is worth using if we can.

Using Nitrogen - the bio doesn't matter.
The atmosphere has about 4 × 1018 kg(N). 3.5 × 1014 kg inorganically in ocean, 3 × 1014 as dead critters in ocean, 1.5 × 1014 inorganically in soil, 1 × 1014 as dead critters in soil. Basically, the atmosphere more than 1000 times bigger than the rest of the surface stores. And for a people related timescale (say a million years), my guess is removal, by seafloor crust being subducted down into the mantel, is trivially small. Thus, the atmosphere has so much more N than anyplace else, we can ignore the rest.

Atmosphere numbers
The atmosphere is 5.14 × 1018 kg, and has 0.755 kg(N) / 1 kg(air). So there are 3.88 × 1018 kg(N). N's average atomic mass is 14.01 (almost all N's are 14N), so with 1.66 × 10-27 kg/nucleon, thats 23.3 × 10-27 kg/atom(N). So the atmosphere has about 1.7 × 1044 Nitrogen atoms. (1018 / 10-26 = 1018 - -26 = 1044).

Breath numbers
Lung volume is around 4 or 5 liter for an adult female or male. Lets just take a breath volume of 0.5 l. STP air density is 1.3 kg/m3. A liter is 10-3 m3. So that's 0.65 × 10-3 kg(air). N mass fraction is 0.75, so that's 0.49 × 10-3 kg(N). At 23.3 × 10-27 kg/atom(N), that's 2.1 × 1022 Nitrogen atoms.

Mixing breath back into atmosphere
[Cow p237] gives mixing times of
vertically in troposphere order 10 days
between hemispheres order 1 year
with stratosphere order 10 years
So what mixing time to use?
10 years seems safe from most purposes. A year or few may sometimes also serve.

The result
A "marked" breath of 2 × 1022 atoms(N) after 10 years is well mixed with the atmosphere's 2 × 1044 atoms(N). So the odds of any randomly sampled atom(N) being "marked" is 10-22. So a sampling breath, again of 2 × 1022 atoms(N), can be expected to have on average 2 or 3 "marked" atoms.
(One can also bypass the atom mass calculation - the atmospheric 3.88 × 1018 kg(N) divided by the breath 0.49 × 10-3 kg(N) gives 7.9 × 1021. Remember that the breath volume might vary by ×2, so great precision isn't useful here.)

I get nervous when long calculations with big rough numbers come down to tight tolerances around one. So let's escape. The easiest way is to add breaths. At say 4 sec/breath, we get ×10 in a minute, ×100 in 10 minutes, and ×1000 in an hour. As long as the accumulated breaths are roughly independent. And any time the breather isn't in an enclosed space with still air, this seems a good bet (though it would be nice to run the numbers to make sure). One can play this game at both or either end. If the Phoenician sailor journeyed for days, even a tiny sip of your breath is going to comfortably intercept some of the atoms. If the sailor "huff"ed once jumping off the boat at journeys end, a few hours of your breathing should be sufficient. Also, breathing deeply rather than shallowly gives almost ×10.


Moles for Noles and its solution.[links broken] A back-of-the-envelope calc using Ar.

A View from the Back of the Envelope
Comments encouraged. - Mitchell N Charity <mcharity@lcs.mit.edu>

  The current "clean" state of the above notes doesn't reflect the flailing
    that I went through first.  Thus misleading and potentially dispiriting to
    others trying something similar for the first time.
    So I note - the first draft was a mess.

  Improve intro - it's still rough and doesn't capture the concept.
  Encourage use, describe as tool.
  Give use parameters up front rather than buried in notes.
  Probability of direct hop atoms.
  Better mixing time.
  Atom's perspective, lung hopping.
  Pretty this up
     trip, hand, signing, sunset, sail, personally applicable, 
     trip begin/end, Phon. sailor, Rocky trade, Alps cross, 
     sistine, franklin, magna carta,
     how to choose?
  Where does this belong on the site?
    Is link to `Deep understanding' really appropriate?
  Discuss as example of big numbers interacting.
  Mention von C's `Fermi Solution'.  Did Paulos also have something like this?

  2003-Feb-03   Repaired links - 2 flagged.
  1998.May.11   Cleaned up.
  1998.May.09   Initial draft.