# 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 *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.

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