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Lecture 19 Lattice Statistics - PowerPoint PPT Presentation

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

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Presentation Transcript

Lattice Statistics

• The model

• Partition function

• Free energy and equilibrium vacancy concentration

The partition function

In each cube we place an atom

Empty cube means vacancy

For N atoms the partition function in the Einstein model

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

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

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

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

N atoms adsorbed on a surface of M sites

Total partition function

Helmholz free energy

Chemical potential

Where is the coverage

Entropy

Adsorbed gas in equilibrium with gas in vapor

Which gives

Or defining

We get the Langmuir adsorption isotherm