Unit Cell Calculations

Question

Calculate the density (in Τ𝑔 𝑐𝑚3) of a metal that

crystallizes in a face centered cubic unit structure

with an edge length of 559 pm and a molar mass

of 40.078 g/mol

Calculate the molar mass (in Τ𝑔 𝑚𝑜𝑙𝑒𝑠) of a metal

that crystallizes in a body centered cubic unit

structure with an edge length of 338 pm and a

density of 0.789 g/𝑐𝑚3.

Answer 1

Answer to First part of question:

Given: Molar mass of metal 40.078 gm/mole and edge length, one crystal of metal

Face centered cube 559 pm = 559 x 10^-10 cm

Calculate density of the metal in T gm/cm³

 

One mole of any substance contains Avogadro number (AN) of atoms

 N_A = 6.022 x 10²³ atoms

Therefore, mass of one atom of metal in this question m = molar mass/(N_A)

                                                                                         = 40.078/(6.022 x 10²³)

                                                                                         = 6.6553 x 10^-23 gm

                                                                                         

For face cubic center one cube of crystalline structure contains

(1/2 atoms x 6 faces) + (1/8 atoms x 8 corners) = 4 atoms

URL: https://www.corrosionpedia.com/definition/1592/face-centered-cubic-fcc

Given that edge length, a cube of metal crystal is 559 pm = 559 x 10^-10 cm

Volume of one cube = (559 x 10^-10)³ cm³

                                   = 1.7468 x 10^-22 cm³

Density of metal = mass one atom/volume one atom

                             = 6.6553 x 10^-23 gm/[(1.7468 x 10^-22 cm³)(1/4)]

                             = 1.524 x 10^-1 gm/cm³

                             = 1.524 x 10^-13 T gm/cm³

Answer to Second part of question:

Given: Density of metal 0.789 gm/cm³ and edge length, one crystal of metal

body centered cube 338 pm = 338 x 10^-10 cm

Calculate molar mass of the metal?

For Body centered cube crystal number of atoms per cube

(1 atom at center) + (1/8 atoms x 8 corners) = 2 atoms

URL: https://www.corrosionpedia.com/definition/1593/body-centered-cubic-bcc

Mass of single metal atom = density x volume of one atom

                                          = 0.789 gm/cm³ x (338 x 10^-10)³/2

                                          = 1.5233 x 10^-23 gm

One mole of any substance contains Avogadro number (AN) of atoms

 N_A = 6.022 x 10²³ atoms

Therefore, molar mass of the metal = mass of 1 atom x N_A  

                                                        = 1.5233 x 10^-23 x 6.022 x 10²³ gm

                                                       = 9.1734 gm

I hope this helps.

Sincerely,

Shailesh (Sky) Kadakia, Expert tutor at WYZANT and Former Professor

University of Pittsburgh, Johnstown PA

Electrical Engineering Technology Department