Question

The density of mercury is 13.6 g/mL. Calculate the diameter of an atom of mercury assuming that each atom of mercury is occupying a cube of edge-length equal to the diameter of the mercury atom.
(Atomic mass of mercury = 200)

Answer 1

Hint: Density is defined as mass of a unit volume of a material substance. The density formula is d = M / V, where d is density, M is mass, and the volume is V. Edge-length is the distance that lies on the same face between two opposite sides of the cube.

Complete answer:
A unit cell is the smallest portion of a crystal lattice that shows the three-dimensional pattern of the entire crystal. We can think of a crystal as repeating the same unit cell in three dimensions over and over again.
Avogadro’s number $({{N}_{A}})=6.023\times {{10}^{23}}$
Atomic weight of mercury (Hg) = 200
Therefore, in 1gm of mercury number of atoms present = $\dfrac{6.023\times {{10}^{23}}}{2\times {{10}^{2}}}$
$=3.012\times {{10}^{21}}$ atoms present in 1gm of mercury.
Mass of $3.012\times {{10}^{21}}$ atoms = 1gm
Therefore, mass of 1 atom = $\dfrac{1}{3.012\times {{10}^{21}}}$
Density of mercury = 13.6 g/mL or 13.6 g/c.c
$volume=\dfrac{mass}{density}$
Volumes of 1 atoms of mercury (Hg),
$\dfrac{1}{3.012\times {{10}^{21}}\times 13.6g|c.c}$ = $\dfrac{1\times {{10}^{3}}\times {{10}^{1}}}{3012\times {{10}^{21}}\times 136}$
$\dfrac{1\times {{10}^{-17}}}{3012\times 136}$ = $\dfrac{1\times {{10}^{-17}}}{409632}$
$=2.44\times {{10}^{-23}}$ c.c
Since, each mercury atom occupies a cube of edge length equals its diameter.
Diameter of one Hg atom = ${{\left( 2.44\times {{10}^{-23}} \right)}^{1/3}}cm$
$=2.905\times {{10}^{-8}}cm$
Hence, the diameter of an atom of mercury assuming that each atom of mercury is occupying a cube of edge-length equal to the diameter of the mercury atom is $=2.905\times {{10}^{-8}}cm$ .

Note:
To avoid calculation mistakes, convert all quantities into SI units before calculation.
Atoms in opposite corners of this unit cell come into contact with each other, so this cell's edge length is equal to two atomic radii, or one atomic diameter.
Density is commonly expressed in units of grams per cubic centimeter.
Avogadro’s number $\left( {{N}_{A}} \right)$, the number of atoms or molecules in one mole of a substance, equal to $6.023\times {{10}^{23}}$