Approximately how many atoms are in a grain of salt?

Asked by: Roger Bevels (teacher)

Answer

This is mainly an estimation problem -- there is no
exact measure of how big a 'grain of salt' is. So,
the answer will be only correct within an order of
magnitude (a factor of 10).

The first step is to estimate how large a grain of salt
is. I spilt some salt onto the table, and visually estimated
that about three grains of salt placed end-to-end
are about a millimeter long. (Your mileage may vary.)
So, as a simple estimate, I'll assume that salt grains
are 0.3 millimeters long.

The next assumption is that salt crystals are cubes,
with 0.3 millimeter sides. In fact they tend to be, since
the crystal structure of NaCl is cubic. However, most have
broken corners, but we will ignore that.

The density of NaCl is about 2.165 gr/cm^{3}. With the cube
assumption, we find that a grain of salt is about
5.85x10^-5 grams. (We could have arrived at this result by
weighing an individual grain, or by weighing a gram and
then counting the number of grains in it, but this is
left as an exercise for the PhysLink reader.)

The next thing we need is the weight of a 'salt atom'.
There is no such thing as a salt atom, it consists of
Na (sodium) and Cl (chlorine) atoms. So, we need to use an
average value. The atomic mass of Na is 23 gr per mole,
and the atomic mass of Cl is 35.5 gr per mole. So, the
average 'atomic weight of salt' is 29.25 grams per mole.

Now it is a simple matter to find how many atoms there
are in a grain of salt. (Note that one mole contains
Avogadro's number of atoms, which is 6.02x10^{23}.)

So a grain of salt contains about:

5.85x10^{-5} gr/ (29.25 gr / 6.02x10^{23})
= 1.2x10^{18} atoms, half of which are sodium atoms.
(The other half is chlorine atoms, of course.)
Answered by: Yasar Safkan, Ph.D. M.I.T., Software Engineer, Istanbul, Turkey

'The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.'