Lecture 19 Lattice Statistics

1. Lecture 19 Lattice Statistics • The model • Partition function • Free energy and equilibrium vacancy concentration • Langnuir adsorption

2. 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

3. 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

4. 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

5. Energy and entropy Energy And entropy Which can be seen as a sum of two terms, configurational and vibrational entropy

6. 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

7. Langmuir model of adsorption N atoms adsorbed on a surface of M sites Single adsorbed site partition function Total partition function Helmholz free energy

8. Chemical potential and entropy Chemical potential Where is the coverage Entropy

9. Langmuir adsorption Adsorbed gas in equilibrium with gas in vapor Which gives Or defining We get the Langmuir adsorption isotherm