Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof Wisconsin - Green Bay

Density is a good illustration of the way macroscopic properties of materials can bedescribed in accurate detail by their atomic properties. Halite or table salt, NaCl, givesus a particularly simple illustration. The atoms in halite are arranged in a cubic patternas shown below:

The spacing between Cl atoms is 5.64 Angstrom Units or 0.564 nm

It's obvious that the most fundamental cube (unit cell) of halite contains more thanone Na and one Cl atom, but how many? If we zoom in for a closer look we see that atomsare shared between adjacent cubes, complicating matters further. There are three atomiclayers in each cell and they look as shown below. In the diagrams, the Na and Cl symbolsrepeat ** only to show how atoms are shared**. There are only nine atoms in each diagram: atthe corners, midpoints of edges, and center of each square.

- The front layer layer also shares atoms with the layer in front of it. Thus we have 1/8 of the four corner Cl atoms plus half of the center one: 4(1/8)+1/2 = 1. Also, we have 1/4 of the four shared Na atoms: 4(1/4)=1.
- In the middle layer we have 1/4 of the four corner Na atoms plus the central one: 4(1/4)+1=2. Also we have half of four Cl atoms: 4(1/2)=2.
- The rear layer also shares atoms with the layer behind it. Thus we have 1/8 of the four corner Cl atoms plus half of the center one: 4(1/8)+1/2 = 1. Also, we have 1/4 of the four shared Na atoms: 4(1/4)=1.

Thus we have one Na and Cl atom in the front layer, two in the middle, and one in theback, for a total of four. Our tendency to draw unit cells with atoms at thecorners actually complicates matters. If we draw the unit cell boundariesbetween the atoms, we actually get a much simpler picture as shown below.However, sometimes it's necessary to count atoms as shared, so it's important tobe able to do it either way.

Here it is obvious we have four sodium atoms and four chlorines in a unit cell. |

The volume of the unit cell is (5.64 x 10^{-8}) cm, cubed, or1.794 x 10^{-22} cubic centimeters. The atomic weight of Na is 23 and Cl is 35.45. The totalmolecular weight of the atoms in a unit cell is 4(23 + 35.45) = 233.8.

How do we convert that to grams? A mole of sodium weighs 23 grams and contains 6.02 x10^{23} atoms (Avogadro's Number) so a single atom weighs 23/(Avogadro's Number) grams. Thusthe mass of all the atoms in a unit cell is 233.8/(Avogadro's Number) = 3.884 x10^{-22 }grams.

**Fact worth Remembering: **The mass of any atom or molecule in grams is its atomic or molecular weight divided by Avogadro's Number.

The density of halite, then, should be mass divided by volume, or

(3.884 x 10^{-22})/(1.794 x 10^{-22}) = 3.884/1.794 = 2.165 grams per cubic centimeter. **This agrees very closely with the tabulated value. **Actual densitiesmay vary slightly because of impurities but for pure materials, the calculated densityshould agree with the observed density to very high precision.

The ionic radius of Na(+1) is 0.98 Angstrom units, that of Cl(-1) is 1.81. The width ofa cube is two ionic radii of Na plus two of Cl, or 2(0.98+1.81) = 2(2.79) = 5.58 Angstromunits. This is a gratifyingly close match to the measured value. It's not exact becausethe effective radius of an ion varies a bit depending on the other atoms around it, andsome ions, especially the semi-metals, behave as if they're ellipsoidal. At the atomiclevel we begin seeing quantum effects that complicate our macroscopic concepts of reality.Nevertheless, it's amazing how far you can go in crystallography by treating atoms asrigid spheres.

**"Theory" is not some vague, arm-waving notion divorced from reality.Theories are supposed to match observation exactly. **If they don't, it'sbecause the theory is imperfect, or the observational data is inaccurate. When both thedata and the theory are sound, they should match to high precision, as they do here.

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*Created 18 September 1998, Last Update 31 May 2020*