Thermodynamics of mixtures containing alkanes

FluidPhaseEquilibria, 20 (1985)27-45 Elsevier SciencePublishers B.V., Amsterdam

27 -Printed inThe Netherlands

THERMODYNAMICS OF MIXTURES CONTAINING ALKANES

S-N. Bhattacharyya, M. Costas, D. Patterson* and H.-V. Tra Chemistry Department, McGill University, 801 Sherbrooke St-W. Montreal, Quebec H3A 2K6, Canada

ABSTRACT Binary mixtures of various liquids (components 1) with the n-alkane series (components 2) have given many results for the following excess or mixing functions (XE or AX). Of particular interest are: (1) the trend of XE with n; (2) change of XE when the n-Cn are replaced by the corresponding series of highly-branched alkane isomers, and (3) comparison with theoretical predictions. Certain theories, e.g. van der Waals 1, fail in their present form due to neglecting the Prigogine number of external degrees of freedom, c, whereas the Flory, incorporating c, gives useful predictions which form a "base-line" against which to see special effects. For instance, when component 1 is a quasi-spherical inert molecule, e.g. cyclohexane or CC1 anomalous positive E4' AC, and AqVT, dVE/dT contributions occur in HE SE and VE, but negative in Cp, and AyVT, while GE and dVd /dP seem little affected. These effects are usually attributed to a net decrease of structure during mixing due to a destruction of short-range orientational order in n-C,.,.When component 1 is a plate-like molecule, e.g. 1,2,&trichlorobenzene or chloronaphthalene these effects are decreased or reversed in sign. This has been ascribed to a net increase of structure due to a change of molecular conformation or a hindrance of molecular motion. When component 1 is an alcohol, the n-C, are relatively inert while component 1 is associated in both the pure and solution states leading to + or effects in Cp depending on whether the solution or pure component 1 is the more structured. Recently Grolier, Wilhelm and Inglese have shown that when component 1 is an ether or ketone, CE may show an unusual concentration dependence with a double minimum, pe:haps again due to molecular association or to conformational changes. INTRODUCTION Thermodynamic data have been accumulating in recent years for systems of type A + n-C, composed of a liquid A mixed with the series of normal alkanes, n-Cn. (A special case occurs when A is another n-alkane). interest for at least two reasons-

The systems have been of

(1) The n-alkane molecules are chains of

almost identical segments which, by reason of their different lengths confer different degrees of thermal expansion or free volume on the liquids.

The

thermodynamic properties of the liquids thus change regularly, and to a first approximation, this is due to the change of free volume alone.

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0 1985 Elsevier SciencePublishers B.V.

Thus A + n-C,

28 systems provide excellent tests of equation of state or free volume theories, e.g. the Prigogine (1957) or Flory (1970).

Other interesting series of liquids

would be the silicones or fluorocarbons, but the n-alkanes seem to be uniquely popular. (2) It has turned out that free volume is not the only factor to be taken into account when dealing with the n-alkanes.

There are special effects

associated with various types of order (orientational, conformational or rotational) which may be decreased or enhanced on passing from the pure to the solution state.

First, however, we should mention the various thermodynamic

quantities at our disposal.

Excess and Mixing Functions While the mixing quantity, AX, is merely the observed change of X occurring during the mixing of the components, expressed per mole, the excess quantity XE is defined as the difference between the molar X for the experimental solution and its value for an "ideal" solution formed through an ideal mixing process where AV and AH = 0.

For H, U, and CP, X E = AX but this is not the case for

instance with Cv and cr(vT. For these quantities the transformation of AX into XE is complex (Benson and Kiyohara, 1979) and we choose to retain AX which is in any case more convenient for comparison with theory. The excess or mixing quantities determined for n-alkane systems have included GE and the first-order quantities HE (= UE at negligible P), SE and VE. However, second-order quantities are now being increasingly studied, particularly C:.

This is sometimes resolved into the contributions ACv and AoryVT which

correspond to changes at P = 0 of UE with T, due to, respectively, the increase of thermal motion and of thermal expansfon.

This separation could be useful in

differentiating between effects due to order since one type of order might be sensitive to packfng, i.e. volume, and another might reflect thermal motion. Other second-order quantities almost as important as Ci, but less studied are dVE/dT and the directly related dHE/dP, and dVE/dP.

The mixing function -AyvT

corresponds to the change on mixing of -VdU/dV i.e. of the variation of U with relative compression of the liquid.

Thia quantity should be particularly

interesting (see below) but apparently has never been used in the literature. Thus, the second-order quantities, particularly Cf; are useful as indicators of order or structure which is temperature- and pressure-dependent.

Much of this

work has been made possible through the use of the Picker instruments - the dynamic microcalorimeter

(Picker et al., 1971) for measurement of Cpand the

vibrating-cell densimeter (Picker et al., 1974) also available from Anton Paar.

29

Order Contributions These have been discerned experimentally through three effects: unexpected dependence of XE on n.

(1) an

Thus HE per unit volume of solution might be

expected to be independent of n.

In terms of the Flory theory, the Xl2 para-

meter derived from HE by subtracting off the free volume term would be expected series, reflecting only the difference of n chemical nature between A and an alkane segment. To a better approximation, one to be constant for the A + n-C

might differentiate between two kinds of alkane segments, methyls at the ends and methylenes in the interior of the chain.

However, it appears that such end-

effects are small (Lam et al., 1974), leaving Xl2 a constant.

In fact,

experiment shows that X 12 may increase (Lam et al., 1974) or decrease (Wilhelm, 1977; Wilhelm et al., 1978; Grolier et al., 1981; Inglese et al., 1980) with n, and this has been taken as indicative of order effects. isomer effect, i.e.

(2) an unexpected

difference between XE obtained for the normal alkane and

for a branched isomer of the same carbon number.

The normal and branched alkane

liquids will be similar in all respects except for molecular shape and flexibility which affect order.

(3) an unexpected difference between the

experimental XE value and that predicted by a reliable theory, e-g. theory.

the Flory

The XI2 parameter is usually fitted to HE and other excess quantities

are predicted using this value.

Since order manifests itself in Xl2 at that

temperature, it will have some effect in the predicted XE.

However, order has a

characteristic temperature and pressure dependence which will not be accounted for in Xl2 and hence not appear in the predicted value of XE.

We should now E refer to Flory predictions for A + n-C, systems, particularly of V .

FREE VOLUME EFFECTS ANU PRIGOGINE-FLORY PREDICTIONS E Apart from the combinatorial S , XE or AX depend on two effects:

(1) The

first is due to the interaction between the segments of the two molecules, associated with the Xlz parameter. small in VE, C"p and dV /dT.

This term is important in HE but relatively

Fig. 1 shows the VE contributions calculated for

cyclopentane + n-C, and taken from Costas and Patterson (1982).

The Xl2 para-

meters were fitted to the equimolar HE(n) then used to calculate X12contributions to the equimolar VE which parallels He itself in fig. 1.

(2)

The second

effect denoted free volume or equation of state, is associated with the difference of reduced volumes (v1-v2) of the two liquids (v"is a measure of the degree of free volume of the liquid and is directly related to a the thermal expansion coefficient). The free volume term in HE is small, negative and 2 proportional to (Y1- Y2) , but as discussed in Van and Patterson (1982) the

30 free volume effect in VE is large.

Further, it

d ivides into two terms:

the

first, the 7 curvature term is again negative and proportional to (cl- i2)' as

in

WE. It

may be seen in fig. 1 going to zero for n-C, which happens to have

the same CL or ? as c-C Note that eq. (15) of Van and Patterson (1982) which 5' -l/3 gives expressions for the three contributions to VE has an error in VEurV; V should be ?-1'3.

The second free volume contribution, the P* term, is a

peculiarity of VE , and can be + or - being proportional to (v1-v2)(PT-PG).

The

reduction parameter P* is a measure of internal pressure but is independent of a, and remains roughly constant for all the n-alkanes, i.e. 400-450 J cm-3 . * * -3 Cyclopentane has Pi- 509 J cm and hence Pl-P2 is positive while (vl-7,) and the P* term itself change from - to + as n increases (see fig. 1).

There is

good agreement in fig.1 between the experimental and theoretical total VE.

The

10

0.6 %

0.4

5 1

0.2

2 II

0

z %

-0.2

-04

-0.6

‘\

\

\

‘h Fig. 1: Experimental and theoretical values of the molar VE at equimolar composition and 25-C for cyclopentane mixed with the series of n-alkanes. * The three contributions (X12, P , and v curvature) to the theoretical VE are indicated as well as equimolar values of the molar excess enthalpy HE(from Costas and Patterson, 1982).

P*

Fig. 2: Experimental (3O'C) and theoretical (25OC) values of molar VE at equimolar composition for the linear siloxane, decamethyltetrasiloxane. mixed with n-alkane series. The three theoretical contributions to VE and the equimolar HE are also indicated. (from Costas and Patterson, 1982).

31

*. 0.4

---__

-L---1 x12

--------*_

“-

0.0

T a E :

.EXP

-0.4

-0.8 a i

“>

i i i i

-1.2

\ i i \ I ;“ow~;,

-1.6

n

Fig. 3: Experimental (Grolier et a1.,1981) and theoretical values of the equimolar VE at 25OC for lchloronaphthalene+n-alkanes,showing the three theoretical contributions.

.-

._

Fig. 4: Theoretical and experimental values of VE for cyclohexane +n-alkanes. The Flory theory (F) is applied using the c parameter and also with c=l. The van der Waals 1 theory (VDW) is applied in its original form (lower curve), replacing the combining rule (3) by eq. (4) (upper curve) and finally using eq. (4) and introducing the c parameter.

shape of the Vg(n) dependence is largely due to the P* contribution, but also interesting is the decrease of VE for large n where the VfuLv term becomes important. Systems composed of a linear siloxane, decamethyltetrasiloxane + n-Cn are of interest since aI= (I at nz 7 but, unusually, P* is lower than for the n-alkanes -5 being only 354 J cm . Hence Pr- Pi is now negative and the P* term has the opposite slope against n to that for the cyclopentane set of systems, as shown in fig. 2 taken from Costas and Patterson (1982). n-dependence of the P* term the totalVE(n) image of that in fig. 1 for c-C5.

Due to the opposite

dependence now resembles a mirror

32

*

*

Returning to the P1>P2 situation, we consider the interesting data of Grolier et al. (1981) for 1-chloronaphthalene+n_Cn. Here HE is positive and fairly -1 large, e.g. 655 J mol for chloronaphthalene+n46. Yet the VE are all negative and become very large for short n-alkanes, as shown in fig. indicates the successful predictions.

3, which

also

First, however, note that the Xl2

contribution decreases as n increases while in figs. 1 and 2 the Xl2 contribution increases.

Following Grolier et al (1981), this apparently

indicates increasing creation of order in the chloronaphthalene systems, while in figs. 1 and 2 there is increasing destruction of order as n increases. For the -3 using our velocity of sound measurement and data

P* contribution, PF = 566 J cm from Grolier et al. (1981).

However, cc is extremely small being only 0.66 x103

i7l at 2S°C, i.e. much less than for n-C

In fig. 3 the P*term never becomes 16' positive even for large n, leading to the observed VE values. Before looking at some second-order mixing functions we examine the reasons for the success of the

Flory theory with polyatomic systems.

The Number of External Degrees of Freedom (3~) A number of corresponding states theories are available based on different equations of state, e.g. van der Waals 1, Frisch, etc. (see Ewing and Marsh (1977) for applications to mixtures of cycloalkanes).

However, the success of

the Flory theory for the n-alkane systems seems quite unique.

This is because

the theory uses the Prigogine concept of the number of external degrees of freedom (3c), a vital concept when dealing with polyatomic molecules.

It is

absent from most theories of liquid mixtures which are therefore restricted to simple fluids. The central quantity of the Prigogine-type theories, of which the Flory is one, is the reduced temperature, y, of a liquid. volume, and directly related to y and a-

It is

This is a measure of the free given as a dimensionless ratio

of two energies, viz

f-

thermal energy (external)/molecule cohesive energy/molecule

(1)

Here the denominator favours contraction of the liquid, and the numerator promotes expansion.

But in the numerator only the external degrees of freedom

count in the thermal motion of the molecule.

Eq. (1) may be rewritten:

(2)

with 3c being the number of external degrees of freedom of the molecule.

The

33 importance of the c parameter may be realized from fig. 4 where the Va of the c-c6 + n-C, are first predicted with good results using the Flory theory.

They

are then recalculated using the Flory theory, but with the c parameter fixed at unity, i.e. the molecules are structureless.

Of course compressibility or y

data are not then required in applying the theory, whereas they are normally necessary to determine the value of c.

Again Xl2 is fitted to the HE data.

In

fig. 4 VE predictions are again satisfactory for very low alkanes, but as n increases they fail completely.

The reason for this may be seen in eq. (2) * which shows that it is the ratio P /c which is fitted to the free volume or thermal expansion coefficient of a pure component.

The c parameters for

cyclohexane and for a long n-alkane are quite different.

By forcing the two to

be unity, the relation between the P* parameters for the components is falsified * E with serious consequences in the P contribution to V . Fig. 4 also shows the predictions of the van der Waals 1 theory based on the simple van der Waals equations of state. (Xl2 is again fitted to HE).

This

theory also uses a different combining rule for volumes, V* of the solution being

v* = x; v; + x; v; + l/4

(3)

xl x2(v1 *II3 + V2*1'3)3

instead of the simple

VA= XpT;+

x2

v;

used in the Flory theory.

Fig. 4 shows that the van der Waals predictions are

incorrect, and indeed it is known in work with simple liquids that eq. (3) and the van der Waals theory cannot be used when molar volumes of the components are too different.

Replacing eq. (3) by the superior eq. (4) gives similar results

to those obtained with the Flory theory with c = I, i.e. still incorrect. Finally, however, one can introduce the c parameter into the van der Waals 1 theory giving the good results seen in fig. 4.

Thus the reason for the success

of the Plory theory seems to be, not a better equation of state, but the inclusion of the Prigogine concept of external degrees of freedom and the use of eq-

(4).

Much testing shows that the Flory theory gives reasonable predictions

of XE and AX which constitute a base-line against which to see special effects.

DESTRUCTION OF PURE LTQUTD ORDER Certain special effects are manifested by the thermodynamics of mixtures containing the n-alkane series.

The first is associated with "correlations of

molecular orientations" (CMO) in the longer n-alkanes in their pure state.

34 While this orientational order between segments 1s short-range in character, it can affect the thermodynamic properties which depend on nearest neighbour interactions.

The order is equivalent to a molecular cohesion which lowers H, S

and V and since the order is found to fall off rapidly with increase of T, it increases C and dV/dT. Since (BU/BP)T = -(aV/aT)P the (negative) pressure P dependence of the energy will also be increased in magnitude. These changes are small and difficult to isolate in the pure liquids, but they manifest themselves in the mixing functions of systems of type A + n-C,.

If A is a spherical

molecule incapable of entering into CM0 with an n-alkane chain, mixing will cause a net disruption of order and hence contributions to the AX which are of opposite sign to the order contributions in the pure alkane X. They are E therefore positive in HE and S , positive but small in GE (enthalpy-entropy compensation), negative in C 5, dVE/dT and positive in dHg/dP and dVE/dP.

Since

order only appears in n-C n of large n, one expects HE(n) to increase rapidly with n.

Furthermore, for any n one can select alkane isomers with increasing

degrees of chain-branching which reduces the ability of the alkane to order either in the pure, or solution states, essentially eliminating the order contribution for a highly-branched isomer (br-C,).

Thus the order contribution

should be revealed by the difference:

AX(order) = AX(n-C,)-AX(br-Cn)

(5)

Further, the Flory theory should in principle give good results for the br-C, case where order is absent, and hence one may hope to obtain

AX(order) = AX(n-Cn)-AX(Flory)

(6)

E Note, however, that by fitting Xl2 to H , one is already building the effect of order into the theory when applied to other first-order quantities like VE.

A

second-order quantity will depend on an effective change of X12 with T or P, and this is not taken account of by theory.

Hence eqn. (6) should indicate the

contribution of order to a second-order mixing quantity. Order effects have been observed for many systems, the most extensive measurements being when the order-destroying molecule is cyclohexane.

Heintz

and Lichtenthaler (1982) have recently reviewed this area, including effects of pressure and much of the extensive spectroscopic evidence for 0iO in the n-C,. Here we wish to add a description of recent work (Bhattacharyya and Patterson, 1985) on the effect of order in the equation of state.

Fig. 5 shows experi-

mental and predicted values of dVE/dT for C-C, + n-Cnand + br-C,.

In the

35 case

of the br-C the Flory theory predicts dVC/dT very well. However, for n-C n n E similar dV /dT are predicted whereas experiment yields quite different results, as seen in fig. 5.

These may be attributed to the presence of temperature-

dependent order in the pure longer n-alkanes which enhances dV/dT, and hence on destroying order in the mixture, a negative contribution appears in dVE/dT. Similar effects should be visible for other quasi-spherical molecules, e.g. CC14 and benzene mixed with the n%:,.

These molecules, like cyclohexane, are

2ooc

A(rv-nncn

P15Gi k g P

loot

E “I

500

6

tl

lb

1‘2 14

16

18

n

Fig. 5: Equimolar dVE/dT at 25°C against carbon number for cyclohexane mixed with the n-alkane series (n-C,) and a series of highly branched isomers (br-C,): 2, 2_dimethylbutane, 2,2,4-trimethylpentane, 2,2,4,6,6-pentamethylheptane and 2,2,4,4,6,8,8_heptamethylnonane. Dashed curves, Floty theory predications. At n-16: l experimental and theoand n retical'points for, in descending order 2,4, and 6-methylpentadecane. At n=19 points for 2,6,10,14_tetramethylpentadecane of intermediate branching.

Fig. 6: Equimolar values at'25OC of -AyVT for cyclohexane mixed with the n-C, and br-C, series of fig. 5. At n-19: 2,6,10,14-tetramethyl entadecane. % Theoretical values (H ) are also shown.

36 predicted to give rather small and positive dVb/dT values with the higher n-alkanes, hence show up clearly a negative contribution.

With A molecules of

lower P*, e.g. II-C6, dVE/dT is predicted to be large and negative making the order contribution hard to isolate. well-predicted

Interestingly, p-xylene + n-C, are

by the Flory theory suggestfng that this flat molecule does not

significantly disrupt alkane order (Gonzalez Fernandez et al., 1984). Since order is enhanced by pressure, presumably making dV/dP more negative, one would expect a positive order contribution in the quantity dVF/dP for quasi-spherical A molecules mixed with the n-C . Such systems have been studied n by Aicart et al. (1980; 1981a,b; 1983 a,b) where the Flory theory gives good results.

Costas et al. (1985) shows that the n-C, series with cyclohexane give

sfmilar dVE/dP values and the Flory theory predicts both sets reasonably well. This is in itself interesting but disappointing from the order viewpoint.

A

simple argument can in fact show that order effects in dVF/dP are relatively small (Costas et al., 1985).

However, another equation of state quantity

promises (Costas et al., 1985) to be a sensitive indicator of order, viz. -AyVT.

The quantity yVT=

-VdU/dV is the change of energy with relative

compression and fs therefore negative.

The Flory theory, and all others based

on the van der Waals volume dependence of the energy predict generally that the E change of this quantity on mixing is equal to H , i.e. -AyvT

=

HE

(7)

Thermal pressure data in the literature suggest that this relation is followed by various systems indicating that the components and the mixtures are van der Waals liquids.

However, if a liquid contains order which may be imagined as a

cohesion that increases with compression, then one would expect that -VdU/dV would be increased in magnitude, i.e. more negative and hence that -AyVT should be larger than HE if there is a net destruction of order during mixing.

Fig. 6

(from Costas et al. 1985) shows that for cyclohexane mixed with the series of br_C,, eq.

(7) is followed so that these liquids may be considered as van der

Waals liquids, as assumed by the Flory theory. eq. (7) are seen for c-C6+ n*:,. of the expected sign.

However, strong deviations from

These increase with increasing n, and they are

We believe therefore that -AyVT constitutes another

useful indicator of orientational order and that y measurements, or velocity of sound measurements leading to y should be useful when treated in this way. Of special interest is Raman work by Merajver and Wunder (1981) indicating that there may be a shift of n-Cl6 conformers from trans to gauche when the molecule moves from the pure state to solution in CUC15.

Thus here it is

37 conformational order which is apparently lost on passing into solution, and such an effect would contribute to both HE and CE . Snyder (1982) however finds the P effect to be at most very small. Furthermore CF is found (Costas et al. (1985) to be less a reflection of CE v, i.e. change of energy with thermal motion increase, than of AayVT, i.e. change of energy with thermal expansion.

This is

in line with the effect of order on equation of state, and seems more consistent with an intermolecular packing-dependent orientational order than with an intramolecular conformational order. Also significant is the effect of pressure on the trans-gauche populations of n-alkanes as determined by Raman spectroscopy of n-alkanes compressed in the diamond anvil cell.

While pressure leaves the trans-gauche population of n-Cl6

essentially unaffected , there is a shift to gauche in n-C, (Schoen et al. 1977 and to trans for n-C4C and polyethylene (Wunder et al. 1979). al. (1983). solution.

Also see Wong et

It is tantalizing that results are not available for n-alkanes in Nevertheless, the insensitivity to pressure of the n-Cl6 conformer

equilibrium suggest that a disturbance of this equilibrium is not responsible for the large effects of pressure (Heir&z and Lichtenthaler, 1982) on HE in systems containing n-Cn.

CREATION OF ORDER IN THE MIXTURES Mixtures of the type A f n-C, yield surprising results when A is a highly sterically-hindered branched alkane, e.g. 3,3_diethylpentane, or a plate-like cycloalkane like trans-decalin or dimethycyclohexane in equatorial-equatorial configuration.

When n< 10, HEis typically negative and too large to be

explained as a free volume effect, while CE may be inexplicably positive or very slightly negative (de St. Romain et al., 1979a).

With a sterically-free

branched isomer like 2,2,5_trimethylhexane or a more globular equatorial-axial dimethylcyclohexane the signs are normal, i.e., HE is positive and CE negative. P The anomalous signs of the excess quantities suggest that order of some sort is being created in these mixtures.

When the normal alkane is long-chain, e.g.

n-C16, HE may become positive as expected from destruction of CMO. However, -1 while with the with 3,3-diethylpentane the equimolar HE is only 18 J mol -1 sterically-free nonane isomer 2,2,5_trimethylhexane HE is 318 J mol . Spectroscopic measurements have indicated that the order in the longer n-C, is indeed destroyed (TancrPlde et al., 1977).

The small HE values then suggest that the

destruction of CM0 in the pure n-alkane is balanced by another effect corresponding to the creation of order in the solution.

The same

negative HE and

positive Cp* are found if the same sterically-hindered alkanes or plate-like cyclics are mixed with cyclopentane (de St. Romain et al., 1979b) rather than with an n-C,.

However, if cyclohexane, cycloheptane or cyclooctane is used the

HE results are positive as usual.

Here it seemed natural to speculate that the

38 order created in the solution was rotational in nature, associated with a hindrance of the rotation of the flat cyclopentane molecule by the flat or sterically-hindered

A molecule.

The same hypothesis was made for n-alkanes,

i.e. order was created by the A molecules through hindering rotational motion of segments Ln the flexible n-alkane chain. Independently of the above, Grolier, Wilhelm, Inglese and their collaborators have obtained a large body of data indicating an ordering in the solution.

They

have measured VE, HE and CpE for A + n-C, where A is a plate-like, usually polar, molecule:

1,2,4-trichlorobenzene,

chlorobenzene, bromobenzene, dlchloro-

benzenes, cyclic ethers and chloronaphthalene

(Wilhelm, 1977; Wilhelm et al.,

1978, 1981; Grolier et al., 1981; Inglese et al., 1980, 1983). results can be more spectacular.

Here the

In almost all the cases HE(n) actually

decreases as n increases, and a stronger decrease is found for the related Xl2 parameter.

This is of course in contradiction to CM0 increasing with n and

consistent with an ordering in the solution for the higher n.

The VE can be

extremely negative when n is small, but these values are not anomalous and are explained by the Flory theory (as in fig. 3).

However, Ci can again be positive

and anomalous, as in the very interesting 1-chloronaphthalene + n-C n systems (Grolier et al., 1981), and the effects here are large, e.g. the equimolar CE P -1 for n-Cl2 is 1.7 J K-Lmol . It is significant that when br-Cl2 is used the corresponding Cf value is small and negative, and it seems clear that when n-C

n

are involved there is a creation of order in the solution. Grolier and collaborators suggest that this order is essentially i ntramolecular associated with a change in the trans-gauche population of the n-alkane.

Although a full treatment of this conformational ordering has not

been given, presumably the trans conformers must interact favourably with the _ flat A molecule (the origin of the negative HE effect) while an unfavourable interaction with the gauche form increases the energy gap between trans and gauche forms.

The resulting change in trans-gauche population can bring about a E of the alkane and in C . No doubt rotational and P P conformational ordering are not very different mechanisms, but still it might be

positive contribution in C

possible to distinguish between them, for instance if CT were split into the two contributions AC v and AayVT.

The former is the change of UE due to the increase

of thermal motion, while the latter corresponds to the change due to the expansion with increase of T.

If the order %s primarily intermolecular and sensitive

to packing one would expect the second contribution to be the greater, while if the order is intramolecular, e.g. due to trans-gauche changes, one might expect AC, to be dominant.

A further test of a model will be to give the correct ratio

of TCi(order)/HE(order) which is large, -5.

Finally, If the creation of order

39

is more inter- than intramolecular in nature, it might show up in positive contributions to dVE/dT and negative in -AyVT, i.e. opposite to those in the previous section for destruction of order. Recently Grolier et al. (1984) and Inglese et al. (1984) have found even more interesting C:(x) curves for p-dioxane mixed with cyclohexane and the n-alkanes, and the unusual W-shaped concentration dependence is seen in fig. 7.

The same

dependence has been published for another ether, 2,5,8,-trioxanonane mixed with n-C7 (Kimura et al., 1983) and for alkanones + n-C, (Grolier and Benson, 1984). The origin of the W-shaped concentration dependence has been tentatively attributed to conformational changes in either of the two component molecules, but we return to W-shaped CE's below. A clear message of this work is that C:, P now measured conveniently through use of the Picker dynamic microcalorimeter, can reveal many unusual and significant effects.

I 0.2

I 0.4

I

I

0.6

0.8

x

Fig. 7: Excess molar heat capacities, Ci, at 25OC of p-dioxane n-C7, n-CL0 and n-C14 against dioxane mole fraction. (fig. taken from Grolier et al., 1984).

Fig. 8: Molar excess heat capacity, CE at 10, 25 and 40°C of 1-decanol in-Cl0 against decanol mole fraction (increasing to the left). At very low decanol concentration points have been omitted due to crowding.

40 ASSOCIATED 6 INERT LIQUIDS Many results have been obtained for A + n-Cn where A is an associated liquid such as an alcohol.

Again order or structure is important, but here the alkane

may be taken to be inert and it is the alcohol which is structured.

The

literature indicates that both RR and $' are strongly positive and dCEIdT>O. Positive HE reveals a net breaking of hydrogen-bonds on mixing, but positive Ciindicates more "structure" in the solution, so the fewer H-bonds in solution correspond to bringing together alcohol molecules over longer distances than in the pure alcohol.

At extremely low alcohol concentration, however, only dis-

sociated single molecules can exist in solution and mixing must then correspond to a net breaking of structure, and Ci must be negative.

This behaviour has

been found recently (Costas and Patterson, 1985) as shown in fig. l-decanol mixed at low concentration with n-Clo.

8 for

At xROH -0.1 C; , dC;/dT>O,

as usually found, but with decrease of alcohol concentration we pass from this concentration region I to region II, where C

0, dCi/dT
region III at lowest x where C"P, dC;/dTC 0. omitted in fig.

(Negative Ci points have been

8 due to crowding, but are seen in Costas and Patterson

(1985). A simple model can predict the CF behaviour in the three concentration ranges starting from

c;=~oH($c- c;,

(8) 0

with 0, the apparent C

of alcohol in solution and C the value for the pure P P state. In using eq. (8) we need only be concerned with that part of C which P is due to association since the internal C of the alcohol is eliminated and P 0 other intermolecular contributions will be small. The associational $c and C P are given satisfactorily by a simple association theory due to Treszczanowicz

and Kehiaian

(TK), (Kehiaian and Treszczanowicz,

1969; Treszczanowicz and

Kehiaian, 1970, 1979) in which each molecule is capable of two energy levels corresponding to the associated state and the dissociated monomer.

With

increase of T, the molecules move to the higher energy level, but the wease" with which this dissociation takes place increases with increasing dilution of the alcohol.

This is indicated in fig. 9 which shows the associational U(T)

and Cp(T) for the alcohol in the pure state (curve 0) and in solution.

As the

concentration of the alcohol is lowered, curve 0 is displaced to lower T giving curves I, II and III.

The Cp(T) curves have the form of "Schottky peaks" with

maxima occurring to a very rough order of magnitude at:

kT-AH°KRoR

(11)

41

where AH0 is the association energy -25000 J mol

-1

for H-bonding.

The

experimental temperature T' must lie far below the maximum of curve 0 since for hydrogen-bonding at SOOK, AH'/kT - 10, but it will lie far above the maximum for the Schottky peak III corresponding to very low concentration.

Thus following

eq. (8) and subtracting curve III and 0 at the temperature T', we find C:XO and on increasing T' in fig. 9 clearly dCt/dT
With increase of concentration the

Schottky peaks move to II and I and again using eq. (8) fig. 9 shows that C>O, dCE/dT
dCi/dT>O in region I.

Thus the three combinations

G of signs I, II and III are obtained by scanning concentration.

The C:(x) is

shown schematically through the concentration range in fig. 10 where curve a represents the alcohol + inert system at ordinary temperature and curve b at a higher T.

T'

T-

T’

T-

L Fig.

9: Schematic representatfon of energy (upper diagram) and heat capacity (lower diagram) against T for an alcohol in the pure state (0) and at increasing dilution in an inert solvent (I, II III). T' represents the experimental temperature.

Fig. 10: Schematic of C!(x) as predicted by the Tresaczanowicz-Kehiaian theory for an alcohol + inert solvent at room temperature, a and successively increasing temperatures, b, c and d.

42 One of the attractive features of fig. 9 is that it also provides a picture of the thermodynamic effects of n-alkane orientational order in A + n-C, systems. "E There experiment indicates that CiO, i.e. we have a fourth combination of signs, IV.

Correlated and uncorrelated alkane segments

correspond to the two energy levels, but since the associating forces are now much weaker than H-bonding AH'/kTCl.

Hence T' in fig. 9 will now lie on the

high-T side of the pure alkane Schottky peak 0 as well as of peaks I, II and III.

It is now clear that for all concentrations, corresponding to I, II and

III peaks, Ci is negative, and that dCi/dT is positive.

Thus CE corresponds

here to an alcohol + n'Cn system at extremely high temperature, i.e. fig.

10.

curve c in

With increase of T' in fig. 9 CFp becomes less negative as found for

systems where CM0 is destroyed.

In fig. 10, Ci moves to curve d as T is

increased with dCi/dT>O.

Of course, for n-alkane order fig. 9 can only be taken E in a qualitative sense, and to obtain correct values of C one would require a P' temperature and volume-dependent energy of association. Fig. 10 also suggests a general result:

for any fixed x, all four combina-

tions of CE dC;/dT signs, I, II, III and IV are scanned as the temperature is P' raised from a very low to a very high value. At a fixed T' lying below the maximum of the pure Schottky peak (0), only the three combinations of signs, I, II and III are scanned by varying x. alcohol + inert systems.

This was seen experimentally in the

At a fixed T' lying above the maximum of the pure

Schottky peak, the single combination of signs, IV, exists at all x, as found in an n-alkane + inert system. The H-bond and CM0 corr;spond to respectively extremely strong and extremely weak association, where AH /kT>>l and AH'/kT
Associations of intermediate

strength might be furnished by polar solutes and here we would expect C:(x) to have an S-shaped concentration dependence negative at low concentration of the associated component, as in curve b in fig. 10. and the Treszcxanowicz-Kehiaian inert systems. solvent.

We believe that figs. 9 and 10

p for polar + theory are useful in dealing with CE

However, the TK theory assumes ideal mixing of the multlmers and

The combination of an associated component and a non-random solution

caused by a large enthalpy of mixing, could give CE curves which are negative at P both ends of the concentration range and positive in the middle, i.e. W-shaped curves such as found by Grolier et al (1984) and seen in fig.

7.

Although

"unusual" in shape they will probably become recognized as a quite general phenomenon.

43 ACKNOWLEDGEMENTS We are grateful to the Natural Sciences and Engineering Research Council of Canada for support to which the Ministsre de 1'Education du QuBbec also contributed, and also thank the Universidad National Autonoma de M6xico for a fellowship to M.C.

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