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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.
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Solution
Suppose, the edge length of the unit cell = a
Number of atoms present per unit cell = Z
Atomic mass of the element = M
For a cubic unit cell,
Volume of the unit cell = a3 ...(i)
Density of the unit cell
ρ' = `"Mass of unit cell"/"Volume of unit cell"` ...(ii)
Mass of unit cell = Number of atoms per unit cell × Mass of one atom
= Z × m ...(iii)
where m is the mass of a single atom. This is given by
`m = "Atomic mass"/"Avogadro’s number"`
`m = M/N_A` ...(iv)
Substituting the value of m in eq. (iii), we have
Mass of unit cell = `Z xx M/N_A`
Substituting the corresponding values in eq. (ii), the density of the unit cell (ρ') is given by
ρ' = `"Mass of unit cell"/"Volume of unit cell"`
ρ' = `(Z xx M/N_A)/a^3`
or, ρ' = `(Z xx M)/(a^3 xx N_A)`
The density of a crystal (ρ) is the same as the density of its unit cell (ρ'), i.e., ρ' = ρ. Therefore the density ρ of a crystal is given by
`rho = (Z xx M)/(a^3 xx N_A)`
∴ `M = (rho xx a^3 xx N_A)/(Z)`
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