About two-thirds of the metallic elements have closed-packed structures. Each atom is surrounded by — Physical Chemistry Chemistry Question
Closed-Packed Structures
About two-thirds of the metallic elements have closed-packed structures. Each atom is surrounded by as many neighbouring atoms as possible. All the atoms in the structure are identical.
Draw a two-dimensional model of a closed-packed assembly of spheres.
Model Answer
In the two-dimensional model each indistinguishable atom is surrounded by six other atoms.
Change this model into a three-dimensional one. How many different possibilities are there of stacking a) three or b) an infinite number of layers? What is the coordination number of each atom?
Model Answer
A transformation into a three-dimensional model can be achieved by stacking the 2D closed-packed layers (I). Each atom has six neighbours in the plane surrounding it, and three further atoms located in the holes above the atom and three atoms located in the holes below the atom.
a) Looking at the second layer, there are two possibilities of putting a third layer on top. Either the atoms are put into the holes such that there is no atom directly beneath them in the first layer (a1), or into the same positions they occupy in the first layer (a2). These possibilities create the two different closed-packed structures, ABCABC (cubic closed-packed) and ABAB (hexagonal closed packed).
b) In principle, an infinite number of stacking patterns can be generated by the combination of these two basic stacking possibilities.
Atoms packed together are closed-packed when they occupy the minimum volume possible (assuming they are incompressible spheres). They have the maximum possible packing efficiency, defined as the ratio of volume of atoms to volume of space used.
The following arrangement is called 'cubic-F':
Insert the closed-packed layers into this illustration.
Calculate the packing efficiency and compare it with that of a cubic-primitive packing of spheres.
Model Answer
In this illustration, the atoms touch on the face diagonals. The length of the edges of the cube is 2r∙√2. There are 4 complete atoms in the cube (8 corners with one eight of an atom in each and 6 sides with one half of an atom in the middle of each).
So the packing efficiency is:
(4 × 4/3 π r^3) / (2r√2)^3 = (16/3 π r^3) / (16 r^3 √2) = π / (3√2) = 0.74 or 74%
A cubic primitive packing has a packing efficiency of:
(1 × 4/3 π r^3) / (2 r)^3 = 4/24 π = 0.52 or 52%
Insert the tetrahedral and octahedral cavities into a cubic closed-packed structure.
Model Answer
The elemental cube of a face centered cubic structure contains 4 packing atoms (one at the corner and three on the faces of the cube), eight tetrahedral holes (one in each octant of the cube) and 4 octahedral holes (one in the centre of the cube, 12 additional holes in the middle of the edges of the cube, each shared of 4 cubes).
The arrangements of ions in a crystal depend to a great extent on the relative sizes of the ions as shown in the table below.
The radius of the particles X that form the holes is r.
The radii of the largest particles M that fit into the holes without distorting them are 0.225×r for a tetrahedral hole and 0.414×r for an octahedral hole.
Show that the ideal rM/rX value for the cation-anion and anion-anion contacts of a tetrahedral arrangement of anions around a cation is 0.225.
One edge of a tetrahedron with two anions touching and the cation in the center of the tetrahedron. 2 θ = 109.5°.
Model Answer
A line perpendicular to the edge divides the tetrahedral angle into two halves. The length of the edge is 2 rX. The distance from a tetrahedral vertex to the center is rM + rX. The angle is 109,5° / 2.
sin θ = rX / (rM + rX)
sin (109,5°/2)∙ (rM + rX) = rX
0.816 rM = 0.184 rX
rM/rX = 0.225
Calculate the ideal rM/rX ratio for cation-anion and anion-anion contacts of an octahedral arrangement of anions around a cation as illustrated in one plane in the figure below.
Cation-anion and anion-anion contacts in one plane of an octahedron.
Model Answer
(2rx)^2 = (rM + rx)^2 + (rM + rx)^2
4rx^2 = 2(rM + rx)^2
√2 rx = rM + rx
rM/rX = (√2 - 1) = 0.414
rM/rX = 0.414