An element crystallizes in a body-centered cubic lattice. The edge of the unit cell is \(2.86 \AA,\) and the density of the crystal is \(7.92 \mathrm{~g} / \mathrm{cm}^{3} .\) Calculate the atomic weight of the element.

We are given the edge of the unit cell, which is equal to \(2.86 \AA\). To calculate the volume, we need to cube this value: Volume = \(a^{3}\) where "a" is the edge of the unit cell. However, we must first convert the edge length from Angstroms to centimeters (1 Å = \(1\times10^{-8}\) cm). \(a_{cm} = 2.86 \AA \times (1\times10^{-8} \frac{\mathrm{cm}}{1\AA}) = 2.86 \times 10^{-8} \mathrm{cm}\) Now, we can calculate the volume of the unit cell: \(V = (2.86 \times 10^{-8} \mathrm{cm})^{3} = 2.34 \times 10^{-23} \mathrm{cm}^{3}\)

In a body-centered cubic lattice, there are two atoms per unit cell. One atom is at the corners, and the other one is at the center of the unit cell.

Density can be expressed as: Density = \(\frac{\mathrm{Mass}}{\mathrm{Volume}}\) We are given the density of the crystal as \(7.92 \frac{\mathrm{g}}{\mathrm{cm}^{3}}\) and have calculated the volume as \(2.34 \times 10^{-23} \mathrm{cm}^{3}\). We can now use these values to find the mass of the unit cell: Mass = Density × Volume \(M = 7.92 \frac{\mathrm{g}}{\mathrm{cm}^{3}}\times 2.34 \times 10^{-23} \mathrm{cm}^{3} = 1.85 \times 10^{-22} \mathrm{g}\)

Now that we have the mass of the unit cell and the number of atoms per unit cell, we can find the atomic weight of the element. The atomic weight can be expressed as: Atomic weight = \(\frac{\mathrm{Mass~of~unit~cell}}{\mathrm{Number~of~atoms~per~unit~cell}}\) Substitute the corresponding values into the formula: Atomic weight = $\frac{1.85 \times 10^{-22} \mathrm{g}}{2\ atoms\ per\ unit\ cell} = 9.25 \times 10^{-23}\ \frac{\mathrm{g}}{1\ atom}\) Converting the result to atomic mass units (1 amu = \(1.6605\times10^{-24}\ \mathrm{g}\)): Atomic weight = \(\frac{9.25 \times 10^{-23} \mathrm{g}}{1.6605 \times 10^{-24}\ \mathrm{g/amu}} = 55.7\ \mathrm{amu}\) The atomic weight of the element is approximately 55.7 atomic mass units.