J. CHEM. SOC. FARADAY TRANS., 1993, 89(1), 89-93 89

Van der Waals Liquids, Flory Theory and Mixing Functions for Chlorobenzene with Linear and Branched Alkanes

Asuncion Dominguez, Gloria Tardajos and Emilio Aicart" Departamento de Quimica-Flsica , Facultad de Ciencias Quimicas , Universidad Complutense , Madrid 28040, Spain Silvia Perez-Casas, Luis M. Trejo and Miguel Costas* Departamento de Fisica y Qulmica Teorica , Facultad de Qulmica , Universidad Nacional Autonoma de Mexico, Mexico D.F. 04510, Mexico Donald Patterson Chemistry Department, McGill University, 801 Sherbrooke St. West, Montreal P.Q. H3A 2K6, Canada Huu van Tra Departament de Chimie, Universite du Quebec a Montreal, C.P. 1888 Succ. A , Montreal P.Q. H3C 3P8, Canada

The basic assumption of the Flory theory of solution thermodynamics, van der Waals behaviour of the com- ponents, is contravened by chlorobenzene and other complex liquids. Their internal pressure decreases, instead of increasing, with decreasing volume. Using chlorobenzene with a series of normal alkanes and four highly branched isomers the following quantities have been measured at 25 "C and equimolar concentration : H E , VE, dVE/dT, dVE/dp, A(yVT), CE, A(ayVT) and ACv. Against expectation, the Flory theory predicts the main trends of all these data. dVE/dT, A(yVT) and CE for n-C, systems show deviations from the theory which are readily explained by temperature-sensitive order in the long-chain pure n-C,,, liquids. W e conclude that the Flory theory remains useful, at least for chlorobenzene, in spite of the breakdown of its van der Waals assumption.

Second-order thermodynamic mixing quantities, viz. d VE/d T , dVE/dP, A(yVT), CF, A(ayVT) and AC, have been measured recently for cyclohexane'-* and l-~hloronaphthalene~ mixed with the normal alkanes (n-C,) and their highly branched isomers (br-C,) and for ethylben~ene,~ benzene, toluene and p-xylene, mixed with the n-C, series. The experimental results have been compared with the theoretical predictions of Flory theory which is able to predict the main trends observed for the br-C, series. However, with the n-C, series Flory theory fails in its predictions for several properties, the discrepancies having been associated with special effects ignored by the theory: (1) there is short-range orientational order between long-chain n-C, molecules in the pure state; when a long n-alkane is mixed with cyclohexane, n-CJn-C, order is destroyed so that mixing involves a net destruction of order; (2) when the br-C, series is used, order effects are absent since orientational order cannot form between these globular-shaped alkanes; (3) when the n-C, series is mixed with toluene, ethylbenzene, p-xylene and chloronaphthalene the destruction of order in the pure n-C, is increasingly counterbalanced by the formation of order between the n-C, and these plate-like aromatics; the effects of order thus diminish and finally change sign. More quantitatively, order in the n-alkanes is considered to change the volume depen- dence of the internal energy causing a small deviation from the van der Waals (vdW) dependence assumed by the Flory theory:

u = -a/V"' (1)

For any liquid following the vdW equation [eqn. (l)], (6U/6V), must increase as V decreases, i.e. (62U/dV2)T < 0. This is the case for benzene where (dU/6V), was determined experimentally by Gibson and Loeffler.6 Here, at 25°C (6U/6V) , increases by 2.5% for a volume decrease of 6 cm3 mol-' (a 1 kbar change of pressure). However, ref. 6 also indicated that for other liquids, uiz. chlorobenzene, bromo- benzene, nitrobenzene and aniline, (S2U/S V 2 ) , is positive; for chlorobenzene at 25"C, (6U/6V), decreases by 4% for a

volume decrease of 5.5 cm3 mol-' (a 1 kbar change of pressure). Gibson and Loefl'ler emphasized that the simple vdW model was invalid and suggested that with these liquids it is necessary to include a repulsion term in eqn. (1). Predic- tions of corresponding-states theories, such as the Flory, depend on (d2U/6V2),. The question must, therefore, be asked whether the Flory theory can make reasonable predic- tions for systems where its basic assumption fails. In this context, the present work reports H E , V E , dVE/dT, dVE/dp, CF, A(yVT), A(ayVT) and AC, for chlorobenzene mixed with the n-C, and br-C, series at 25 "C and equimolar concentra- tion and compares the results with those predicted by the Flory theory which are surprisingly good.

Experimental Data at 25 "C were obtained for the following equimolar mix- tures: chlorobenzene with n-C, (m = 6, 8, 10, 12, 14 and 16) and with the series of highly branched alkanes (br-C,,,) [2,2-dime thy1 butane (br-C,), 2,2,4- trimet hylpen tane (br-C,), 2,2,4,6,6-~entamethylheptane (br-C, 2 ) and 2,2,4,4,6,8,8-hep- tamethylnonane (br-C16)]. Materials were from Aldrich Chem. Co., Wiley Chem. Co. or Fluka Chem. and their stated purities were 99% or better. All liquids were used without further purification.

Molar excess heat capacities, CE, were measured using a Picker flow microcalorimeter (Sodev Inc, Sherbrooke, Canada). The instrumentation and procedures have been described in the literature., The volumetric heat capacities were converted into molar heat capacities by determining the density of each mixture. Density data were obtained with a vibrating-tube densimeter (Sodev Inc.) leading also to excess volume data, VE. Both C: and V E data throughout the con- centration range are reported elsewhere., Excess volume data have been previously reported for n-C, ,9 n-C," and n-C,;" these data are in good agreement with the value for n-C6 and the VE(m) dependence found here. The equimolar HE values were measured using a C-80 calorimeter (Setaram, Lyon,

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90 J. CHEM. SOC. FARADAY TRANS., 1993, VOL. 89

France). Previously reported HE data for n-C,"-'3 and n-C,13 are in satisfactory agreement with the HE(rn) depen- dence found here. However, for n-C,, the value reported here is 11% higher than that found in ref. 13. Expansion coeffi- cients, a, and isothermal compressibilities PT were determined by direct piezometric measurement of (d V/d T), and (dV/dp),. Details of the techniques have been decribed pre- v iou~ ly . '~ Both ct and PT at various temperatures are re- ported elsewhere.' ' The thermal pressure coefficient, y, for pure chlorobenzene found here is in very good agreement (0.6%) with that reported earlier.16

Table 1 reports the molar volumes V , expansion coeffi- cients a, isothermal compressibilities PT , thermal pressure coefficients y and heat capacities C , for chlorobenzene and the equimolar mixtures of chlorobenzene with the linear and branched alkanes. Data for pure n-C, and br-C, liquids were reported previ~usly.~ We estimate the accuracies to be the following: +O.O02 cm3 mol-' for VE, f 4 x 1 0 - ~ cm3 mol-' K-' for dVE/dT, + 4 x 10, cm3 mol-' TPa-I for dVE/dp, +20 J mol-' for A(yVT), + 15 J mol-' for HE, fO.l, 0.2 and 0_3 J K - ' mol-I for CF, A(ayVT) and ACv, respectively.

Results and Discussion First- and second-order mixing and excess quantities calcu- lated from the experimental data in Table 1 are shown in Table 2 together with HE values.

Excess Volumes and their Temperature and Pressure Dependence

Experimental equimolar values from Table 2 for VE are shown in Fig. 1 together with Flory theory predictions calcu- lated in the manner described in ref. 17. Parameters required

-0.8 1 m

Fig. 1 Experimental (a) and Flory theory predictions (-) of the molar VE at equimolar concentration and 25 "C for chlorobenzene mixed with the n-C, and br-C, series of alkanes

by the theory, i.e. P*, a, p and s for the alkanes are from Table 1 (a and p ) and ref. 18 and 19 (P* and s), and for chlorobenzene in Table 1 (a and p ) ; the values for P* and s for chlorobenzene are 596 J cm-3 and 1.0 (estimated), respec- tively. Values for the solution parameter, X,, , given in Table 2 were obtained by fitting the theory to experimental HE values at equimolar concentration. The theory successfully predicts the variation of VE with the carbon number rn for both the n-C, and br-C, series. Similar VE(rn) dependences for the n-C, series mixed with liquids of different internal

Table 1 Experimental data for pure components and equimolar mixtures at 25 "C"

V/cm3 mol-' a/10-3 K - 1 B,ITPa - ?/lo-, TPa K - ' CJJ mol-' K - '

chloro benzene chlorobenzene with : n-C, n-C, n-C10 n-C 12

n-c,, n-C 16 br-C, br-C, br-C br-C ,

102.146

1 16.406 132.766 149.238 165.620 182.098 198.454 1 17.294 133.800 165.959 196.23 1

0.982

1.171 1.084 1.023 0.978 0.946 0.92 1 1.190 1.096 0.985 0.900

755

1145 1042 97 1 917 878 844

1231 1132 996 885

1.2998

1.0224 1.0399 1.0534 1.0656 1.0779 1.0917 0.9669 0.9683 0.9892 1.0173

153.78

174.92 202.74 232.99 262.67 292.96 323.48 172.04 197.68 252.27 306.10

" Pure component data for n-C, and br-C, given in ref. 3.

Table 2 First- and second-order mixing quantities for chlorobenzene with n-C, and br-C, series at 25 "C and equimolar concentration

(d V:/d T) ( - d VE/dp) VE a /lo- cm3 /103cm3 A(yVT)' A(ayVT) C;a ACV HE X1Zb

/cm3 mol-' mol-' K - ' mol-' TPa-' /J mol- /J mol-' K - ' /J mol-' K - ' /J mol-' K - ' /J mol-I /J c n P 3

chlorobenzene with n-C, - 0.448 n-C, - 0.070 n-C 10 0.205 n-C 12 0.258 n-c,, 0.356 n-C1, 0.355 br-c, - 0.627 br-C, -0.301 br-C 0.076 br-C , 0.238

-4.7 - 0.7

0.6 0.4 0.7 0.1

- 8.2 - 2.4

1.2 2.7

- 15.1 - 4.9 -0.8

0.2 2.2 2.4

- 26.7 - 12.6 - 4.0

0.4

- 540 - 679 - 722 - 897 - 1108 - 1316

- 668 -681 - 223

5

- 0.30 -0.39 - 0.42 - 0.97 - 1.12 - 1.51

- 0.64 - 0.34

0.09 0.22

- 0.80 - 1.19 - 1.56 - 2.26 - 2.94 - 3.52

- 0.79 - 0.45 -0.11 -0.19

- 0.50 - 0.80 - 1.14 - 1.29 - 1.82 - 2.01

-0.15 -0.11 - 0.20 - 0.41

63 5 690 738 792 838 892

744 788 8 50 885

27.6 26.8 26.8 27.5 28.1 29.0

35.4 33.2 30.7 30.4

Values throughout the concentration range are given in ref. 8. Fitted to HE at x1 = x 2 , according to ref. 17.

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pressures, e.g. cyclohexane,' 1-~hloronaphthalene,~ ethyl- b e n ~ e n e , ~ benzene, toluene and p-xylene,' cyclopentane, carbon tetrachloride and 1,4-dioxane,' * have been previously discussed and rationalized through the Flory theory. Its success with all these systems occurs in spite of the presence of order in the long-chain n-alkanes. For the chlorobenzene-br-C, mixtures in Fig. 1, although the Flory theory VE(m) prediction is qualitatively correct, the theoreti- cal equimolar values are seen to be much larger than the experimental ones, particularly with br-C,, and br-C,, . A similar situation was found for cyclohexane-br-C, in ref. (1). It has been suggested,' that the high predictions of the theory may be due to a packing effect where cyclohexane, and in the present case chlorobenzene, would fit into the 'empty regions' around br-C,, molecules which are taken as cylinders. Arguing against this is the success of the Flory theory in predicting VE for mixtures of cycloalkanes of very different molecular size.,

Fig. 2 shows experimental equimolar values from Table 2 for dVE/dT together with Flory theory predictions using eqn. (6) and (7) in ref. (1). For the br-C, series, Flory theory is able to predict well the main trend of dVE/dT, i.e. the change from negative to positive values as rn increases. Note that the theory gives a satisfactory rendering of the large negative d VE/d T value for chlorobenze-br-C, which arises from the P* and contributions to VE,17 both of which become more negative with increasing T . As with VE in Fig. 1, the theoreti- cal dVE/dT predictions for br-C,, and br-C,, deviate con- siderably from the experimental values, this same behaviour having been observed before with cyclohexane.' For the n-C, series, Fig. 2 indicates that the experimental dVE/dT values start in good agreement with the theoretical value for n- hexane, but as rn increases dVE/dT becomes only slightly positive and then decreases to reach a very small value for n-hexadecane. This decrease in dVE/dT for long-chain n- alkanes, and its progressively larger deviation from Flory theory predictions, is also present with ethylben~ene,~ benzene, toluene and p-xylene5 mixed with the n-C, series and much more evident with cyclohexane,' where dVE/dT for n-hexadecane is ca. -8 x cm3 mol-' K - ' while the theoretical prediction is 2 x cm3 mol-' K-I . The dVE/dT behaviour, and the failure of the Flory theory to reproduce it, is attributed to the presence of temperature- dependent order in the higher n-C, liquids. This corresponds to a cohesion which decreases with increasing T ; thus, dV/dT for the pure n-alkane is enhanced and order destruc- tion on mixing causes a negative contribution in dVE/dT not

8 /

m

Fig. 2 Experimental (0) and Flory theory predictions (--) of dVE/dT at equimolar concentration and 25 "C for chlorobenzene mixed with the n-C, and br-C, series of alkanes

accounted for by the theory with a temperature-independent X , , parameter.

Experimental equimolar values from Table 2 for - d VE/dp = (j?TV)E together with Flory theory predictions using eqn. (7) and (8) in ref. 2 are shown in Fig. 3. Here the carbon number dependence is similar for both series of alkanes. Qualitatively similar dependences were found for cyclohexane,2 ethylben~ene,~ benzene, toluene and p-xylene5 and 1-~hloronaphthalene,~ as in those cases predictions from the theory are in good agreement with the experimental values. It appears that the effect of destroying the n-C, order in -dVE/dp is insufficient to show up against the back- ground of other effects in this quantity and, as reported earlier,,, the Flory theory is able to give reasonable predic- tions for - d VE/dp without consideration of any additional effects.

Excess Enthalpies and the Quantity - A(y VT)

Amongst the second-order quantities, the mixing function A(yVT) is of prime importance since it reflects directly the effect of structure in the equation of state. The change during the mixing process of yVT can be related to the internal energy, U, through the thermodynamic equation of state:

(2) (6U/6V), = T(dp/dT), - p = yT - p

- A(? V T ) = n'HE

which gives at p = 0

(3) or with n' = 1, i.e. vdW type behaviour, -A(yVT) = HE. The agreement between theory and experiment seen in Fig. 1-3 is less good in Fig. 4 where equimolar HE and -A(yVT) values from Table 2 are shown. Here, for br-C,, and br-C,,, -A(yVT) is much lower than the corresponding HE values, i.e. these mixtures deviate from the simple vdW or Flory theory which predicts -A(yVT) = HE. The explanation for this deviation could lie in the non-vdW behaviour of chloro- benzene reported by Gibson and Loeffler6 who measured (6U/6V) , and found it to decrease as V decreases. Hence for chlorobenzene, eqn. (1) does not hold and it is necessary to express the configurational energy in a more general form

10

5 r

I

E o

g -5

5 -10

z 3

r I -

m

.-. -15

-20 4

I I1

-25 S s c -30

-35

I m

Fig. 3 Experimental (0) and Flory theory predictions (-) of -dVE/dp = (& V)E at equimolar concentration and 25 "C for chlo- robenzene mixed with the n-C, and br-C, series of alkanes

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J. CHEM. SOC. FARADAY TRANS., 1993, VOL. 89

n -C, '*

I ;

1500

1000

500

0

7 \FC,

6 10 14 m

Fig. 4 Experimental equimolar values of - A(y V T ) (0) and HE (0) at 25°C for chlorobenzene mixed with the n-C, and br-C, series of alkanes

such as

u = -a/V"' + c / P ' (4)

which was proposed by Hilderbrand and Using eqn. (4) together with the condition (d2U/6V2)T = 0, which holds for chlorobenzene at 25 "C and atmospheric pressure,6 y VT and the ratio yVT/( - U) are obtained as

yVT = (an'/V"')[I - (n' + l)/(mf + I)] = w n d w C 1 - (n/ + l)/(m/ + 1)1 ( 5 )

] (6) [ ;:I: ;]I[ rn'(m' + 1) 1 - n'(n' + 1)

yVT/( -U)=n' 1 --

with m' being larger than n'. Eqn. (5) indicates that yVT for a liquid following eqn. (4), uiz. chlorobenzene, will be less than that given by vdW behaviour and hence, assuming vdW behaviour for the solution, there must be a positive contribu- tion in A(yVT) and a negative one to -A(yVT) . The devi- ations from the equality HE = - A ( y V T ) observed in Fig. 4 for br-C,, and br-C,, are consistent with the failure of vdW behaviour of chlorobenzene. Using y V T for chlorobenzene in Table 1, - U = (AH,,, - RT) from ref. 24 and a given n' value, eqn. (6) can be solved for m'. The experimentally found result, -A(yVT) - HE = -890 J mol-' for chlorobenzene-br-C,, in Fig. 4 and Table 2 is reproduced with n ' = 1.03 and m'= 47. Note that the present - A(y V T ) c HE would also follow from a creation of order in the solution.

The success of eqn. (4) in qualitatively explaining the devi- ation of - A ( y V T ) from HE for br-C,, and br-C,, in Fig. 4 is not repeated in the case of dVE/dT and -dVE/dp. For these two properties, it is predicted that the non-vdW behaviour of chlorobenzene should manifest itself in a positive contribu- tion which would increase their values above those given by the vdW behaviour of Flory theory. This is not seen in Fig. 2 and 3, where the experimental dVE/dT and -dVE/dp values are smaller than Flory theory predictions. Furthermore, -A(yVT) = HE for br-C, and br-C,, whereas the failure of vdW behaviour should result in - A ( y V T ) c HE for these systems as well. We conclude that present evidence against the vdW assumption is unconvincing. We are unable to provide an explanation for the discrepancy found in the br-CI2 and br-C,, cases, while at the same time allowing for the success of the Flory theory with br-C, and br-C, .

The relation between HE and -A(yVT) for the n-C, series shown in Fig. 4 is similar to that previously found for other A-n-C, m i x t u r e ~ , ~ , ~ * ~ i.e. HE and -A(yVT) values are close for short n-alkanes but as m increases - A ( y V T ) becomes larger than H E . The origin of this departure can be found in the presence of order in the higher n-C,, causing an increase of yVT in the pure liquid and hence, on destruction of order in the solution, a negative contribution to A(yVT) and a posi- tive one to -A(yVT) . The discrepancy between - A ( y V T ) and HE for n-hexadecane is of different magnitude depending on the order-destroying liquid used: [ - A ( y V T ) - HE]/ J mo1-l are 1500 for cyclohexane,2 1200 for b e n ~ e n e , ~ 950 for t ~ l u e n e , ~ 770 for p-~ylene ,~ 940 for ethylbenzene [HE from ref. 25 and - A ( y V T ) from ref. 41 and 420 for chlorobenzene in Fig. 4 or Table 2. In ref. 5, the small difference between - A ( y V T ) and HE for p-xylene was attributed to an ordering in solution taking place between the plate-like p-xylene and the n-C, molecules which competes with the n-C, order destruction. This order creation in solution is more pro- nounced for 1 -chloronaphthalene with long-chain n-alkanes, where for n-hexadecane - A ( y V T ) - HE = -375 J mol-'.3 In the present case, this difference is smaller than for p-xylene and it can be assigned to either the non-vdW behaviour of chlorobenzene or an ordering in solution, both of which, as seen above, have an equivalent effect on -A(? VT) .

Excess Heat Capacities and the Quantities A(ay VT) and AC,

The molar excess heat capacity C; and the mixing functions A(ayVT) and A C , = CF - A(cryV/T) from Table 2 are plotted against m in Fig. 5. According to Flory theory, CF = A(ayVT) , A C , being set to zero in this theory as in similar vdW-type theories. The non-vdW behaviour of chloroben- zene increases CI and y of the solution and hence should dis- place A(ayVT) to less negative values. However, the CF and A(ayVT) seen in Fig. 5 are consistent with the predictions of Flory theory plus the presence of order in the higher n-C, pure liquids. The Flory theory predictions for CF are seen in Fig. 5 to be identical for both the n-C, and br-C, series, their values being small and positive. While for the br-C, series the CF(rn) dependence is qualitatively well predicted, for the n-C, series the experimental values are seen to depart considerably from the theoretical ones. For the n-C, series, Cp" becomes increasingly negative as rn increases. This large discrepancy, which has been found earlier for other A-n-C, mixture^,^.^'^ is evidence for a volume- and temperature-dependent order

- I 1 2 1 ( a ) * \ t-1

t -3

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in higher n-C,. The cohesion that order brings about in the pure n-C, liquid decreases in strength as the temperature increases, so that C, = dH/dT is enhanced. Upon mixing, the order is destroyed, causing a negative contribution to CF. Strong negative CF values are then due to a decrease of order in the n-C, liquid as T increases, an effect which is absent in the vdW-type Flory theory. Fig. 5 shows that for the br-C, series the two contributions to Cp", i.e. A(cryVT) and ACv, are small, displaying only a slight m dependence. In contrast, for the n-C, series, the carbon number dependences of A(cryVT) and ACv are similar, each of these quantities having approx- imately the same magnitude. ACv is then clearly an impor- tant contribution to Cp" which is ignored by Flory theory and also by theories using eqn. (4) where, since the energy U depends only on the volume V , ACv is equal to zero. Removal of this approximation would require the intro- duction of both volume and temperature dependences for the energy.

Amongst all first- and second-order thermodynamic func- tion, CF is the most sensitive to structural changes occurring in the mixing process. Hence, CF values for A-n-C, mixtures can be used as a measure of the order-breaking ability of different liquids, A. Previous work2v5 has shown that cyclo- hexane and benzene are the most effective order-breakers, their equimolar CF values with hexadecane being -6 and -7 J mol-' K - ', respectively. Comparing the CF values for chlorobenzene--n-C, in Fig. 5 with those for toluene in ref. 5 and ethylbenzene in ref. 26, it is seen that the three sets of data are very similar, their Cp" values with n-hexadecane being less negative than for cyclohexane or benzene. Further support for the similarity between toluene, ethylbenzene and chlorobenzene in their order-breaking ability is as follows : (a) the experimental d VE/dT values for chlorobenzene-n-C, in Fig. 2 are very close to those observed previously for toluene5 and ethylben~ene,~ (b) their experimental A(? V T ) values are similar and (c) the heat capacities of transfer from br-c,, to n-CI6, i.e. from a non-ordered environment to an ordered one, are virtually identical.27 It is concluded that the order- breaking ability of toluene, ethylbenzene and chlorobenzene is the same. Since the substitution of a methyl group or an ethyl group by a chlorine atom on going from toluene or ethylbenzene to chlorobenzene does not change significantly the order-sensitive mixing functions, it appears that the ability to break the n-C, order is mainly a function of the shape of the order-breaking molecule rather than of its polarity.

Conclusion The mixing functions of chlorobenzene with the linear and branched alkanes can be almost entirely explained by the Flory theory coupled with ordering either in the pure n- alkanes or the mixtures. In spite of the failure of the vdW U ( V ) relation found by Gibson and Loeffler the excess properties provide little evidence for this. This, to us, sur- prising and inexplicable result justifies continued use of the Flory theory with systems containing complex liquids.

Tecnologia de Mexico and the Natural Sciences and Engin- eering Research Council of Canada for financial support.

References 1

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We thank the Ministerio de Educacion y Ciencia de Espaiia (DGICYT PB89-0113), the Consejo Nacional de Ciencia y Paper 2/03238D; Received 19th June, 1992

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