ENGR-1100 Introduction to Engineering Analysis

1
Lecture 19 Lattice Statistics

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

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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
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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
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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
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Energy and entropy
Energy And entropy Which can be seen as a
sum of two terms, configurational and vibrational
entropy
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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
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Langmuir model of adsorption
N atoms adsorbed on a surface of M sites Single
adsorbed site partition function Total
partition function Helmholz free energy
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Chemical potential and entropy
Chemical potential Where is the
coverage Entropy
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Langmuir adsorption
Adsorbed gas in equilibrium with gas in
vapor Which gives Or defining We get the
Langmuir adsorption isotherm