Previously in Chem 104:

• •

types of solids Unit Cell

•

TODAY 3 types of cubic cells

•

contents of unit cell, Z

•

quantify relationship between cell and density

•

ionic solid unit cells

•

solid stability

Cubic Unit Cells have a = b = c

a = b = c

= 90 deg b c a

Three Types of Cubic Unit Cells b c a Simple Cubic Body Centered Cubic Face Centered Cubic

These Three Cubic Unit Cells are Structures of most Metallic Elements (also hexagonal, hcp, to be seen later) Cu, Ag, Au are all fcc Cr, Mo, W are all bcc Only Po is simple cubic (

rare — why?

) Simple Cubic Body Centered Cubic Face Centered Cubic

What is one result of a metal’s “choice” to adopt a cubic, bcc or fcc lattice?

Simple Cubic Body Centered Cubic Face Centered Cubic

What is one result of a metal’s “choice” to adopt a cubic, bcc or fcc lattice?

Simple Cubic

Z = 1 atom/cell Least Dense

Body Centered Cubic

Z = 2 atom/cell How did we get Z???

Face Centered Cubic

Z = 4 atom/cell Most Dense

To get Z, add ups the “bits” of atoms inside unit cell:

Simple Cubic Body Centered Cubic Face Centered Cubic

Simple Cubic Body Centered Cubic Face Centered Cubic

Z = 1 Z = 2 Z = 4 Knowing the unit cell structures can be used with other physical data and relationships: Cell volume, V = a 3 or l 3 , l

is cell length

Cell mass, m = Z x A at.wt.

Cell Density = solid density = mass = Z x volume A x at.wt.

a 3

Cell edge, a or cell length, l is related to the atomic radius but “how?” depends on which structure:

Simple Cubic

Z = 1

Body Centered Cubic

Z = 2

Face Centered Cubic

Z = 4 a or l = 2r

Cell edge, a or cell length, l is related to the atomic radius but depends on which structure:

Simple Cubic

Z = 1 a or l = 2r

Body Centered Cubic

Z = 2

Face Centered Cubic

Z = 4 Diagonal 4r =

√

2 a or

√

2 l Solve for edge: 4r /

√

2 = a or l a or l = 2

√

2 r

Cell edge, a or cell length, l is related to the atomic radius but depends on which structure:

Simple Cubic

Z = 1 a or l = 2r

Body Centered Cubic

Z = 2

Face Centered Cubic

Z = 4 4r =

√

3 a a = 4r /

√

3 a = 2.3 r Diagonal 4r a = 2

√

2 r a = 2.8 r

Rh metal crystallizes in a cubic lattice where a = 380.34 pm.

What is the crystal structure of Rh? Find Z: defines if simple, bcc or fcc Density = Z A x x at.wt.

a 3 This is a summary of the relationships What do we need? Z What do we have? Nothing here, but can’t we look up Density of Rh metal ?

Web Elements: at. weight = 102.91 g/mol Density = 12450 kg m-3 Atomic radius = 173 pm

Packing a Square Lattice: Makes a simple cubic cell

Can you pack spheres more densely?

The Rhomb is the Unit Cell Shape of Hexagonal Lattices

Closest Packing: hexagonal layers build up 3D solid

Find the triangular gaps in the Pink layer

Note how layers “sit” on top of each other: The Cyan layer covers the “up” triangles of the Pink layer The Yellow layer covers the “down” triangles of the Pink layer

This packing sequence is A B C A B C, Where B and C cover different “holes” in A A B C A B C

A C B A C B A

A B Packing direction C A B C

ccp

C

ubic

C

losest

P

acking: A B C A B C …

A C B A C B A Packing direction

ccp

C

ubic

C

losest

P

acking: A B C A B C …

A B C CCP viewed as packing layers C B A CCP viewed unit cell; LOOK! It’s face centered cubic!!! CCP = FCC!!

….mmmMMM

Packing direction A B A B A . . . . Packed towards you

A B A B A B A Packing direction

hcp

Hexagonal Closest Packing: A B A B … ….mmmMMM

From Metals to Ionic Solids Will ionic solids pack exactly like metallic solids?

Na bcc unit cell as metal NaCl unit cell?

From Metals to Ionic Solids Build up Ionic Solids conceptually like this: • assume Anions are larger than Cations, r- > r+ • pack the Anions into a cubic lattice: ccp, simple or bcc • add Cations to the interstitial spaces (“

Mind the gap!

”) r- + r+ 2 x r 2 x r-

How to draw this

The Simplest Ionic Solid is CsCl, simple cubic Start with simple cubic Unit cell of Cl- ions Then add one Cs+ in center

Z = C. N. (Cs) =

How to make NaCl: start with fcc unit cell of Cl- ions

Add Na+ in between

Add Na+ in between, everywhere

Z = C. N. (Na) =