Question

A certain element exists in a face-centered cubic structure. The atomic radius of an atom of this element is 185 pm. What is the density of this element in g/cm3?(Assume the element has a molar mass of 70.9 g/mol.)

A certain element exists in a face-centered cubic structure. The atomic radius of an atom of this element is 185 pm. Calculate the density of this element in g/cm3. (Assume the element has a molar mass of 70.9 g/mol.)

Answer 1

The density of the element is `"3.29 g/cm"^3"`.

Explanation:

Calculate the mass of a unit cell

An fcc unit cell contains

`8 color(red)(cancel(color(black)("corners"))) × (1/8 "atom")/(1 color(red)(cancel(color(black)("corner")))) + 6 color(red)(cancel(color(black)("faces"))) × (1/2 "atom")/(1 color(red)(cancel(color(black)("face")))) = "1 atom + 3 atoms = 4 atoms"`

`"Mass" = 4 color(red)(cancel(color(black)("atoms"))) × (1 color(red)(cancel(color(black)("mol"))))/(6.022 × 10^23 color(red)(cancel(color(black)("atoms")))) × "70.9 g"/(1 color(red)(cancel(color(black)("mol")))) = 4.709 ×10^"-22"color(white)(l) "g"`

Calculate the volume of a unit cell

The diagonal along a face is `d = 4r`.

`d^2 = a^2 + a^2 = 2a^2 = (4r)^2 = 16 r^2`

`a^2 = 8r^2`

`a = 2sqrt2r = 2sqrt2 × 185 × 10^"-12" color(red)(cancel(color(black)("m"))) × "100 cm"/(1 color(red)(cancel(color(black)("m")))) = 5.233 × 10^"-8" color(white)(l)"cm"`

`V = a^3 = (5.23 × 10^"-8" color(white)(l)"cm")^3 = 1.433 × 10^"-22"color(white)(l) "cm"^3`

Calculate the density

`ρ = m/V = (4.709 ×10^"-22"color(white)(l) "g")/(1.433 × 10^"-22"color(white)(l) "cm"^3) = "3.29 g/cm"^3`