Heat Capacity
Vibrational Heat Capacity
Atoms in solid materials are constantly vibrating at very high frequencies and with relatively small amplitudes. 
- vibrations are coordinated since adjacent atoms have bonds between them.
- what results are waves, like sound waves, traveling at the speed of sound.
- set number of waves (or phonon modes)
- the higher the number of waves, the higher the heat capacity
Specific Heat

Essentially the energy required to heat material:

Relation to electronic conduction 
As heat increases, more vibrational waves cause more scattering of free electrons.
Relation to thermal conduction
Waves also participate in the transport of energy during thermal conduction.
Relation to temperature

Rises rapidly, then plateaus.

Other heat capacity contributions
Electric contribution (very minor): electrons absorb energy by increasing their kinetic energy.  
- only possible for free e- 
Thermal Expansion
Formula

alpha l - the linear coefficient of thermal expansion (units of reciprocal temp).  For volume:

(with diff constant alpha v)

Thermal Conductivity
Atomic Perspective
Thermal expansion = increase in average distance between atoms.
Graph
As temperature rises (E1 -> E5), our average bond distance rises (R1 -> R5). Interestingly, if this trough was symmetric, then no thermal expansion would occur!
Relation to atomic bonding energy
The greater the atomic bonding energy, the deeper and more narrow the trough in the graph is, so an increase in temperature will give rise to a smaller increase in interatomic separation.  Thus, the greater the atomic bonding energy, the smaller the value of alpha l (thermal expansion coefficient) 
Metals, ceramics, polymers
Ceramics - very strong bonds, so small coefficent
Metals - intermediate
Polymers - weak bonds, so very large coefficient (big thermal expansion)
Defined as

Where q = heat flux (per unit time per unit area perpendicular to flow direction)

k = thermal conductivity 

dT/dx = temperature gradient throughout the conducting medium 

Main Idea
Atomic perspective
Thermal conductivity k 
k = k(l) + k(e)
k(l) = k from lattice vibration waves
k(e) = k from free/conducting electrons (they carry energy through kinetic energy to colder regions).  Much more effective in transport of heat energy. 
Metals
Have lot of free e, so electron mechanism of heat transport much more efficient, and are really good conductors of heat.  
- Alloying causes impurities, which act as scattering centers, lowering the efficiency of e motion. 
Ceramics
Nonmetallic materials are thermal insulators (lack large # of free electrons, and phonons not really as effective as free e).
- Furthermore, glass & other amorphous ceramics have lower conductivities (bc they have highly disordered/irregular structures, so phonon scattering)
Polymers
Good thermal insulators (low thermal conductivities). 
- energy transfer accomplished by vibration and rotation of chain molecules.  
- The more ordered and crystalline structure, the higher the thermal cond. (more coordinated vibrations)
Thermal Stresses
Stress Resulting from Change in Temp

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