There is an old quip ... \"About every 40 th breath, each human being on the ear

Question

There is an old quip ... "About every 40th breath, each human being on the earth takes in one of the Argon atoms from Julius Caesar's last dying breath." Let's see if this assertion could be true.

One gram-mole of a substance contains about 6 x 1023 molecules. This number is called Avogadro's number. Although Avogadro's number seems quite large, it doesn't represent such a large quantity of matter. The air in the room in which you are now sitting has about one mole of air molecules (mostly oxygen and nitrogen molecules) for every volume of 22 liters (= 0.022 m3). This is the volume of a cube roughly 28 cm on each side. Argon makes up about 1% of the atmosphere's volume. Assume that all the Argon atoms in Julius' last breath are now evenly distributed over the earth. (It requires a bit of statistical mechanics to show that the molecules have had plenty of time to become evenly distributed by now.) Here's your task.

Estimate the volume "occupied" by one of these Argon atoms today (that would be volume per atom) and estimate the number of breaths that that volume represents.

Be sure to write out your assumptions and calculations in your response. Try to do this estimate on the spot! Remember you are doing a rough estimate, so = 3, .022 is about 1/50, etc. Here's some useful info to make your estimate easier.

The radius of the earth is about 6 x 103 kilometers.

At 50 km above the earth's surface, you are essentially in outer space. To estimate the atmosphere's volume, treat the atmosphere as a very thin shell of thickness 50 km.

Your hand is about 10 cm wide and your lungs are certainly no bigger than your upper torso.

Explanation / Answer

so a normal person's breath in volume is equivalent to V = 0.5 l
this means a normal person exhales 0.5 l air every breath
and 22.4 l of air makes one mole of air
so number of moles of air in last breath of julius ceaser, n = 0.5/22.4 = 0.022321 moles
now, argon makes up of 1 percent of atmosphere's volume,
so number of moles of argon in last breath of julius ceaser, n' = 0.01*n = 2.2321*10^-4 moles
now, total volme of atmosphere = 4*pi*[(R+d)^3 - R^3]/3
where R is radius of earth, R = 6371,000 m
d = depth of the atmospheric shell, d = 50 km = 50,000 m
so, Volume of atmoshpere, V = 4*pi*[(6571,000+50,000)^3 - (6571,000)^3]/3 = 2.73365*10^19 m^3
so if all the atoms of argon are evenly spread, volume occupied by each atom = 2.73365*10^19/ 2.2321*10^-4 * 6.022*10^23 = 0.2033709 m^3

this volume is equivalent to m breaths, where m = 0.2033709/0.5*10^-3 = 406.741

so every 400th breath a person takes has atleast one atom of argon and not 40th

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