J. Phys. Chem. 1992, 96, 7510-7517
7510
ARTICLES Molecular Theory of Chromatography for Blocklike Solutes in Isotropic Statlonary Phases and Its Application to Supercrtticai Fluid Chromatographic Retention of PAHs Chao Yant and Daniel E.Martire* Department of Chemistry, Georgetown University, Washington, D.C. 20057-2222 (Received: January 13, 1992)
A molecular theory of chromatography for blocklike solutes in anisotropic stationary phases has been recently developed and applied to gas, liquid, and supercritical fluid chromatography (GC, LC, and SFC, respectively). That work is extended here to treat blocklike solutes in isotropic stationary phases. Similar to our previous treatment of blocklike solutes in anisotropic phases, statistical thermodynamics and a lattice model are utilized to describe the equilibrium partitioning of the blocklike solutes between the mobile and stationary phases. A retention equation is derived and discussed. The solute distribution coefficient, K,is represented by In K = VJ' + A,&, where V, and A,f are, respectively, the van der Waals volume and effective contact area of the solute molecule, and P and E are related to state variables and molecular parameters. An SFC experiment mimicking the model system has been carried out and the results are used to test the theory. In agreement with the predictions, it is found that (a) the isotropic stationary phase has no shape selectivity for isomeric polycyclic aromatic hydrocarbon (PAH) solutes, (b) the logarithm of the capacity factor, In k'(or In K ) , is a linear function of the reciprocal temperature (1/Z') at constant mobile phase density, and (c) for any given solute, the plot of In k'vs mobile phase density for the isotropic stationary phase is essentially parallel to the plot for the anisotropic stationary phase at the same temperature.
1. Introduction The retention behavior of blocklike solutes, such as polycyclic aromatic hydrocarbons (PAHs), in anisotropic and isotropic stationary phases has been the subject of several studies.14 Although some thermodynamic investigations attempting to define the solute retention mechanism(s) have been done,+* a complete, informative, molecular-level description is still a challenging task. In our previous studies a molecular theory of chromatographyfor blocklike solutes in anisotropic, rodlike stationary phases was developedgand tested using data from a PAH/CO,/liquid crystal experimental SFC system.1° That work is extended here to treat a system composed of blocklike solutes with an isotropic, chainlike stationary phase. In common with our previous treatment of blocklike solutes in anisotropic stationary phases, statistical thermodynamics and a mean-field lattice model are utilized to describe the equilibrium partitioning of the solute between an isotropic mobile phase and an isotropic stationary phase. The entropic or steric term of the configurational partition function is rigorously derived by packing the molecules into the threedimensional lattice one by one, and the energy term is obtained by considering attractive interactions and shielding effects. From the chemical potential of the solute in both the mobile phase and the stationary phase under the equilibrium condition, a retention equation, applicable to GC, LC, and SFC, is derived. The solute distribution coefficient, K,is represented by In K = V> &E, where V, and Ad are, respectively,the van der Waals volume and the effective contact area of a solute molecule and P and E are related to state variables (temperature and density) and molecular parameters (interaction energies and molecular dimensions). The cross-linked octylmethylpolysiloxane stationary phase studied here and other chainlike stationary phases are widely used for separation and identification in GC and SFC.2,4These kinds of stationary phases cannot be modeled as anisotropic, rodlike phases because the chainlike molecules are quite flexible and are generally believed to be isotropically distributed in the liquid state. Therefore, in attempting to mimic the model system, a capillary
+
* Corresponding author.
'Present address: Analytical Research and Development, Building 360/ 1034,Sandoz Pharma, Ltd., CH-4002Basel, Switzerland.
0022-365419212096-7510$03.00/0
SFC experiment was designed by using PAH solutes with supercritical fluid (SF) C 0 2 as the mobile phase and octylmethylpolysiloxaneas the stationary phase. The retention behavior of PAH solutes in the chromatographicsystem over a wide range of densities and temperatures is determined and the data are used in testing the theory. In addition to the prediction, interpretation, and analysis of the solute retention, a molecular-level understanding of the retention behavior assists the informed selection and manipulation of operating conditions and the selection of the right combination of mobile and stationary phases for a particular separation problem. 2. Theory The model employed in this study is a three-dimensional, simple-cubic, mean-field lattice model, based on an extension of a statistical "counting* procedure by DiMarzio." In common with our previous paper? the configurational partition function, Q, in the Bragg-Williams approximation, is given by
Q,(N,M,T) = Q,(N,M) exp[-~,(N,M)/kT1 (1) where Q,(N,M) is related to the average configurational entropy (steric or packing term), E,(N,M) is the average configurational energy, N is the number of molecules, M is the total number of cells or sites, T i s the absolute temperature of the system, and k is the Boltzmann constant. The configurationalHelmholtz free energy, A,, can be derived in terms of Q, A,(N,M,T) = -kT In Q,(N,M,T) (2) Then, the chemical potential for a component i in the system can be obtained from the partial derivative of the Helmholtz free energy with respect to the molecular number of that component (3)
2.1. Chemical Potential for Blocklike Solutes in an Isotropic, Blocklike Mobile Phase. The model system consists of No holes (each occupies a unit cell of volume vo),N,,, blocklike mobile phase molecules (each has qqr segments), and Nt solute molecules (each 0 1992 American Chemical Society
Molecular Theory of Chromatography for Blocklike Solutes has abc segments). All of them are assumed to be isotropically distributed among the M cells ( M = No + qqrN, abcN,). We have already derived the chemical potential for the solute component in the mobile phase, pt(,) (see eq 60 in ref 9)
+
r
1
The Journal of Physical Chemistry, Vol. 96, No. 19, 1992 7511 density of the solute in the stationary phase. Note that the uptake of the mobile phase molecules by the stationary phase (swelling) is assumed to be negligible. 2.3. Retention Equation. The chromatographic distribution constant or partition coefficient, K,is defined as the ratio of the equilibrium concentration of solute in the stationary phase, c!(~), to that in the mobile phase, c,,), in the limit of infinite dilution of the solute (pt 0). This ratio is also equal to the ratio of the respective p,‘s? i.e.
-
This intensive thermodynamic distribution constant is related to the extensive capacity factor, k’, by K = k{
where 0, is the occupied volume fraction (segmental density) of the pure mobile phase, e, is the attractive interaction energy between a segment of a solute molecule and a segment of a mobile phase molecule, and pt(,) is the molecular density of solute in the mobile phase. 23. Statistical Mechanics for Mean Field Lattice Model for Isotropic Mixture of Blocklike and Flexible, chainlike Molecules ( S t a t i ~ ~ rPhase y System). The model system consists of N , completely flexible, chainlike stationary phase molecules, each occupying m sites, Nt blocklike solute molecules, each occupying abc sites, and No holes, each occupying a unit cell of volume uo. We assume that the stationary phase molecules and the solute molecules are isotropically distributed. The blocklike solute molecules have six possible orientations9 and the number of molecules lying in each orientation is the same, i.e.
N,, = Nt2 = Nt3 = Nt4 = Nt5 = N16 = N1/6
+ In gt3 + In gt4 + In gt5 + In gl6 -
Rt%t
+ Rsk3 + Rtt‘tt kT
(6) Substituting the relevant terms into eq 6 , using Stirling’s approximation, taking the partial derivative of Helmholtz free energy with respect to N,, then assuming infinite dilution of the solute in the stationary phase, one obtains the chemical potential for the blocklike solute in the isotropic, chainlike stationary phase
--h s ) kT
= k@
(9)
where V, and V, are, respectively, the total volume of the mobile and stationary phases and 9 is the phase ratio. The capacity factor can be determined from
where t R is the retention time of the solute and to is the holdup time (or void retention time). Assuming abc >> 1, we obtain the chemical potential for the blocklike solutes in the isotropic, chainlike stationary phase from eq 7
r
1
(5)
The detailed methodology for deriving the steric term and attractive interaction-energy term of the configurational partition function has been given in our previous paper? Similarly, we can obtain the total configurational partition function, Qc(s), for the present stationary phase system (see Appendix). Taking the logarithm of QN,), one obtains the configurational Helmholtz free energy for the system
= In g, + In g,, + In gt2
2)
The equilibrium condition at the operational temperature and density is given by
where i denotes the ith component distributed between the stationary and the mobile phase. Applying the equilibrium condition to the solute component along with eqs 4 and 11, one obtains the distribution coefficient for blocklike solutes with a chainlike stationary phase and a blocklike mobile phase
abcln(1-0,)+(1-abc)ln
+
where abc and ab + ac bc are, respectively, the scaled van der Waals volume and the effective contact area (obtained by scaling dimensions relative to the unit volume, uo, and unit area, ao.of the lattice9)of a solute molecule, 0, is the occupied volume frachon (segmental density) of the pure stationary phase, Gt is the attractive interaction energy between a segment of a solute molecule and a segment of a stationary phase molecule, and P , ( ~ is ) the molecular
3. Experimental Section 3.1. Experimental Procedure. The solutes, apparatus and
Yan and Martire
7512 The Journal of Physical Chemistry, Vol. 96, No. 19, 1992 TABLE I: Natural Logarithm of Capacity Factor (In k') of PAHs with SB-Octyl-50 Column density (g/mL) 0.15
naphthalene biphenyl phenanthrene anthracene pyrene benzo [c] phenanthrene triphenylene p-terphenyl benz[ a] anthracene chrysene
0.54 1.28
naphthalene biphenyl phenanthrene anthracene pyrene benzo [c] phenanthrene triphen ylene p-terphenyl benz[ a]anthracene chrysene
0.25 0.95
naphthalene biphenyl phenanthrene anthracene pyrene benzo[c] phenanthrene triphenylene p-terphenyl benz[o] anthracene chrysene
0.03 0.68
naphthalene biphenyl phenanthrene anthracene pyrene benzo [c] phenanthrene triphenylene p-terphenyl benz [a] anthracene chrysene
-0.21 0.41 2.07 2.09
naphthalene biphenyl phenanthrene anthracene pyrene benzo[c)phenanthrene triphenylene p-terphenyl benz[ a] anthracene chrysene
-0.42 0.15 1.77 1.80
0.20
0.25
-0.02 0.60 2.24 2.26
-0.51 0.02 1.51 1.53
2.55
-0.30 0.29 1.91 1.93
-0.82 -0.29 1.19 1.22 2.24 2.13
-0.51 0.05 1.60 1.61
-1.02 -0.5 1 0.92 0.93 1.96 1.82
-0.76 -0.01 1.33 1.34
-1.20 -0.76 0.67 0.68 1.67 1.50
-0.94 -0.43 1.04 1.06 2.04 1.97
-1.39 -0.94 0.39 0.41 1.37 1.93 2.16 1.19 2.16 2.18
0.30 0.35 107 OC -1.05 -1.56 -0.58 -1.11 0.83 0.21 0.83 0.22 1.82 1.14 2.31 1.66 2.65 1.90 1.68 0.93 2.64 1.93 2.67 1.93
0.40
0.45
0.50
0.55
-1.90 -1.51 -0.29 -0.27 0.61 1.05 1.28 0.32 1.29 1.29
-2.21 -1.83 -0.78 -0.78 0.08 0.44 0.68 -0.31 0.69 0.70
-2.66 -2.41 -1.27 -1.27 -0.43 -0.09 0.15 -0.84 0.15 0.15
-3.22 -2.66 -1.61 -1.61 -0.87 -0.58 -0.30 -1.27 -0.30 -0.30
117 OC -1.27 -1.66 -0.84 -1.31 0.52 -0.01 0.54 0.00 1.50 0.89 2.10 1.40 2.32 1.57 1.34 0.65 2.33 1.61 2.34 1.64
-2.04 -1.71 -0.51 -0.51 0.39 0.76 1.01 0.03 1.02 1.03
-2.41 -2.12 -0.97 -0.97 -0.15 0.18 0.44 -0.51 0.44 0.46
-2.53 -1.39 -1.39 -0.65 -0.27 -0.01 -1.02 -0.01 0.01
-1.77 -1.77 -0.99 -0.73 -0.56 -1.43 -0.56 -0.56
127 "C -1.47 -1.08 0.31 0.31 1.27 1.81 2.01 1.04 2.01 2.04
-1.83 -1.47 -0.25 -0.25 0.63 1.12 1.32 0.39 1.32 1.33
-2.21 -1.90 -0.73 -0.73 0.08 0.51 0.69 -0.22 0.70 0.69
-2.53 -1.20 -1.20 -0.42 -0.05 0.12 -0.78 0.13 0.14
-1.56 -1.56 -0.87 -0.54 -0.39 -1.24 -0.37 -0.37
-1.97 -1.97 -1.24 -0.97 -0.87 -1.66 -0.87 -0.84
137 OC -1.61 -1.24 0.09 0.10 0.99 1.52 1.74 0.79 1.74 1.76
-2.04 -1.71 -0.48 -0.46 0.39 0.85 1.04 0.1 1 1.05 1.05
-2.41 -2.12 -0.97 -0.97 -0.16 0.25 0.43 -0.48 0.43 0.43
-2.41 -1.39 -1.39 -0.63 -0.27 -0.12 -0.99 -0.12 -0.12
-1.77 -1.77 -1.08 -0.76 -0.63 -1.47 -0.63 -0.63
147 OC -1.77 -1.50 -0.16 -0.16 0.72 1.22 1.43 0.48 1.43 1.43
-2.04 -1.83 -0.69 -0.69 0.15 0.59 0.77 -0.15 0.77 0.77
-2.30 -1.20 -1.20 -0.34 0.01 0.20 -0.69 0.20 0.20
-1.61 -1.61 -0.84 -0.51 -0.36 -1.24 -0.36 -0.36
experimental procedures have been described in detail in our previous work.I0 The capillary column used in this study contained a film of SB-Octyl-50 (50% octyl and 50% methyl polysiloxane). The structure of the stationary phase polymer is shown in Figure 1. The column was 10 m long with a 50 pm internal diameter and a film thickness of 0.25 Mm (Lee Scientific, Inc.; Salt Lake City, UT). 3.2. Results. The capacity factor, k', was obtained from eq 10. The values of the natural logarithm of the capacity factor, In k', of the solutes are provided in Table I. 4. Test of Theory and Discussion Equation 13 is applicable to systems composed of blocklike
solutes isotropically distributed between a chainlike stationary
0.60
-1.24 -1.05 -0.82 -0.82 -0.82
-1.43 -1.08 -0.99 -0.99 -0.99
0ctylmothylpolyrllox.n.
0 Cb- SI-
C&-
Cy- C&-CHz-CHz-C&-C&-CHs
I
0 I
Ch- SI-
C&-Cl-&-C&-C&-Cl+-C&-
C&-CHs
I
Figure 1. Structure of the stationary phase polymer.
phase and a blocklike mobile phase. To test the theoretical model, an experiment was designed to mimic the model system by using PAHs as solutes, SB-Octyl-50 as the stationary phase, and SF C 0 2as the mobile phase.
Molecular Theory of Chromatography for Blocklike Solutes
The Journal of Physical Chemistry, Vol. 96, No. 19, 1992 7513
1
4,
aaI-
% P
I-
4
Figure 2. Linear dependence of In k’on Aef (scaled, solute effective contact area) for planar, four-ring PAH isomers on the SB-Octyl-50 column with SF C02 as the mobile phase at 117 “C.
4.1. Effect of Solute Characteristics. Equation 13 can be written as In K = VwP AefE (14)
+
+
where V, = abc and Acf ab ac + bc are the scaled van der Waals volume and the effective contact area of a solute molecule, respectively
(15) is related to the packing (entropic) terms, and
is related to the energetic terms. For a specific chromatographic system (fixed mobile and stationary phases) under fixed operating conditions, P and E are constant. Aef is almost the same for planar PAH isomers with identical V, (see Table I1 in ref 10). Therefore, eq 14 predicts that there should be virtually no isomeric separations with a truly isotropic stationary phase. Retention data for PAH compounds on the SB-Octyl-50 column show that anthracene coelutes with phenanthrene and the four-ring PAH isomers have almosr the same retention, except for benzo[c]phenanthrene (nonplanar). However, the contact areas for the four-ring PAH isomers are not exactly the same. According to eq 14, we would expect that the very small differences in contact area may cause a little difference in retention times. As the slope of In k’vs Acf, E normally should be positive and increase with decreasing mobile phase density (6, p,), since Os 2 Om and e < 0 (attractive); therefore, it is more likely to see some separation at lower mobile phase density. Illustrated in Figure 2 are plots of In k’vs Acf for three of the four-ring PAH isomers on the SB-0ctyl-50 column at 117 OC. Clearly, we see a very small positive slope at mobile phase densities of 0.30, 0.35, 0.40,and 0.45g/mL, respectively. However, the slopes are virtually zero at densities of 0.50 and 0.55 g” For simplicity, we assume that the contact area is proportional to the van der Waals volume for PAH compounds with different numbers of rings Q:
Vw
(17)
+ intercept
(18)
constant Acf Then, eq 14 becomes In k’ = Acf slope
J
(19) The mobile phase contribution (first two terms) to the solute retention in the isotropic stationary phase is the same as that in the anisotropic stationary phase although the stationary phase contribution (In KO) is different. Therefore, all the discussion about mobile phase effects in the anisotropic system is, in general, applicable to the isotropic system.l0 The general dependence of In k’(or in K ) on mobile phase density, Om, is as follows: (a) in k’decreases with increasing 0, in the operational region. (b) The slope of In k’vs Om is governed predominantly by the energetic term in the usual density region. Although the entropic term is not large enough to change the declining tendency of In k’in this region, it makes the slope less negative and eventually zero. (c) In the very high density region, where the entropic term becomes more important, the highly compressed mobile phase makes the packing of the solute molecules into the mobile phase more Micult and would therefore tend to drive the solute toward the stationary phase, more so when the solute molecule has a larger van der Waals volume. As a consequence, In k’increases with increasing 0, in that region. (d) Although a minimum in In k’vs Om is predicted, it would be difficult (if not impossible) to attain in practice, since it is predicted to occur at such a high mobile-phase density.I0 Interestingly, for any given solute, no matter how different the stationary phase may be, the slope of In k’vs mobile phase segmental density, Om (which is proportional to pm in g/mL), is predicted to be the same (of course, the intercept will be different). Equation 13 also predicts that the intercept of the In k’vs Om should be the same for all the planar PAH isomers in the isotropic phase since their contact areas are approximately the same. The experimental results confirm these predictions. The In k’vs 8, plots
7514 The Journal of Physical Chemistry, Vol. 96, No. 19, 1992
Yan and Martire
for the four-ring PAH isomers (except the nonplanar one, ben~~[clphenanthrene) overlap at each temperature in the mobile phase density range studied. Therefore, the isotropic phase is not effective in separating PAH isomers. Figure 4 demonstratesthat the plots of In k'vs 0, for the four-ring PAH isomers in the anisotropic and the isotropic systems are indeed parallel. However, the intercepts of the plots for different solutes are different in the anisotropicsystem but the same in the isotropic one, which is in agreement with the predictions. 4.3. Effect of Stationary Phase Parameters. With the mobile phase variables fixed, we obtain from eq 13
r
1
+(Our-dnp Pw.
+
+I+ W.]M"mm8 ah-
Figure 4. Comparison of dependence of In k' on mobile phase density for planar, four-ring PAH isomers on SB-Smectic and SB-Octyl-50 stationary phases at 107 OC.
where the constant = the mobile phase contribution - In @. The first term in cq 20, which arises from the configurational entropy of the solute in the stationary phase, is independent of temperature. It leads to a decrease in In k' (or In K ) with increasing stationary phase density (8,). The second term in eq 20, which arises from stationary phasesolute attractive interactions, is temperature dependent. It leads to an increase in In k'with increasing 8, and decreasing T. The overall contribution from the stationary phase should be a combination of the entropic and energetic effects. Since the coefficient of the first term is V, (which is identical for PAH isomers) in the isotropic system compared with &,, (which is quite different for the isomers) in the anisotropic system, there should be no shape selectivity in the isotropic stationary phase. This is just what is observed experimentally. The dependence of In k'on stationary phase density should have the same trend as in the anisotropic stationary phase,1° Le., with increasing e,, In k'increases in the operational 0, region, gradually approaches a glaximum, then decreases in the very high 8, region. (Note that the maxima for four-ring PAH isomers in the anisotropic stationary phase are predicted to appear at 0, = 0.95 segments/cell, a value which is hardly accessible.I0) 4.4. Effect of Temperature. Equation 13 can be rearranged as
Recalling eq 9 and that In K = -(AH/RT) from a linear regression of In k'vs 1 / T
AH W-H"') slope = -- = = 2(ab R R
intercept
+ In
AS
C$
+ ( S I R ) ,we obtain
+ ac + bc) x
s=-sm -
= -= R R
(24)
abc In
[
]); [-1 -
is-es
1- -(I
);
-
- abc In
-
1
1-8,
-em
I----
39
3r
Comparing this equation with eq (78) in ref 9, we can see that the basic form for the isotropic system is similar to the anisotropic system. The differences are (a) the coefficient of the energetic term (second) is ab + ac + bc instead of ac + bc and (b) the coefficient of the packing term is V, instead of Ami,,. Therefore, qualitatively, the general trends should be similar. Equation 21 can be written as 1
In K = slope T
+ intercept
(22)
where AH and AS are, respectively, the molar enthalpy and entropy of solute transfer from the mobile phase to the stationary phase, and H and S refer to (absolute) solute partial molar quantities in the respective phases. The value of H"' in the PAH/C02/SB-0ctyl system should be identical to the value of H"' in the PAH/C02/SB-Smectic system,'O but the value of Pishould be much lower than the value of Hs,, since the coefficient is ab + ac + bc instead of ac + bcS9 Therefore, we anticipate that the van't Hoff plot would produce a more positive slope in the isotropic system. Another difference is the coefficient of the S term, which is V, in the isotropic phase but Aminin the anisotropic one. Since S is always negative, we expect that ASi (the intercept) in the isotropic system should be more negative than ASain the anisotropic system under the same conditions. Provided in Table I1 are the values of the slope and intercept obtained from linear regression analyses; the correlation coefficients are in excess of 0.99. Figure 5 shows van? Hoff plots for both stationary phases under identical conditions. It is evident that all three predictions are confirmed. 5. Conclusions
DiMarzio lattice statistics were extended to treat isotropic mixtures composed of blocklike molecules (solute and mobile phase) and flexible, chainlike molecules (stationary phase). A
Molecular Theory of Chromatography for Blocklike Solutes
The Journal ofPhysical Chemistry, Vol. 96, No. 19, 1992 7515
TABLEII: Valws of Slope and Intercept from ra’t Hoff Plots at SF COZ b i t y Of 0.35 g/mL and in the Temperature R8we of 117-147 ‘C‘
solute chrysene benz[a]anthracenc triphenylene bcnzo[c]phcnanthrcne
SB-Smectic phase slope intercept 3.87 3.33 2.35 2.06
-1.27 -6.41 -4.84 -4.52
SB-0ctyl-50 phase slope intercept 4.73 -10.50 4.57 4.39 4.42
aslope. = - U / R and intercept = (AS/R) - In
-10.10 -9.66 -9.94
where the other symbols are defined in the text. Using a method analogous to that described in section 2 of ref 9, we obtain the configurational partition function for the solute molecules (subscript t). Orientation 1: Ntl(a,,b,,~,,)
a. t
l
=
where V I A E
WIAE
-
8 a4ady+m+fipk.ny*lr
WOlBT
=
WlBT + -)1.)* chyrr* *bmz[.)m”armtckyrrn
F’igwe 5. van’t Hoff plots for planar, four-ring PAH isomers on both the SB-Smectic (anisotropic) and the SB-Octyl-50 (isotropic) stationary phases.
retention equation, applicable to GC, LC, and SFC, was derived using the chemical potentials obtained for the system. The preliminary tests, using our SFC experimental data, are promising, though most of the predictions need to be tested quantitatively. The present theory of chromatography describes, at the molecular level, the general dependence of solute retention on the mobile phase density, temperature, stationary phase density, and solute geometric parameters. It also allows the microscopic understanding of thermodynamic properties, such as the Helmholtz free energy (A), the chemical potential (p,), and the solute entropy (AS) and the enthalpy (AH) of transfer. With a carefully designed and well-controlled experimental system, the retention equation may be used for the determination of molecular parameters, such as the segmental interaction energies (eij). However, it should be kept in mind that several simplifications were made: (a) a clear boundary between the stationary and the mobile phases; (b) a completely isotropic system; (c) infinite dilution of the solute. The current theoretical approach could be extended to treat many kinds of chromatographic systems made up of a variety of solutes and stationary and mobile phase molecules with different sizes, shapes, and degrees of flexibility. In the framework of the theory, the uptake of the mobile phase molecules by the stationary phase (swelling) and partial alignment of the stationary phase molecules can also be taken into account. Work in these directions is in progress.
Acknowledgment. This material is based upon work supported by the National Science Foundation under Grant CHE-8902735. Appendix
Derivation of the C ~ n f i g u r a t i Partition ~~l Function for the S t r t i o ~ r yPhase System. Steric Term. To evaluate the steric
configurational partition function a,, we place all molecules in the stationary phase into the lattice, one by one, segment by segment. The packing of completely flexible chains has been done by Dowell and Martire.I2 The steric part g, for the pure chainlike molecules is
WICT
WID,
M
- mN,
= M - mN, - Ntlabc
WlcT
=M
=M
1 W O ~ D T= M - ‘ j ( m - 1)N,
- -31( m - 1)N, - Ntlac(b- 1) 1 - -(m 3
- 1)N, - Ntlbc(a - 1)
1 = M - -(m - 1)N, - Ntlab(c- 1) 3 Orientation 2: Nt2(ay,b,,c,)
7516 The Journal of Physical Chemistry, Vol. 96, No. 19, 1992
Yan and Martire
where
w03= ~M ~ - mNs - abC(N11+ N12) W ~ A=Em - mNs - abc(Nt, W3BT
M
+ Nt2 + Nt3)
=
1 - -(m - l ) N s- Ntlac(b- 1 ) 3
- Nt2bc(a- 1 ) - Nt3ub(c- 1 )
1
- l ) N s- Ntlac(b- 1 ) - N&(a - 1 ) w 0 3 ~k ~f - -(m 3 = 1 M - -(m - l ) N s- Ntlbc(a- 1 ) - Nt2ac(b- 1 ) - Nt3bc(a- 1 ) 3 1 WO3cT = M - -(m - l ) N s- N,lbc(a - 1 ) - N , ~ u c -( ~1 )
w3CT
3
= 1 M - -(m - l ) N s- NIlab(c- 1 )
w3DT
3
- NI2ab(c- 1 ) - Nt3uc(b- 1 )
1 W O ~ D= T M - -(m - l)Ns- Ntlab(c- 1 ) - NI2ab(c- 1 ) 3 Orientation 4: Nt4(uy,bZ,cx)
The average number of solute-stationary phase, pairwise segmental interactions is given by
J. Phys. Chem. 1992, 96, 7517-7523
los, = 2N,(ab + ac + bc)Ps,
(1-10)
The average number of solutesolute segmental interactions will be
7517
Total Configurational Partition Function. The total configurational partition function for the system is
References and Notes (1) Lee, M. L.;Novotny, M. V.;Bartle, The average number of stationary phase-stationary phase, painvise segmental interactions will be
The average configurational energy E,(,,is given by J%,
