Calculate the molar mass of a metal having a density of $7.8 \ g \ cm^{-3}$ that crystallizes in a $bcc$ structure with an edge length of $288 \ pm$.

Copper crystallizes in an $fcc$ lattice with an edge length of $3.61 \times 10^{-8} \, cm$. If the measured density is $8.92 \, g \, cm^{-3}$,calculate the theoretical density of the crystal in $g \, cm^{-3}$. (Atomic mass of $Cu = 63.5 \, g \, mol^{-1}$)

In an ionic compound with a $bcc$ structure, the distance between the two nearest ions is $173 \, pm$. What is the edge length of the unit cell in $pm$?

Element $X$ crystallizes in a $12$ coordination face-centered cubic $(fcc)$ lattice. On applying high temperature,it changes to an $8$ coordination body-centered cubic $(bcc)$ lattice. Find the ratio of the density of the crystal lattice before and after applying high temperature. The atomic radius of $X$ is the same in both crystals.

In a body-centred cubic $(bcc)$ lattice of potassium,the correct relation between the atomic radius $(r)$ of potassium and the edge-length $(a)$ of the cube is:

Calculate the volume of the unit cell of an element having a molar mass of $27 \ g \ mol^{-1}$ that forms an $fcc$ unit cell. Given: $\rho \cdot N_{A} = 16.0 \times 10^{23} \ g \ cm^{-3} \ mol^{-1}$.