> If we assume that ... these molecules are preserved over time (a reasonable assumption—nitrogen is relatively inert),
But they are not inert. Single UV photon can break single N2 molecule bond.
Elemental N is highly reactive and will form new N2 molecule pretty fast, but that is NEW and different molecule!
N2 is not stable over period of 2000 years under constant exposure to solar UV radiation!
So what's the rate of this photodisassociation?
I found it weirdly hard to Google an answer on this. Firstly, rates are given in terms of decays per second instead of in half-life which would be more relevant for our purposes. Secondly, it seems to be well studied in the interstellar medium than in atmospheric conditions.
Anyway, the most relevant measurements I could find [0] say photodisassociation of N2 in the interstellar medium happens at a rate of approximately 10^-10 s^-1 - i.e. every 10 billion seconds on average.
Caesar died about 60 billion seconds ago [1] so at that rate, many of the molecules would still be alive.
However, we don't live in the interstellar medium. By interstellar standards, we pretty much live on the surface of the sun. The average point in the ISM is maybe 2 light years from the nearest star [2] but we are only 10^-5 ly away. They're all the same photons, but radiation intensity diminishes with the square of the distance, so our nitrogen molecules should disassociate every 1 second instead. If that's true, Caesar's last breath had its last surviving molecules persist for only a minute or two after Caesar himself.
[0] https://www.aanda.org/articles/aa/full_html/2013/07/aa20625-... https://www.aanda.org/articles/aa/full_html/2013/07/aa20625-...
[1] https://math.answers.com/math-and-arithmetic/How_many_second...
[2] https://www.livescience.com/space/how-far-apart-are-stars
If even 1% of N2 reacts every year, then there will be effectively none of the original molecules left after 2000 years. (0.99^2000 = 1.86e-9)
Not to mention that O2 (21% of the atmosphere) is even more reactive.
It's interesting how often fermi estimation problems are used as proxy's for "intelligence". Something like: 'let's assess how well "they can think" - how many golf balls fit in a baseball stadium?' etc.
Often, doing well in these kinds of problems can more than makeup for a lack of specific knowledge in something someone is interested in assessing!
There are about 13,500 taxi medallions.
edit: itchy trigger finger, think i subconsciously wanted to be the first to comment. it is stated quite early that molecules preservation is assumed. still think it would be more correct and just as interesting to discuss atoms, not molecules.
edit 2: quick research has taught me that nitrogen gas, n2, and naturally occurring isotopes do not even have a half life. they do not radioactively decay. til.
You may be right, but according to quantum mechanics, you can't really meaningfully talk about the "same" atoms, or any particles, because they don't have identities. There was a particle here, now there's a particle there, but we can't say exactly where it was at all the times in between, and it may not have been at any particular place: its amplitudes may have passed through two doors at once.
The O-O and N-N bonds are much stronger than H-O bonds, but there are still atmospheric processes that can break them. For instance, O2 undergoes photodissociation under ultraviolet light and recombines into O3 ozone, and N2 likely also undergoes photodissociation. And obviously, the fact that living beings breathe O2...
I don't know how often the average water CO₂/H₂O molecule gets dismantled this way, but there can't be many left since 44 BC.
Also, bismuth was once thought to be the most massive "fully" stable element, but turns out does decay with a half life of 10^19 years, compared to the universe's age of ~10^10 years.
Neutrons decay into a proton/electron pair after 15 minutes when not part of a nucleus.
Protons appear to be fully stable for any practical considerations, however they might decay after 10^30 years.
Yes, I would agree. Perhaps too many. But it's a fun exercise.
Similarly, there is a sensation from Adenosine for chemical cardioversion that creates a hot flushing feeling inside your body as it spreads, and it's quite the sensation to feel it going from your chest down to your extremities in a few seconds.
2k years is a long time for gas dispersion in such a "small" volume as the earth's atmosphere. early weather behaviour probably affected the distribution unevenly, but by now it should be relatively evenly distributed across the globe. no more or less in rome or italy. this is, however, as we say in sweden, a "guy's guess".
Ex - we see consistent, long term, patterns in weather that make it unlikely that this dispersion is anything close to "ideal gas in a chamber" style dispersion.
Further - we have all sorts of compounding effects. Ex - atmospheric escape is a real thing, plants do nitrogen fixation, hydrogen and oxygen can be bound up in the oceans, etc...
Maybe 2000 years is enough time for real random dispersion, maybe it's not. But it's a huge assumption baked into this that doesn't feel especially reasonable to me.
All we have is this:
>If we assume that a breath diffuses evenly throughout the atmosphere and that these molecules are preserved over time (a reasonable assumption—nitrogen is relatively inert)
Which... I challenge is likely not a particularly reasonable assumption to base this on.
https://en.wikipedia.org/wiki/File:Aerial_Superhighway.ogv
The jetstream blows at around 110 mph, and Earth's circumference at mid-northern latitudes is around 12500 miles, so it takes 12500/110=114 hours or just under 5 days for the jets to complete a lap around the planet, assuming we choose a molecule that doesn't take a diverging path on that lap. That's 73 laps per year, so 2000 years is nearly 150,000 times that the faster parts of the atmosphere have circled the globe, twisting, breaking, and reconnecting paths the whole time.
It's still an atmosphere mostly made of nitrogen, on a scale vastly exceeding 2000 years.
I don't have an intuition for how molecules actually disperse, but I do know that general climate trends certainly aren't "random dispersion".
Big volcano eruptions make for pretty sunsets across the world. Nuclear testing fallout is detectable in everything since atmospheric nuclear testing began. Everywhere we find the K-P boundary, we find iridium. The counter-assumption (which may well be true!) is the counter-intuitive one.
But on the other 9 breaths, you get to breath quite a lot of his other farts... So... your breath is really never Caesar-fart-free.
But as a consolation, most of humanity will breath your farts on every breath, so...
In other words, let’s hand-wave away the most interesting part of the question, and then come up with a trivial answer to whatever’s left.
It’s of course possible to track a single molecule if you really try hard. But this hasn’t been done since Caesar's time and the molecules have mixed. Even if we knew the exact state of the universe right now and could play back time perfectly it would be impossible to say that some particular molecules were part of his last breath.