1. Intermolecular Potentials and the Evaluation of the Second Virial Coefficient

2. Second virial coefficient
In Chapter 7, the following relationship was established for spherically symmetrical molecules:
We will now apply it to selected potentials

3. Interaction potentials for spherical molecules
Hard sphere potential:
This is the simplest potential, given by:
(source

4. Interaction potentials for spherical molecules
Hard sphere potential:

5. Interaction potentials for spherical molecules
Hard sphere potential:
For hard spheres, the second virial coefficient is:
always positive;
temperature independent.

6. The 2nd virial coefficient for Helium
Molecules repel each other at short distances, and attract each other at larger
intermolecular separations. At low T, long range attractive forces important;
at high T, short-range repulsions dominate.

7. Interaction potentials for spherical molecules
Point centers of repulsion potential:

8. Interaction potentials for spherical molecules
Point centers of repulsion potential:
Gamma function
n > 4
If n < 3, B2 is infinite

9. Interaction potentials for spherical molecules
Point centers of repulsion potential:
For this potential, the second virial coefficient is:
always positive;
temperature dependent (decreases with increasing T, in part correct).

10. Interaction potentials for spherical molecules
Coulombic potential:
This is a special case of :
However, it diverges and gives infinite virial coefficients. Charged species need a special approach.

11. Interaction potentials for spherical molecules
Square-well potential:
This is the simplest attractive-repulsive potential, given by:

12. Interaction potentials for spherical molecules
Square well potential:

13. Interaction potentials for spherical molecules
Square well potential:
For this potential, the second virial coefficient:
may be positive or negative;
is temperature dependent.

14. Interaction potentials for spherical molecules
Square well potential:

15. Interaction potentials for spherical molecules
Square well potential:
The temperature in which the second virial coefficient is equal to zero is called Boyle temperature. For the square-well potential:

16. Interaction potentials for spherical molecules

17. Interaction potentials for spherical molecules
Lennard-Jones potential:
Very commonly used
Lennard-Jones potential
(source: Wikipedia)

18. Interaction potentials for spherical molecules
Lennard-Jones potential:

19. Interaction potentials for spherical molecules
Lennard-Jones potential:

20. Interaction potentials for spherical molecules
Lennard-Jones potential:
HG1F1: hypergeometric function F1 as defined in Mathematica (first of Appell series)

21. Interaction potentials for spherical molecules
Lennard-Jones potential:
Reproduces experimental behavior very well

22. Interaction potentials for spherical molecules
Yukawa potential:
Commonly used to model colloidal and protein solutions
Parameter that determines the range of interaction
Yukawa potential
Fluid Phase Equilibria, Volume 285, Issues 1–2, 15 November 2009, Pages 36–43

23. Interaction potentials for spherical molecules
Yukawa potential:
An alternative way of expressing the formula
Parameter that determines the range of interaction
Yukawa potential
Fluid Phase Equilibria, Volume 285, Issues 1–2, 15 November 2009, Pages 36–43

24. Interaction potentials for spherical molecules
Yukawa potential:

25. Interaction potentials for spherical molecules
Yukawa potential:
all values are comparable
and also compare well
to experiments  we can’t
use experimental B2 to
obtain potential models
Comparison plot of for square-well fluid with well-extent 1.5, Lennard-Jones fluid and Yukawa fluid at λ = 1.8.
Fluid Phase Equilibria, Volume 285, Issues 1–2, 15 November 2009, Pages 36–43

26. Second virial coefficient in mixtures
The second virial coefficient of a mixture is given by:

27. Second virial coefficient in mixtures
The second virial coefficient of a mixture is:
This is a theoretically rigorous result, valid for any intermolecular potential.
Similarly, the third virial coefficient is given by a triple summation that depends on the triple product of mole fractions, and so forth for higher order coefficients.

28. Second virial coefficient in mixtures
The second virial coefficient of a mixture is:

29. Second virial coefficient in mixtures
For pairs with equal indexes, e.g.:
the calculation uses the parameters of each pure component.
For pairs with unlike indexes, e.g.:
cross parameters are necessary. They are calculated using “combining rules”

30. Second virial coefficient in mixtures
The most common combining rules are the Lorentz-Berthelot rules:

31. Mixing Rules for Simple Equations of State
The rigorous result for the mixing rule for the second virial coefficient can guide the formulation of mixing rules for simple equations of state.
Consider the van der Waals equation of state:

32. Mixing Rules for Simple Equations of State
Let us now expand the compressibility factor around the condition of zero density:

33. Mixing Rules for Simple Equations of State
But there is no rule a priori set to compute the values of a and b for mixtures. At this point, we can recall the result for the virial equation of state:

34. Mixing Rules for Simple Equations of State
A solution (but not the only one) is:
We now define:
These are known as van der Waals one-fluid mixing rules.

35. Mixing Rules for Simple Equations of State
To use these equations, it is necessary to adopt combining rules. The most common are:
Empirical, adjustable parameter
These rules resemble the Lorentz-Berthelot rules used in the virial equation of state for mixtures.

36. Mixing Rules for Simple Equations of State
Using the combining rule for b in the mixing rule:

37. Interaction potentials for complex molecules
The term “complex molecules” here means those that have multiple atoms, are non-spherical, large, etc… - in essence, anything that deviates from the spherical symmetry assumed in the first part of this slide set.
Zurich, Switzerland, 27 February 2012—IBM (NYSE: IBM) scientists were able to measure for the first time how charge is distributed within a single molecule.
As reported in the journal Nature Nanotechnology, scientists Fabian Mohn, Leo Gross, Nikolaj Moll and Gerhard Meyer of IBM Research succeeded in imaging the charge distribution within a single molecule by using a special kind of atomic force microscopy called Kelvin probe force microscopy at low temperatures and in ultrahigh vacuum.

38. Interaction potentials for complex molecules
More than one approach is possible:
Use atom-atom interactions: possible but very complicated because of the number of interactions that need to be considered plus the fact multiple atom molecules can have internal rotations and bending.
Use site-site interactions: groups of atoms are lumped into “sites” – for example: CH2, CH3 sites in a n-alkane molecule.
Assume an idealized non-spherical shape for the molecules: e.g., cylinders, spherocylinders, etc…

39. Interaction potentials for complex molecules
More than one approach is possible:
Calculate intermolecular interactions as the sum of a spherical and of a non-spherical contribution – usually the contribution of a permanent dipole. A permanent dipole exists when there is an uneven charge distribution in the molecule.

40. Interaction potentials for complex molecules
Hard sphere with dipole
Permanent dipole moment of the molecule.

41. Interaction potentials for complex molecules
Stockmayer potential
Permanent dipole moment of the molecule.

42. Interaction potentials for complex molecules
Stockmayer potential
Computing the second virial coefficient involves averaging over the possible orientations to get it only as function of intermolecular distance. When this is done (see the textbook for details):

43. Second virial coefficient in mixtures
Show that the second virial coefficient of a mixture is:

44. But now recognizing that particles 1 to n are of type species A, n+1 to N are of type species B, since uAB (r) =uBA (r) and looking at only the first two terms, this equation can be written as

46. The rest follows as for the pure fluid virial coefficient analysis
The rest follows as for the pure fluid virial coefficient analysis. So, generalizing