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## Material Information- Title:
- Thermodynamic properties of multicomponent mixtures from the solution of groups approach to direct correlation function solution theory
- Creator:
- Telotte, John Charles, 1955- (
*Dissertant*) O'Connell, John P. (*Thesis advisor*) Hoflund, Gar B. (*Reviewer*) Dufty, James W. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1985
- Copyright Date:
- 1985
- Language:
- English
- Physical Description:
- xi, 167 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Argon ( jstor )
Correlations ( jstor ) Isotherms ( jstor ) Mathematical independent variables ( jstor ) Modeling ( jstor ) Molecular theory ( jstor ) Molecules ( jstor ) Parametric models ( jstor ) Perturbation theory ( jstor ) Thermodynamics ( jstor ) Chemical Engineering thesis Ph. D Dissertations, Academic -- Chemical Engineering -- UF Statistical mechanics ( lcsh ) Thermochemistry ( lcsh ) Thermodynamics -- Tables ( lcsh ) - Genre:
- Tables ( lcsh )
bibliography ( marcgt ) non-fiction ( marcgt )
## Notes- Abstract:
- A solution of groups technique was developed for use with fluctuation solution theory. The general expressions for calculation of pressure and chemical potential changes from some fixed reference states have been shown. A new corresponding states theory correlation for direct correlation function integrals was proposed and used with the group contribution technique for calculation of pressure changes during compression for several n-alkanes and methanol. This work gives a detailed analysis of the RISM theory of liquids. Shown are new results for perturbation theory and a generalized compressibility theorem for RISM fluids. The use of the RISM theory for calculation of thermodynamic properties of real fluids also is given. The use of hard sphere reference fluids for development of equations of state has been explored. A generalized hard sphere equation of state was developed. It was shown that the most accurate hard sphere equation of state is not the best reference system for construction of a liquid phase equation of state of the van der Waals form.
- Thesis:
- Thesis (Ph. D.)--University of Florida, 1985.
- Bibliography:
- Bibliography: leaves 163-166.
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by John Charles Telotte.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000872594 ( alephbibnum )
AEG9857 ( notis ) 014558075 ( oclc )
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THERMODYNAMIC PROPERTIES OF MULTICOMPONENT MIXTURES FROM THE SOLUTION OF GROUPS APPROACH TO DIRECT CORRELATION FUNCTION SOLUTION THEORY By JCILH CHARLES TELOTTE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1985 To Bonnie, for her support and encouragement and patience and love ACKNOWLEDGEMENTS The author would like to thank Dr. John P. O'Connell for his guidance and understanding throughout the years. Thanks also go to Dr. Randy Perry for many helpful discus- sions and the members of the supervisory committee. TABLE OF CONTENTS PAGE ACKNOWLEDGEMENTS................................ iii LIST OF SYMBOLS ................................. vi ABSTRACT........................................ x CHAPTER 1 INTRODUCTION ............................... 1 2 THERMODYNAMIC THEORY................ ...... .. 4 Thermodynamic Properties of Interest....... 5 Calculation of Liquid Volumes.............. 6 Chemical Potentials and Fugacities......... 7 Fluctuation Solution Theory................ 9 Summary .................................... 13 3 GROUP CONTRIBUTIONS ........ ............... 14 "Reactive" System Theory ................... 18 RISM Theory. ................................ 19 Differences between RISM Theory and "Reactive" Solution Theory............... 20 Thermodynamic Properties from Group Functions. ................................ 21 Summary................... ................... 24 4 FORMULATION OF MODELS FOR DIRECT CORRELATION FUNCTIONS ...................... 25 Thermodynamic Consistency .................. 25 Basis for Model Development................ 28 Analysis of the Model....................... 30 Property Changes Using the Proposed Model.. 32 5 MODEL PARAMETERIZATION..................... 34 Corresponding States Theory................ 35 Choice of a Reference Component............ 39 Method of Data Analysis..................... 40 Analysis of Argon Data..................... 44 Analysis of Methane Data................... 49 Use of the Correlations to Calculate Pressure Changes. .................. ...... 53 6 USE OF THE GROUP CONTRIBUTION MODEL FOR PURE FLUIDS ................................ 59 Extension of the Model to Multigroup Systems.................................. 59 Designation of Groups ...................... 61 Pressure Change Calculations............... 61 Summary .................................... 68 7 DISCUSSION ................................. 73 Fluctuation Solution Theory................ 74 Approximations ............................. 74 Comparison Calculation.......... ............. 82 Summary .................................... 83 8 CONCLUSIONS ................................ 84 APPENDICES 1 FLUCTUATION DERIVATIVES IN TERMS OF DIRECT CORRELATION FUNCTION INTEGRALS ............. 86 2 PERTURBATION THEORY FOR DIRECT CORRELATION FUNCTION INTEGRALS USING THE RISM THEORY... 92 3 THREE BODY DIRECTION CORRELATION FUNCTION INTEGRALS .................................... 96 4 NONSPHERICITY EFFECTS...................... 98 5 HARD SPHERE PROPERTIES..................... 110 6 COMPUTER PROGRAMS........ .................. 119 7 COMPRESSIBILITY THEOREM FROM RISM THEORY... 156 REFERENCES...................................... 163 BIOGRAPHICAL SKETCH ............................. 167 LIST OF SYMBOLS an,aB nth order expansion coefficient in perturbation function for pair aB C matrix of direct correlation function integrals CO matrix of short-ranged group direct correlation function integrals C matrix of purely intermolecular group direct correlation function integrals C.. molecular direct correlation function integrals C05 group direct correlation function integral c.. molecular direct correlation function cas group direct correlation function Cijk three body direct correlation function integral c.. group direct correlation function f temperature dependent function g temperature dependent function gij pair correlation function H matrix of total correlation function integrals h supermatrix of group total correlation functions h.. total correlation function 1J h group total correlation function ij3 HE excess enthalpy I, I identity matrix and supermatrix i molecule K matrix of differences between direct correlation function and its angle average k wave vector k Boltzmann's constant M general property M general partial molar property N total number of moles, number of components N. number of moles of species i 1 O lowest order P pressure R gas constant Position vector r distance r position vector T temperature V, V volume, molar volume V. partial molar volume 1 W matrix of intramolecular correlation functions W intramolecular correlation function integral X matrix of mole fractions X independent set of mole fractions X. mole fraction of molecule i 1 Z compressibility factor a generalized hard sphere parameter = 1/k T, dimensionless inverse temperature Yi activity coefficient 5(.) Dirac delta function vii 6.. Kroniker delta E energy n packing fraction i. chemical potential v matrix of stoichiometric coefficients 5X th order packing fraction p total density p vector of densities P matrix of densities pipa density of species i or group a 0 supermatrix of intramolecular correlation functions o. diameter 1 pa8 perturbation function 2 orientation normalization factor Q matrix of correlation functions 2 correlation function integral _,] matrix or supermatrix of intramolecular correlation functions Subscripts i,j,... species quantity n nth order term in series expansion a,B,... group quantity Superscripts E excess property HS hard sphere property viii mix mixture property o reference state property PYC, PYV using Percus-Yevick compressibility or virial forms r residual property ref reference T transpose of a matrix or vector Special Symbols vector matrix supermatrix partial molar property < > ensemble average or angle average fourier transform Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THERMODYNAMIC PROPERTIES OF MULTICOMPONENT MIXTURES FROM THE SOLUTION OF GROUPS APPROACH TO DIRECT CORRELATION FUNCTION SOLUTION THEORY By John Charles Telotte May 1985 Chairman: John P. O'Connell Major Department: Chemical Engineering A solution of groups technique was developed for use with fluctuation solution theory. The general expressions for calculation of pressure and chemical potential changes from some fixed reference states have been shown. A new corresponding states theory correlation for direct correla- tion function integrals was proposed and used with the group contribution technique for calculation of pressure changes during compression for several n-alkanes and methanol. This work gives a detailed analysis of the RISM theory of liquids. Shown are new results for perturbation theory and a generalized compressibility theorem for RISM fluids. The use of the RISM theory for calculation of thermodynamic properties of real fluids also is given. The use of hard sphere reference fluids for development of equations of state has been explored. A generalized hard sphere equation of state was developed. It was shown that the most accurate hard sphere equation of state is not the best reference system for construction of a liquid phase equation of state of the van der Waals form. CHAPTER 1 INTRODUCTION The rational design of chemical process equipment requires knowledge of the thermodynamic properties of the substances involved. More specifically, volumetric properties are needed to size a piece of equipment, enthalpies are needed to determine heat duties, and fugaci- ties are used to decide feasibility of reactions and separations. The general problem in physical property correlation and estimation is then to determine a procedure for calcula- tion of volumes, enthalpies, and fugacities, for multi- component, multiphase mixtures. This has been accomplished for some pure components using an equation of state with a large number of parameters. Here, a more modest problem is addressed. A formalism is developed that allows the volumetric properties and chemical potentials for liquid mixtures to be expressed in terms of a single group of parameterized functions. The approach used here tries to recognize the essential differences between liquid and vapor phase properties. A thermodynamically consistent formulation for the volumetric properties and chemical potentials is developed using an equation of state. Here, unlike many other schemes, the 1 equation of state uses a liquid reference state. This is all made possible by utilization of the Kirkwood-Buff (1951) or fluctuation solution theory. In an effort to allow for extrapolation of present results to state conditions not examined, the correlations are put into a corresponding states form. This can be justified from analysis of the microscopic theory that is the basis for this macroscopic approach. Finally, to allow for actual physical property estima- tion, the theory is cast in a "group-contribution" form. This is easily accomplished formally because the fluctuation solution theory relates molecular physical properties to microscopic correlations. These quantities are well defined for groups, or sites, as they will be often termed. The next chapter in this work goes into detail in discussing the thermodynamic aspects of the problem and the possible approaches that are available to express the required physical properties. Then, the flucutation solution theory is outlined. Chapter 3 is devoted to the discussion of group con- tribution modeling. Two possible means of linking the group contribution ideas to the fluctuation theory are presented and their differences expounded upon. This chapter concludes with the formulation of the physical property relationships from group correlation function integrals. Approaches to modeling correlation functions are the emphasis of Chapter 4. General forms, as suggested by microscopic perturbation theories, are outlined and discussed. The thermodynamic consistency requirements for model formulation are given along with the final expres-. sions for the thermodynamic property changes from the chosen form. Because the correlation function integrals contain unknown functions, expressed in corresponding states form, some experimental data are required for the model parameteri- zation. In Chapter 5 possible data for this parameterization are discussed critically, and two sets of general correla- tions are developed. One of these correlations is then decided upon based on the problem of interest and accuracy. Chapter 6 then presents calculations for several n-alkane molecules' compressions. Both correlations and predictions are shown. Chapter 7 is a general discussion of all of the previously reported work, and Chapter 8 offers several conclusions and recommendations for future work. There are also several appendices which give greater detail on developments presented in the body and listings of several useful computer programs. CHAPTER 2 THERMODYNAMIC THEORY Thermodynamics is a science that has its basis on certain empirical observations that have become known as laws. The simplest of these observations, and the ones most often used in practice, are that mass and energy are conserved quantities. These statements allow for the con- struction of useful mass and energy balance equations. These balances, when augmented with constitutive relations such as the conditions of phase or reaction equilibria, can be used for design of-chemical process units. However, the power of these balance expressions cannot be fully utilized unless the physical properties that appear in the expressions are available. In the following discussion one other empirical obser- vation is required. This is not known as a law but is often listed as a postulate of thermodynamics (Modell and Reid, 1974). This result may be stated in many ways, but the simplest formulation is as follows: For a simple, homogeneous system of N components, with only P-V work and thermal interactions with its surroundings, that N+l independent, intensive variables are required to specify the state of the system. The simplest use of this observa- tion is the existence of equations of state, such as P = f(T,V,X) Thermodynamic Properties of Interest As stated previously, this work is concerned with the volumetric properties and chemical potentials of the components in liquid mixtures. The chemical potentials are of interest for phase or chemical equilibrium calcula- tions because of the constraints that are imposed. For equilibrium between phases a and B the constraints are vi = 1. V. in both a and B (2-2) 1 1 1 and the chemical equilibrium constraints are I v. v. = 0 V reaction j (2-3) iJ 1 These equilibrium conditions are used to determine both feasible operating conditions for chemical processes and minimum work requirements for some processes. To size a piece of process equipment one must have knowledge of volumetric properties of the fluids involved. The molar volume of a mixture is found from knowledge of the component partial molar volumes by V = V x. V. (2-4) -- i 1 (2-1) An important aspect that must be considered in con- struction of thermodynamic models is the relationship between chemical potentials and partial molar volumes ( ) = (2-5) P T,X When both the chemical potentials and partial molar volumes are determined from the same equation of state, then equation (2-5) will always be satisfied. However, if different correlations are used for calculating these two different properties, then a thermodynamic inconsistency exists and can cause computational difficulties. Calculation of Liquid Volumes Reid, Prausnitz, and Sherwood (1977) give an extensive review of techniques for correlating liquid molar volumes. Most effort has been placed on calculation of saturated liquid volumes because of the insensitivity of liquid volumes to pressure. Most of the correlations are in corresponding states form. Calculations of compressed (subcooled) liquid volumes often are based on a knowledge of the saturated liquid volume at the system temperature. In general, the volumes are found from expressions of the form V = V(T,P) (2-6a) Vmix = V(T,P,X) (2-6b) These are calculationally convenient forms. If a complete equation of state is used, the volumes are found from solu- tion of the implicit relation P = P(V,T,X) and care is often necessary to ensure that the proper volume is calculated. Equation 2-1 can be satisfied for several values of the volume at a fixed temperature and composition. Partial molar volumes are found from the definition V = (2-7) 1 T,P,Njfi For most correlations this is evaluated by using one of the pure component correlations with a set of mixing rules applied to the parameters. Chemical Potentials and Fugacities Chemical potentials are used for solution of phase or reaction equilibria problems. However, the chemical potentials themselves are not often used because the equilibrium expressions can be written in terms of the fugacities, defined by S= 1 + RT Zn (fi/f ) (2-8) ii where the superscript o refers to a given reference state of a specified pressure, composition, and phase at the system temperature, T. The fugacity can be found from two general approaches based on different reference states. For the ideal gas reference state a complete equation of state is required, one that can reasonably predict the system volume. Another, more common, approach is to use a liquid reference state and an expression for the excess Gibbs free energy. In this formulation the fugacity is written as f. = X. Y f (2-9) 1 1 1 1 The Y. term, known as the activity coefficient, is found 1 from the free energy model and is used to correct for com- position nonideality in the liquid solution. Mathias and O'Connell (1981) have proposed a slight variant on this liquid reference state scheme. Their approach is based on using the temperature and the component densities, 0i = N./V as the independent variables to describe the state of the system. Then, at constant temperature the following relation holds: 3 Zn f. d Zn f = ) dp (2-10) 1 3 j J T 'Pk/j Fugacity ratios can be calculated based on any reference state if the partial derivatives in equation 2-10 are known. Expressions for these derivatives in terms of integrals of microscopic correlation functions are presented in the next section. Fluctuation Solution Theory Fluctuation solution theory (Kirkwood and Buff, 1951; O'Connell, 1971, 1982) is a bridge that connects the thermo- dynamic derivatives to statistical mechanical correlation function integrals. The basic relation of fluctuation theory is 3 I-L = j T,V,P j 1 1 (2-11) where the brackets denote an average over an equilibrium grand canonical ensemble and B = 1/k T. These averages are related to correlation function integrals by 1 3 13 1 Q2 ] where fdl is an integration over all phase space coordinates required for molecule 1 and Q is the normalization constant for the orientation dependence. The function g.j(1,2) is known as a pair correlation function and is directly related to a two molecule conditional probability density. It is often more convenient to work with the total correla- tion function, h. (1,2), defined by h..(1,2) = gi. (1,2) 1 (2-13) Combination of equations 2-12 and 2-13 leads to 3 ---! = .. STV,,kj 13 1 1 3 13 where we have defined H.i = --1 I dld2 hi (1,2) (2-15) Now because of the translational invariance of an equilibrium ensemble not subject to external fields hij(l,2) h (R1,R2, ) = h ((R-R2, 2) (2-16) and thus we can also write 1 dRd dh (, (2-17) ij V I dRd2 2 1 2 To simplify the further analysis we rewrite equation 2-14 using matrix notation as =1 ) = [N + NHN]i (2-18) j T,V,Ik j where the elements of the N and H matrices are (N) = .ij (H)i = H. (2-19b) To calculate changes in chemical potentials, one is interested in the inverse of equation 2-18 j) = {[N + NHNI- j (2-20) j kpj This is most easily expressed in terms of integrals of the direct correlation functions, c.i(1,2), introduced by Ornstein and Zernike (1914). These direct correlation functions are defined by h. (1,2) = c. (1,2) + V- J d3cik(l,3)hkk(3,2) (2-21) 3 k Because this integral is not of full convolution form, it may seem that it is not possible to relate the fluctuation derivatives to integrals of the direct correlation functions but that is incorrect. Appendix 1 gives the details of the relationship required. Using equations 2-20 and Al-23 we find 3Bu. -1 ( i) = [Nh /v]j (2-22) 3 T,V, k j and if the independent variables used are the component densities the result is 3ui -1 p ) = [e C]i (2-23) If equation 2-23 is combined with equation 2-10 one finds that Snf ) -Cij (2-24) TPkfj These relations are most useful because they are required to derive the differential equation of state. If the Gibbs-Duhem equation (at constant T) is written as dPS = I pi [- d p. (2-25) i j pj T,pkj then it is shown that the equation of state can be found through knowledge of the direct correlation function integrals. Combining equations 2-25 and 2-23 yields dPB = [ [1 C p. C..]d p (2-26) A knowledge of the C.. then allows for the calculation of both pressure changes and fugacity ratios relative to any chosen reference state. These results can also be used to determine partial molar volumes of all components in a mixture. 13 Summary This chapter has dealt with some of the properties of interest for process design. The relation, required by thermodynamic consistency, between the partial molar volumes and the chemical potentials has been emphasized. Mention was made of the common forms of correlations for these quantities, asserting that often liquid reference state approaches are used for both. The final section presented the fluctuation solution theory that allows for calculation of the partial molar volumes and chemical poten- tials, based on any reference state, to be expressed simply in terms of ope set of functions. CHAPTER 3 GROUP CONTRIBUTIONS One of the more powerful tools developed for physical property estimation has been the group contribution concept. The term group normally refers to the organic and inorganic radicals but can be more specific. A molecule of interest can be described by the number of the different types of groups of which it is composed. There are two general methods for using the group contribution concept. In the first approach some molecular property is written in terms of the state variables and a set of parameters, 6, M = M(T,P; 8) (3-1) and the parameters for a given set of substances are found as sums of group contribution i a 9 ia (3-2) all groups a The most common forms of this type of formulations have been for predicting critical properties (Lydersen, 1955) and ideal gas specific heats (Verma and Doraiswamy, 1965). "In some cases this technique can be justified on molecular grounds. The second use of group contribution ideas has been to assume that the groups actually possess thermodynamic properties and that the molecular properties are then a sum of these group properties, M. = M 1 a l a (3-3) groups a in i This idea is the basis for two popular activity coefficient correlations, ASOG (Derr and Deal, 1968) and UNIFAC (Fredendslund et al., 1975). A model for the group proper- ties must be developed for this technique to be useful. The true utility of the group contribution approach stems from its predictive ability. Because all of the molecules in a homologous series are formed of the same groups, only in different proportions, the data for several of the elements of the series can be used to establish the group property correlations. These can then be used to predict the properties of the other series members. This has even greater scope for mixtures. Consider for example mixtures of n-alkanes and n-alkanols. They can be considered to be made of only three groups, -OH, -CH2, and -CH3. Thus, any mixture of the alkanes and alkanols can be described by the concentra- tion of these three groups. If a viable theory exists for some property in terms of the group functions, then the properties of all mixtures of these groups are set. Figure 3-1 shows an application of these ideas for calculation of the excess enthalpy of alkane-alkanol mix- tures. The figure shows the surface of the excess enthalpy for all mixtures of the hydroxyl, methyl, and methylene groups calculated using the UNIFAC equation (Skjold- Jirgensen et al., 1979). Any compound made of the three groups-is represented by a point in the base plane. For example, the point (XCH = 1/2, XCH = 0, XOH = 1/2) is that for methanol. The possible group compositions for any mixture are found along the line connecting two molecular points. In the figure these lines are drawn in for methanol- pentane, ethanol-pentane, and pentanol-pentane. The predic- tion of the excess enthalpy is then found by the intersection of a vertical plane along the composition path and the property surface. To obtain the enthalpy prediction of the molecular system, the ideal solution value must be subtracted from the group estimate. The ideal solution line simply connects the property surface at the points of the pair molecules. The enthalpy prediction for a molecular system of methanol and pentane and at (XME = M 2/3, XE = 1/3) is shown as the value Ho in the figure. The power here is that this one diagram can be used to find excess enthalpies for all alkane-alkanol mixtures. 17 \ LE-X- ", - -- L jOLrz ;O -00 Z-- --- --- ----------- systems at 298 K calculated using UNIFAC theory. "Reactive" System Theory Equation 3-3 can also be written as Mi = v ui Ma (3-4) where via the stoichiometric coefficient, represents the number of groups of type a in molecule i. This expression is completely analogous to that found for a system of groups "reacting" to form the molecule I Via" i (3-5) With this idea the group contribution expressions have a physical interpretation and the thermodynamics of "reactive" systems (Perry et al., 1981) can be applied to obtain many results. The motivation here has been to use the fluctuation solution theory in terms of the direct correlation function integrals. It is then desirable to determine the relation- ship between the molecular and group direct correlation functions. This has been accomplished in a very general fashion by Perry (1980). The development is lengthy, but the salient features are presented below. The most important aspect of this approach is that even if the groups do not have a thermodynamics, there are still well-defined correlations between groups. This allows for the development of the Kirkwood-Buff theory 19 in terms of group fluctuation, with the constraints offered by equation 3-5. The result is i T -1 j ~ = [MT (e' c' -]6) jp T kj 1 (3-6) where p' is the matrix of group densities and C' is a matrix of group direct correlation function integrals. While this theory is formally exact for the "reactive" system, it can be difficult to apply. The correlation function integrals contain both inter- and intramolecular contributions. For the total correlation functions these effects can be separated, but no known analogous result exists for the direct correlation functions (Lowden and Chandler, 1979). This is a problem that becomes of great importance in attempting to model the correlation function integrals. RISM Theory Chandler and Andersen (1973) have formulated a molecular theory for hard sphere molecules known as RISM. The mole- cules are assumed to be composed of overlapping hard spheres or groups. They show how the molecular Ornstein-Zernike equation 2-18 can be reduced to a group form with explicit separation of the intermolecular and intramolecular correla- tions. This allows them to define group direct correlation function integrals that have only intermolecular 20 contributions. The development is detailed but the essential result is cij(1,2) = i Vje C B,(ri r (3-7) Note here that the molecular correlation function has orien- tation dependence even though the group functions, (-a 6 caB(ri,rj), are written for spherically symmetric inter- actions. Chandler and Andersen discuss many attributes of these group functions but put no emphasis on the thermo- dynamic ramifications of these findings. Differences between RISM Theory and "Reactive" Solution Theory The major difference between the RISM theory and the "reactive" solution theory is the nature of the direct correlation functions. In Perry's formulation we have 4- c B(r = direct correlation function (3-8) between group a and group B and the RISM theory uses c "(ri,r) = direct correlation function (3-9) Between group a on molecule i and group B on molecule j This shows that the RISM functions contain less information but may be easier to model because they seer. analogous 21 to molecular functions for which models have been developed. Chandler and Andersen have also presented a variational theory which can be used to obtain the direct correlation functions for hard body systems. Figure 3-2 shows the results obtained using data from Lowden and Chandler (1973) for a system of hard diatomic molecules. The diatomic molecules are treated as overlapping spheres of diameter o separated by a distance L. The figure shows that the calculations agree to a reasonable degree with the Monte Carlo calculations for this type of system. Another important aspect of the Chandler-Andersen theory is that they discuss how the c.B functions could be written for real molecules. In general, they show that perturbation theories that would be valid for molecular systems would also apply to RISM group system. The RISM formulation will be employed in the present work. Thermodynamic Properties from Group Functions Equations 2-23 and 2-26 can be combined with equation 3-7 to express the molecular thermodynamic differentials in terms of group correlation function integrals. The first quantity required is C.. = i(y ) dld2 cij(1,2) (3-10) 1: s 1 Using equation 3-7 this is 45 30 15 I I 0.0 0.3 0.6 0.9 3 Figure 3.2. Comparison of RISM (---) theory prediction and Monte Carlo (e* ) for a hard, homonuclear diatomic molecule with separation to diameter ratio of 0.6. c. f dld2 c Cr., r.) C'3 = r r Vi vjs ( 12 2) I dld2 ce6 (+a : (3-11) To evaluate this quantity, a coordinate transformation is required. dld2 = dR1dR2dSi, d 2 drl, dr2d~id 2 (3-12) Here, i., represents the set of angles needed to specify the orientation of molecule i with respect to a fixed coordinate system. The above transformation is canonical, and the angular integration can be performed to yield C.. = I v, ( ) dr dr c (?, r.) ii] ]B i 2 i j aO 1 ] (3-13) Now, the systems under that consideration are homogeneous so c (ri?, rt) = c a(ri ) c t(r) 06 i ] j B 16 Then Cj = .i vj6 Ce where CaB = ( d cB() This leads to (3-14) (3-15) (3-16) (3BiJ_5 6. i Ti n Uia B. 3 C 3 (3-17a) j T,Pk7j i a B or in differential form do. d. i v (j C dp ) (3-17b) i p aB and the corresponding expression for the pressure variation is dPB = dP D CaF dpg (3-18) a b Summary Group contribution approaches are valuable for predict- ing thermodynamic properties. In this chapter two general approaches for incorporating group contributions into fluc- tuation theory have been presented and analyzed. Finally, the thermodynamic property differentials for molecular systems have been expressed in terms of group direct correla- tion function integrals. Actual property changes can be formulated once models are expressed for the integrals. This is the subject of the next chapter. CHEAPER 4 FORMULATION OF MODELS FOR DIRECT CORRELATION FUNCTIONS In previous chapters it has been shown that changes in thermodynamic properties of molecular systems can be expressed in terms of integrals of site direct correlation function. Expressions for these integrals are required before the calculations can be performed. The purpose of this chapter is two-fold. First, thermo- dynamic requirements on models for the direct correlation functions will be presented and examined. With these restrictions on model form established, a feasible model for the correlation function integrals will be presented. Thermodynamic Consistency One aspect that has been emphasized throughout this work is the necessity of the formalism to meet thermodynamic consistency requirements. Because our proposed calculational scheme employs direct correlation function integrals, the models that are developed for these quantities are subject to consistency tests. The easiest check that can be employed is one of equality of cross partial derivatives of the dimensionlesss) chemical potentials. 2 2 ( ) = ( )a ] V i,j,k (4-1) aPjSpk p kj This can be expressed in terms of the direct correlation functions integrals because ) = C. (4-2) j T,Pkj Combination of equations 4-1 and 4-2 leads to aC. aC DC-3 ik) (4-3) r___ = (aplk Pk T,pZ/k j Tr,/j It must be noted that this is equivalent to the equality of three-body direct correlation function integrals (Brelvi, 1973) C.. = C. (4-4) ijk = Cikj (4-4 where the subscripts can take on all values associated with the species in the mixture. The case of interest here is that in which the direct correlation function integrals for the molecular species are written in terms of group quantities Cij = ( I vi vj CB (4-5) a 8 Then equation 4-3 takes the form [ k B B T, ik nTT,p p S a [K r Vio VkB CB ) (4-6) j a B T,p j This requirement can be expressed in terms of group prop- erties if the chain rule is used for the derivatives _P i YaP (4-7) yV 1 P y y Combination of equations 4-6 and 4-7 leads to C ac aBy iy kB y Bp nny i- Y UkVji (a --] (4-8) This relation will only be satisifed in general if a3C 3C 0y) T6,p (4-9) T' -Y P Z- Thus, the consistency requirements for group direct correla- tion function integrals are equivalent to those of the molecular quantities. This constraint will be employed in formulation of the working models for the group direct correlation function integrals. Basis for Model Development It has already been shown that the thermodynamic con- sistency tests for a group formulation and a molecular formulation are equivalent when written in terms of direct correlation function integrals based on the RISM theory. This should not be surprising considering that stability conditions for fluids were shown to be equivalent for the two approaches by Perry (1980). Thus, in developing models for group direct correlation function integrals, the same considerations should apply as those used by Mathias (1979) in developing molecular models. The general philosophy that will be employed is to use as much theoretical information as possible to develop these models. This requires use of some concepts of statis- tical mechanical perturbation theory to obtain approximate forms for the correlation function integrals. The analysis of Chandler and Andersen (1972) has shown that for RISM theory direct correlation function results of molecular perturbation theory are easily extended to the group func- tions. Appendix 2 contains a complete development of an exact perturbation theory for the direct correlation func- tions based on the RISM theory. In this chapter we shall only deal with the important ramifications of these results. It is always possible to write B ref + a ref (4-10) C =a6 + (Cc a (4-10) where the superscript ref refers to some reference system. The purpose of perturbation theory is to determine a refer- ence system so that an approximate form for the perturbation (second on the right-hand side) term can be made that yields useful results. Because this work is concerned with dense fluids, we require a reference system that can adequately represent the behavior at high densities. In liquids the optimal choice of a reference system seems to be one with only repulsive forces (Weeks, Chandler, and Andersen, 1971). Values of Ca3 are not available for this type of model system. However, the system with purely repulsive forces can be well represented by a system of hard spheres if the hard sphere diameters are chosen as functions of tempera- ture (Barker and Henderson, 1967). This approach will be followed in this work. Even with this choice of reference system the perturbation term cannot be exactly identified. A further approximation used is that the zero density limit of this term is adequate. Thus, the model employed here is -HS lim HS C CB + lim (CaB C) (4-11) Appendix 2 shows the evaluation of the required limit which gives the working relation Ca C H + (a) /Tn (4-12) n=o aB 30 The constants in equation 4-12 are dependent on the inter- molecular potential (written as a sum of group of interac- tions). No explicit calculation of these terms is attempted here because we choose to determine the expressions on the basis of experimental data. This model form is com- pletely analogous to the model used by Mathias (1979). Analysis of the Model Several interesting features of the direct correlation function model seem to merit mention. Equation 4-12 is highly similar to the RISM form of the mean spherical approximation (Lebowitz and Percus, 1966; Chandler and Andersen, 1972) and is essentially equivalent if the hard sphere diameters are chosen as functions of temperature only. The assumptions involved in obtaining the present approximant and the mean spherical form are different, but it seems that for our purposes this difference is immaterial. Of greater interest here is the relationship suggested by the present model for the three-body direct correlation function integrals. These can be found from 3C. C = (--) (4-13) ijk 8k T,p /k The form for the C.. proposed in equation 4-12 then suggests that HS C CHS (4-14) ijk ijk This is probably not a very accurate approximation, but the flexibility in the model form inherent due to the determination of the parameters from experimental data may make this adequate. A better interpretation of this analysis is that the density dependence of the three-body direct correlation function is assumed to be approximated by that of the hard sphere quantity. This analysis also suggests an extension of the RISM theory to three-body correlation functions, a derivation of which can be found in Appendix 3. Cjk i ji jky CaBy (4-15) Finally, thermodynamic consistency of the proposed model must be considered. If equation 4-12 is rewritten in explicit form HS C (pT) = C (p,T) + B (T) (4-16) then the consistency requirement is met if ECHS aCHS ( "B ( y) py Tp T,pp (4-17) HS In this work, the Cae are determined from the generalized hard sphere equation of state, which is consistent, then this must be true. Note that if terms of higher order in density had been included in the perturbation term, they also would have to have been chosen to be thermodynami- cally consistent. Also of interest here is the functional dependence of the hard sphere diameters. If they are chosen to be functions of density, as is normally done in the Week- Chandler-Andersen perturbation scheme, then this dependence would have to be such that equation 4-17 was satisfied. This is an aspect not always appreciated. Property Changes Using the Proposed Model This chapter has presented and analyzed a feasible model for group direct correlation function integrals. The utility of this model can only be tested by its use to calculate changes in molecular thermodynamic properties. The required formulas are developed below. Chemical Potential Equation 3-17b expresses the differential of a molecular chemical potential in terms of the group direct correlation function integrals. This is conveniently rewritten in terms of the residual chemical potential as dBf = i vj CaB dPg (4-18) Insertion of equation 4-16 and integration leads to Ari = I )i [A HS + r A Ap] (4-19) aa a where ',HS is the residual chemical potential calculated from the hard sphere equations for group a in a solution of groups. Pressure Change To calculate pressure changes, it is again easiest to work with residual properties. The required differential form is dP = p CB dp (4-20) ca a Insertion of the model and integration leads to Apr = APr,HS + (1) Y A(p p ) (4-21) 6 5 2 a a aS where A r,HS is the change in the residual pressure (divided by k T) calculated from the hard sphere equation for the solution of groups. It should be noted that the ideal gas state used for the residual property changes on either side of equation 4-21 are different. This should be expected for the group equation of state is written in terms of the group densities which sum to a larger value than the molecular densities. However, because this form is not used as a complete equa- tion of state, this should not present a problem. CHAPTER 5 MODEL PARAMETERIZATION This work has been concerned with formulating the thermodynamic properties of molecular systems in terms of integrals of group direct correlation functions. In the last chapter a form for these terms was developed C = CRE + (5-1) and further, the reference state was identified as that of a hard sphere system. The full dependence on state variables is written as C (p,T) = CHS (P,T; ) + 0a (1/T) (5-2) where U is the vector of hard sphere diameters for the groups in the system. The purpose of this chapter is to develop empirical forms for the functions U'aS and to estab- lish the dependence of the hard sphere diameters on state parameters (Weeks,Chandler, and Andersen, 1971). Corresponding States Theory The utility of the form proposed in equation 5-2 can be enhanced if the perturbation functions and hard sphere diameters could be written in corresponding states form. This is a two-step process: 1. Determine if a corresponding states principle exists for the direct correlation function integrals for liquids. 2. Determine if the functional form proposed in equation 5-2 can represent the observed correspondence. To answer these questions simple molecules will be considered first. The molecules considered here can be considered to consist of only one group. The relation to thermodynamics used in the analysis is ( ) = 1/(l-pC) (5-3) T Figure 5-1 shows a corresponding states correlation for the quantity on the left-hand side of equation 5-3 for simple molecules, using only one parameter. Further, Brelvi and O'Connell (1972) showed that this behavior can be found for polyatomic molecules also, if the density is large enough (p > 2Pc). These results suggest that for large densities a one-parameter corresponding states formulation may be valid for the direct correlation function integrals. -2 in o CCl4 -3 4 0 CH m CO2 -N 0.7 0.8 0.9 3 p3 Figure 5-1. Bulk modulus versus reduced ensity for simple fluids. The line is argon data and o is Lennard-Jones potential parameter. Note that the temperature dependence seems to have little effect. Inclusion of temperature dependence in these cor- relations should be able to enhance the accuracy. Figure 5-2 shows another test of this correspondence. This is in an integrated form of equation 5-3. PB = PBef + S (1-pC) dP (5-4) Pref Here, a characteristic volume for each molecule, V*, and a characteristic temperature, T*, have been found to yield the correspondence as P = Pref' f(, T; ref) (5-5) where P = PV*/RT = pV* T = T/T* The function, f, in equation 5-5 is a corresponding states correlation for the density integral of the direct correlation function. This is what is desired. The data presented seem to suggest that a corresponding states formu- lation can be found for pure component direct correlation function integrals for dense fluids. A detailed discussion 100 argon 50 methanol V acetic acid 2 n-Cl6H34 n-C16 H 34 A 25 38 20 10 T = T/T* P = PV* P =-+C RT 2 T=0.92 T=0.60 T=0.50 T=0.80 Figure 5-2. Corresponding states correlation for liquids. Figure 5-2. Corresponding states correlation for liquids. 39 of the origins of this correlation is presented in Appendix 4. Mathias (1979) has shown that these concepts can be extended to mixture-direct correlation function integrals with the proper choice of mixing rules. He used a form similar to equation 5-2, but the group contribution concept was not employed. The applicability of the two-parameter corresponding states principle seems valid for the direct correlation function integrals. The question of the use of the func- tional form must now be addressed. This is again most easily accomplished by analysis of a set of experimental data for a simple molecule. After the noble gases are used for this type of testing, however, the major interest here is for hydrocarbons so that a hydrocarbon reference system may prove to be more useful. Choice of a Reference Component Mathias (1979) had great success in developing a corresponding states correlation for direct correlation function integrals, using a form similar to equation 5-2, for molecular systems, using argon as a reference component to determine model parameters. This work is concerned with hydrocarbon properties so that it seems reasonable to examine the difference in direct correlation function integrals between argon and a simple hydrocarbon, methane. The comparison is performed here using the dimensionless quantity 1-pC. Table 5-1 shows this comparison for two isotherms, in the dense fluid region. The behavior is similar on both isotherms. It is seen that the fluids act quite similarly at lower densities but behave differently as the density becomes larger. Methane is increasingly more compressible as the density increases. This can be explained by the polyatomicity of methane. Both molecules are essentially spherical, but at the higher densities the methane seems to allow for interlocking of the hydrogens. These effects can be important for the development of the group contribution correlations. If the groups were chosen as monatomic, then argon would probably be the better choice of reference substance due to physical similarity. However, if the groups are chosen as the common organic radicals, then methane may be the better choice of reference substance. The actual choice of a reference substance will be made after it has been determined whether equation 5-2 is a viable functional form for the direct correlation function integrals. Method of Data Analysis This section addresses the question of whether the form proposed in equation 5-2 can be used to fit the direct COMPARISON TABLE 5-1 OF DIRECT CORRELATION FUNCTION INTEGRALS FOR ARGON AND METHANE T/T = 0.76 T/TC = 2.0 pVc (1-pC)argon ( 1-pC)methane pVc (1-pC)argon (1-pC)methane 2.26 6.44 6.35 2.1 7.23 7.14 2.31 7.97 7.82 2.2 8.50 8.35 2.37 9.68 9.44 2.3 9.96 9.73 2.43 11.51 11.25 2.4 11.65 11.29 2.48 13.57 13.24 2.5 13.58 13.07 2.54 15.83 15.43 2.6 15.79 15.06 2.59 18.30 17.83 2.7 18.29 17.30 2.65 21.05 20.46 2.8 21.13 19.91 2.70 23.88 23.33 2.9 24.33 22.59 2.76 27.10 26.45 3.0 27.94 25.68 42 correlation function integrals of real fluids. The analysis will be performed in terms of the dimensionless quantity, C, definedby PBS C pC = 1 ( (5-6) The thermodynamic derivative is evaluated from correlated experimental data. From equation (5-2) this is C CHS (p, T; o) + p6 (5-7) For this work the hard sphere diameter is treated as a function only of temperature. Thus, along an isotherm, if the proposed functional form is proper, for the proper choice of hard sphere diameter, the value of 6 should be constant given by S= (CHS C)/p (5-8) for any value of p. The suitability of the proposed model is determined by how well this is obeyed. The hard sphere properties are determined using the generalized hard sphere expressions as presented in Appendix 5. It should be noted that all of the hard sphere equations are generated by the same microscopic form of the direct correlation function. Here the hard sphere direct correlation function 43 integrals are found using equation 5-6 with the corresponding form of the hard sphere equation of state. For these pure component cases we have -HS (1+a)n4 4(1+a)n3 + 2 8 C 4 (5-9) (1-n) where n = V/6.po and in corresponding states form n = [T/6 o3/V*][pV*] (5-10a) = X pV* = Xp (5-10b) Here, the characteristic volume, V*, has been introduced as a reducing parameter for both the hard sphere volume and the density. On each isotherm there are actually two parameters that can be used to vary the model, a and X. The analysis here has been done sequentially, a value of a is chosen and then the optimum X has been found for that value of a. The characteristic volume has also been used to reduce the perturbation term 0 as u = (t,/V*)V* = V*f(T) (5-11) The model can then be written as C(p,T) = CHS(n; T,X,a) + pf (5-12) Figure 5-3 gives a simplified flowchart of how the calcula- tions are performed for each isotherm for a fixed value of a. The subscript i denotes the number of data points on the isotherm, NP. The optimization routine, RQUADD, can be found in Appendix 6. Analysis of Argon Data 3P8 The values of ( needed for this analysis were generated from the equation of state of Twu et al. (1980). These authors claim to be able to reproduce dense fluid pressures to within the experimental accuracy. Eleven temperatures, evenly spaced between the triple and critical points, were chosen to generate liquid phase data. Each isotherm con- sisted of 16 density-correlation function integral pairs evenly spaced between the vaporizing and freezing densities. Supercritical data were also used. The isotherms were for values of T/T from 1.1 to 3.0. The densities used were in 0.1 increments of P/P from 1.5 to 3.0, or the freezing density, whichever was lower. A listing of the program used to generate the data is given in Appendix 6. Figure 5-3. Flowchart for calculation of X at a fixed a. For the initial analysis values of a of 0, -1, -2, -3, -4, and -5 were used. Table 5-2 shows the average absolute deviation in the direct correlation function integrals found for the optimum value of x at each isotherm. A noticeable minimum exists in the range of a = -4. Note that the Carnoban-Starling equation (a = -1) does not yield the best predictions, even though it is the "best" hard sphere equation of state. Several isotherms were chosen to examine the sensitivity of the results to the value of a, on a finer scale. These results are given in Table 5-3. For the liquid phase iso- therms a value of a = -4.3 seems to be optimal. The super- critical data are best fit with a of -4.0 or -4.1. It was decided to use a constant value of a = -4.2 to fit all of the argon data. With a set at -4.2 the value of X and f could then be determined. The optimum values of X found were correlated as a polynomial in inverse temperature. For this analysis the characteristic parameters were chosen to be equal to the critical parameters. The form that fit X best was X = 0.12342 + 0.15069 _0.18545 + 0.15947 T 2 3 T T 0.070261 0.012111 (5 + (5-13) 4 -5 T T TABLE 5-2 REGRESSION ANALYSIS OF ARGON DATA T Average Absolute Deviation in C for = a 0.5966 0.6297 0.6629 0.6960 0.7292 0.7623 0.7954 0.8286 0.8949 0.9280 0.8617 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.2 2.4 2.6 2.8 3.0 0.0101 0.0392 0.0785 0.1225 0.1651 0.2062 0.2445 0.2758 0.3104 0.3053 0.2994 0.2840 0.2413 0.2175 0.1990 0.1837 0.1707 0.1611 0.1530 0.1498 0.1403 0.1310 0.1242 0.1192 0.1152 0.1126 0.0102 0.0395 0.0789 0.1228 0.1651 0.2059 0.2436 0.2736 0.3056 0.2990 0.2963 0.2764 0.2344 0.2088 0.1899 0.1740 0.1611 0.1504 0.1411 0.1344 0.1286 0.1196 0.1131 0.1084 0.1047 0.1023 0.0094 0.0361 0.0714 0.1100 0.1461 0.1798 0.2094 0.2304 0.2408 0.2374 0.2430 0.2142 0.1835 0.1580 0.1383 0.1227 0.1101 0.1003 0.0920 0.0854 0.0798 0.0713 0.0651 0.0610 0.0582 0.0566 0.0058 0.0219 0.0427 0.0642 0.0830 0.0983 0.1085 0.1104 0.1122 0.1280 0.1071 0.1260 0.1058 0.0912 0.0801 0.0713 0.0643 0.0536 0.0495 0.0425 0.0371 0.0324 0.0290 0.0261 0.7266 1.3274 1.7500 2.0472 2.2372 2.3746 2.3629 2.1992 2.0497 2.2885 1.4495 1.1049 0.8986 0.6364 0.5516 0.4845 0.3864 0.3505 0.2948 0.2536 0.2214 0.1960 0.1749 48 TABLE 5-3 EFFECT OF HS EQUATION ON FITTING ARGON DATA Average Absolute Deviation in C for = a -3.8 -3.9 -4.0 -4.1 -4.2 -4.2 -4.4 .7954 .1435 .1277 .1085 .0848 .0547 .0214 .0401 .8286 .1516 .1329 .1104 .0831 .0535 .0263 .0595 .8617 .1507 .1297 .1071 .0829 .0592 .0399 .0866 .8949 .1477 .1294 .1122 .0916 .0732 .0667 .1264 .9280 .1581 .1434 .1280 .1144 .1049 .1152 .1851 1.1 .1484 .1368 .1309 .1235 .1258 1.2 .1344 .1303 .1260 .1263 .1364 1.3 .1126 .1092 .1058 .1064 .1065 0590 .0585 .0618 .0698 1.8 .0607 The maximum error in calculated x values from equation 5-13 was 0.105% with an average error 0.022%. With x values from equation 5-13 and a set at -4.2 the optimal values of f were then found. These values were also fit to a polynomial as f = -14.70 23.237 +25.221 11.636 f = -1.4098 + + 2 ~3 ~4 T T T T 2.0463 + 2 (5-14) T-5 This polynomial form reproduced the f values with an average error of 0.15% for the isotherms analyzed. The temperature dependence of X and f is shown in Figure 5-4. Table 5-4 gives a summary of the values of X and f for argon along with the average error in the prediction of the direct correlation function integrals. In general, the correlation function integrals were correlated to within the experimental accuracy. The only range of temperatures for which the fit was not excellent was near the critical point, which is to be expected. Analysis of Methane Data There is less high quality P-V-T data available for dense methane than for argon. In this work the equation 9.0 l/V* 5.0 Figure 5-4. .0.20 /I 3 - /V* 6 0.18 0.16 1.0 2.0 T/T* Dimensionless hard sphere diameter and perturbation function based on argon reference. TABLE 5-4 CORRELATION OF ARGON DCFI USING a = -4.2 Average Error T X f V inC c 0.5966 0.6297 0.6629 0.6960 0.7292 0.7623 0.7954 0.8286 0.8617 0.8949 0.9280 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1. 9 2.0 2.2 2.4 2.6 2.8 3.0 0.21159 0.20916 0.20692 0.20482 0.20283 0.20094 0.19915 0.19745 0.19583 0.19430 0.19286 0.18649 0.18348 0.18086 0.17852 0.17643 0.17453 0.17280 0.17120 0.16971 0.16833 0.16582 0.16359 0.16159 0.15978 0.15814 890.79 820.09 759.79 707.47 661.13 620.05 583.39 550.40 520.61 493.31 467.98 373.74 335.43 306.28 282.27 262.19 245.04 230.17 217.10 205.48 195.05 177.04 161.94 149.05 137.88 128.10 .0035 .0128 .0525 .0372 .0466 .0532 .0562 .0547 .0596 .0732 .1076 .1264 .1377 .1166 .1037 .0936 .0844 .0764 .0700 .0648 .0605 .0527 .0458 .0393 .0343 0308 52 of state of Mollerup (1980) was used to generate nine sub- critical isotherms and one supercritical isotherm, as was done for argon. These supercritical isotherms measured by Robertson and Babb (1969) were also analyzed. The pro- cedure used for the analysis of the methane data was the same as for argon, except some positive values of a were also investigated. For the liquid phase isotherms a value of a = -4.2 was clearly the best. For the supercritical data values of a from -3.0 to -4.3 could produce almost equivalent results, but a = -4.0 was the optimum. It should be noted that for the supercritical isotherm values of a = 9 could also offer a reasonable representation of the data, but not as good as -4.0. Thus, a value of -4.2 was chosen to represent the methane data. Proceeding as was done before for the argon analysis, the values of X and f were fit to polynomials in inverse temperature. The resultant forms were 0.026446 0.080054 0.089994 X = 0.13832 + + -2 -3 T T T S0.039085 0.0060984 -4 ~5 (5-15a) T T 23.21 50.827 40.662 15.818 f = 5.0851 -+ -2 3 + T T T T 2.3173 -5 (5-15b) T where the critical properties were used as the reducing parameters. These functions are plotted in Figure 5-5. Table 5-5 gives the summary of results for the methane correlation. The proposed model was able to reproduce the data to within the experimental uncertainty. Use of the Correlations to Calculate Pressure Changes The development of the DCFI models for the pure com- ponents was first tested by performing equation of state calculations for the base substances, argon and methane. The method of calculation was to choose the lowest density point on each isotherm as a reference and then to calculate pressure changes for all other points on the isotherm from that reference. There are thus two criteria for goodness of fit, the average percent error in pressure changes, defined by ND LPcalc Aexp aP = [ ] / ND (5-16) i=1 APexp and the sum squared percent error, defined by ND APcalc Aexp 2 SSEAP = [ ] (5-17) i=l APexp where in both cases ND is the total number of data points for a compound. When parameters were found to best "fit" a set of data, the quantity SSEAP was minimized using a TABLE 5-5 CORRELATION OF METHANE DCFI USING a = -4.2 Average Error T X f V in C c .4986 .22182 1545.26 .0015 .5511 .21596 1309.43 .0155 .6036 .21095 1131.07 .0188 .6561 .20671 996.35 .0157 .7085 .20302 890.93 .0140 .7610 .19974 805.32 .0335 .8135 .19675 733.53 .0612 .8660 .19400 671.78 .0946 .9185 .19145 617.18 .1413 1.1 .18381 473.60 .1484 1.6173 .16916 260.80 .2262 1.9585 .16316 204.86 .1437 2.4833 .15703 168.74 .0637 13.0 9.0 )/V* 5.0 3.0 1.0 2.0 T/T* Figure 5.5. Dimensionless hard sphere diameter and perturbation functions based on methane reference. 1.20 113 -3/V* . 18 . 16 56 nonlinear regression routine and the program GROUPFIT listed in Appendix 6. However, the data comparisons will normally be made in terms of the quantity AP. This procedure is common and is used to eliminate bias from the error mini- mization. The first test performed was to calculate AP for argon using the characteristic parameters as the critical parameters and the functions given by equations 5-13 and 5-14. The average error was found to be 1.42%. This is larger than expected, but if only the liquid phase data reconsidered, the error is just 0.629%, and if the isotherm that is within 10% of the critical temperature is neglected, the error for the liquid isotherms is only 0.34%. This is within the accuracy of the equation of state for this region of the phase diagram. On the whole the largest errors are found in the low density data near the critical temperature. The equation is able to reproduce the highest temperature isotherms (T/Tc = 2.8 and 3.0) with an average error of 0.47%. So the fit of the argon data seems satis- factory for the desired usage. In an attempt to improve the argon representation, a regression was performed to find the set of characteristic parameters that minimized the sum square error in the pres- sure changes as per equation 5-17. The parameters found were T* = 144.45 K and V* = 74.31 cm3/gmole as compared to T = 150.86 K and V = 74.57 cm /gmole. The average c c 57 error for all of the data increased to 1.53%. This worsening of the percentage error occurred because the fit was made more even; that is, SSEAP was decreased from 4.549 x 103 to 3.034 x 10 but on the average the error was larger. This is due to the larger localization of error in the critical region. A calculation was also performed on the methane data using the argon functions and methane's critical parameters as the characteristic parameters; the error was 3.13%. And when a regression was performed on the methane data, the characteristic parameters were found to be T* = 190.54 K and V* = 98.40 cm3/gmole (compare T = 190.53 K and V = 98.52 cm /gmole) with an error of 2.98%. This shows that the trends found for the methane DCFI as compared to argon carries over to the pressure change calculations. Similar calculations to those described above were performed using the hard sphere diameter and perturbation term functions as given by equations 5-15. The average error in pressure changes for all of the methane data was 0.57% with the largest error from the isotherm at 175 K (within 9% of the critical temperature). Even at this temperature the high density data are well described. In the reduced temperature range of 0.5 to 0.89 the average error was only 0.18%, well within the accuracy of the data. As for argon, when the GROUPFIT program was employed to find optimal characteristic parameters for methane, the 2 2 SSEAP was reduced from 4.13 x 10 to 2.44 x 10 but the average error was increased to 0.727%. Again .the localiza- tion of the error causes this phenomenon. A calculation of the pressure changes for the argon data using the methane functions gave an error of 4.65%, and when optimized, the argon characteristic temperature only changed 1.3% and the error reduction was only to 4.56%. This again shows that there are noticeable differences in the compressive behavior of argon and methane at high densities. In summary, the temperature dependence of the hard sphere diameter and perturbation term in the DCFI model are seen to adequately represent the data from which they were developed. The difference in DCFI noticed for the atomic species argon and the polyatomic methane are notice- able in calculations of compressions. The rest of this work will deal with calculations of hydrocarbon compressions using the group contribution formulation. The groups of interest are methyl (-CH3) and methylene (-CH2). These are physically more similar to methane than to argon. For this reason all of the group contribution calculations will be performed using the methane reference functions. CHAPTER 6 USE OF THE GROUP CONTRIBUTION MODEL FOR PURE FLUIDS The development thus far has been limited to the formal expressions for calculating property changes of mixtures based on the group contribution model of direct correlation function integrals and the parameterization of the model using data for argon and methane. In both cases the mole- cules were assumed to consist of only one type of group. This chapter presents the calculations of pressure changes for pure fluids composed of different types of groups. The first section discusses the extension of the cor- relations required for systems of several groups. The remainder of the chapter presents and discusses the results of the calculations performed for several n-alkanes and one n-alkanol. Extension of the Model to Multigroup Systems The expressions developed in Chapter 4 for calculation of pressure changes involve two contributions. APB APBS + APPERTURBATION (6-1) 60 The hard sphere contribution is calculated for the solution of groups using the results shown in Appendix 5. To cal- culate the perturbation contributions for multigroup systems, the form of the 'P functions must be established for the terms with a 3 B. In Chapter 5 a corresponding states expression was developed a = V* f (T/T ) (6-2) where f is a universal function of the reduced temperature. This result is extended to the unlike terms as S= V* f (T/T* ) (6-3) and the characteristic parameters are found from 1 1/3 1/3) V = (V* )1/3 + (V )3) (6-4a) aB 8 at B6 T* = (T* T )1/2 (-k) (6-4b) where k is an empirically determined binary interaction parameter for groups a and 3. It is important to note that this binary parameter can still be determined from analysis of pure component volumetric data. Designation of Groups Before the group contribution model can be used for calculation of property changes, the decision as to which collections of atoms in a molecule comprise a group must be made. The simplest choice would be to consider each atom as a group, and thus the properties of all molecules could be described with minimal information. However, that is an overly ambitious approach. Even for the molecules considered here that contain only carbon, hydrogen, and oxygen atoms, this is unworkable. Typically, the common organic radicals are designated as the groups. For this work it must be decided if the n-alkanes are composed of only one type of group, or two. If they are considered as being comprised of only one type of group, then the model could never predict excess volumes for n-alkane mixtures. This is reasonable at low pressures, but for highly compressed systems, noticeable excess volumes do exist (Snyder, Benson, Huang, and Winnick, 1974). Thus, the n-alkanes will be considered to be composed of two groups, methyl (CH3) and methylene (CH2). To analyze methanol the hydroxyl moiety (OH) will also be considered as a separate group. Pressure Change Calculations All of the molecules analyzed in this work will be considered to be composed of only two types of groups. The equations required for the calculation of the pressure changes are P6 (P)ref = PHS (pHS)ref PB -(PB) PB (PB 1 p2_ 2 ref) 2 (6-5) 2 111+2 v212222) (6-5) where here, v. is the number of groups of type i in the molecule. The individual terms are found using PBHS 1 1 3- 11 + 3623 1 (1-53) (1-63) 4.2 2 3 } (6-6a) (1-3) with 1 = P[vlo + 2c2 ] (6-6b) and the oi terms are found using 3 o /V* = f(T/Tf) (6-6c) 6 i l 1 where the function of reduced temperature required in equation 6-6c is given in Chapter 5. The perturbation contributions are found using equations 6-3 and 6-4 with the required function of reduced temperature again given in Chapter 5. These functional forms are dependent only on the reference component chosen for their development, either argon or methane. For the two group molecules there are then only five independent parameters, V*, V*, T*, T*, and kl2. Analysis of n-Alkane Compressions In this section we consider the ability of the proposed formulation to calculate changes in pressure during compres- sion of four n-alkane molecules: ethane, propane, n-pentadecane, and n-octadecane. Calculation of pressure changes using the equations given above were performed for both the argon- and methane-based reference functions as derived in Chapter 5. The initial plan of attack was to determine the methyl parameters using ethane data and ethylene and cross parame- ters from the propane data. The model capabilities would then be examined by using these same parameters for calcula- tions on the long chain molecules. The analysis was per- formed both with and without a binary interaction coeffi- cient. The characteristic parameters for the methyl and methylene groups as determined by regression analysis are shown in Table 6-1 along with the errors in calculated pressure changes. Several conclusions can be drawn from these results that will be of future interest. First, the best fit of all of the data was obtained using the methane-based correlations with the inclusion of a binary or 0 -4 4. .a N^ N CN N 421 H0 m -4: Z Z 4 z LO < C) a z mz 04 < o 204 u a U E e n uen men EH M 0u I-4 04 a0 2- UH 3 C 4O CC 0 ., 13 < 0 0 a C C np t 4 0 0 C c o u S44 Ln rn -H ( 41 0 0 0 PO 905 Oo U Ou -H 4-) 4-3 4C1 0 4 a 4- uCO .0 p e 10 Co 0)444 Co Co U o re0 440 er N* 32 U 9r N m * U mC N o0 o0 4 .0 I-l M P *= U u ri 0 rI Q -) U) 0 0 o 0 0 D m a 04 Co c S en m O -4- H .0 0 0 71 0 0 H H an 0 10 l C 42 42, Ln N C o oo a- interaction coefficient. This binary constant was much larger than usual and thus gave a large value of the T* 12 parameter. This results from the fact that the model is much more sensitive to the values of the characteristic volumes than the characteristic temperatures so that large adjustments to the binary constant are required to signifi- cantly affect the calculations. It was also seen that 2V* = 0.9 Ve and CH C,ethane 2V* + V*C = 0.9 V This suggests an approach CH 3 CH2 c,propane for estimating characteristic volumes of the groups from critical properties. The characteristic temperatures might also be obtainable using critical values with some sort of mixing rule applied. While the results are not as good as the molecular fit, they could be satisfactory. A more important question is their ability to predict compression data for other hydrocarbons. For n-pentadecane and n-octadecane results were rather poor, the best agreement being an average error in the pressure changes of 16.6% using parameter set 1. This is discouraging but not unexpected. The hard sphere contribution does not behave properly at high densities as explained in Appendix 5. In addition, the improper ideal gas limit of the equation of state affects the results. In an effort to improve the model performance, the data for the two long chain hydrocarbons were used to deter- mine the methylene and interaction parameters. The results of interest are shown in Table 6-2. The methyl parameters were those found from analysis of the ethane data. The results are rather encouraging in that either the four- or five-parameter models could give an adequate representation of data for both short- and long- chain hydrocarbons. However, it should be noted that none of the parameter sets could give a reasonable representation of the propane pressure changes. These results do yield two interesting conclusions. First, that when the larger hydrocarbons are considered for the parameter estimation that the binary parameter becomes very important. Noticeable improvement in the pressure change calculations occurs when the binary parameter is included. This suggests that the methyl and methylene groups should be considered as distinct entities. However, the magnitude of the binary parameter is much greater than would be expected. It can also be seen that the methane- based correlations are more apropos for modeling the alkane compounds because the groups are physically more akin to methane than to argon. After studying the aforementioned results, it was decided to establish a final set of model parameters for the methyl and methylene groups from an analysis of all of the n-alkane data simultaneously. The methane-based reference functions were used, and the calculations were performed both with and without a binary parameter. These (N 0 rn El mn n enNm N mL o ef N 0 n H * 0 N N V ul E MW z a4 U Hg O i H Z 0 o WE-E < Z 0 IEi Z z WC I z H a OZ HOC M0: HZ HU U 01 N N 00 C C n* * Lfl m (n e en en rN Lf Lf3 (0 '1 L (D c 00 . \0 0 0 o a 0 u0 000o 040 0o00 u0 -H o .m E (0 m 0 1 -H> - 11 O C o > 41 1 4J 40 00) N N - 0U 0 0 -H 0 0 0 0 O U o 0 0 r 0 Q) a) S4 0 o 1 4J (a E 1 r results are shown in Table 6-3. It can be seen that use of a binary parameter does improve the results and that the order of magnitude of this parameter is reasonable. Using only five adjustable parameters, the average absolute error in pressure changes calculated for all four hydro- carbons was 5.9%. Analysis of Methanol Compressions A set of compression measurements for methanol were analyzed to determine if the present model could adequately calculate pressure changes for molecules other than n-alkanes. Methanol was considered to be composed of one methyl group and one hydroxyl (OH) group. The methyl group characteristic parameters used are those listed under parameter set 11 in Table 6-3. The regression results are shown in Table 6-4. The characteristic volume for the hydroxyl group appears to be quite reasonable because V* + VH 0.9 Vmethan which is the same pattern CH OH c,methanol seen for the alkanes. The characteristic temperature and binary parameter found are surprising, but these values were necessary to obtain a fit of the data about as good as that for the alkanes. Summary While the calculations presented here are based on recognized assumptions that are believed to be sound, there O 0 o m; LO 0U I z1 aZ ME W Z Z 0H 0 I 0 MO Urn 0 < 42 0 H2 0 0U in cn 0r in N * 0 < O * o ^ 2 O-, 0) 0 (U 0 -J Mi 0) 0 012-- 40 100 OJ- n 0 0 0 0 M -H .-I a 40 O 04 - 4 C Co C) 004a Cow, CO 12< N CN Co. CDo m * m H i- i i- 6iT 0 o o 0 OE z )- 0 H 0u >4 =; 4 1 0 P E z 04 EM a O o CL4 H < 0 U M a- O 0 OH 0 00 2-1 2-l 0 C)0 U) -4 tL - aC 0 I 0 4-) -4 OI 0 4 > c 0 4-1 H 4- ) 4-- X oE O 000 E X H 4E 0 00 > U 4 4 K 0 O 4 -Hc) - C)a N 0 S C o o o H N P~ N N- C N- i0 N Cl --4 .- are certain disconcerting aspects. In particular, the equation of state developed does not properly approach the ideal gas limit. Even though the calculations are always performed for pressure differences from a liquid, or dense fluid, reference state this may be important. Also, the percentage errors in the pressure change calculations appear to be rather large. In most process design calculations the temperature and pressure are known and one wishes to calculate the volume. It would then be of interest to know how the present model would perform in calculating volume changes for given pressure changes. A simple error analysis shows how this can be estimated. At constant temperature errors in pressure calculations and volume calculations are related by 3V AV = (- ) AP (6-7) T Then average absolute errors in volumes and pressures must be related by / / = PT /P/ (6-8) where 8T is the isothermal compressibility. For liquids the product PBT is almost always less than 0.2 and even then is only that large at very high pressures. This sug- gests that if the present correlation were used to calculate 72 a volume change for a given pressure change that the average absolute errors would be less than 20% of the errors reported here for the pressure changes. This shows that the present model compares reasonably with existing correlations. One reassuring result of regression analysis is the magnitude of the group characteristic volumes that were calculated. Ratios of the group characteristic volumes found here are very nearly equal to the ratios of the van der Waals volumes for these groups. This may allow for estimation of these group parameters. This type of regu- larity was not seen for the group characteristic tempera- tures. Also, whereas the characteristic volumes for the different groups did not vary much when determined from different data sets, the characteristic temperatures did. As a result, the binary constants did not follow a pattern. The binary constant was necessary in all cases to obtain an optimal representation of the data, but the improvement obtained by introducing this additional parameter was minimal. No calculations have been reported for chemical poten- tial changes or for pressure changes for multicomponent systems. The necessary formulae have been presented in Chapter 4 and Appendix 5. Pressure change calculations for multicomponent systems are no more difficult than those reported here because the molecules considered contained more than one group. CHAPTER 7 DISCUSSION This work has centered on the development of a group contribution liquid phase equation of state. Fluctuation solution theory was used to develop the exact relationships between thermodynamic derivatives and direct correlation function integrals. Two approximations were made to actually construct the equation of state. The first approximation made was the use of the interaction site formalism, or RISM theory. The -last stage in the model formulation was the choice of an approximate form for the group direct correlation function integrals. After that development, the general expressions for calculation of pressure and chemical potential changes were written and used for several compounds. In this chapter, the important aspects of each stage in the development described above will be examined in detail. Great emphasis will be placed on the approximations embodied in the model development as they pertain to the numerical results. Fluctuation Solution Theory The relationship between integrals of molecular correla- tion functions and thermodynamic derivatives have come to be known as fluctuation solution theory (O'Connell, 1981). Kirkwood and Buff (1951) originally reported these relationships though they have not been extensively used. The original results, using total correlation function integrals, are valid for both spherical and molecular sys- tems. With the introduction of the direct correlation functions, through the Ornstein-Zernike equation, the analy- sis is not as simple. Appendix 1 details a procedure which surmounts all difficulties encountered. This result is not new, but the rigorous proof appears to be. This result could also be derived through the use of the operator tech- nique of Adelman and Deutch (1975). Approximations While equations 2-24 and 2-25 are exact results for fluctuation derivatives, one must have values for the direct correlation function integrals for these to be useful. Because exact values of the required integrals are not generally available, some approximations must be used to obtain these values. The approximations used will determine, to a large degree, the success of any thermodynamic calcula- tions. In this work, two levels of approximation are used and they will be discussed separately below. Group Contributions One of the major goals of this research was the develop- ment of a group contribution formulation of fluctuation solution theory. This appears to be a realistic and rational goal. This is true because fluctuation solution theory requires knowledge of correlation functions and correlations among the groups are well defined. The major problem encountered was the ability to separate the inter- and intramolecular correlations. The RISM theory (Chandler and Andersen, 1972) purports to affect this separation and offer an approximate relationship between group direct correlation functions and molecular correlations. The calculations reported here are based on this original version of the RISM theory. Recently, the direct correlation functions calculated using the RISM theory have become the objects of considerable attention (Cummings and Stell, 1981, 1982b; Sullivan and Gray, 1981). It has been shown that in many situations for neutral molecules that the site-site direct correlation functions are necessarily long-ranged. This means that these functions are nonintegrable and thus no compressibility relation involving these functions may exist. Cummings and Stell (1982a) have shown how a general compressibility relation can be derived in terms of a limiting operation, but this is not a practical solution. However, for the cases that have been considered in the literature it seem this problem can be overcome. If the compressibility rela- tion is viewed as the k o limit of a result in Fourier space, where all of the correlation function transforms exist, then all of the mathematical manipulations required may be performed. Then as the long wavelength limit is taken, we find (Appendix 7) the following relationship: ~PX = l-p XiXji iviB C, (o) (7-1) DP T,X ij ab While the individual functions C a(r) may be long-ranged, it appears that the collection of terms wcy is well defined in the k o limit. It is known (Cummings and Stell, 1982a) that for diatomic molecules this is true, and thus a simple compressibility theorem applies. Also for triatomics, as examined by Cummings and Stell (1981), we find that this summation of terms is nondivergent. As has now become known, the possible divergencies in the C a(r) functions are due to intramolecular effects that are not shown explicitly. It appears that the projection by the v matrix removes the divergent terms. This is analogous to the ionic solution case (Perry and O'Connell, 1984) where the charge neutrality constraint removes the divergent part of the direct correlations. Chandler et al. (1982) formulated the proper integral equation theory for site-site correlations to offer an exact formulation involving nondivergent CB(r) functions that here are labeled C (r). These new functions are related to the RISM C B(r) functions and are made nondiver- gent by removal of intramolecular correlations involving only sites a or 6. This requires use of auxiliary functions SaB(r) that have a complicated density dependence. These functions do possess the interesting property eaB(o) = 0 Va,B (7-2) It is then possible to derive a compressibility relation involving only C (r) function integrals and 0 (1P 1 X.X. v. j ST,X ij a s J dr C C (r) (7-3) In this case, the integrals always exist. However, we have no knowledge of the functional dependence of 0 on temperature or density. One further aspect of the group contribution approach used here needs to be mentioned. This formulation works on the Ansatz that cij(l,2) = I jia Ij c Cs B) (7-41) jS a which is only to be considered an approximate relation. However, the compressibility relation obtained from the above starting point and that found using the proper integral equation formalism (as shown in Appendix 7) are the same. The only difference is that all forms of the RISM theory assume that each group, as opposed to each type of group, have separate correlation, i.e., that the set of functions cij r) should be considered, not just caB(r). This is definitely true at the level of the correlation functions themselves. However, it is always possible to define a -)* aS -* set of c C(r) functions that are averages of the c (r) functions (Adelman and Deutch, 1975) and retain the same compressibility relations. Modeling of Direct Correlation Function Integrals (DCFI) As the DCFI are functions of temperature and density, it would be possible to model them using polynomial expan- sions for the pure fluids and a solution theory for the mixture quantities. In this work, we attempted to use fluctuational forms that have a theoretical basis. This approach seems to have both merits and demerits. For mole- cules, Mathias (1979) had success in modeling DCFI using an approach suggested by perturbation theory. In the present case this is not of the greatest value because analytic forms are available for group DCFI in only some limited cases (Morris and Perram,1980; Cummings and Stell, 1982a), 79 and even then, the solutions are not in closed form. Thus, some reasonable assumptions must be made to proceed with the modeling. It is known that the RISM correlation functions can be related to Percus-Yevick correlation functions for molecular species in some limiting cases (Chandler, 1976). This led to the use of the form for the DCFI presented in Chapter 5. The important aspect of this approach is that the groups are dealt with as if they were all indepen- dent. This is not physically the situation. The dependence is caused by the intramolecular correlations, and they must be properly accounted for to yield an accurate model. Originally it was believed that the RISM theory took account of this behavior, but as was mentioned in the preceding section, this is not true. Even the proper integral equation direct correlation functions, equation 7-3, while short ranged, still involve intramolecular correlations due to third body effects. Cummings and Sullivan (1982a) explicitly show that while the h (r) functions are purely intermolecu- lar quantities that the c (r) are not. The exact resummation of the cluster expansion for h o(r) shown by Chandler et al. (1982) suggests a way around the aforementioned difficulties. It should be possible to define a set of direct correlation functions, labeled here as ca (r), that include only intermolecular effects. Chandler's (1976) analysis can be examined to see that these diagrams can be isolated. And then, in the same fashion as was used to eliminate the long-ranged behavior of the original RISM direct correlation functions, it would be possible to write an exact proper integral equation of the form S+ Ph = [I Pc ] (7-5) where the elements of the W matrix would contain all infor- mation about intramolecular effects. It is then easy to show the relationship between the different classes of direct correlation functions, for example y 1 -1 -1 = c + -( ) (7-6) The elements of the W matrix would be functions of temperature and density, but the exact functionality could not be determined in general. It is possible to identify some of the properties of the new functions, such as olim W = (7-7) and klm = W V (7-8) 0 ( a3 a,8 Using equations 7-5 and 7-8, in conjunction with the techniques of Appendix 7, a compressibility relation can be derived in terms of these new group direct correlation functions (3P 1 p . ST,X ij a i J dr C (r) (7-9) All of the above analyses show that there is no simple solution to the modeling problem. One may either work with a simple compressibility relation written in terms of DCFI whose behavior is not certain, or in terms of well behaved DCFI but have another unknown quantity present. The former approach was taken for this work and is discussed below. DCFI Model The rationale behind the development of the DCFI model has been presented in Chapter 5. It is of interest to note that the proposed form is a group variation of a van der Waals, or mean field, model. Appendix 2 details how higher order correction terms could be appended to the proposed form. Because the DCFI model was formulated as a correspond- ing states correlation, the constants in the temperature dependence of the hard sphere diameters and perturbation terms had to be determined using experimental data. Volu- metric data for both argon and methane were used to accom- plish this task. The results of the compression calculations show that the methane-based correlations were definitely superior for representation of the properties of the larger molecules. Comparison Calculation Chapter 6 presents the results of the group contribution compression calculations for the n-alkane and methanol. The results seem to indicate that the theoretical basis is feasible but that the accuracy is not that desired. It is an accomplishment to be able to perform the compression calculations for all of the n-alkanes using only five parameters, but the overall accuracy is not high. One would hope to be able to calculate pressure changes with an accuracy of about 1% (this allows for density change calculations accurate to -0.2%), and this cannot be assured. The theoretical basis behind the equation of state development is now on firm footing. Our compressibility relation is exact; one must simply have proper models for the functions involved. Here, this appears to not be the case. Because we use the RISM approximation, the c a(r) functions must contain intramolecular effects that have no analogy at the molecular level. Thus, our model that 83 is based on an analogy to a previously successful molecular approach appears inadequate. Note, however, that we seem to have little recourse. Because if we were to introduce easily modelable cB8(r)'s, then the aspect of determining the W function must be addressed, and this is still an unsolved problem. Summary This study has revealed many interesting aspects about site-site correlation functions and their relationship to thermodynamic derivatives. A group contribution liquid reference state equation of state has been formulated and tested. The results are encouraging enough to warrant further investigation but not accurate enough for practical density calculations. It appears that the major problem associated with the present approach is the inability to model the required functions, and suggestions have been made as to how one can alleviate that problem. CHAPTER 8 CONCLUSIONS The goal of this work was the development of a group contribution technique for calculation of liquid phase properties of mixtures. The general results are available but have only been tested on a few pure components. The calculated pressure changes for four n-alkanes and methanol are in general in error by less than 7%. This is reasonable but not of high enough accuracy for process engineering calculations. While the results of this study are not outstanding, there are several interesting conclusions that can be drawn. It has been shown that it is possible to derive thermo- dynamic property derivatives from a RISM theory. A general- ized compressibility theorem was proven, and this was used along with a model, based on a rigorous perturbation theory, to develop a van der Waals equation of state. The liquid phase equation of state was applied to the representation of compression measurements of argon and methane. It was shown that proper choice of the repulsive contribution to the equation of state is important. Also, that an optimum repulsive contribution could be determined from a new hard sphere equation of state. The use of the group contribution model was limited. Pressure changes could be calculated for both long- and short-chain alkanes using a five-parameter model. The results were of reasonable accuracy for these cases and when the model was applied to methanol. While the volumetric parameters in the model appeared to correlate well with previous molecular results (Mathias, 1979), the temperature parameters followed no pattern. An analysis of the present work also shows several areas that require future work. First, the present model should be applied for calculation of chemical potential changes for n-alkane mixtures. This, along with mixture volumetric calculations, would be a strong test of the model's ability. Secondly, some effort to identify the intramolecular correlation functions of the form shown in Chapter 7 may be required to improve the present model. This appears to be the weakest aspect of the present work. APPENDIX 1 FLUCTUATION DERIVATIVES IN TERMS OF DIRECT CORRELATION FUNCTION INTEGRALS To eliminate the difficulties associated with the form of the general O-Z equation we define an angle averaged total correlation function, and then we implicitly define a new set of direct correlation functions, the + j k dR3 Now, because the full h.. (1,2) is translationally invariant, the average function must also be, and thus the 13 function must also possess this property. Then if we integrate equation Al-2 over one coordinate we have l] k ik And if this is then integrated once more we find 86 k where we define 1j V i] (Al-4) (Al-5a) (Al-5b) and when these terms are collected in matrix form we find (Al-6) Here, the terminology is the same as that used in Chapter 2 for the definition of the matrix elements. This relation becomes most useful when it is realized that (Al-7) as can be shown by direct substitution. And now rearrange- ment leads to [N + NHN]- = N-1 (Al-8) which shows that the fluctuation derivatives can be written in terms of the integrals of the 1] relation between cij and derived. By definition and, if the O-Z equation is angle averaged, we find + -k f d3 ddd2 c i(1,3)h k(3,2) (Al-9) kv-7 1 2 ik kj k VQ Now, when these two relations are combined, the link between the angle averaged c.. and the as + [ 3 dihld cik(l,3)hkj(3,2) k v3 2 J dR3 And now insert the definition of + 3 I d3d~ld 2 cik(1,3)hkj(3,2) k v1 2 N2 dd2 k V'2 2 ik 3 kj and the terms can be grouped as <= N d3d2hkj (3,2) {1 dn c (1,3)- (Al-12) Now, use the fact that to rewrite equation Al-12 as 11 1 d { = --- + d3d(2 hkj(3,2) {- I dl cik(1,3) k VQ It is now advantageous to define |