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

Question:

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

Solution:

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 ab 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.