One reason that copper is so valuable is its high malleability, which is related to its structure. Suppose you have been asked to examine how different types of heat treatment affect the crystal structure of copper. One way to detect a change in structure is from the density. After one heat treatment the density of copper is 8.93 g · cm^–3. Is the metal

(a) Body-centered cubic or

(b) Close-packed? The atomic radius of copper is 128 pm.

PLAN Calculate the density of the metal by assuming first that its structure is bcc and then that it is ccp (fcc). The structure with the density closer to the experimental value is more likely to be the actual structure. The mass of a unit cell is the sum of the masses of the atoms that it contains. The mass of each atom is equal to the molar mass of the element divided by Avogadro’s constant. The volume of a cubic unit cell is the cube of the length of one of its sides. That length is obtained from the radius of the metal atom, the Pythagorean theorem, and the geometry of the cell.