In modeling solid-state structures, atoms and ions are most often modeled as sph

Question

In modeling solid-state structures, atoms and ions are most often modeled as spheres. A structure built using spheres will have some empty, or void, space in it. A measure of void space in a particular structure is the packing efficiency, defined as the volume occupied by the spheres divided by the total volume of the structure. Given that a solid crystallizes in a face centered cubic structure that is 3.51 Å on each side, answer questions (a)-(e). (1 Å = 1. 10-10 m.) (a) How many total atoms are there in each unit cell? atom(s) (b) What is the volume of one unit cell in Å3? WebAssign will check your answer for the correct number of significant figures. Å3 (c) Assuming that the atoms are spheres and the radius of each sphere is 1.24 Å, what is the volume of one atom in Å3? (Vsphere = 4/3r3.) WebAssign will check your answer for the correct number of significant figures. Å3 (d) Therefore, what volume of atoms are in one unit cell? WebAssign will check your answer for the correct number of significant figures. Å3 (e) Based on your results from parts (b) and (d), what is the packing efficiency of the solid expressed as a percentage? WebAssign will check your answer for the correct number of significant figures. %

Explanation / Answer

edge length = 3.51 Angstrom = 3.51 x 10^-10 m

(a) For an FCC structure we have 4 atoms per unit cell

(b) Volume of unit cell = (3.51 x 10^-10)^3 = 4.32 x 10^-29 m^3

(c) radius = 1.24 Angstrom = 1.24 x 10^-10 m

Volume of one atom = 4.3 pi r^3 = 4/3 x 3.1416 x (1.24 x 10^-10)^3

= 7.99 x 10^-30 m^3

(d) Total volume of 4 atoms in one unit cell

= 4 x 7.99 x 10^-30 = 3.196 x 10^-29 m^3

(e) packing efficiency = d/b = 3.196 x 10^-29/4.32 x 10^-29 = 0.74

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