Answer to Question #254201 in General Chemistry for Kimmay17

Question

Osmium has a density of 22.6 g/mL. Calculate the radius of an Os atom expected if osmium crystallizes in the face-centered cubic structure.

A.701 pm

B. 101 pm

C. 338 pm

D. 135 pm

Answer 1

In a face-centered cubic unit cell there is total 4 atoms per cell. (8*1/8 + 2*1/2 + 2*1/2 + 2*1/2 = 4)

In 22.6 grams of osmium there is 22.6/190.23 = 0.1188 mol or 0.1188*6.02*10^23 = 7.152*10^22 atoms.

This means that 1 cm3 of osmium has 7.152*10^22 atoms.

We need to find the volume that contains 4 atoms.

V = 4/(7.152*10^22) = 5.59*10^-23 cm3 = 5.59*10^7 pm3. This is the volume of out face-centered cubic unit cell.

It is clear from the picture, that the radius of osmium atom is 1/4 of the cube diagonal. The cube edge is $\sqrt[3]{5.59*10^7}$ = 382 pm. The diagonal is $\sqrt{2}*382$ = 540.8. 1/4 of the diagonal is 540.8/4 = 135.2 pm.

Answer: 135 pm.