Answer:
Elemental gold to have a Face-centered cubic structure.
Explanation:
From the information given:
Radius of gold = 144 pm
Its density = 19.32 g/cm³
Assuming the structure is a face-centered cubic structure, we can determine the density of the crystal by using the following:
[tex]a = \sqrt{8} r[/tex]
[tex]a = \sqrt{8} \times 144 pm[/tex]
a = 407 pm
In a unit cell, Volume (V) = a³
V = (407 pm)³
V = 6.74 × 10⁷ pm³
V = 6.74 × 10⁻²³ cm³
Recall that:
Net no. of an atom in an FCC unit cell = 4
Thus;
[tex]density = \dfrac{mass}{volume}[/tex]
[tex]density = \dfrac{ 4 \ atm ( 196.97 \ g/mol) (\dfrac{1 \ mol }{6.022 \times 10^{23} \ atoms})}{6.74 \times 10^{-23} \ cm^3}[/tex]
density d = 19.41 g/cm³
Similarly; For a body-centered cubic structure
[tex]r = \dfrac{\sqrt{3}}{4}a[/tex]
where;
r = 144
[tex]144 = \dfrac{\sqrt{3}}{4}a[/tex]
[tex]a = \dfrac{144 \times 4}{\sqrt{3}}[/tex]
a = 332.56 pm
In a unit cell, Volume V = a³
V = (332.56 pm)³
V = 3.68 × 10⁷ pm³
V 3.68 × 10⁻²³ cm³
Recall that:
Net no. of atoms in BCC cell = 2
∴
[tex]density = \dfrac{mass}{volume}[/tex]
[tex]density = \dfrac{ 2 \ atm ( 196.97 \ g/mol) (\dfrac{1 \ mol }{6.022 \times 10^{23} \ atoms})}{3.68 \times 10^{-23} \ cm^3}[/tex]
density =17.78 g/cm³
From the two calculate densities, we will realize that the density in the face-centered cubic structure is closer to the given density.
This makes the elemental gold to have a Face-centered cubic structure.
