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/cm³. 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 6.02x10²³.)

So a grain of salt contains about:

5.85x10^-5 gr/ (29.25 gr / 6.02x10²³) = 1.2x10¹⁸ 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