THE SOLUTION TO THE NECKLACE OF DEMOCRITUS
Giorgio Carboni, May 1999
Tranlsation edited by Ron Wickersham, Santa Rosa, California, USA
Revised in February 2009
PROBLEM
Determine the length of the row that is obtained when the atoms in a 1 mm^{3} grain of table salt are placed one next to the other.
DATA
(from the CRC Handbook of Chemistry and Physics)
Density of salt | d(NaCl) = 2,165 g/cm^{3} | |
Atomic weight of sodium | M(Na) = 22,9898 g | mass of 1 mol of sodium |
Atomic weight of chlorine | M(Cl) = 35,453 g | mass of 1 mol of chlorine |
Avogadro's number | N = 6,0221 × 10 ^{23} | Number of atoms in 1 mole |
SOLUTION N° 1
On the basis of the density of salt, the weight of one mole of NaCl and Avogadro's number, calculate the number of atoms in the grain.
On the basis of the number of atoms present in the grain and of their bond distance, calculate the length of the row which results when you place the atoms side by side.
Calculation of the number of atoms in the salt grain:
gram-molecule of salt | M(NaCl) = M(Na) + M(Cl) |
M(NaCl) = 22,9898 g + 35,453 g | |
M(NaCl) = 58,4428 g | |
mass of the salt grain | grainmass = d(NaCl)/1000 |
grainmass = 2,165/1000 g | |
grainmass = 2,165 × 10 ^{-3 }g | |
number of molecules present in the grain | n°molecules = N × grainmass/M(NaCl) |
n°molecules = 6,0221 × 10 ^{23 }× 2,165 × 10 ^{-3 }/ 58,4428 | |
n°molecules = 2,2309 × 10 ^{ 19} | |
number of atoms present in the grain | n°atoms = n°molecules × 2 |
n°atoms = 2,2309 × 10 ^{19} × 2 | |
n°atoms = 4,4618 × 10 ^{ 19} |
Calculation of the length of the row of the atoms of the grain:
bond distance | bd = 1 / cubic root of the number of atoms present in the grain |
bd = 1 / cubic root of 4,4618 × 10 ^{ 19} | |
bd = 2,8194 × 10 ^{-7 }mm | |
bd = 2,8194 × 10 ^{-10 }m | |
Length of the necklace | L = bd × number of atoms |
L = 2,8194 × 10 ^{-10 }m ´ 4,4618 × 10 ^{19} | |
L = 12,58 × 10 ^{9 }m | |
L = 12,58 × 10 ^{6 }km |
SOLUTION N° 2
Considering atoms to be like spheres, since we know their diameter, it is possible to determine by geometry how many of these spheres are contained in a volume of 1 mm^{3}.
In the salt crystals, the atoms occupy the nodes of a cubical mesh. Let's make the assumption that the atoms are all of the same dimension. Actually the chlorine atom is double the diameter of the sodium atom, but this difference does not influence the result at all. In fact, if the bond distance stays the same, the nodes of the crystal lattice remain unchanged too.
In order to think about this method, we make the hypothesis that the atoms have the diameter D = 0,1 mm. In this case, the number of atoms present in the cube would be equal to: 10 x 10 x 10 = 1000. Placing the 1000 atoms in a row, we calculate the length: 1000 x 0,1 = 100 mm.
From this consideration we can create the following formula:
L = sl/D × sl/D × sl/D × D
From which we obtain: L = sl^{3}/D^{2}
where sl is the length of the side of the cube and D is the mean diameter of the atoms and the bond distance also. The fraction sl/D expresses the number of atoms present along a side of the cube. Therefore, in a very simple way, replacing D with the bond distance of atoms in the salt and expressing all the dimensions in millimeters, we obtain:
Length of the necklace | L = sl^{3}/bd^{2} |
L = 1/(2,8194 × 10 ^{-7} )^{2} mm | |
L = 1,258 × 10 ^{13} mm | |
L = 12,58 × 10 ^{6} km |
CONCLUSION
The Necklace of Democritus is about 12.58 million kilometers long! This dimension is more than 33 times the distance between the Earth and the Moon (384,400 km). Far from negligible is this necklace! When Democritus tells the result to Leucippus, he will astonish him. And we are dumbfounded too. Dumbfounded not only for the length of the necklace of Democritus, but also and above all for the dimension of the atoms that up to now we had never imagined to be so minute, not to speak of the astronomical number of them that comprise a grain of table salt. The next time that you see a grain of salt on the table cloth, take it between your fingers and look at it for some moments: it deserves your contemplation!
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Some information about Leucippus: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Leucippus.ht
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Some information about Democritus: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Democritus.
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The atomic theory of the matter: http://www.neutron.anl.gov/l
eucippus.htm
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