The principle of congruence and its application to compressible states

/RE ELSEVIER

Fluid Phase Equilibria 105 (1995) 193-219

The principle of congruence and its application to compressible states C.J. Peters a.,, L.J. Florusse a, J.L. de R o o a, j. de S w a a n A r o n s ~', J . M . H . Levelt Sengers b " Delft University of Technology, Faculty of Chemical Technology and Materials Science, Laboratory q[" Applied Thermodynamics and Phase Equilibria, Julianalaan 136, 2628 BL Delft, The Netherlands b National Institute of Standards and Technology, Thermophysics Division, Gaithersburg MD 20899, USA Received 2 April 1994; accepted in final form 11 September 1994

Abstract

The principle of congruence claims that certain thermodynamic and transport properties of a mixture of n-alkanes are the same as those of the pure n-alkane of the mole-fraction-averaged carbon number. We discuss the origin of the principle, and attempts at theoretical justification and validation with respect to experimental data. We demonstrate that the principle applies even in cases where no fundamental justification exists. Whenever the principle applies, prediction of thermodynamic behavior of a mixture of n-alkanes is inherently simpler and usually more accurate than that based on empirical equations of state, molecular interaction parameters, and mixing and combining rules. Our experimental verification serves to pinpoint those properties that can be safely and accurately predicted on the basis of the principle of congruence. The solubility of hydrogen in mixtures of long-chain n-alkanes is used as a prime example.

Keywords: Theory; Experiment; Application; Excess enthalpies; Excess volume; Vapor liquid equilibrium; Compressible states; Hydrocarbons; Principle of congruence

1. Introduction

The principle of congruence was formulated by Bronsted and Koefoed (1946) as a very simple way of relating the thermophysical properties of a mixture of n-alkanes to those of a representative pure n-alkane. They based the principle on their own measurements of the activity coefficient of the volatile component in a mixture of two n-alkanes, and on preceding

* Corresponding author. 0378-3812/95/$09.50 @ 1995 - Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 3 8 1 2 ( 9 4 ) 02614-9

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experimental results and theoretical considerations mostly from the Hildebrand school. In their own words, "The theory of congruence assumes that the thermodynamic properties of normal paraffins are determined by the index of the mixture, and that congruent solutions therefore will show the same values, e.g. for the activity coefficients of dissolved substances, irrespective of the individuality of the components". They define the index of the mixture as the mole-fractionaveraged carbon number. First, we note the usefulness of the principle. It allows us to predict properties of a mixture by equating them to the properties of a pure fluid at the same pressure and temperature. It is not necessary to know the critical parameters, equation of state, mixing and combining rules. Formulations based on the principle of congruence are of utter simplicity compared to any other method devised for describing fluid mixtures. We note that there are various loose ends in this definition that will also plague later work. The variables to be held constant, when the mixture and the pure fluid are compared, are not defined. F r o m the context, it appears that the temperature is kept constant and that the pressure is close to ambient. Secondly, it is not defined which properties are supposed to follow the principle of congruence. Vapor pressures of liquid mixtures of n-alkanes are the property referred to in the paper of Bronsted and Koefoed (1946). In generalization to other properties, and in application to phases that are not dense and incompressible, these points need to be specified. In what follows, we plan to examine alternatingly arguments for the validity of the principle, experimental verifications, and cases where the principle does not hold. We will discuss a wide collection of thermodynamic properties, including excess enthalpies, virial coefficients, dew and bubble curves, lengths of three-phase lines and solubility of gases in n-alkane mixtures. As a conclusion, we will give guidelines as to the applications in which the principle can be expected to hold.

2. Theoretical justification for the principle of congruence Bronsted and Koefoed (1946) give no theoretical justification for the principle of congruence; it is therefore no more than a hypothesis consistent with certain experimental facts. LonguetHiggins (1953) was the first to reason that the principle follows from the statistical mechanics of chain molecules. He argued that in a mixture of long-chain molecules, the joining of the ends of two chains gives rise to a factor in the partition function that depends on the interaction energy of the two ends, and on the pair distribution function of the two ends. The physical assumptions are then made that this factor, being a local property, is independent of individual chain length, and depends only on the number of chain segments per unit volume (the overall density). Therefore the configurational partition function and the configurational thermodynamic properties should be unaffected by the process of cutting chains and rejoining ends differently, which implies the validity of the principle of congruence for configurational properties. In this proof, the thermodynamic variables are specified, but they are temperature and volume instead of temperature and pressure. Longuet-Higgins (1953) does remark that these two sets of variables can be used interchangeably if the mass density of the mixsture and those of the components are all the same at a given pressure. The properties are also specified: they are the

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configurational ones. This will turn out not to be a good choice if generalization to conditions other than those of dense liquid states is desired.

3. Hijmans scaling A large step forward in the theoretical and experimental justification of the principle was made in a sequence of papers by Hijmans (1958,1961, Holleman and Hijmans (1962,1965) and Hijmans and Holleman (1969). These authors were inspired by earlier work by van der Waals and Hermans (1950a,b), van der Waals (1951), Prigogine et al. (1957) and Desmyter and van der Waals (1958). Here we will give a summary of the arguments of Hijmans and Holleman, limiting ourselves to n-alkane molecules with identical segments, for which we take the individual -CH2- groups. Hijmans and Holleman considered the end CH3 group as different from the middle segments. This distinction affects the proof of congruence only in a minor way. It is not made in the simplified perturbed hard chain theory (SPHCT) to be discussed later in this paper. In order to connect with contemporary work, we use the notation customary in SPHCT. Hijmans (1961), generalizing from lattice models of polymers introduced earlier by Prigogine, compares the properties of a fluid of chain molecules with those of a fluid of monomers. To the monomer, a length parameter ~r and an energy parameter E are ascribed, which represent the characteristic diameter of the hard core and the interaction energy of the m o n o m e r respectively. The author now calculates the interaction parameters of segments in chain molecules. In Fig. 1, an artist's view of the repulsive and attractive regions of such a chain are depicted. The number of segments in the chain (the carbon number) equals n. For each chain molecule, the hard core volume is smaller than that of n separate segments because of shielding effects. The number of independent monomers that would have the same excluded volume as one n-mer is indicated by a number s(n) which one would expect to be

Fig. 1. An artist's view of the repulsive(inner shell) and attractive (outer shell) regions of an aliphatic chain molecule.

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proportional to n with an amplitude Sm independent of n and smaller than unity. If the excluded volumes of the two ends (e) are different from those of the middle groups (m), the net effect is a constant 2Se in the postulated relation for s(n):

s(n) = 2se + (n - 2) Sm

( 1)

The number s(n) is part of the scale factor for the volume, v* = Ns(n)a 3, which represents the effective excluded volume of the collection of N n-mers. Likewise, a certain fraction of the volume available to attractive interactions is excluded from attractive interactions with segments of other chains, again because of shielding effects. The number of independent monomers having the same interaction volume as the n-mer is indicated by q(n), which is expected to be proportional to n, with an amplitude smaller than unity. Here the end effects are more subtle, since there will be differences in interaction of end-middle segments, compared to e n d - e n d and middle-middle segments. Hijmans shows that if every type of segment experiences an average surrounding of segments of the two types (mean-field approximation), then for long chains the net dependence on n is similar to that for the excluded volume, so that

q(n) = 2qe + (n - 2)qm

(2)

with qe and qm constants being independent of n and qm smaller than unity. The scale factor for energy is therefore Nq(n)e. Finally, the motions of parts of the chain that require rearrangement in space are not as many as those of n independent monomers, because of the fact that the segments are connected by rigid bonds. This requires the introduction of a third number c(n) which indicates the number of monomers that would have the same number of external degrees of freedom as the n-mer. Again, c(n) is expected to be proportional to n, apart from some departures for short chains:

c(n) = 2c~ + (n

-

2)Cm

(3)

with c¢ and Cm constants independent of n and Cm smaller than unity. Hijmans calls the molecular degrees of freedom that require spatial rearrangements and that therefore will depend on the density, "external" degrees of freedom. From the perspective of monomers, the chain molecule has fewer than the full complement of 3n degrees of freedom. From the perspective of the chain molecule, it has its full complement of internal motions, such as vibrations and rotations, but some of these motions also contribute to the configurational partition function, because they are density dependent. According to the principle of corresponding states, all substances whose intermolecular force laws are identical, apart from an energy and a length scale factor, have the same universal equation of state in properly reduced (dimensionless) variables. In an application to n-alkanes, even if all segments have the same values of E and o-, there are still three adjustable parameters s(n), q(n) and c(n); therefore a law of corresponding states is not found to be valid and cannot be expected to hold. Nevertheless, in this case it is still possible to formulate a universal "external" reduced Helmholtz energy Aext(l?,i~) and equation of state /Dext('V,'F), from which properties of individual n-alkanes are obtained by means of what we will call Hijmans scaling: Aext = Aext( 1~,T)NEq(n)

(4a)

U~xt = (7extNEq(n)

(4b)

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197

V = ~'Ncr3s(n)

(4c)

Sext = S~xtNkc(n)

(4d)

T = TEq(n)/[kc(n)]

(4e)

P = PEq(n) ~[ks(n)]

(4f)

Hijmans scaling is a consequence of the assumption that the "external" energy, entropy and volume of the system can be obtained by summing the contributions from individual segments that experience the average surroundings (mean-field approximation). In the generalization to n-alkane mixtures, Hijmans assumes that all segments still have the same values of E and a. The index i denotes a particular component of the mixture, with mole fractions xi. Each component has its own set of parameters qi, si and c~. The scale factors for the mixture are then composed as follows: qmixt = 2 qixi

(5a)

Smixt = Z i

SiXi

(5b)

Cmixt = 2 i

cixi

(5c)

i

The assumed linearity in n of the functions q(n), s(n) and c(n) then leads immediately to the principle of congruence, because qmixt = Z q ixi = 2 i

[2qe + (Eli - - 2)qm]X/ = 2qe + ( { n ) -- 2)qm

(6)

i

and similarly for Smixtand Cmixt.Here ( n ) is the mole-fraction-averaged chain length {n ) = E~ n~xi. Eq. (6) shows that the three parameters characterizing the mixture are the same as those of the pure fluid of mole-fraction-averaged chain length. By virtue of Eq. (6), the extensive properties of the mixture are the same as those of this pure fluid. This is the content of the principle of congruence, with the important caveat that it is derived for fixed temperature and volume, not pressure. Hijmans argues that for liquid n-alkane applications this distinction is irrelevant. Hijmans' derivation is a good example of the number of assumptions that need to be made in order to justify the principle of congruence: independent additivity of volumetric, energetic and entropic contributions from the segments in the mixture; mean-field assumption about the environment of each segment; linearity in carbon number of the effective volume, interaction energy and external degrees of freedom of a chain molecule; application to cases where constant pressure and constant volume conditions are not very different. In Hijmans's justification of congruence, the independent variables are temperature and volume, and the properties that obey congruence are the "external" properties which the authors appear to identify with configurational properties (apart from the ideal-mixing term, and from some terms depending on temperature only, which do not effect the pressure or the phase equilibria). In the applications Hijmans and Holleman had in mind, namely liquid alkanes below their boiling points, the difference between residual and configurational properties, or between excess Helmholtz energy and excess Gibbs energy is irrelevant. It is, however, easy to see that

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198

in generalization of the principle to compressible states, the application cannot be to configurational properties, since Hijmans scaling, Eq. (4), does not preserve the ideal-gas law. A simple way to cure this problem is to apply Hijmans scaling to residual properties. This is the path we have taken.

4. Experimental tests

4.1. Excess properties Holleman (1963,1964) and Holleman and Hijmans (1965) devised ingenious experimental tests of the principle of congruence for excess volumes and excess enthalpies measured by Holleman (1964). The excess volume is the volume of the mixture compared with the mole-fraction average of the volume of the two pure components at the same pressure and temperature. The reduced temperature, however, are not the same, and therefore the path followed in the theoretical justification discussed before, including the unspecified universal functions T, P and l? in Eq. (4), is not helpful. Here is where the strength of the empirical principle of congruence shows. Suppose one measures, as Holleman did, excess enthalpies or volumes for a number of binary n-alkane systems at various compositions, but all at ambient pressure and a fixed temperature. The excess properties measured for the mixture of the shortest and longest n-alkane in the family are then used to trace out the excess enthalpy or volume curve as a function Of average carbon number. Next, another, intermediate pair is chosen, the chord is drawn between the appropriate carbon numbers on the excess property plot, and the measured excess properties are plotted at the average carbon numbers, with the chord as reference. The principle of congruence states that all enthalpies or volumes so plotted should fall on the same curve. This is indeed what Hijmans and Holleman found. For the excess enthalpies this phenomenon is shown in Fig. 2(a), whereas Fig. 2(b) illustrates the principle for the excess volumes.

4.2. Virial coefficients and congruence The first excursion into the realm of compressible fluids is a test of the principle of congruence for the second virial coefficient. We have a choice here between the second pressure and the second volume virial coefficient, which, however, are exactly proportional. In this work we will talk about the second volume virial coefficient. The principle of congruence then states that at fixed temperature a mixture of n-alkanes of mole fraction x has the same second virial coefficient as a pure component of the mole-fraction-averaged carbon number. Note that the principle of congruence is not identical with the rule of Lewis and Randall, which states that the second virial coefficient of the mixture is the mole fraction average of those of the two components. The prediction of the second virial coefficient of a mixture, based on the exact principle of quadratic mixing Bmixt(T)

=

(1 -- x)2B,, (T) + 2x(1 - x)Blz(T) + x2B22(T)

(7)

where x is the mole fraction of component 2, forms the starting point of a more sophisticated prediction of mixture second virial coefficients from the interactions of the different pairs of

C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193-219

199

0.00

150

-0.50 100

Vqcm3.mot-1

Ht/j.mot-1 I

- 1.00

-1.50

- 2.00

-50 (a)

,

,

,

h

12

18

2/+

30

-----

n

Fig. 2. (a) H E vs. carbon number n-C6-i-n-C16; O, n-fg-[-n-C36; V, carbon number n at T = 3 7 9 . 1 5 K n-Cs + n-C24; II, n-C9 + n-C36; ~,

-2.50

36

8 (b)

n at T=349.15 K and P = 0 . 1

,

,

,

,

k

,

12

16

20

2t~

28

32

,,-

n

36

MPa: A n-C6+n-C36; A, n-Cs+n.C32; ©,

n-C7--[-n-C36 , [~, n-C8 +n-C24; m, n_C16_~_n.C36; ~ , n_C6..~.n.C24. (b) V E vs.

and P = 0 . 1

MPa: O, n-f8-k-n-C36; A, n-fs-I-n-Ci6; A n_C9+n_C24; V,

n-C8 + n-C32.

component molecules. Here B1~ and B22 are the second virial coefficients of the two pure components and B12 is a mixed second virial coefficient that is not known a priori. Even for conformal two-parameter pair potentials, which are inadequate for chain molecules, it is necessary to know the potential parameters or critical parameters of the pure components, and to postulate pseudocritical parameters or unlike-pair potentials if the second virial coefficient of the mixture is to be predicted. Only very specific assumptions about the dependence of those interactions on the carbon number, including those of the representative pure substance, would ever lead to a principle of congruence. The principle is therefore much simpler than its potential justification. It seems of interest to inquire how well the principle of congruence predicts the second virial coefficients of a mixture of n-alkanes, compared to the Lewis and Randall rule or more complex results from statistical mechanics. A first test of this kind was performed by Barker and Linton (1963) on the basis of highly accurate second virial coefficients for n-alkanes, with 3 < n _< 8, measured by McGlashan and Potter (1962) in the range from 298 to 413 K. The latter authors had already shown that the Lewis and Randall rule gave a very poor approximation to the virial coefficient of mixtures with results at least 20% more negative than the experimental data (see Fig. 3). McGlashan and Potter (1962) also developed a new correlation for the pure-fluid and mixture data in the following way. They fitted the temperature dependence of the second virial coefficient of the six pure alkanes by an empirical expression which obeyed simple corresponding states (requiring input of two critical parameters per substance) except for one temperature-dependent term which was linear in the carbon number: This expression was shown to represent the second virial coefficients of pure n-alkanes to within experimental error. The expression was generalized to mixtures by assigning pseudocritical parameters and substituting the mole-fraction-averaged

200

C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193-219 o

B~2/c~mo[ -~

t

- 51)0-

/ /././ / /" -1000

-

300

~ IlK

L,O0

SO0

Fig. 3. Experimental second virial coefficient data as a function of temperature (McGlashan and Potter, 1962): [], pure pentane; ©, equimolar mixture of propane + heptane; -. , Blz of an equimolar mixture of propane + heptane, calculated from the correlation of Tsonopoulos (1978) with k~2 = 0; - - - , application of the principle of congruence on the correlation of Tsonopoulos (1979) for the equimolar mixture of propane + heptane, i.e. (n) = 5; , B~2 of an equimolar mixture of propane + heptane, calculated from the Lewis and Randall rule. carbon number. The resulting expression represents the second virial coefficients for the system p r o p a n e - h e p t a n e with no more than 5% systematic error, the mixture data falling above the calculated curve. Barker and Linton (1963), however, showed that the mixture data agreed to within 4% with the measured second virial coefficients of pure pentane (see Fig. 3), thus demonstrating that the principle of congruence performed at least as well as the more complex correlation o f M c G l a s h a n and Potter (1962). Barker and Linton (1963) also showed that the second virial coefficient of an equimolar mixture o f methane and propane equals that of ethane to better than 1 cm 3 mo1-1 at temperatures from 377 to 510 K. We n o w turn to the more recent correlation of second virial coefficients by Tsonopoulos (1979). This author used the P i t z e r - C u r l relation which is a generalized corresponding-states treatment using the acentric factor as a third parameter. The main effort in Tsonopoulos' paper was directed at generalizing this expression to mixtures. This involved devising combining rules for the pseudocritical parameters of the equimolar mixture, and of its acentric factor. The pseudocritical pressure and the acentric factor of the mixture were calculated from pure-component values without further adjustable constants, whereas for the pseudocritical temperature an additional binary interaction constant k~2 was used to correct the geometric-mean rule. Much effort was devoted towards optimizing k~2. In Fig. 3, we show the experimental second virial coefficients for the equimolar mixture of propane and heptane and for pure pentane, as measured by McGlashan and Potter (1962), the correlation of T s o n o p o u l o s for pure pentane, and his prediction for the second virial coefficient o f the mixture, based on k12 = 0 (we have checked that the prediction is very insensitive to the choice o f k~2). We note that Tsonopoulos's mixture prediction falls almost as far above the experimental curve as the Lewis and Randall rule falls below it. Fig. 3 vividly demonstrates the power of the principle of congruence. It gives a reliable estimate of the second virial coefficient

C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193-219

201

of the mixture based solely on pure-fluid data and requires no knowledge of critical or pseudocritical parameters. Critique of the principle of congruence for this application has been voiced by Dantzler et al. (1968). These authors measured excess second virial coefficients, which express the difference between the mixed virial coefficient B~2, Eq. (7), and that according to the Lewis and Randall rule: B E = B.2-- (B~I + B22)/2

(8)

They measured B E for 15 binary mixtures of normal alkanes, methane through hexane, at temperatures from 198 to 373 K. Dantzler et al. (1968) showed that their data could be accurately represented by the three-parameter representation of McGlashan and Potter (1962), discussed earlier. They then proceeded to test the principle of congruence by transforming their excess second virial coefficient data to excess volume (V E) data by means of the approximate relation V E = Vmi×(P,T) - [ V I ( P , T )

+ V 2 ( P , T ) ] / 2 = 2x~x2B E

(9)

where V~ and V2 are the molar volumes of the two pure components. Dantzler et al. (1968) find reasonable adherence to the principle of congruence but remark that there are large departures, especially for the higher hydrocarbons and at the lower temperatures. They ascribe the departures to deviations from random mixing of segments that should be more pronounced under these circumstances. It appears to us, however, that the use of the approximate expression (Eq. (9)) should first be questioned. The relation between VE and B E is given by V E = (RT/P)[Bm/Vm

- x ~ B ~ / V ~ - x 2 B 2 2 / V 2 --~ O ( V

2) ~_...]

(10)

In order to retrieve the form (Eq. (9)) used by Dantzler et al. (1968), it is necessary to assume that Vm equals V~ and V2 at fixed pressure (about 1 bar) and that higher order contributions are neglegible. F r o m the virial coefficient data it is clear, however, that the nonidealities are large, especially at the lower temperatures and for the higher n-alkanes. The three volumes can easily differ by several hundred cubic centimeters per mole. Thus it is not excluded that the scatter in the calculated VE values that the authors observed is due to the approximate character of Eq. (9) instead of being due to failure of the principle of congruence. Finally, we comment on a recent paper by Schouten et al. (1990), in which the application is to the accurate metering of natural gas. Although these authors see great advantage in characterizing a hydrocarbon mixture by an "equivalent hydrocarbon", they point out some of the drawbacks. One is an obvious one, namely that in general the mole-fraction-averaged carbon number is not an integer. There are various ways of dealing with that problem, as we have seen in the work of Hijmans (1958,1961), Holleman and Hijmans (1962,1965) and Hijmans and Holleman (1969), McGlashan and Potter (1962), and Dantzler et al. (1968), but they all take away from the directness of the application of the principle of congruence. The other drawback is the limited accuracy. Schouten et al. (1990) compared the second virial coefficient of pure ethane with that of an equimolar mixture of methane and propane, and of a 75%:25% mixture of methane and pentane at temperatures from 273 to 343 K. These temperatures are much lower than those of Barker and Linton (1963), and the departures from the principle of congruence are

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C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193 219

of the order of 10 cm 3 mo1-1. Given the fact that the pentane virial coefficient is of the order of - 1 0 0 0 cm 3 mo1-1 at this temperature, however, we consider this error not to be really substantial. At ambient pressure, it will cause an error of 0.05% in the density, which is below the uncertainty tolerated in the metering of natural gas (0.1%).

5. Phase boundaries and congruence

In the following sections, we will be concerned with the phase boundaries of systems containing long-chain n-alkanes, namely dew and bubble curves, three-phase regions, and the solubility of gases in mixtures of n-alkanes. In the process, we will also investigate the behavior of a particular equation of state designed for chain molecules, which obeys the Hijmans conditions for congruence under certain circumstances, the simplified perturbed hard chain theory (SPHCT), and we will demonstrate that although its bubble curve conforms closely to the principle of congruence, the dew curve does not. The principle of congruence equates the residual properties of a homogeneous mixture of n-alkanes at fixed temperature and pressure to those of the representative, carbon-number-averaged pure alkane. Even if the principle holds, properties involving coexisting phases, such as dew and bubble points, cannot be expected to be the same for the mixture as for the representative pure alkane, because the mixture separates for a different reason, i.e. material instability, than does the pure fluid. Thus in P,T space, the phase boundary of a binary alkane mixture of fixed composition encloses a two-dimensional region, while the boundary is a one-dimensional curve for the representative pure alkane. We must therefore expect that there will be differences in location of phase boundaries when the n-alkane mixture is replaced by the representative pure n-alkane. Exceptions may occur when the concentration of the nonvolatile n-alkanes is negligible in one of the phases. This could happen at bubble points of a mixture of long-chain n-alkanes in methane or ethane at low pressure, where the gas phase is almost pure methane or ethane. Also our last and most convincing example of the validity of the principle of congruence, the solubility of hydrogen in mixtures of n-alkanes, will be such a case.

5.1. The simplified perturbed hard chain theory and the principle of congruence Beret and Prausnitz (1975) and Donohue and Prausnitz (1978) have developed an equation of state for chain molecules that follows closely the ideas outlined by Hijmans, discussed earlier in this paper. Here, we give a short summary of this model, based on the papers by Beret and Prausnitz (1975), Donohue and Prausnitz (1978), and Kim et al. (1986). For pure n-alkanes, the SPHCT uses the same three n-dependent factors s(n), q(n) and c(n) which account for the reduction of, respectively, the excluded volume, the attractive energy and the number of degrees of freedom for a chain compared to its constituent monomers. An assumption of linearity in the carbon number is made for each of these factors, and is corroborated by experimental evidence. The theory is formulated very generally, admitting different segment types in different chains, with s, q and c depending on the type of segment. The authors do not make the hypothesis of randomness proposed by Hijmans, because they want to allow for local composition variations as a consequence of differences in interaction between segments of different types. As a

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consequence, the energy no longer obeys Hijmans scaling if different types of segment are present. For the application to n-alkanes, it is assumed that all segments are equal. The pure-fluid residual Helmholtz energy is given by Kim et al. (1986) Ares/NkT = cArep -q- CZM l n [ V / ( V + v* Y]

(11)

where Y = exp(Eq/2ckT) - 1

(12)

and ZM is a constant. The repulsive part Are p is purely a function of V/v* where v* is Hijmans's scale factor for the volume, apart from a factor of ,,]/2, so Arep is a function of ft. Here Y is an exponential function of T solely, and the term within the square brackets in Eq. (11) is a function of T and 1~. Absolute temperature T scales as c/eq, so Ares scales as eq times a function of T and 17. Comparing with Eq. (4), we conclude that SPHCT for pure n-alkanes obeys Hijmans scaling. The interesting point is that SPHCT does not assume that a given segment experiences the average environment, as Hijmans does. In applications to mixtures, the SPHCT assumes that molecule i and molecule j may have not only different chain lengths, but also different segments. The mixing rule for the effective excluded volume 1

v*(mixture) = ~i

xivi* = N/f2 "~ixiNsia3

(13)

is innocuous enough, since the effective repulsive interactions are added over the individual segments. Likewise, the rule for the effective number of degrees of freedom c(mixture) = ~ xici

(14)

i

is innocuous, being another addition over all segments. The quantity Y, however, according to Kim et al. (1986), is averaged as cv* Y(mixture) = ~ ~ i

XiXjCil)j~'i[exp(cijqj/2cikT)

- 1]

(15)

j

Here Eij= Eji is the geometric mean of E;~ and ~jj, and v*, we believe (Peters et al., 1988), is given by 1 vii* = ~f~ N~r3 sj

(16)

and therefore not equal to v*, even if we assume the arithmetic-mean rule for a~j (Kim et al., 1986). The quantity Y will not be a function of a reduced temperature 7~ simply related to T by means of a single scale factor containing mixture-average values of q and E. Instead, Y will be proportional to a sum of three or more exponentials, each with a different prefactor of T. Only in the case where E and o- are equal for all segments will Hijmans scaling result. This is not really a limitation, because, although Hijmans did make a distinction between middle and end groups, he limited himself to n-alkane mixtures where the middle groups are the same. In conclusion, since the SPHCT equations for mixtures of chain molecules with identical segments obey Hijmans scaling, and since SPHCT assumes linearity in carbon number of the

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C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193-219

three scale factors s(n), q(n) and c(n), SPHCT obeys the principle of congruence with volume and temperature as the independent variables, even though the mean-field approximation has not been made. 5.2. Three-phase equilibrium

In this section the relationship between the quasi-binary approximation and the principle of congruence will be discussed. We focus on the three-phase equilibrium (l~ 12g) that may occur in binary and ternary mixtures in which the major component is a volatile n-alkane of low carbon number, such as methane, ethane or propane, and the other component(s) are less volatile n-alkanes of much higher carbon number and low mole fraction, typically a few per cent. These three-phase equilibria display type V phase behavior in the classification of Van Konynenburg and Scott (1980) and they occur in the near-critical region of the volatile component in the binary or ternary mixture. For details on this type of phase behavior in volatile-nonvolatile n-alkane mixtures, we refer to the work of Scott and co-workers, to be discussed below, and to Peters (1986) and Peters et al. (1986,1989). Scott and co-workers initiated experimental work on these systems because of their interest in tricritical phenomena. For a recent review, see Pegg et al. (1990). Since according to the phase rule it is impossible to reach a tricritical point in a binary fluid mixture, the first study, that of Creek et al. (1981), introduced the concept of the quasi-binary approximation. This approximation assumes that a ternary mixture, with the two nonvolatile components only slightly different in chain length or structure, effectively behaves like a binary mixture. The criterion for the validity of the hypothesis is that the mole fraction ratio n~/n2 of the two nonvolatile components is the same in the three coexisting phases. Creek et al. (1981) tested this hypothesis in ternary mixtures of methane +2,2-dimethylbutane+2,3-dimethylbutane, and found that it held to better than 2% in the mole fraction ratio in the different phases. In questioning the relation of the quasi-binary approximation to the principle of congruence, we caution again that, because of the phase rule, the principle of congruence can only exceptionally by applied to phase coexistence. If the quasi-binary approximation holds, a mole-fraction-averaged carbon number can be defined that is the same in the three phases, holding out hope that the principle of congruence may be valid; if the quasi-binary approximation does not hold, it is impossible to define such a single mole-fraction-averaged carbon number, and the principle of congruence cannot be applied. A test of the quasi-binary approximation will therefore lead to a necessary but not sufficient test of the application of the principle of congruence to this type of phase equilibrium. The system used by Creek et al. (1981), with methane as the volatile component, does not fall in the category of n-alkanes, to which the principle of congruence is restricted. An earlier study, by Kim et al. (1967), however, for the system ethane + nonadecane + eicosane, which does fall in the class for which the principle of congruence is formulated, had already demonstrated the mole fraction ratios in the two liquid phases differ by at least 8% and sometimes by as much as 13%, depending on overall composition, temperature and pressure. Other systems studied by Scott and collaborators include that of ethane + octadecane by Specovius et al. (1981) and that of ethane + hexadecane + eicosane by Goh et al. (1987). The latter tested the quasi-binary approximation by measuring the composition ratios of hexadecane

C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193-219

205

and eicosane in the three phases; chosing the carbon numbers of the two nonvolatile components to be further apart obviously increases the stringency and sensitivity of the test. In line with earlier findings by Wagner et al. (1968), deviations between the mole ratios in the different phases were found to be as high as 40%. With departures from the quasi-binary approximation this obvious, it cannot be expected that the three-phase behavior of the ethane + hexadecane + eicosane system is identical with that of ethane + octadecane. Also, the claim of Goh et al. (1987) that in the ternary mixture ethane + heptadecane + octadecane the quasi-binary approximation holds to at worst 3% seems on the optimistic side, given the much larger differences found by Kim et al. (1967) in the related ternary system ethane + nonadecane + eicosane. We test the validity of the quasi-binary approximation, as a necessary condition for applicability of the principle of congruence, by a comparison of the length AT = TucEP-- TLCEP (where UCEP is the upper and LCEP the lower critical endpoints) of the three-phase region 1112g in binary and ternary mixtures of n-alkanes with ethane and propane, respectively, as the near-critical solvent. In the ternary systems the amounts of the two long-chain n-alkanes were chosen in such a way that the mole-fraction-averaged carbon number of both components equals the carbon number of the long-chain n-alkane in the corresponding binary mixture. Fig. 4 shows, for the ethane mixtures, the length of this three-phase region as a function of the carbon number ratio nl/n2 of the two long-chain n-alkanes. In all mixtures the average carbon number ( n ) was 18. Experimental data points from Goh et al. (1987) are also included. Similar results, by Peters et al. (1989), for the ternary mixture of propane with equimolar mixtures of two n-alkanes of average carbon number {n) = 40, are shown in Fig. 5. In both cases, the length of the three-phase region increases rapidly as the ratio of the carbon numbers of the two nonvolatile components increases. This evidence supports the earlier conclusions of Wagner et al. (1968) and of Goh et al. (1987) that as the two nonvolatile components become

5.0 24112 iii I" /*.0' iI

L~T/K /

I

I

/

I 3.0.

/

/

/ // / iI

2.0' /I

I

, ~ 22/14 i/

1.0'

/// //

/ H20/16

1.0

t~ nl/N 2

Fig. 4. Length A T o f the three-phase region 1112g of e t h a n e - b a s e d ternary mixtures of long-chain n-alkanes as a function o f the c a r b o n n u m b e r ratio nt/n2; (22,14) denotes n~ = 22 and n 2 = 14.

206

C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193-219

/

,oP50/30

/ /

AT/K

J / /

T

// / /

lO

/

r

/ I ss /

. ~"td, / 36 s

i

i

i

1

I

1.0

i

1.5 r11/rl

I

I

I

2.0

2

Fig. 5. Length A T of the three-phase region ]tl2g of propane-based ternary mixtures of long-chain n-alkanes as a function of the carbon number ratio n~/n2; (44,36) denotes n~ = 44 and n2 = 36.

more dissimilar, the quasi-binary approximation will become poorer. In addition, a comparison of the ethane-based ternary mixtures and the propane-based ternary mixtures shows that the quasi-binary approximation is worse for the p r o p a n e + ( ~ n ) = 4 0 ) systems than for the ethane ÷ ( ~ n ) = 18) systems. Since the quasi-binary approximation does not hold for these ternary three-phase systems, the principle of congruence, in the sense of Bronsted and Koefoed (1946), cannot be expected to hold either. The failure of the principle of congruence had already been recognized by Goh et al. (1987). 5.3. Bubble and dew points

This section presents the testing of the SPHCT equation of state, proposed by Kim et al. (1986), for predicting bubble and dew points in mixtures of long-chain n-alkanes and methane or ethane. Peters et al. (1988), and also Gasem and Robinson (1990), tested this equation on its applicability to hydrocarbon mixtures with large differences in chain length. In both cases the original parameters given by Kim et al. (1986) were used. We have already demonstrated that Hijmans scaling and the principle of congruence hold for this equation if applied to n-alkanes, with volume and temperature as independent variables. We make use of the observed linearity in carbon number of the model parameters to obtain these parameters for n-alkanes of such high carbon number that data required to evaluate the pure-component parameters are not available. We compare the predicted bubble and dew points of a ternary mixture, containing two different long-chain n-alkanes in a short-chain n-alkane, with those of the equivalent binary mixture, as prescribed by the principle of congruence. We begin with the bubble curve, which originates at zero mole fraction of the short-chain solvent n-alkane. Depending on the pressure, the liquid phase may have mole fractions ranging from 0.08 to 1.0 in the long-chain component. The coexisting gas phase is rich in the short-chain component. In Fig. 6 the principle of congruence is tested for the ternary mixture CI + nCI0 + n-C32, where n-Ci denotes a normal alkane of carbon number i. The long-chain n-alkanes

C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193-219

207

35

35 30

o

25 PIMPn

piMPa 25 o

15

15

10

10 o

o

S

S

0

i

0 CHL.

t\ \ \

Q2

0./~ m X

0.6

0.8

1.0 C16 H3~

i

0

Q2

0./, ~

0.6 X

0.8

1.0 C20H~.2

Fig. 6 (left). Predicted bubble points by the SPHCT equation of state (all k(j = 0.02) at 423.15 K in the binary mixture methane + hexadecane (solid curve) and the quasi-binary mixture methane +decane + dotriacontane with ( n ) = 16 (broken curve). The empty circles represent experimental data for the binary system methane + hexadecane, taken from Glaser et al. (1985). Fig. 7 (right). Predicted bubble points by the SPHCT equation of state (all k u = 0.02) at 313.15 K in the binary mixture methane+eicosane (solid curve) and the quasibinary mixture methane+decane+dotriacontane with ( n ) - - 2 0 (broken curve). The empty circles represent experimental data for the binary system methane + eicosane, taken from van der Kooi (1981).

are mixed in such a way that the mole-fraction-averaged carbon number is ( n ) = 16. The broken curve represents the result for the bubble point curve of the ternary mixture and the solid curve for that of the binary mixture C, + n-C16. The empty circles are experimental data of the binary C, ÷ n-C16, taken from Glaser et al. (1985). From this figure it can be seen that the principle of congruence is close to holding for the SPHCT equation. On the other hand, the predictions obtained from the SPCHT equation are not in complete agreement with the experimental data, be it that this was not the primary objective. Fig. 7 compares the bubble point curve of the ternary mixture C, ÷ n-C,0 ÷ n-C32 (broken curve), with mole-fraction-averaged carbon number of the long-chain n-alkanes ( n ) = 20, with that of the binary C~ + n-C20 (solid curve). The empty circles are experimental data, taken from van der Kooi (1981). Because the primary objective of this study is not so much to obtain optimum agreement between experiment and model predictions as to test the principle of congruence, no special attention was given to optimizing the k u values in the calculations. For the insignificant role of the k u values in the SPHCT phase equilibrium calculations, one is referred to Peters et al. (1988,1992) and Gasem and Robinson (1990). Similar conclusions, as pointed out for Fig. 6, hold for Fig. 7. Figs. 8(a)-8(c) compare the bubble point curves of the ternary mixture C 2 + n C 1 6 ÷ n-C24 (broken curves) for ( n ) = 20 with those of the binary C2 + n-C20 (solid curves) at three different temperatures. The experimental data are taken from Peters et al. (1987). The figure shows clearly that the principle of congruence nearly holds for the SPHCT bubble point

208

C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193-219 15-

(a)

15-

15

(b)

T=350K

T=400K I

i

10

)0

10-

0

(c)

O

~

T= 450K

10

p/HPa

5 -

0

5 -

I

0 0.2 C 2H6

I

I

I

0.~

0.6 X

0.8

0 1.0 C2o H~2

I

0 0.2 CzH6

1

I

I

0.4

0.6

0.8

=

X

0 1.0 C 2oH~2

I

0 0.2 C2H6

I

I

I

0./.

0.6 X

0.8

1.0 C2oH~2

Fig. 8. Predicted bubble points by the SPHCT equation of state (all k u = 0.02) at 350 K (a), 400 K (b) and 450 K (c) in the binary mixture ethane +eicosane (solid curves) and the quasi-binary mixture ethane + hexadecane + tetracosane with { n ) = 20 (broken curves). The empty circles represent experimental data of the binary system ethane + eicosane, taken from Peters et al. (1987).

predictions, and also that the experimental data are in close agreement with the predictions obtained from this equation. Finally, Figs. 9(a)-9(c) show the bubble point curves of the ternary C2 + rt-C20 + r/-Cs0 at a temperature of 432.2 K. In Fig. 9(a) the overall composition of the long-chain n-alkanes was chosen in such a way that {n) = 28 i.e. the ternary bubble point curve has to be compared with the binary C2 + n-C28. Fig. 9(b) shows a similar comparison with the binary C2 + n-C36 and Fig. 9(c) with the binary C2 +/'/-C44. The empty circles in these three figures represent experimental data points, taken from Gasem et al. (1989). Similar conclusions, as pointed out for Fig. 8, hold for Fig. 9. The results presented in Figs. 6-9 indicate that for predicting bubble points of mixtures of long-chain n-alkanes in a short-chain solvent, the principle of congruence in terms of P and T as independent variables is an excellent approximation. This conclusion is not based on experimental observation but on model calculations with the SPHCT equation of state, which, in turn, are corroborated by experimental data. There also exists direct experimental evidence that bubble point curves of mixtures of long-chain n-alkanes fulfill the principle of congruence, be it for a short-chain solvent that is not an n-alkane. Figs. 10(a) and 10(b) show, at two different temperatures, bubble points (circles) of the ternary mixture C2 (ethylene)+n-C20+n-C40, with mole-fraction-averaged carbon number {n) = 26 for the two long-chain n-alkanes. In both figures the squares are experimentally determined bubble points of the binary system C2 ÷ n-C26. All experimental data points are taken from de Loos et al. (1984,1985). The solid curves are best fits to the experimental data. It is striking that up to the near-critical region of the binary C2 + r/-C26, at approximately 25 MPa, the principle of congruence holds.

209

C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193 219

15.0

15.0

PIMPa

PIMPo

(a)

1S.O

I 10.0

I 10.0

5.0

5.0

C2H6

Q2

Or,

X

CzsH~

0 (b)

P/MPo T10.0

5.0

0

0.6 0.8 1.0

\

CzH6

0

02 0.4 0.6 0.8 1•

0 (c)

CzH6

0.2 0.t, 0.6 0.8 X

1.0 E,..~.HgO

Fig. 9. Predicted bubble points by the SPHCT equation of state (all klj = 0.02) at 423.2 K. (a) , The binary mixture ethane + octacosane; , the quasi-binary mixture ethane + eicosane + pentacosane with ~n) = 28; ~, experimental data of the binary mixture ethane + octacosane taken from Gasem et al. (1989). (b) - - - , The binary mixture ethane + hexatriacontane; - - -, the quasi-binary mixture ethane + eicosane + pentacosane with {n) = 36; O, experimental data of the binary mixture ethane + hexatriacontane taken from Gasem et al. (1989). (c) , The binary mixture ethane + tetratetracontane; - - - , the quasi-binary mixture ethane + eicosane + pentacosane with {n) = 44; O, experimental data of the binary mixture ethane + tetratetracontane taken from Gasem et al. (1989).

Finally, we question the applicability of the principle of congruence to the prediction of dew points o f methane-based mixtures of n-alkanes. In the cases to be presented, only the nonvolatile solutes, not the methane solvent, are included in the tests o f congruence. C o m p a r e d to bubble-point data, accurate dew-point data for these mixtures are scarce, even in the case of binary systems. Therefore we base our comparison of dew points o f ternary systems with those of equivalent binary ones principally on the S P H C T equation of state; the few available data for dew points of binary m e t h a n e - n - a l k a n e mixtures are used as a check on the performance of S P H C T for this application. First we compare the predicted dew point curve of methane with equimolar hexane and decane with that of methane and octane, measured by Kohn and Bradish (1964). Figs. 1 l ( a ) - 1 l(c) show the results in the isobaric t e m p e r a t u r e - c o m p o s i t i o n representation, at pressures of 2.0 MPa, 5.0 M P a and 8.0 M P a respectively. The S P H C T equation, which itself obeys the principle of congruence, represents the binary experimental dew points well. However, Figs. 1 l ( a ) - I l(c) show quite clearly that the dew temperatures of the ternary mixture predicted by the S P H C T equation are much higher than those o f the equivalent binary mixture, thus demonstrating the failure of the principle of congruence for the prediction of dew curves. Likewise, in Figs. 1 2 ( a ) - 12(c), the isobaric dew curve of the equimolar mixture of octane and dodecane in methane is compared to that of the equivalent binary mixture o f methane and decane, again in T , x coordinates, and at pressures of 1.0 MPa, 5.0 MPa, and 10.0 M P a respectively. The experimental data for the binary mixture (circles) are from Rijkers et al. (1992). In this case o f lower dew temperatures and longer-chain admixtures compared to the

210

C.J. Peters et al. / Fluid Phase Equilibria 105 (I995) 193 219 35

35

30

30

25

P/HPa

25

]

PIN Pa

20

20

15

15

10

10

0 OO0

0.20

0./,0

0.60

0.80

0 0.00

1.00

0.20

0.4.0

X (a)

0.60

0.80

1.00

•-~ X C~Hs4

CzHz.

(b)

f-2Hz.

C26H~

Fig. 10. Experimental evidence of the principle of congruence for bubble points of mixtures of long-chain n-alkanes. (a) [], Bubble points at 348.15 K in the binary system ethylene + hexacosane; ©, bubble points at 348.15 K in the quasi-binary system ethylene + eicosane + tetracontane with ( n ) = 26; - - - , best fit to the empty squares. (b) ~ , Bubble points at 423.15 K in the binary system ethylene+hexacosane; (3, bubble points at 423.15 K in the quasi-binary system ethylene + eicosane + tetracontane with {n) = 26; , best fit to the empty squares.

450 T(K] 400

T(K) i

1400

I Ill

350

350

300 . . . . . . . . . Ell4 O.OS

(a)

SS/~IsJ

4S0

"Y

//

(b)

/

T(K)

0

400

///

7

350

300 . . . . . . . . . CH~, O.OS

0.10

450

"Y

300 . . . . . . . . . CH~. 0.05

0.10

(c)

0.10

"Y

Fig. 11. Predicted dew points by the SPHCT equation of state (all kij = 0.02) at 2.0 MPa (a), 5.0 MPa (b) and 8.0 MPa (c) in the binary mixture methane + octane (solid curves) and the quasi-binary mixture methane + hexane + decane with ( n ) = 8 (broken curves). The empty circles represent experimental data of the binary system methane + octane, taken from Kohn and Bradish (1964).

C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193-219

350

350 T(K)

211

350

T(K)

T(K) 0

300

T300 /

300

/

0 0

250

200 [Ha

(a)

250

.

.

.

.

.

05.10 -4

'- Y

.

.

.

1.0 10"~'

.

250

.200.

(b)

., , . , .

CH¢

.

200

05"10 -~'

1.0 10-~'

- Y

...... CHt,

(c)

, , , QS.IO -~'

1.0 10-~'

, y

Fig. 12. Predicted dew points by the SPHCT equation of state (all k u = 0.02) at 1.0 MPa (a), 5.0 MPa (b) and 10.0 MPa (c) in the binary mixture methane + decane (solid curves) and the quasi-binary mixture methane + octane + dodecane with = 10 (broken curves). The empty circles represent experimental data of the binary system methane + decane, taken from Rijkers et al. (1992).

previous case, the SPHCT equation gives a poorer description of the dew temperatures, especially at the higher pressures. As in the case of Fig. 11, however, for any but the highest pressure, Fig. 12 demonstrates, for the SPHCT equation, inapplicability of the principle of the congruence to the prediction of dew points of ternary mixtures from those of the equivalent binary one. It is plausible that the dew point is principally determined by the longest-chain alkane present rather than by the mole-fraction-averaged carbon number. We conclude that the use of the principle of congruence for the prediction of bubble points in mixtures of methane and long-chain n-alkanes appears to work quite well. Here the mixture is in a liquid state of low compressibility, while the incipient compressible phase is almost pure methane. The principle of congruence fails, however, for the prediction of dew points, where the mixture finds itself in a compressible vapor phase, the incipient liquid phase being mainly composed of the alkane with the longest chain length. Although we have not investigated this point, we expect that in the reverse case, that of a long-chain component dissolved in a mixture of short-chain molecules, the opposite situation will prevail: the dew point might be well predicted by the principle of congruence while replacing the mixture of short-chain components by a single component, whereas the bubble point might not be well predicted on the basis of the principle of congruence.

5.4. Solubility of hydrogen in mixtures of heavy n-alkanes At Delft University of Technology the solubility of hydrogen has been measured in pure long-chain n-alkanes and in mixtures of two long-chain n-alkanes. The experimental work covers a temperature region from 300 to 450 K and pressures up to 15 MPa were applied. Hydrogen solubilities of up to 30 mol% were measured. The experimental work was carried out

212

C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193 219 n =z.6~ T : ~-00 K

n:28

n=36

n:16

I0 p/MPa

I

I

I

I

I

0.05

01

0.15

0.2

0.25

0.3

XHz Fig. 13. Isothermal solubilities at 400 K of hydrogen in pure long-chain n-alkanes with carbon numbers 10, 16, 28, 36 and 46: ©, experimental; best fit to Eq. (17).

15 T =z+50K n=t.6

n=28~, n=36C)

10 p/i'4Pa

n=16

I

[

1

I

I

005

01

0.15

0.2

0.25

0.3

XH2

Fig. 14. Isothermal solubilities of 450 K of hydrogen in pure long-chain n-alkanes with carbon numbers 10, 16, 28, 36 and 46: (3, experimental results; - - , best fit to Eq. (17).

in the following binary hydrogen + n-alkane systems: H 2 + n-C10 , H 2 + rt-C16, H2 4- n-C28, H 2 +/7-C36 and H2 + n-C46. In addition, hydrogen solubilities were also measured in the ternary systems H 2 4- n-Clo 4- F / - C I 6 , H2 4- r/-Clo 4- n-C2o, H2 -4- n-Ca0 + F / - C 2 8 , H2 + n-Clo + n-C36 and H 2 + n-Clo +/7-C46. For the binary mixtures a total number of 347 data points were collected and for the ternary ones 158 data points were obtained. Details of the applied experimental technique are described by de Loos et al. (1983), Peters (1986) and Coorens et al. (1988). In order to represent the experimental solubility data for the binary systems, an empirical model was developed by one of us (L.J.F). It expresses the pressure P as a poloynomial in temperature T, mole fraction x of hydrogen in the liquid phase, and carbon number n: p=

RuTJ 1 n~ 1+ ~ i=1

=1

k=l

RukTJ-I n ~-1 x k ~ i=1

(17)

=1

In this equation, R u and Ruk are adjustable parameters. Figs. 13 and 14 show the solubility of hydrogen as a function of pressure at 400 K and 450 K respectively, for the five different binary mixtures. The empty circles are experimental data and the solid curves are their representation by Eq. (17).

C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193-219

213

IS -CI°*n-C16

p/MPa

I

I

I

350

t,00

t~50

T/K

Fig. 15. Solubility of hydrogen in mixtures of decane + hexadecane ((3, experimental results; , prediction from Eq. (17)): curve A = 14.56; curve B = 15.12; curve C, = 12.72; curve D, = 10.98; curve E, (n> = 13.78; Curve F, (n> = 10.72; curve G, = 12.80. To our surprise, we have found that Eq. (17), without the use of any experimental information on these ternary systems, is able to represent the solubility in the ternary systems very well. In order to show this, we substituted the mole-fraction-averaged carbon number (n > into Eq. (17). Here = x l n , + x 2 n 2 , were x~ and x2 are the respective mole fractions of the respective hydrocarbons, calculated on a hydrogen-free basis. In Fig. 15, seven isopleths are shown for the ternary H2 + n-C,o -I- n-C16 system. The empty circles are experimental data points and the solid curves are predictions from Eq. (17). Similar results are shown in Figs. 16-19 for the ternary systems H 2 q- n-Cm + n-C20, H2 + n-Clo + n-C28, H2 Jr- n-Cm + Jv/-C36 and H2 + nClo-q-n-C46 respectively. F r o m these results it is clear that the principle of congruence in the sense o f Bronsted and K o e f o e d (1946), with pressure and temperature as independent variables, predicts hydrogen solubilities in n-alkane mixtures with high accuracy on the sole basis of the known solubility in pure n-alkanes.

15

p/MPa

H2"N-Cl0+n- [20

~

A B --~ C

I

I

I

350

~00

450

"~ T/K

Fig. 16. Solubility of hydrogen in mixtures of decane + eicosane, (C), experimental results; - - , prediction from Eq. (17)): curve A, = 12.22; curve B, = 15.95; curve C, = 15.06.

214

C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193 219

15 +n-C2B

p/MPa l 10 -

5

-

i

i

i

350

400

450

T/K

Fig. 17. Solubility of hydrogen in mixtures of decane + octacosane (O, experimental results; - - , prediction from Eq. (17)): curve A, ( n ) = 22.15; curve B, ( n ) = 20; curve C, ( n ) = 14.77.

15 H2+~-C1o*n-[36

p/HPa l

10-

5 --

I

I

I

.350

400

450

-= T/K

Fig. 18. Solubility of hydrogen in mixtures of decane + hexatriacontane (Q, experimental results; - - , prediction from Eq. (17)): curve A, ( n ) = 24.53; curve B, ( n ) = 15.87. The principle of congruence also permits a simple estimate of Henry's constant for hydrogen in a mixture of long-chain liquid n-alkanes: In Hm, l = ~ x / I n H~,,

(18)

i

Here Hi,~ is Henry's constant for the hydrogen solubility in a pure long-chain n-alkane and Hm, I is Henry's constant for the hydrogen solubility in a mixture of liquid long-chain n-alkanes. F o r details of the derivation of Eq. (18), see Peters et al. (1995). By a quite different approach, O'Connell and Prausnitz (1964) obtained a similar expression for gas solubilities in mixed solvents.

6. Discussion We have reviewed the evidence for the validity of the principle of congruence for the thermodynamic behaviour of mixtures of long-chain n-alkanes, which was formulated by

C.J. Peters et al. / Fluid Phase Equilibria 105 (1995) 193-219 15

215

H~+n-C1o+n-C~6

plMPa

I

I

I

350

t+O0

t~50

T/K Fig. 19. Solubility of hydrogen in mixtures of d e c a n e + h e x a t e t r a c o n t a n e ( O , experimental results; , curves prediction from Eq. (17)): curve A, = 35.87; curve B: (n> = 16.02; curve C, (n> =25.34; curve D, = 13.68.

Bronsted and Koefoed (1946) for predicting certain thermodynamic properties of mixtures of n-alkanes at ambient pressure from those of a pure n-alkane of mole-fraction-averaged carbon number and generalized, at various times and by different authors, to states where pressure is a significant variable. We have found that the theoretical justifications by Longuet-Higgins (1953), and by Hijmans and Holleman (1969) require many restrictive assumptions, and demand that the principle be applied at a fixed molar volume and temperature, rather than a fixed pressure and temperature. We show that the simplified perturbed hard chain theory applied to mixtures of n-alkanes obeys the principle at a fixed molar volume and temperature. Experimentally, we find that the principle of congruence is a simpler and better predictor of second virial coefficients than correlations based on the statistical mechanics of fluid mixtures. A priori, it cannot be expected that the principle holds for properties of coexisting phases, and we demonstrate departures for properties such as the length of the three-phase region in mixtures of very unlike n-alkanes. There is evidence from experiment, and also from the simplified perturbed hard chain theory, that the principle of congruence predicts bubble points (but not dew points) of mixtures of long-chain n-alkanes in a short-chain solvent accurately, with pressure and temperature as independent variables. It is to be expected that the reverse would be the case for the prediction of bubble and dew points for a long-chain solute in a mixture of short-chain components, the latter represented by a single component according to the principle of congruence. We have, however, pursued this case. We find that the solubility of hydrogen in mixtures of n-alkanes is predicted within experimental uncertainty by the principle of congruence, solely on the basis of hydrogen solubility in pure n-alkanes at the same pressure and temperature, thus greatly reducing the need for measurements in mixed solvents. Given the simplicity of the principle of congruence, it is advisable to use it as a property estimator for properties of mixtures of n-alkanes, instead of, or prior to, the development of more complicated statistical-mechanical mixture models.

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Acknowledgments A NATO grant has enabled transatlantic collaboration of this project. Professor M.D. Donohue and collaborators (Johns Hopkins University, Baltimore, USA) have provided information and computer codes for the SPHCT model. The authors are grateful to J.S. Gallagher for supporting the computational part of this study. Dr. R. Reijnhart and Dr. J. de Dood (Koninklijke Shell/Laboratorium, Amsterdam, The Netherlands) have graciously permitted us to display and to interpret the hydrogen solubility data. For critical reading of the manuscript, the authors are grateful to Dr. A.H. Harvey and Dr. J. Magee.

List of symbols A B C

g H k kij 1 LCEP n

(n) N P q R Rig Rijk s

S T T* U UCEP V*

V X

Y ZM

Helmholtz energy second virial coefficient number of independent-monomers with the same number of external degrees of freedom as on n-mer; pure component parameter in SPHCT vapor phase Henry's constant Boltzmann constant binary interaction parameter liquid phase lower critical endpoint carbon number mole-fraction-averaged carbon number Avogadro's constant pressure number of independent monomers with the same interaction volume as one n -mer universal gas constant adjustable parameter adjustable parameter number of independent monomers with the same excluded volume as one n -mer entropy absolute temperature SPHCT pure component parameter internal energy upper critical endpoint scale factor for the volume; SPHCT pure component parameter molar volume liquid phase composition exponential in the SPHCT mixing rule coordination number in the SPHCT equation of state

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217

Greek letters A 6 O"

difference interaction energy of one segment diameter of one segment

Subscripts e

ext

i,j,k m

mixt rep res

end segment external summation indices middle segment; mixture mixture repulsive residual

Superscripts E

excess property double bond reduced property

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