Alkane mixtures: Comparisons between theory and experiment for the butane–heptane system

Chemical Physics Letters 458 (2008) 313–318

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Alkane mixtures: Comparisons between theory and experiment for the butane–heptane system Ronald P. White, Jane E.G. Lipson * Department of Chemistry, 6128 Burke Laboratory, Dartmouth College, Hanover, NH 03755, United States

a r t i c l e

i n f o

Article history: Received 20 February 2008 In final form 1 May 2008 Available online 7 May 2008

a b s t r a c t In this work we introduce our continuum integral equation methodology to the study of alkane mixtures. The theory is applied to predict liquid–vapour coexistence properties in the mixed butane–heptane system using a model parameterization based only on the pure fluids. These predictions are in strong agreement with corresponding experimental values available for the mixture, including temperature– composition phase envelopes, and temperature–pressure phase diagrams spanning a wide region of the system’s overall coexistence regime. The alkanes are modelled as square-well chains, marking here our first application of the theory to chain mixtures with attractive interactions. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction The modelling of alkanes and alkane mixtures has received considerable interest over the years, as they are of direct interest to the petroleum industry, in addition to being important solvents in the processing of polymeric and other material systems. Beyond the importance of studying the alkane properties themselves, these systems are the simplest real systems that manifest the fundamental aspects of chain molecule behaviour. They therefore provide a valuable test bed for theories that may be aimed at modeling larger, more complex, chain-like molecules. Theoretical methodologies designed for the study of fluid systems (such as alkanes) cover a diverse range of approaches, including for example, the application of microscopically inspired equations of state, perturbation techniques, various integral equation treatments, and direct computer experiments on the microscopic model (e.g. molecular simulation) [1,2]. In our laboratory, we have developed two related integral equation approaches designed for the study of chain molecule fluids: a lattice theory [3,4], and a more detailed continuum theory [5–7]. The two methodologies share a similar theoretical framework [8,9], wherein integral equations are used to describe site–site distributions and their derivatives in terms of the average force between the monomer units. Given the similar grounding of the two methods, there is the opportunity to make a direct comparison of equivalent lattice and continuum versions of a theory. Some comparisons have already been made for the case of pure fluids [7]. It is our goal in upcoming work to extend this comparison to mixtures. * Corresponding author. Fax: +1 603 646 3946. E-mail address: [email protected] (J.E.G. Lipson). 0009-2614/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.05.002

While previously applied only to pure fluids, in recent work [10], we have adapted the continuum integral equation theory for the case of hard-sphere chain mixtures and demonstrated good agreement on comparison with our MC simulation results. Here, we will apply the theory, for the first time to realistic mixed systems, where we study alkane mixtures using a square-well interaction potential. This is a simple, but effective, choice for modeling nonpolar fluids, and furthermore, it is natural choice for drawing comparisons with our lattice model. Over the years a number of other integral equation approaches have been devised for chain molecules, including PRISM theory [11–13], the PY theory of Chiew [14–16], approaches based on Wertheim’s integral equation theory [17,18], and a density functional theory [19]. These approaches have most often been applied to hard-sphere, or soft repulsive, intermolecular site-site potentials, with the direct modeling of attractive interactions being somewhat less common. In modeling chain molecules with attractions (e.g., square-well chains), perturbative approaches have been widely applied (including SAFT-related, and other approaches). In one family of strategies [20–25] (e.g., SAFT-VR), a hard-sphere monomeric fluid is ‘perturbed’ into square-well monomers, which are then perturbed (in a chain-forming step) into square-well chains. This approach relies heavily on estimating the structure of the relevant monomeric fluids. A second strategy [26–28] (e.g., PCSAFT) would be to start with a hard-sphere chain fluid and apply perturbation theory (for instance using structural information from one of the above integral equation approaches) to link to the square-well chain fluid. Additional approaches include combination simulation–perturbation methods [29,30] where the perturbation results are determined from averages in the simulated hard-sphere chain reference ensemble.

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The strategy employed in this work is similar to the second category of perturbation approaches (mentioned above) in the sense that we employ, as a starting point, the hard-sphere chain reference fluid. (For convenience, we use Chiew’s equation of state [14], however, in principle, we could also apply our own hardsphere chain results.) Rather than applying perturbation theory however, our approach is to model the structure (site–site distributions) of the square-well chain fluid itself, which then leads to the thermodynamics via the energy route. In the remainder of the paper we give a brief description of the theory (Section 2), followed by a discussion of the results in Section 3. We show our theoretical fits to the pure alkane properties, followed by our theoretical predictions for the alkane mixture. Results include selected slices of the mixed system liquid–vapour phase diagram and a comparison with experiment. Concluding remarks are given in Section 4. 2. Theory and implementation 2.1. Integral equation theory for binary chain fluid mixtures In the following, we highlight the fundamental framework of the continuum integral equation (CIE) method, leaving the more involved details to descriptions that are available elsewhere [5– 7,10]. We will consider a binary chain fluid mixture containing Na chain molecules of species a, and Nb chains of species b, at volume, V, and temperature, T. Each chain of species a (b) is comprised of na (nb) sites/monomers. The goal of this integral equation approach is to calculate the set of intermolecular site–site distributions (g) which describe the structure and, ultimately, the thermodynamics, of the system. In a mixed chain fluid we take as an example, the site–site distribuia ;ja tion, g aa ðria ja Þ, which is proportional to the probability that two sites, ia and ja, (on two different a molecules) will be found at some separation distance, ria ja . The CIE formalism begins [8,9] with the derivatives of the site–site distributions. It is important to note that these derivatives express the average force, F ¼ FðrÞ, between ia ;ja the two sites of interest. For the case of g aa ðr ia ja Þ, we obtain a ;ja kB T ria ln g iaa ðr ia ja Þ ¼ F ia ja þ

na X

F ia ma þ ðNa  2Þ

na X

ma 6¼ja

þ Nb

nb X

F ia k b þ

kb

þ ðNa  2Þ

F ia ka

ka na X la 6¼ia na X na X la 6¼ia

ka

F la ja þ

na na X X

F la m a

la 6¼ia ma 6¼ja

F la ka þ Nb

nb na X X la 6¼ia

F la k b ;

ð1Þ

kb

where, in general, kBTln g(r) defines the ‘potential’ of average force

kB Tln gðrÞ ¼

Z

r

Fdr0 :

ð2Þ

1

Eq. (1) expresses the total average force ðFÞ on site ia given the presence of ja. Specifically, with ia and ja fixed (at some distance ria ja ), we average the force on ia over all configurations of the remaining Na  2 and Nb molecules, and, over all sites la (other than ia) on the first molecule, and, over all sites ma (other than ja) on the second molecule. (See also Fig. 1 which gives a diagram of all of the relevant sites.) The various terms in Eq. (1) are the force contributions from the different categories of sites surrounding ia. For example, the first term is the direct force on site ia by site ja. The second term is the average force on ia by all of the sites ma 6¼ ja on the second molecule. The third term is equivalent to the average force on ia by all of the remaining Na  2 a molecules.1 Here, to

1 Note that in Eq. (21) of [10], (Na - 2) and Nb were absorbed into the F terms so the meaning there is somewhat different.

Fig. 1. Diagram showing all of the relevant sites used in the calculation of a ;ja a ;ja kB T ria ln g iaa ðr ia ja Þ, where g iaa ðr ia ja Þ is the intermolecular site–site distribution for a chosen pair of sites, ‘ia’ and ‘ja’ on two different molecules, each of species a. The sites ‘ia’ and ‘ja’ belong to ‘molecule 1’ and ‘molecule 2’ respectively, where, in the figure a dashed line is drawn between these two sites thus emphasizing this chosen site–site distribution, and, the distance, r ia ja , upon which it depends. (Note also that ia ;ja the choice of ia and ja corresponds to a specific g aa ðria ja Þ for the case where the sites ia ;ja are ‘interior-end’.) The derivative kB T ria lng aa ðria ja Þ is equivalent to the average force on site ia given the presence of ja. In obtaining this average, the positions of sites ia and ja are fixed, while all of the other sites in the diagram are moveable. Specifically, we average over all configurations of the remaining sites la on molecule 1, and, over all the remaining sites ma on molecule 2. It is also necessary to include the contribution from all of the remaining Na  2 and Nb molecules in the fluid. The Na  2 a molecules are accounted for by averaging over a single representative a molecule (containing sites ka) which is free to range about the total volume, V. The Nb b molecules are represented by a single freely ranging b molecule containing sites kb. In this diagram, the two species differ by the number of sites per chain, 3 for species a, and 4 for species b. (It is also noted that while the chains modelled in this work are tangent spheres, the diagram above uses a common ball and stick representation.)

P compute nkaa F ia ka , we employ a (single) representative ‘third molecule’ with sites ka, and average its contribution (from all of its sites) as it is moved over the entire volume, V. Similarly, the fourth term is the average force on ia by all of the Nb b molecules and is calculated by considering a single representative b molecule containing sites kb. It is noted that the remaining terms involve averaged forces on all of the la 6¼ ia, the sites other than ia on the first molecule. These terms stand in place of (i.e. they are equivalent to) the total average intramolecular force on ia (which in general would be composed of both bonded and nonbonded force contributions). ia ;ja In an exact formulation of the derivative, kB T ria ln g aa ðr ia ja Þ, each term in Eq. (1) is written as an average over the relevant multi-site distribution. (See Eq. (7) in [10].) For instance, the average in ia ;ja ;ma the second term would involve a multi-site distribution, g aa (a two-molecule, three-site distribution), the third term involves ia ;ja ;ka g aaa (a three-molecule, three-site distribution), and so on for the other terms. However, since these distributions are not known, ‘closure approximations’ are needed, such that the more complex multi-site distributions are rewritten as expressions involving only the simpler site–site (pair) distributions. In so doing, one is essentially expressing the derivative of a site–site distribution (rg(r)) in terms of itself (g(r)). Therefore, starting with an approximate form for the site–site distribution, this ‘self consistent’ expression for the derivative can then be integrated over r (essentially, Eq. (2) with an approximated F) to thus yield a better approximation. This process can then be iterated until a converged solution for the site–site distribution is obtained. Having outlined the general approach above, a few remaining points should be added. One is that a key aspect of the method is how the multi-site distributions are expressed in terms of the

R.P. White, J.E.G. Lipson / Chemical Physics Letters 458 (2008) 313–318

site–site distributions. In some of the cases, the Kirkwood superposition approximation KSA [31] can be applied (g1;2;3 = g1;2g1;3g2;3). In other cases, where some of the sites share bonded connections, the KSA would do very poorly, therefore, we introduce ‘normalized superposition approximations’ [5,32] that effectively preserve the fundamental effects of chain connectivity. Another important point regarding the implementation is that ia ;ja the site–site distributions, g aa ðria ja Þ, are not all identical; some may involve end sites, interior sites, etc. Furthermore, there are ib ;jb two other categories of site–site distributions, g bb ðr ib jb Þ (for two ia ;jb sites on two different b molecules), and g ab ðria jb Þ (one site on an a molecule, the other on a b molecule) which must be considered in the workup. This leads to considerable coupling in the equations, some of which is alleviated by using the averaged site–site distributions gaa ðrÞ; gbb ðrÞ; and gab ðrÞ.

gaa ðrÞ ¼

na X na 1 X a ;ja g iaa ; 2 na i j a

gbb ðrÞ ¼

a

nb X nb 1 X ib ;jb g bb ; 2 nb i j b

b

nb na X 1 X ia ;jb g ab : ¼ na nb i j a

thus the e’s will vary by species in the present continuum modeling. As for r, a single (rather than species dependent) value for k is most appropriate for the lattice comparison. Several values have been tried; a value of k = 1.75 works well for us. We note that our choice of k has not been systematically optimized because it cannot be varied (unlike e and r) after the calculations when fitting to experimental data. The number of sites on the square-well chains will depend on the particular alkane to be modelled (i.e. the longer the alkane, the longer the square-well chain). Given that the present squarewell chain model is composed of tangent spheres, it follows from geometric considerations that the number of sites in the squarewell chain should not be equal to the number of carbon atoms (C) in the alkane. We will take n (na or nb) from a well accepted empirical relation [33,34]

n¼1þ

gab ðrÞ

ð3Þ

b

Indeed, our main results here will be for these averaged distributions, as they are used in the calculation of the thermodynamic properties (discussed below). The integrations (Eq. (2)) are carried out numerically (see [5– 7,10]), where the site–site distributions are discretized on a grid. The grid is spaced in units of 0.05r, and the distributions are solved for all distances in the range r 6 r 6 25r, where at the maximum grid distance, 25r, g(r) is taken to be unity (r is defined below.). After each iteration the new ‘output g(r)’ is mixed (often at a ratio of 10%) with the previous g(r). Typically 150 iterations are sufficient to achieve convergence, though the highest densities can be more challenging, sometimes requiring a smaller mixing ratio and more iterations.

315

C1 : 3

ð6Þ

This relationship is based on the fact that an intramolecular carbon– carbon bond is roughly one third the value of a methane–methane intermolecular distance. It has been commonly applied in other studies that have modeled alkanes with square-well chains [20– 24]. (In [20], where n can be treated as a continuous parameter, free optimizations of n (for any particular alkane) were shown to be consistent with Eq. (6). Other studies [30] have shown a larger number C for each freely jointed segment.) As noted above, we will be modelling mixtures of butane (C = 4) and heptane (C = 7), which corresponds to mixtures of square-well dimers (n = 2) and trimers (n = 3). This choice of butane and heptane is a convenient starting point, because Eq. (6) maps these alkanes to exact integer numbers of tangent spheres. While an integer number of segments is required, the theory can ultimately be extended to chains consisting of fused spheres (see for example [32]) which would alleviate these restrictive ‘jumps’ in atom/carbon number. 2.3. Calculation of thermodynamic properties

2.2. The square-well chain model and relation to alkane modelling The above derivations are general, in the sense that they can be applied to a wide variety of pair interaction models. Here, we employ the square-well model, which as noted above, is a common choice for modelling nonpolar systems such as alkanes. The square-well potential for a pair of sites, both of type a, is given by

8 > < 1; r < raa uaa ðrÞ ¼ eaa ; raa  r  kraa ; > : 0; r > kraa

ð4Þ

where raa is the hard core diameter, eaa is the well depth, and k is the well width. The pair potential, ubb(r), for two sites, both of type b, is defined similarly in terms of rbb and ebb. Standard combining rules are commonly used to obtain rab and eab associated with the ab pair interaction, uab(r). Note that the pair force for the squarewell model is required in evaluating Eq. (1) (and other related expressions). This is accounted for by substituting the relation [1]

d ½uðrÞ=kB T ¼ dðr  rþ Þ þ ð1  ee=kB T Þdðr  krþ Þ: dr

ð5Þ

In this work, r will be the same for all site–site interactions (i.e. raa = rbb = rab = r), and further, the factor, k, will be fixed at 1.75. The two species, a and b, will differ in chain length (na 6¼ nb), and in e (eaa 6¼ ebb). For the latter we will use the geometric mean, pffiffiffiffiffiffiffiffiffiffiffi eab ¼ eaa ebb , for the mixed pair interaction. These choices anticipate future comparison with the analogous lattice-based theory. Specifically, we want the segment size (r) in each component to be the same, just as is the case in a lattice model. On the other hand, the lattice theory does incorporate different energetic interactions, and

Upon solution of the integral equations for the averaged site– aa ðrÞ; g bb ðrÞ; and g ab ðrÞÞ, we can obtain the site distributions ðg thermodynamics of the system via the energy route. This is done by evaluating the distributions over a wide range of the V,T space (i.e. on a grid). At each V,T point, we use the site–site distributions to evaluate the average intermolecular energy, U,

U=N ¼

Z kr 1 2 ðeaa Þgaa ðrÞ4pr 2 dr na xa qa 2 r Z kr 1 ðebb Þgbb ðrÞ4pr2 dr þ n2b xb qb 2 r Z kr ðeab Þgab ðrÞ4pr 2 dr; þ na nb xa qb

ð7Þ

r

where N = Na + Nb, xa = Na/N, and xb = Nb/N, qa = Na/V, and qb = Nb/V. The Helmholtz free energy, A, of the system can then be obtained by thermodynamic integration using the derivative,

  oðA=TÞ ¼ U: oð1=TÞ V

ð8Þ

Eq. (8) is integrated along a line of constant density starting at 1/T = 0 (i.e. T = 1) which corresponds to the hard-sphere chain reference state. The value for (A/T)ref is taken from the work of Chiew [14]. The integration of Eq. (8) is performed at all densities making A(V, T) available over a wide space, and thereby providing a direct route to the other thermodynamic quantities. For example, the pressure is obtained from

  oA P¼ ; oV T

ð9Þ

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and the Gibbs free energy via G = A + PV. While the chemical potential, for a pure fluid, is just l = G/N, in the case of mixtures, we must evaluate, la ¼ ðoG=oN a ÞT;P;Nb and lb ¼ ðoG=oNb ÞT;P;Na . Because we perform calculations at only five explicit compositions, we fit the individual values for G, at these compositions, (all at the same chosen T,P) to a continuous functional form taken from regular solution theory [35].

G=N  xa la þ ð1  xa Þlb þ kB T½xa lnxa þ ð1  xa Þlnð1  xa Þ þ xa ð1  xa Þc;

ð10Þ

where l and l are the chemical potentials of pure a and pure b, and c is an adjustable parameter (which absorbs both energetic, and excess entropic effects). Eq. (10) can be applied separately to the liquid and gas phases. This allows for the analytic evaluation of the chemical potential of each species in each phase, which then allows us to solve for the compositions in each of the phases at coexistence. More details on the fitting of G(x) are available in [36] where we studied coexistence in square-well monomer mixtures. Here we only briefly remark that the fits involve, at most, two unknown parameters (c, and either la or lb ; both may not be available for a single phase), and further note that the resulting quality of these fits is quite good.  a

 b

3. Results and discussion Our focus in this work is on modelling the alkanes butane and heptane as square-well dimers and trimers, respectively. We have performed calculations on this mixed model system at five compositions: xbutane = xdimer = 0 (pure trimers), 0.25, 0.5, 0.75, and 1 (pure dimers). As outlined in Section 2.3, at each composition, we calculate the site–site distributions, and thus obtain U, and ultimately A, over a broad range of V,T space. In particular, the determinations cover a 29  25 grid of V,T points, spaced evenly in 1/T and 1/V. Calculations were first performed on the pure systems (x = 0 and 1) to determine the parameters for modeling the mixed systems (x = 0.25, 0.5, and 0.75). The overall strategy is thus to fit the model only to pure component data (butane and heptane), and then, to predict the properties of the mixture. These theoretical predictions will then be compared with the actual experimental results for the mixed system. For the pure square-well chain model, there are a total of four parameters: r, e, k, and n. Only two of these parameters, r and e, will actually be fit to the (pure) alkane experimental data. (n is determined by Eq. (6), and here, k is always 1.75.) Furthermore, for comparison with our lattice model (noted above), we will use only the single best fit r value which has been optimized for both pure butane and pure heptane; this single value will be used for all pair interactions in the mixture. For e, a separate best fit eaa, and ebb, is determined for each of the pure alkanes and eab is then obpffiffiffiffiffiffiffiffiffiffiffi tained using the geometric mean approximation, eab ¼ eaa ebb . In the present work we concentrate on the study of liquid–vapour coexistence. In our fits to (pure) experimental data, it is therefore sensible to focus on physical properties associated with phase equilibria in the pure systems. In particular, we fit to critical temperatures, liquid and vapour coexistence densities, and vapour pressures. The best fit parameters are obtained by simultaneously optimizing agreement with these experimental data. The results of our fits to the critical temperature and coexistence densities are given in Fig. 2, and the resulting pure vapour pressures (i.e., corresponding to the same pure system parameters) can be seen in Fig. 4. (The experimental values for the pure alkanes are taken from Ref. [37]). It is seen in Fig. 2 that the theory is capable of describing liquid–vapour coexistence over a wide range of temperature.2 The theoretical critical temperatures (extrapolated 2 The results in [7] differ from the present results due to an error in that work in the integration of the average force contribution of type ðF ik Þ.

Fig. 2. Theoretical fits to the liquid and vapour coexistence densities for pure butane and pure heptane. Theoretical results are given by solid circles and solid lines; experiment results are given by open circles and dashed lines. Experimental values are taken from Ref. [37].

from the coexistence curves) are in reasonable agreement with the experimental values for both butane and heptane. While the pure vapour pressures (for both butane and heptane) show very good agreement with the experimental values, the coexistence densities are not as close. The liquid coexistence densities in particular deviate by extremes of roughly +9% at 0.55Tc and 9% at 0.9Tc for the case of heptane, and +7% and 18% respectively for butane (at 0.55 and 0.9Tc). It is noted that if we fit to the coexistence densities, alone, (and disregard the data that are very close to the critical region) the agreement can be improved. (In this case the critical temperature is then over-predicted which is common in theoretical approximations.) However, the aim is for simultaneous agreement with vapour pressure data, and improvement of the coexistence density fit (by increasing e and r) is at the expense of the vapour pressure fit. The resulting parameters from the pure fits are as follows: e/kB = 158 K for butane, 182 K for heptane, and, a single best fit r = 3.7 Å for both. These values are reasonably consistent with other microscopic parameterizations of related systems. For instance, typical Lennard-Jones parameters applied to model methane are about e/kB  150 K and r  3.8 Å. Using the above parameters, we now move to the theoretical predictions for the mixed butane–heptane coexistence properties. We compare the results with experimental liquid–vapour equilibrium data from Ref. [38], wherein phase boundaries of the butane– heptane system are reported over a wide thermodynamic range (including the critical region). The results of that paper are summarized in a variety of representations, including temperature–pressure relationships, as well as a number of temperature– composition phase envelopes (i.e. constant pressure slices through the overall phase diagram). In Fig. 3 we compare theoretical predictions with experimental data for the butane–heptane temperature–composition (T–x) phase envelope at a pressure of 100 psi. The theory is in superb agreement with the experimental data, with the theoretical envelope only just slightly wider than the experimental envelope (also found to be true for T–x envelopes calculated at other pressures). As expected, the phase diagram shows, that, at any given temperature, the composition of the vapour phase is always richer in butane than is the composition of the coexisting liquid phase.

R.P. White, J.E.G. Lipson / Chemical Physics Letters 458 (2008) 313–318

Fig. 3. Theoretical predictions for the butane–heptane mixture. Shown is a temperature–composition phase diagram at a pressure of 100 psi (0.7 MPa). The low T side of the phase envelope is the bubble point curve of the mixed liquid; the high T side is the dew point curve of the mixed gas. Theoretical results are given by solid circles and solid lines; experimental results are given by open circles and dashed lines. Experimental values are taken from Ref. [38].

Typically, the theoretical predictions for these ‘dew point’ and ‘bubble point’ compositions are within about 2% of experiment or better. Liquid–vapour phase envelopes of similar quality have also been predicted (for similar alkane mixtures) by a number of the perturbation approaches discussed in the introduction. Theoretical and experimental results for the butane–heptane mixture are again compared in Fig. 4, which gives the mixed system liquid–vapour phase boundaries in the form of a temperature–pressure (T–P) diagram. Three sets of curves are shown, corresponding to specific mixture compositions of: xbutane = 0.8010, 0.4249, and 0.1590. In each set, the curve at low T and high P corresponds to the mixed liquid at bubble point, and the other curve, at higher T and lower P, corresponds to the mixed vapour at dew point. The theoretical and experimental vapour pressure curves

317

(mentioned above) for pure butane and pure heptane are also included. In Fig. 4 the theoretical predictions for the mixture are again seen to be in very strong agreement with experiment. Similar agreement is observed for two other mixture compositions for which experimental data are available [38]; these are not shown to avoid clutter in the figure. Noting that the entire liquid–vapour phase diagram is bounded between the two pure vapour pressure curves, it is clear that the theoretical predictions are robust over a very wide region of the coexistence regime. The theoretical results do begin to deviate from experiment at the highest temperatures and pressures, as the critical line is approached. (Indeed we lack theoretical results altogether close to the critical line.) To explain this, we mentioned above (and in [36]) that typically, G values to be fit to Eq. (10), are not always found for a single phase, gas or liquid, over the entire composition range, x = 0–1. Here, as T and P are increased, we find that the main limitation is due to the loss of points (G values) which are needed for the fit. It is possible to improve on these results by incorporating calculations at an increased number of explicit compositions. Indeed, in ongoing work involving additional mixtures we will include specific calculations at compositions of x = 0.1 and 0.9, which will allow us to extend the theoretical predictions further toward the critical region. 4. Summary and conclusions This work represents the first application of our continuum integral equation (CIE) theory to the study of experimental mixtures. Our results focus on liquid–vapour coexistence properties in the butane–heptane system and include a corresponding comparison with experimental data. Modeling the alkanes as squarewell chain molecules, our CIE method is applied to determine the fluid structure (site–site distributions), and ultimately, its thermodynamic properties. We have successfully applied the theory to predict the equilibrium compositions in the coexisting liquid and vapour phases. In doing so we have required only the model parameters obtained from fits to the pure fluid experimental data (along with the usual geometric mean approximation for eab). The theoretical predictions were shown to be in excellent agreement with the experimental properties of the mixture, including T– x phase envelopes and T–P phase diagrams covering a wide region of the butane–heptane coexistence regime. In upcoming work we will extend our reach in considering other mixed alkane systems as well as a wider range of thermodynamic properties where we will again compare with experiment, and with corresponding results from our lattice theory. Acknowledgements We gratefully acknowledge financial support by the National Science Foundation (Grant No. DMR-0502196), and partial support by the Donors of the American Chemical Society Petroleum Research Fund. We also thank Dartmouth Computing Services for use of the Discovery cluster. References

Fig. 4. Theoretical predictions for the butane–heptane mixture. Shown is a temperature–pressure phase diagram given at three constant composition slices corresponding to a mole fraction of butane of 0.8010, 0.4249, and 0.1590. The high P, low T side of each slice is the bubble point curve of the mixed liquid; the low P, high T side is the dew point curve of the mixed gas. Also shown are the vapour pressures of pure butane and pure heptane. Theoretical results are given by solid circles and solid lines; experimental results are given by open circles and dashed lines. Experimental values are taken from Ref. [38].

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