Question

The density of mercury is $13.6g/mL$. 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 mercury atom. (Atomic mass of mercury = 200)

Answer 1

Hint: The density of an atom is equal to the ratio of mass of the atom and the volume of the atom. One mole of mercury contains $6.023 \times {10^{23}}$ number of atoms. Here the diameter of one atom has to be determined.

Complete step by step answer: Let the diameter of one atom of mercury be \[x\].
As given each atom of mercury is occupying a cube of edge length equal to the diameter of mercury atom, so the volume of mercury atom = volume of cube = \[{x^3}\]
The atomic mass of mercury = 200
The density of mercury = 13.6 g/mL
One mole of mercury contains \[6.023 \times {10^{23}}\]atoms.
As $density = \dfrac{{mass}}{{volume}}$
Volume of one mercury atom = \[{x^3}\]
So volume of \[6.023 \times {10^{23}}\] atoms = \[{x^3} \times 6.023 \times {10^{23}}\]
Inserting the values,
$13.6 = \dfrac{{200}}{{{x^3} \times 6.023 \times {{10}^{23}}}}$
${x^3} = \dfrac{{200}}{{13.6 \times 6.023 \times {{10}^{23}}}}$
${x^3} = 24.416 \times {10^{ - 24}}$
$x = 2.9 \times {10^{ - 8}}cm$
Hence the diameter of an atom of mercury is $2.9 \times {10^{ - 8}}cm$.

Note: Atomic weight of a substance is equal to the Avogadro’s number for the substance. Thus it is said that one mole of a substance is taken or used that means the amount of weight equal to the atomic or molecular weight is considered. Density of a substance is a relative property with respect to density of water which is \[1.00{\text{ }}g/mL\]. Mercury is silvery white liquid metal used in thermometers, barometers etc. due to its high density. Mercury is very toxic if inhaled.