1. Monatomic Crystals

2. Motivation
So far, we considered the statistics of low to moderate density gases of non-interacting particles (ideal gases) and interacting particles (truncated virial equation of state).
Predicting properties of liquids is of utmost importance in Chemical Engineering. They are dense fluids of interacting particles, i.e., the molecules are within short distances from one another. Molecular translation plays a significant role in the thermodynamic properties, but it is more constrained than in a gas.

3. Motivation
Liquid behavior can be thought of as an intermediate between gas and solid behavior.
In solids, as in liquids, the intermolecular distances are small.
In crystalline solids, the positions occupied by the molecules are fixed according to well-defined regular geometrical patterns – called lattices.
This chapter is about crystalline solids. For simplicity, monoatomic crystals will be considered.

4. Einstein model of a crystal
To derive the model, some assumptions are needed:
We will analyze one atom (of the monatomic crystal structure), assuming all other atoms occupy fixed, equilibrium positions in the lattice.
The energy interactions of this atom are spherically symmetric. Note that because the other atoms occupy fixed positions in space, this sort of geometrical averaging imposes a severe approximation.

5. Einstein model of a crystal
To derive the model, some assumptions are needed:
Translation is severely constrained: atomic motion appears in the form of small wavelength vibration. The three translational modes (x,y,z) of a gas molecule are replaced here by three vibrational modes (x,y,z), taken to be identical, and modeled using harmonic oscillators.

6. Einstein model of a crystal
The energy levels in the three dimensional Einstein model are given by:
In this equation:
force constant

7. Einstein model of a crystal
The canonical partition function is:
There are two contributions to the energy in state i:
Interactions with all other atoms in the lattice, which has a fixed value Eint;
vibrations

8. Einstein model of a crystal
The canonical partition function is:

9. Einstein model of a crystal
Einstein vibrational temperature:

10. Einstein model of a crystal
After these identifications, the canonical partition function is:
Energy of interaction between the atom and all other atoms in the lattice

11. Einstein model of a crystal
The Helmholtz energy is:
“zero point energy per atom”

12. Einstein model of a crystal
Using the relationships of classical thermodynamics, other properties can be derived:

13. Einstein model of a crystal
Using the relationships of classical thermodynamics, other properties can be derived:

14. Einstein model of a crystal
Limiting behavior at low temperature (as it goes to 0 Kelvin) of the thermodynamic properties of an Einstein crystal:

15. Einstein model of a crystal
Limiting behavior at low temperature (as it goes to 0 Kelvin) of the thermodynamic properties of an Einstein crystal:
To conclude about the limiting behavior for the entropy and heat capacity at constant volume, let us define a new variable x. Note that when the temperature goes to zero, x goes to infinity.

16. Einstein model of a crystal
The limiting behaviors of the entropy and heat capacity at constant volume at low temperature (as it goes to 0 Kelvin) depend on the following limits, both equal to zero:
Therefore:

17. Einstein model of a crystal
In agreement with the third law of thermodynamics, as the temperature goes to zero, entropy of an Einstein crystal goes to zero.
Experimental evidence about heat capacities is that they also go to zero as the temperature approaches 0 Kelvin, what is also predicted for an Einstein crystal:

18. Einstein model of a crystal
Limiting behavior at high temperature of the thermodynamic properties of an Einstein crystal. First note that:

19. Einstein model of a crystal
Limiting behavior at high temperature of the thermodynamic properties of an Einstein crystal:

20. Einstein model of a crystal
Limiting behavior at high temperature of the thermodynamic properties of an Einstein crystal:
Law of Dulong and Petit

21. Debye model of a crystal
In the Einstein model, we analyzed the vibration of a single molecule in an averaged environment that provided spherically symmetric intermolecular interactions.
In the Debye model of a crystal, all molecules are allowed to vibrate simultaneously.
In a N atom crystal, there are 3N degrees of freedom:
3 translational modes corresponding to the motion of the center of mass;
3 rotational modes for rotations of the whole crystal;
the remaining 3N-3-3=3N-6 modes are for vibrations.

22. Debye model of a crystal
The canonical partition function of such crystal is:

23. Debye model of a crystal
The next step is to assume there is a probability distribution of frequencies such that the number of vibrational modes between and is equal to Also, by integrating over all possible frequencies, we have that:

24. Debye model of a crystal
The canonical partition function of such crystal then becomes:
But we are left with the task of finding a suitable expression for the probability distribution, which is function of the frequency.

25. Debye model of a crystal
Probability distribution:
large wavelength (low frequency) vibrations are insensitive to the details of the crystal – taken to be the same for all crystals and equal to the frequency distribution of elastic waves in a 3D continuum:
This distribution is assumed to be valid up to the highest frequency corresponding to the motion of a single atom with all other atoms fixed at their equilibrium positions.

26. Debye model of a crystal
As we are interested in systems with many particles:

27. Debye model of a crystal
Skipping the algebraic details (please refer to the book), the canonical partition function of such crystal then becomes:
With the following definition of function G:

28. Debye model of a crystal
Skipping the algebraic details (please refer to the book), the internal energy of the Debye crystal is:

29. Debye model of a crystal
In the previous slide:

30. Debye model of a crystal
Skipping the algebraic details (please refer to the book), the heat capacity at constant volume of such crystal then becomes:
With the following definition of function K:

31. Debye model of a crystal
The chemical potential and entropy are:

32. Debye model of a crystal
Limiting behavior at low temperature (as it goes to 0 Kelvin) of selected thermodynamic properties of a Debye crystal (algebraic details omitted; please refer to the textbook for additional properties):

33. Debye model of a crystal
Note that at low temperature (as it goes to 0 Kelvin) the entropy and heat capacity of the Debye crystal tend to zero, in agreement with the third law of thermodynamics and with experimental evidence.
Unlike the Einstein crystal, CV depends of the third power of temperature at low temperatures, which is the behavior observed in experiments.

34. Sublimation pressure and enthalpy of crystals
At the sublimation condition, the chemical potentials in the crystal and in the gas phase are equal.
Sublimation generally occurs at low pressure and we will assume the gas phase behaves as an ideal gas and consider an Einstein crystal.

35. Sublimation pressure and enthalpy of crystals
Solving for the pressure, which will be the sublimation pressure, we obtain:
The sublimation enthalpy can be obtained using classical thermodynamics, as follows:

36. Sublimation pressure and enthalpy of crystals
For the Einstein crystal:
Corresponding expressions for the sublimation pressure and enthalpy of the Debye crystal can be found in the textbook, following analogous derivations.

37. Comment on the third law of thermodynamics
At absolute zero temperature, the Einstein and Debye models for crystals predict that the entropy goes to zero, in agreement with the third law of thermodynamics.
However, not all crystals are perfectly ordered at zero temperature. This disorder gives origin to what is called residual entropy of the crystal. Examples of substances include carbon monoxide and water.