solids and surfaces prof. john foord eight lectures 3rd year ptcl ‘core’ (for more (!), see ptcl...

Solids and Surfaces

Prof. John Foord

Eight lectures

3rd year PTCL ‘core’

(for more (!), see PTCL option 1 “Interfaces” in HT )

What is the interest in “Solids”?

• “Solids” are an important application of chemistry e.g. – Structural; steel, concrete, glass, plastics

etc. etc.– Functional; semiconductors,

superconductors, magnetic, optical, coatings, sensors etc. etc.

The properties of solids are controlled by the chemical

bonding

• Strength, hardness, melting point etc.

• Electrical properties

• Optical and Magnetic

• Changes with temperature and pressure Crown jewels

“Great Star of Africa” 530 Carats

Solid Surfaces

•Adhesion and coatings•Functional solid state devices •Nanotechnology•Electrochemistry•Detergent Action•Catalysis•Corrosion

Etc. Etc.

Aims of the course

1. Understanding some properties of solids in a physical chemical framework.

{Revision of some statistical mechanics}

2. An introduction to surface chemistry • Chemistry at the solid-gas interface

{Foundations for PTCL Option 1}

Books (Solids)

• Solid state inorganic texts, West etc.

• Smart and Moore, “Solid State Chemistry”

• Mandl, “Statistical Physics”

• Elliott, “The Physics and Chemistry of Solids”

Solids

1. Heat capacity of solids– Solids possess a significant heat capacity

even at low T– Must have a different origin to the that for

gases, which mainly stems from translation and rotation at room T

– Lattice vibrations– [Electronic excitations in metals]

Early ideas - Dulong and Petit

• C for many monatomic solids is around 3R

[Aside: for a gas Cp-Cv = R ; for a solid the two heat capacities are virtually identical since solids have small expansion effects]

• Each atom is vibrating in the x,y,z direction

• From energy equipartition expect C= 3R

• Deviations at low T later ascribed to “quantum” effects

Element Cp Element Cp

Al 24.4 S 22.7

Au 25.4 Si 19.9

Cu 24.5 C (diamond)

6.1

Formal model - Einstein theory for a pure monatomic solid

• Assume each atom in the solid vibrates independently in three independent directions x,y,z

• So that there are 3NAvo oscillators…

• ..and they will all have the same frequency

• Assume the vibrations are harmonic

• Reminder about stat. mechs

€

U = NkT 2 ∂ lnq

∂T

⎛

⎝ ⎜

⎞

⎠ ⎟

where

q = gi exp −E ikT

⎛ ⎝ ⎜ ⎞

⎠ ⎟≡ mol.partition function

i

∑

and

C = ∂U∂T( ) ≡ heat capacity

• For a single oscillator

€

E i = v + 12( )hν or w.r.t. zero point energy (easier)

E i = v( )hν v = 0,1,2 .......

qvib = e0hv

kT + e−hv

kT + e−2hv

kT + .....

= 11− e

−hvkT ⎛

⎝ ⎜ ⎞

⎠ ⎟

• And need the differential of q w.r.t. T

€

q = 11− x where x = e

−hvkT and dx dT =

hv

kT 2

⎛

⎝ ⎜

⎞

⎠ ⎟x

∂q∂T =

1

1− x( )2

dx

dT

⎛

⎝ ⎜

⎞

⎠ ⎟=

1

1− x( )2

hv

kT 2

⎛

⎝ ⎜

⎞

⎠ ⎟x

So we know what q and its differential is.

We can now get the energy

€

U = NkT 2 ∂ lnq

∂T

⎛

⎝ ⎜

⎞

⎠ ⎟= 3NAvokT

2 1q

⎛ ⎝ ⎜ ⎞

⎠ ⎟ ∂q

∂T ⎛ ⎝ ⎜ ⎞

⎠ ⎟

substitute in the previous expressions

for q and the differential gives

U =3NAvo hv( ) e

−hvkT ⎛

⎝ ⎜ ⎞

⎠ ⎟

1− e−hv

kT=

3NAvo hv( )

ehvkT −1

• And finally differentiate U w.r.t. to T to get the heat capacity.

€

C = 3NAvohv( )−1

ehvkt −1 ⎛

⎝ ⎜ ⎞

⎠ ⎟2

−hv

kT 2ehvkt

= 3NAvokhv

kT

⎛

⎝ ⎜

⎞

⎠ ⎟2

ehvkt

ehvkt −1 ⎛

⎝ ⎜ ⎞

⎠ ⎟2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

= 3RθET

⎛

⎝ ⎜

⎞

⎠ ⎟2

eθ ET

eθ ET −1

⎛

⎝ ⎜

⎞

⎠ ⎟

2 where θE = hv /k

Notes

• Qualitatively works quite well

• Hi T 3R (Dulong/Petit)Lo T 0

• Different crystals are reflected by differing Einstein T (masses and bond strengths)