How can you determine the atomic mass of an unknown metal if you know its density and the dimension of its unit cell?

By knowing the density of an unknown metal and the dimension of its unit cell, the atomic mass of the metal can be determined.

Let ‘a’ be the edge length of a unit cell of a crystal, ‘d’ be the density of the metal, ‘m’ be the mass of one atom of the metal and ‘z’ be the number of atoms in the unit cell.

Now, density of the unit cell $=\frac{\text { Mass of the unit cell }}{\text { Volume of the unit cell }}$

$\Rightarrow d=\frac{z m}{a^{3}}$   ...(i)

[Since mass of the unit cell = Number of atoms in the unit cell × mass of one atom]

[Volume of the unit cell = (Edge length of the cubic unit cell)3]

From equation (i), we have:

$m=\frac{d a^{3}}{z}$  ...(ii)

Now, mass of one atom of metal $(m)=\frac{\text { Atomic mass }(M)}{\text { Avogadro's number }\left(N_{A}\right)}$

Therefore, $\mathrm{M}=\frac{d a^{3} \mathrm{~N}_{\mathrm{A}}}{z}$    ...(iii)

If the edge lengths are different (say a, b and c), then equation (ii) becomes:

$M=\frac{\mathrm{d}(\mathrm{abc}) \mathrm{N}_{\mathrm{A}}}{\mathrm{z}} \quad(\mathrm{iv})$

From equations (iii) and (iv), we can determine the atomic mass of the unknown metal.