Molecular simulation provides a numerical connection between the intermolecular potential model and the macroscopic properties, but it does so one state point at a time. For an equation of state, we need an equation that makes this connection over all state points. The key to making this kind of connection is to consider the average number of neighbors for each molecule within range of the potential model. We alluded to this in Example 1.1(e), and simply called it “four,” but this number must vary with density and strength of attraction and with the precise distance between molecules. Therefore, we must define a quantity representing the average number of molecules at each distance from the center of an average molecule, and study its dependence on density and temperature. To get the configurational internal energy,18 multiply this average number of molecules by the amount of potential energy at that distance and integrate over all distances. To get the pressure, multiply this average number of molecules by the amount of force per unit area at that distance and integrate over all distances. The average number of molecules at a particular distance from an average molecule is characterized by the “radial distribution function,” which is discussed in detail below. If you have ever seen a parking lot, you already know more about radial distribution functions than you may realize.
The Energy Equation
The ideal gas continues to be an important concept, because it is a convenient reference fluid. To calculate the internal energy of a real gas, we simply need to compute the departure from the ideal gas. In this way, the kinetic energy of the gas is included in the ideal gas internal energy, and we calculate the contribution to internal energy due to the intermolecular potentials of the real gas,
where u is the pair potential and g(r) ≡ the radial distribution function defined by Eqn. 7.55. This is often called the configurational energy to denote that it relates to summed potential energy of the configuration. This equation can be written in dimensionless form as
The Pressure Equation
We also may choose to solve for the pressure of our real fluid. Once again it is convenient to use the ideal gas as our reference fluid and calculate the pressure of the real fluid relative to the ideal gas law. Since intermolecular force is the derivative of the intermolecular potential, we note the derivative of the intermolecular potential in the following equation.
This equation is typically derived by determining the product PV, but we have multiplied by density to show the pressure.19 This equation can also be written in dimensionless form, recalling the definition of the compressibility factor:
Note in both the energy equation and the pressure equation, that our integral extends from 0 to infinity. Naturally, we never have a container of infinite size. How can we represent a real fluid this way? Look again at the form of the intermolecular potentials in Chapter 1. At long molecular distances, the pair potential and the derivative of the pair potential both go to zero. Long distances on the molecular scale are 4 to 5 molecular diameters (on the order of nanometers), and the integrand is practically zero outside this distance. Therefore, we may replace the infinity with dimensions of our container, and obtain the same numerical result in most situations. This substitution makes a single equation valid for all containers of any size greater than a few molecular diameters.20
An Introduction to the Radial Distribution Function
As a prelude to a general description of atomic distributions, it may be helpful to review the structure of crystal lattices like those in body-centered cubic (bcc) metals, as shown in Fig. 7.10. Such a lattice possesses long-range order due to repetitive arrangements of the unit cell in three dimensions. This close-packed arrangement of atoms gives a single value for the density, and the density correlates with many of the macroscopic properties of the material (e.g., strength, ductility). These are some of the key considerations fundamental to materials science, and more details are given in common texts on the subject. One goal of introducing the radial distribution function is to generalize the concept of atomic arrangements so that non-lattice fluids can be included.
Figure 7.10. The body centered cubic unit cell.
The distribution of atoms in a bcc crystal is fairly easy to understand, but how can we address the distribution of atoms in a fluid? For a fluid, the positions of the atoms around a central atom are less well defined than in a crystal. To get started, think about the simplest fluid, an ideal gas.
The Fluid Structure of an Ideal Gas
Consider a fluid of point particles surrounding a central particle. What is the number of particles in a given volume element surrounding the central particle? Since they are point particles, they do not influence one another. This means that the number of particles is simply related to the density,
where dNV is the number of particles in the volume element, N is the total number of particles in the total volume, V is the total volume, dV is the size of the volume element, and dNV = NAρ dV
If we would like to know the number of particles within some spherical neighborhood of our central particle, then,
dV = 4π r2 dr
where r is the radial distance from our central particle,
where R0 defines the range of our spherical neighborhood, Nc is the number of particles in the neighborhood (coordination number).
The Fluid Structure of a Low-Density Hard-Sphere Fluid
Now consider the case of atoms which have a finite size. In this case, the number of particles within a given neighborhood is strongly influenced by the range of the neighborhood. If the range of the neighborhood is less than two atomic radii, or one atomic diameter, then the number of particles in the neighborhood is zero (not counting the central particle). Outside the range of one atomic diameter, the exact variation in the number of particles is difficult to anticipate a priori. You can anticipate it, however, if you think about the way cars pack themselves into a parking lot. We can express these insights mathematically by defining a “weighting factor” which is a function of the radial distance. The weighting factor takes on a value of zero for ranges less than two atomic radii, and for larger ranges, we can consider its behavior undetermined as yet.
The hard-sphere fluid has been studied extensively to represent spherical fluids.
Then we may write
where g(r) is our average “weighting function,” called the radial distribution function. The radial distribution function is the number of atomic centers located in a spherical shell from r to r + dr from one another, divided by the volume of the shell and the bulk number density.
This is a lot like algebra. It helps us to organize what we do know and what we do not know. The next task is to develop some insights about the behavior of this weighting factor so that we can make some engineering approximations.
As a first approximation, we might assume that atoms outside the range of two atomic radii do not influence one another. Then the number of particles in a given volume element goes back to being proportional to the size of the volume element, and the radial distribution function has a value of one for all r greater than one diameter. The approximation that atoms outside the atomic diameter do not influence one another is reasonable at low density. An analogy can be drawn between the problem of molecular distributions and the problem of parking cars. When the parking lot is empty, cars can be parked randomly at any position, as long as they are not parked on top of one another. Recalling the relation between a random distribution and a flat radial distribution function, Fig. 7.11 should seem fairly obvious at this point.
Figure 7.11. The radial distribution function for the low-density hard-sphere fluid.
The Structure of a bcc Lattice
Far from the low-density limit, the system is close-packed. The ultimate in close-packing is a crystal lattice. Let’s clarify what is meant by the radial distribution function of a lattice. The radial distribution function of a bcc lattice can be deduced from knowledge of Nc and the defining relation for g(r).
If we assume that the atoms in a crystal are located in specific sites, and no atoms are out of their sites, then g(r) must be zero everywhere except at a site. For a body-centered cubic crystal these sites are at r = {σ, 1.15σ, 1.6σ,…} g(r) looks like a series of spikes. In the parking lot analogy, the best way of parking the most cars is to assign specific regular spaces with regular space between, as shown in Fig. 7.12.
Figure 7.12. The radial distribution function for the bcc hard-sphere fluid.
The Fluid Structure of High-Density Hard-Sphere Fluid
The distributions of atoms in a fluid are most conveniently referred to as the fluid’s structure. The structures of these simple cases clarify what is meant by structure in the context that we will be using, but the behavior of a dense liquid illustrates why this concept of structure is necessary. Dense-liquid behavior is something of a hybrid between the low-density fluid and the solid lattice. At large distances, atoms are too far away to influence one another and the radial distribution function approaches unity because the increase in neighbors becomes proportional to the size of the neighborhood. Near the atomic diameter, however, the central atom influences its neighbors to surround it in “layers” in an effort to approach the close packing of a lattice. Thus, the value of the radial distribution function is large, very close to one atomic diameter. Because liquids lack the long-range order of crystals, the influence of the central atom on its neighbors is not as well defined as in a crystal, and we get smeared peaks and valleys instead of spikes. Returning to the parking lot analogy once again, the picture of liquid structure is considerably more realistic than the assumption of a regular lattice structure. There are no “lines” marking the proper “parking spaces” in a real fluid. If a few individuals park out of line, the regularity of the lattice structure is disrupted, and it becomes impossible to say what the precise structure is at 10 or 20 molecular diameters. It is true, however, that the average parking around any particular object will be fairly regular for a somewhat shorter range, and the fluid structure in Fig. 7.13 reflects this by showing sharp peaks and valleys at short range and an approach to a random distribution at long range.
Figure 7.13. The radial distribution function for the hard-sphere fluid at a packing fraction of bρ = 0.4.
The Structure of Fluids in the Presence of Attractions and Repulsions
As a final case, consider the influence of a square-well potential (presented in Section 1.2) on its neighbors. The range r < σ is off-limits, and the value of the radial distribution function there is still zero. But what about the radial distribution function at low density for the range where the attractive potential is influential? We would expect some favoritism for atoms inside the attractive range, σ < r < λσ, since that would release energy. How much favoritism? It turns out to be simply related to the energy inherent in the potential function.
This exponential function, known as a Boltzmann distribution, accounts for the off-limits range and the attractive range as well as the no-influence (r > λσ) range. Referring to the parking lot analogy again, imagine the distribution around a coffee and doughnut vending truck early in the morning when the parking lot is nearly empty. Many drivers would be attracted by such a prospect and would naturally park nearby, if the density was low enough to permit it.
The low-density limit of the radial distribution function is related to the pair potential.
As for the radial distribution function at high density, we expect packing effects to dominate and attractive effects to subordinate because attaining a high density is primarily affected by efficient packing. At intermediate densities, the radial distribution function will be some hybrid of the high and low density limits, as shown in Fig. 7.14.
Figure 7.14. The square-well fluid (R = 1.5) at zero density and at a packing fraction of bρ = 0.4. The variable β ≡ 1/kT.
A mathematical formalization of these intuitive concepts is presented in several texts, but the difficulty of such a rigorous treatment is beyond the scope of our introductory presentation. For our purposes, we would simply like to understand that the number of particles around a central particle has some character to it that depends on the temperature and density, and that representing this temperature and density dependence in some way will be necessary in analyzing the energetics of how molecules interact. In other words, we would like to appreciate that something called “fluid structure” exists, and that it is described in detail by the “radial distribution function.” This appreciation will be of use again when we extend these considerations to the energetics of mixing. Then we will develop expressions that can be used to predict partitioning of components between various phases (e.g., vapor-liquid equilibria).
The Virial Equation
The second virial coefficient can be easily derived using the concepts presented in this section, together with a little more mathematics. Advanced chemistry and physics texts customarily derive the virial equation as an expansion in density:
The result of the advanced derivation is that each virial coefficient can be expressed exactly as an integral over the intermolecular interactions characterized by the potential function. Even at the introductory level we can illustrate this approach for the second virial coefficient. Comparing the virial equation at low density, Z = 1 + βρ, with Eqn. 7.52, we can see that the second virial coefficient is related to the radial distribution function at low density. Inserting the low-density form of the radial distribution function as given by Eqn. 7.57, and subsequently integrating by parts (the topic of homework problem 7.29), we find
This relationship is particularly valuable, because experimental virial coefficient data may be used to obtain parameter values for pair potentials.
Example 7.12. Deriving your own equation of state
Appendix B shows how the following equation can be derived to relate the macroscopic equation of state to the microscopic properties in terms of the square-well potential for λ = 1.5.
Apply this result to develop your own equation of state with a radial distribution function of the form:
where x = r/σ, b = πNAσ3/6, and S is the “Student” parameter. You pick a number for S, and this will be your equation of state. Evaluate your equation of state at ε/kT = 0.5 and bρ = 0.4.
Solution
At first glance, this problem may look outrageously complicated, but it is actually quite simple. We only need to evaluate the radial distribution function at x = 1 and x = 1.5 and insert these two results into Eqn. 7.60.
Supposing S = 3, Z(0.5,0.4) = 1 + 4·0.4·1.649/(0.2·2.4) – 13.5·0.4·0.648/(0.789·1.211) = 2.83.
Congratulations! You have just developed your own equation of state. Have fun with it and feel free to experiment with different approximations for the radial distribution function. Hansen and McDonalda describe several systematic approaches to developing such approximations if you would like to know more.
a. Hansen, J.P., McDonald, I.R. 1986. Theory of Simple Liquids. New York:Academic Press.
We conclude these theoretical developments with the comment that a similar analysis appears in the treatment of mixtures. At that time, it should become apparent that the extension to mixtures is primarily one of accounting; the conceptual framework is identical. The sooner you master the concepts of separate contributions from repulsive forces and attractive forces, the sooner you will master your understanding of fluid behavior from the molecular scale to the macro scale.
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