91Ó°ÊÓ

X rays of wavelength 2.63 Ã… were used to analyze a crystal. The angle of first-order diffraction \((n=1\) in the Bragg equation) was 15.55 degrees. What is the spacing between crystal planes, and what would be the angle for second- order diffraction \((n=2) ?\)

Calcium has a cubic closest packed structure as a solid. Assuming that calcium has an atomic radius of \(197 \mathrm{pm},\) calculate the density of solid calcium.

Nickel has a face-centered cubic unit cell. The density of nickel is \(6.84 \mathrm{g} / \mathrm{cm}^{3} .\) Calculate a value for the atomic radius of nickel.

A certain form of lead has a cubic closest packed structure with an edge length of \(492 \mathrm{pm} .\) Calculate the value of the atomic radius and the density of lead.

You are given a small bar of an unknown metal X. You find the density of the metal to be \(10.5 \mathrm{g} / \mathrm{cm}^{3} .\) An X-ray diffraction experiment measures the edge of the face-centered cubic unit cell as \(4.09 Ã…\left(1 Ã…=10^{-10} \mathrm{m}\right) .\) Identify X.

Titanium metal has a body-centered cubic unit cell. The density of titanium is \(4.50 \mathrm{g} / \mathrm{cm}^{3} .\) Calculate the edge length of the unit cell and a value for the atomic radius of titanium. (Hint: In a body-centered arrangement of spheres, the spheres touch across the body diagonal.)

Barium has a body-centered cubic structure. If the atomic radius of barium is \(222 \mathrm{pm},\) calculate the density of solid barium.

The radius of gold is \(144 \mathrm{pm},\) and the density is \(19.32 \mathrm{g} / \mathrm{cm}^{3}\) Does elemental gold have a face-centered cubic structure or a body-centered cubic structure?

What fraction of the total volume of a cubic closest packed structure is occupied by atoms? (Hint: \(V_{\text {sphere }}=\frac{4}{3} \pi r^{3} .\) ) What fraction of the total volume of a simple cubic structure is occupied by atoms? Compare the answers.

Iron has a density of \(7.86 \mathrm{g} / \mathrm{cm}^{3}\) and crystallizes in a bodycentered cubic lattice. Show that only \(68 \%\) of a body-centered lattice is actually occupied by atoms, and determine the atomic radius of iron.