Question

X-ray diffraction studies show that copper crystallizes in an fce unit cell with a cell edge of $3.608 \times {10^{ - 8}}\;{\text{cm}}$ . In a separate experiment, copper is determined to have a density of $8.92\;{\text{g}}/{\text{c}}{{\text{m}}^3}$ . Calculate the atomic mass of copper. $\left( {{{\text{N}}_{\text{A}}} = 6.023 \times {{10}^{23}}} \right)$

Answer 1

Hint : X-ray crystallography (XRC) is an experimental science that determines the atomic and molecular structure of a crystal by causing a beam of incoming X-rays to diffract in many different directions due to the crystalline structure. A crystallographer may create a three-dimensional image of the density of electrons within the crystal by measuring the angles and intensities of these diffracted beams. The mean locations of the atoms in the crystal, as well as their chemical bonds, crystallographic disorder, and other information, may be derived using this electron density.

Complete Step By Step Answer:
Face-centered cubic lattice (fcc or cubic-F), like other lattices, includes lattice points at each of the unit cell's eight corners, as well as extra points in the centre of each face. The simplest crystal formations are those in which each lattice point has only one atom. The lattice points at both corners and on each face of the Face-centered Cubic (FCC) unit cell are indicated. Because the atoms are more densely packed than in a Simple Cubic unit cell, this is a more frequent form of unit cell.
In FCC Lattice, number of atoms/unit cell =4
We know that
${\text{M}} = \dfrac{{{\text{d}}{{\text{N}}_{\text{A}}}{{\text{a}}^3}}}{{\text{Z}}}$
Substituting all the values we get,
${\text{M}} = \dfrac{{{\text{d}}{{\text{N}}_{\text{A}}}{{\text{a}}^3}}}{{\text{Z}}} = \dfrac{{8.92 \times 6.023 \times {{10}^{23}} \times {{\left( {3.608 \times {{10}^{ - 8}}} \right)}^3}}}{4}$
Hence Atomic mass of Cu is 63.1 g/mol.

Note :
X-ray diffraction is a process in which the atoms of a crystal generate an interference pattern of the waves present in an incoming beam of X rays due to their equal spacing. The crystal's atomic planes act on the X rays in the same way as an evenly governed grating acts on a beam of light.