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Gleanings Gleanings: The necklace of Democritus

Sommario:
+ Compito in classe
– La collana di Democrito
       Una pagliuzza d'oro...
       ... ne farò una collana!
       Quanto sarà lunga?
       Soluzione

Una pagliuzza d'oro...

L'altro giorno passeggiavo lungo il fiume, quando ho visto luccicare qualcosa: era una pagliuzza che sembrava d'oro. L'ho raccolta e, dopo averla fatta esaminare dal mio orafo di fiducia, ho scoperto che si trattava proprio di oro puro, e che aveva un volume pari a esattamente un millimetro cubo. Era una frazione di grammo (circa un cinquantesimo), quindi aveva un valore irrisorio; ma mi sono messo a fantasticare su cosa ne avrei potuto fare.    ▲

... I'll make a necklace!

I thought a necklace for my wife, maybe thin. Elaborating 'I could get a little more so long as more subtle, and I thought that in theory I could make a thread though infinitely thin infinitely long: the necklace invisible, but still a necklace!

But then it occurred to me Democritus, who was the first to argue that matter is made of atoms: tiny parts that can no longer divide. Then my necklace could not be infinitely thin, but would had to stop at a minimum thickness equal to the diameter of an atom. So the question: When my necklace would be a long, if I could put in a single row all the atoms of that speck of gold?

might proceed as follows: dividing my cubic millimeter in eight parts (mid-high two slices, each divided into four squares - more or less as you do with potatoes) I got a series of thickness 0.5 mm long and 4 mm (8 cubes of 0.5 mm each). Repeating the operation I got 64 cubes of 0.25 mm, a length of 16mm, then 64mm, 256mm, 1024mm (which are already more than a meter) ... until you reach a series in which each cube would contain a single atom of gold. ▲



How long will?

I did a little 'calculations and have come to determine the theoretical length of my necklace. For now, I propose three alternatives, try to answer: How long can, at most, a necklace made from a cubic millimeter of gold? ▲

A - B
15 km - 15,000 km
C - 15 million kilometers

Solution

For those curious about how I did the math, I post them here. Who is interested in the result only go to see the last line ...



Note: I did a lot of rounding and simplifications, in particular, I assumed that the interatomic distance is simply equal to the cube root of the volume occupied by each atom. However, what I wanted to show is the order of magnitude, which I really incredible: a cubic millimeter of gold covering 15 million kilometers, approximately forty times the distance from Earth to the Moon. With a quantity of gold is covered only ten times the distance from Earth to the Sun ..

Ultima cosa: la collana è lunghissima, ma il suo spessore? Beh, coincide con la lunghezza di un atomo, quindi circa 0,26 milionesimi di millimetro, ovvero 260 picometri!    ▲