Question

1. Rubidium metal has a body-centered cubic structure. The density of the metal is 1.532 g/cm³. Calculate the radius of the rubidium atom. Assume that rubidium atoms are spheres. Then note that each corner sphere of the unit cell touches the body-centered sphere.

2. Copper metal has a face-centered cubic structure. The density of the metal is 8.93 g/cm³. Calculate the radius of the copper atom. Assume that copper atoms are spheres. Then note that the spheres on any face of a unit cell touch along the diagonal.

Answer 1

1)

density of rubidium metal = 1.532 g / cm^3

in bcc structure number of atoms = 2

molecular mass of Rb = 85.46 g/mol

mass of Rb in grams = 85.46 / 6.023 x 10^23

                                  = 1.42 x 10^-22 g

there are 2 atoms in bcc so = 1.42 x 10^-22 x 2

                                    = 2.84 x 10^-22 g

mass of 2 atoms = 2.84 x 10^-22 g

volume of metal = mass / density

                           = 2.84 x 10^-22 / 1.532

                           = 1.85 x 10^-22 cm^3

volume = a^3

a = (volume)^1/3

   = 5.7 x 10^-8 cm

find radius r

r = sqrt(3) a / 4

= sqrt(3) x 5.7 x 10^-8 / 4

= 2.47 x 10^-8 cm

radius = 2.47 x 10^-8 cm

2)

density of Cu metal = 8.93 g / cm^3

molecular mass of Cu = 63.546 g/mol

mass of Rb in grams = 63.546 / 6.023 x 10^23

                                  = 1.055 x 10^-22 g

there are 4 atoms in fcc so = 1.055 x 10^-22 x 4

                                    = 4.22 x 10^-22 g

mass of 4 atoms = 4.22 x 10^-22 g

volume of metal = mass / density

                           = 4.22 x 10^-22 / 8.93

                           = 4.73 x 10^-23 cm^3

volume = a^3

a = (volume)^1/3

   = (4.73 x 10^-23)^1/3

   = 3.615 x 10^-8 cm

find radius r

4r = sqrt(2) a

4r = sqrt(2) x 3.615 x 10^-8

r = 1.278 x 10^-8 cm

radius = 1.278 x 10^-8 cm