Question

Barium has a density of 3.59 g/cm3 and crystallizes with the body-centered cubic unit cell. Calculate the radius of the barium ion.

Answer 1

Solution :-

Barium crystallizes in the body center unit cell therefore it had two barium atom ((1/8*8)+1)=2

Density = 3.59 g/cm3

Atomic weight of the Barium is = 137.33 g/mol

Therefore lets calculate the mass of the 2 atoms of barium

Mass of 2 atoms of Ba = (2 atoms * 137.33 g per mol)/ 6.02*10^23 atoms per mol = 4.56246*10^-22 g

Now we have the mass and the density of the mass of the unit cell

Now lets calculate the volume of the unit cell

Volume = mass / density

              = 4.56246*10^-22 g / 3.59 g per cm3

              = 1.271*10^-22 cm3

Now using the volume of the unit cell lets calculate the radius of the unit cell

Volume = (edge length )^3

1.271*10^-22 cm3 = (edge length )^3

Taking cube root of both sides we get

5.028*10^-8 cm= edge length

Now lets calculate the diagonal using the edge length

d= edge length * 3^(1/2)

d= 5.028*10^-8 cm * 3^(1/2)

d= 8.69*10^-8 cm

now lets calculate the radius

4r= d

r=d/4

r=8.69*10%-8 cm/4

r= 2.173*10^-8 cm

now lets convert cm to angstrom

2.173*10^-8 cm * 1*10^8 A/ 1cm = 2.17 A^o

Therefore the radius of the Barium atom = 2.17 A^o