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Lecture 19 Lattice Statistics

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Lecture 19

Lattice Statistics

- The model
- Partition function
- Free energy and equilibrium vacancy concentration
- Langnuir adsorption

Lattice model of a solid

The partition function

In each cube we place an atom

Empty cube means vacancy

For N atoms the partition function in the Einstein model

Lattice model of a solid - II

Since atoms next to vacancies have lower binding energy the partition function due to binding and vibrations has to be modified to

In the above expression we made an assumptions that i) there is no relaxation near the vacancy lowering the energy cost of introducing a vacancy and ii) vibrations of atoms next to vacancies are the same as in the perfect crystal

In addition we need to recognize that there are many ways of placing vacancies thus

Helmholz free energy

The logarithm of the partition function

The thermodynamic function F

Which differs from that developed for the Einstein model by the additive factor

Energy and entropy

Energy

And entropy

Which can be seen as a sum of two terms, configurational and vibrational entropy

Equilibrium vacancy concentration

In equilibrium (M equivalent of V)

Which gives (prove!)

Accounting for vibrational entropy change leads to a modification

Where is the change of vibrational entropy per one vacancy

Langmuir model of adsorption

N atoms adsorbed on a surface of M sites

Single adsorbed site partition function

Total partition function

Helmholz free energy

Chemical potential and entropy

Chemical potential

Where is the coverage

Entropy

Langmuir adsorption

Adsorbed gas in equilibrium with gas in vapor

Which gives

Or defining

We get the Langmuir adsorption isotherm