Lecture 19
Objectives:
1. Be able to list the assumptions and methods associated with the Virial Equation of state. 2. Be able to compute the second virial coefficient from any given pair potential.
3. Identify the origins of cubic equations of state. 4. Describe the need for higher order equations of state.
5. Describe the physical significance of the pair correlation function or radial distribution func-tion,g(r).
6. Be able to write the equations for computing the equation of state properties from the pair correlation function.
1. Virial Equation of State.
We assume that we can write the pressure as an infinite series in the density
P =ρkT +kT
∞ X
n=2
Bn(T)ρn
whereBn(T) is thenth virial coefficient, which is only a function of temperature. This is just a Maclaurin expansion aboutρ= 0 with coefficients given by
Bn+1(T) = 1 n! ∂n(P/ρkT) ∂ρn ρ=0
The virial coefficients can be rigorously calculated from statistical mechanics. The process is much too long for this class but the result is that we can prove thatBndepends only on inter-actions between nmolecules. The second virial coefficient, B2 depends only on interactions between two molecules,B3 is a function of the interaction between three molecules, etc. The second virial coefficient is given by
B2(T) = −21V Z Z f12d~r1d~r2=−21V Z Z f12d~r1d~r12 = −1 2 Z ∞ 0 Z π 0 Z 2π 0 drdθdφr2sinθf12=−2π Z ∞ 0 f12r2dr
Where fij = e−βuij
−1 and is called the Mayer f-function. So B2 depends only on the potential between an isolated pair of molecules.
The virial EOS is exact in the limit of infinite number of terms. The behavior of the gas and vapor phases can be modeled accurately with a few virial coefficients, but the series does not converge for the liquid state.
The virial coefficients are functions of temperature, but not density. The temperature func-tionality comes through the Boltzmann factor in the Mayerf-function. The virial coefficients can be expressed as a function of the potential and the temperature,Bi=Bi(u(r), T). Thus, if you know the potential and the temperature you can calculate Bi. For example, calcu-late the second virial coefficient for the following potentials: hard sphere, square well, and Lennard-Jones.
(a) Hard sphere. B2 = 2π Z ∞ 0 [1−exp(−βu(r)]r 2dr with u(r) = ( ∞ forr < σ 0 forr > σ e−βu(r) = ( 0 forr < σ 1 forr > σ
The integral can be split into two parts
B2 = 2π Z σ 0 (1−exp(−βu(r))r2dr+ Z ∞ σ (1−exp(−βu(r))r2dr
Plugging in the potential, we get
B2 = 2π r3 3 σ 0 + 0 ! = 2πσ 3 3
Note that the hard sphere second virial coefficient is independent of temperature. This is to be expected because the hard sphere fluid is athermal, i.e., its properties do not depend on temperature.
(b) Square well potential.
B2 = 2π Z ∞ 0 (1−exp(−βu(r))r2dr, using u(r) = ∞ forr < σ −ǫ forσ < r < λσ 0 forr > λσ e−βu(r) = 0 forr < σ eβǫ forσ < r < λσ 1 forr > λσ B2(T) = 2π " Z σ 0 r2dr+ Z λσ σ (1−eβǫ)r2dr+ 0 # = 2π " σ3 3 + (1−e βǫ)Z λσ σ r2dr # = 2πσ 3 3 [1 + (1−e βǫ)(λ3 −1)]
Note thatB2 from the square well potential is a function of temperature. (c) Lennard-Jones potential. Using u= 4ǫ " σ r 12 − σ r 6#
we get B2(T) = 2π Z ∞ 0 ( 1−exp −β4ǫ " σ r 12 − σ r 6#!) r2dr
This integral does have an analytic solution, but it is complicated and must be evaluated numerically. You can write a computer program to compute B2 for the LJ potential, either by evaluating the analytic solution or the numerical integral.
B2 for more complicated potentials are evaluated numerically.
Note that the potential is not a function of density or temperature. Therefore, if we look at inverting the process and compute the potential parameters from the second virial coefficient we can devise a method of measuring only low density data over a small temperature range, and deduce the pair potential at all temperatures and densities. This typically involves least-squares fitting of experimental data.
2. Cubic Equations of State.
The virial equation of state is not especially useful for liquid phase calculations. Cubic equations were developed to compensate for this.
Most cubic equations are based on the van der Waals equation of state. This equation can be derived from statistical mechanics. We will do this later on in the course. Recall for now that the van der Waals EOS has two parameters, which in molecular terms is written as
p= N kT
V −N b−
N2a
V2
where b is the excluded volume per molecule, and a is a term related to the mean-field potential felt by a central molecule.
Many cubic EOS try to improve in some way on theaandbterms. A few are: Soave-Redlich-Kwong, Peng-Robinson, etc.
3. Higher-order Equations of State.
Cubic equations are inadequate when very high accuracy is required. Therefore, complex EOS with many parameters have been developed. Many are based on a virial expansion, e.g., Benedict-Webb-Rubin (BWR): p = RT ˜ V + BRT −A− C T2 1 ˜ V2 + bRT −a− d T 1 ˜ V3 +α a+ d T 1 ˜ V6 + c T2V˜3 1 + γ ˜ V2 exp −˜γ V2
The Modified Benedict-Webb-Rubin (MBWR) equation of state is very accurate and has been used to correlate the properties of many real fluids. The following discussion is taken from Johnson, Zollweg, and Gubbins, Molecular Physics, 78, 591-618 (1993). The MBWR equation of state used here is the same as that used by Nicolas et al. [1], and contains 32 linear parameters and one nonlinear parameter. We start by writing the expression for the Helmholtz free energy. We work in reduced units so A∗
Helmholtz free energy of the fluid (Ar(N, V, T)≡A(N, V, T)−Aid(N, V, T), whereAidis the
ideal gas value). A∗
r is given by A∗ r = 8 X i=1 aiρ∗i i + 6 X i=1 biGi (1)
where the coefficientsai and bi are functions of temperature only. These coefficients contain
the 32 linear parameters in the MBWR equation. The Gi functions contain exponentials of the density and the one nonlinear parameter. The functional forms of the ai, bi, and
Gi are given in tables 1, 2 and 3, respectively. From equation (1) we can derive all other
thermodynamic properties. The pressure is given by
P∗ =ρ∗ T∗ +ρ∗2 ∂A∗ r ∂ρ∗ T∗,N , (2) where P∗
=P σ3/ǫ includes the ideal gas contribution. Substituting equation (1) into equa-tion (2) gives P∗ =ρ∗ T∗ + 8 X i=1 aiρ∗(i+1) +F 6 X i=1 biρ∗(2i+1) , (3) whereF = exp(−γρ∗2
),γ is the nonlinear adjustable parameter, and the coefficientsai andbi
Table 1: The ai temperature dependent coefficients for the Helmholtz free energy equation, (1).
The xj’s are the adjustable parameters in the equation of state.
i ai 1 x1T∗+x2 √ T∗+x 3+x4/T∗+x5/T∗2 2 x6T∗+x7+x8/T∗+x9/T∗2 3 x10T∗+x11+x12/T∗ 4 x13 5 x14/T∗+x15/T∗2 6 x16/T∗ 7 x17/T∗+x18/T∗2 8 x19/T∗2
Table 2: The bi temperature dependent coefficients for the Helmholtz free energy equation, (1).
The xj’s are the adjustable parameters in the equation of state.
i bi 1 x20/T∗2+x21/T∗3 2 x22/T∗2+x23/T∗4 3 x24/T∗2+x25/T∗3 4 x26/T∗2+x27/T∗4 5 x28/T∗2+x29/T∗3 6 x30/T∗2+x31/T∗3+x32/T∗4
Table 3: TheGi density dependent coefficients for the Helmholtz free energy equation, (1), where
F = exp(−γρ∗2
), and γ is the nonlinear adjustable parameter. We have chosenγ = 3 in this work.
i Gi 1 (1−F)/(2γ) 2 −(F ρ∗2 −2G1)/(2γ) 3 −(F ρ∗4−4G2)/2γ) 4 −(F ρ∗6 −6G3)/(2γ) 5 −(F ρ∗8 −8G4)/(2γ) 6 −(F ρ∗10−10G5)/(2γ)
4. Pair Correlation Functions.
Recall that the configuration integral, Z, is an average over the relative location of all the molecules in a system. Another way to express the same information is in terms of the structure of the fluid. If we knewon averagewhere molecules are likely to be in the fluid, then we could evaluate the configuration integral with this information to get the total partition function.
The pair correlation function is a measure of the structure of a fluid. It gives the probability of finding a molecule at a specific location from a given molecule in the fluid.
g(r1, r2, ω1, ω2) = N(N −1) ρ2Z Z . . . Z e−βU(rN−2) dr3. . . drN
We can angle average the above expression to obtain what is commonly known at theradial distribution function (rdf). g(r) = 1 Ω Z Ω g(r1, r2, ω1, ω2)dω1dω2
The rdf is related to an experimentally measurable quantity called the structure factor. This is measured from neutron scattering experiments. The rdf can also be computed for model potentials from computer simulations. An example is shown in Fig. 1.
The configurational properties of a system can be computed from the rdf. See the book for details Uc = 2πN ρ Z ∞ 0 u(r)g(r)r2dr p=ρkT −2πρ 2 3 Z ∞ 0 du(r) dr g(r)r 3dr Ac = 2πN 2 3 Z V ∞ Z ∞ 0 du(r) dr g(r)r 3drdV V2
References
0 1 2 3 4 5 r/σ 0 1 2 3 g(r)
Figure 1: The radial distribution function,g(r) for the Lennard-Jones fluid at a reduced tempera-ture of T∗
= 1.0 and a reduced density of ρ∗
= 0.9. This statepoint corresponds to a compressed liquid.
