Thermodynamics of Mixtures Containing Oxaalkanes. 4. Random

Ind. Eng. Chem. Res. 2010, 49, 9511–9524

9511

Thermodynamics of Mixtures Containing Oxaalkanes. 4. Random Mixing and Orientational Effects in Ether + Alkane Systems‡ Juan Antonio Gonza´lez† GETEF, Departamento de Fı´sica Aplicada, Facultad de Ciencias, UniVersidad de Valladolid, 47071 Valladolid, Spain

Molar excess enthalpies, HEm, and volumes, VEm, of linear or cyclic ether + n-alkane systems have been discussed in terms of the effective dipole moment, µ j , of the ether, differences of standard enthalpies of vaporization of the ether and of the isomeric alkane, the number and relative positions of the oxygen atoms, the shape of the ether, and the relative size of the mixture compounds. The mentioned solutions have also been studied using the Flory model and the Kirkwood-Buff formalism. Both theories provide consistent results. At 298.15 K, the random mixing hypothesis is a good approximation for mixtures including linear or cyclic monoethers or linear diethers. Orientational effects become stronger in solutions with 2,5,8-trioxanonane, 2,5,8,11tetraoxadodecane, or 2,5,8,11,14-pentaoxapentadecane. In the case of 1,3-dioxolane mixtures, this type of effect is more relevant than in systems with 1,4-dioxane. This is supported by W-shaped CEp,m curves, large variations of the Flory interaction parameter, X12, with x1, the oxaalkane concentration, and local mole fractions of the ether-ether type that are higher than the bulk ones, particularly at lower x1 values. The latter means that orientational effects are more important at this condition, and this is confirmed by large X12 variations with x1 in the region of low x1 values. At equimolar composition, X12 values of solutions containing 2,5,8,11tetraoxadodecane or cyclic ethers remain nearly constant with the number of C atoms of the alkane, which reveals that systems with such oxaalkanes are similar from an interactional point of view. From values of the internal excess energies at constant volume, UVEm, it is shown that interactions between like molecules are E usually overestimated. Nevertheless, the general trends observed are independent of the Hm or UVEm data considered. Structural effects are present in mixtures with components that differ largely in size (e.g., dipropyl ether + hexadecane or 2,5,8,11,14-pentaoxapentadecane + hexane). 1. Introduction 1

It is well-known that the Flory model is commonly used to E E and Vm of systems formed by describe simultaneously Hm nonpolar (benzene2) or slightly polar compound (N,N,Ntrialkylamine3 or monoether4) and alkane, or involving two polar compounds as 1-alkanol + 1-alkanol5 or 2-methoxyethanol + hydroxy ether.6 The theory has been also applied to predict the isobaric expansion coefficient, RP, isentropic, κS, or isothermal, κT, compressibilities, and speeds of sound, u, of simple systems, as those involving two alkanes, or of the type cyclohexane or benzene + n-alkane.2,7,8 The good results obtained from the Flory model imply that random mixing hypothesis, a basic assumption of the theory, is attained to a large extent for these systems. This may be because such solutions are characterized by dispersive interactions (benzene + alkane) or by weak dipolar interactions (monoether + alkane) or due to the identical chemical natures of the two mixture compounds (1-alkanol + 1-alkanol). The Flory model has been also widely used to investigate order creation and order destruction processes in B + Cn mixtures9-12 (B is usually a nonpolar or slightly polar compound, with spherical or platelike shape). As any order effects are ignored in the theory (random mixing hypothesis), deviations from this behavior lead to differences between E E E E and Vm , Cp,m , (∂Vm / experimental values for magnitudes as Hm E /∂P)T and theoretical results, which are ascribed ∂T)P, or -(∂Vm to order effects. The main conclusion of these studies is the existence of a short orientational order in long chain alkanes,

which does not appear in highly branched isomeric alkanes, or short chain alkanes. In recent works, we have shown that the model application can be extended in two ways. First, the theory provides rather accurate predictions on RP, κS, κT, and u of complex mixtures as alkoxyethanol + dibutyl ether or + 1-butanol.13 Second, it is possible to investigate the existence of orientational effects in these complex mixtures by studying the variation of the X12 interaction parameter with the composition.14,15 Here, the same method is applied to investigate the validity of the random mixing hypothesis in ether + alkane mixtures. For a better understanding of the orientational effects present in these solutions, they are also studied, when the needed experimental data are available, by means of the Kirkwood-Buff formalism.16-18 Previous works19-22 on ether mixtures in terms of the Flory model have paid special attention to the correlation of X12 with the number of C atoms or with the acentric factor of the n-alkane in homologous series containing a given ether. The correlation of the interaction parameter with the molecular structure based on the molecular surface interactions has also been examined.22,23 It should be mentioned that the Nitta-Chao24 and the Victorov-Smirnova25 models have been applied to the prediction of the thermodynamic properties of linear ether + alkane systems.26,27 Similarly, DISQUAC28 has been also applied to this type of mixtures29 and to those involving branched monoethers,30 or cyclic oxaalkanes.31 2. Theories

E-mail: [email protected]. Fax: +34-983-423136. Tel: +34-983423757. ‡ This work is part of a series of two articles dedicated to the memory of Henry V. Kehiaian. †

2.1. Flory Model. In this section, a brief summary of the model is presented. More details are given in the original works.1,32-35 The main features of the theory are as follows:

10.1021/ie101264p  2010 American Chemical Society Published on Web 08/19/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010

(i) Molecules are divided into segments. A segment is an arbitrarily chosen isomeric portion of the molecule. The number of segments per molecule of component i is denoted by ri and the number of intermolecular contact sites per segment by si. (ii) The mean intermolecular energy per contact is proportional to - η/Vs (where η is a positive constant characterizing the energy of interaction for a pair of neighboring sites and Vs is the volume of a segment). (iii) When the configurational partition function is stated, it is assumed that the number of external degrees of freedom of the segments is lower than 3, to take into account the restrictions on the precise location of a given segment by its neighbors in the same chain. (iv) Random mixing is assumed: the probability of having species of kind i neighbors to any given site is equal to the site fraction, θi (θi ) siriNi/srN, where N ) N1 + N2 is the total number of molecules and r and s are the total number of intermolecular segments and contact sites per segment, respectively). For very large total numbers of contact sites, the probability of formation of an interaction between contact sites belonging to different liquids is θ1θ2. Under these hypotheses, the Flory equation of state is given by jV j P V 1 ) 1/3 jT j j j VT V -1 j 1/3

(1)

j ) V/V*, P j ) P/P*, and T j ) T/T* are the reduced where V volume, pressure, and temperature, respectively. Equation 1 is valid for pure liquids and liquid mixtures. For pure liquids, the reduction parameters, V*, i P*, i and T* i can be obtained from experimental data, such as RPi and κTi. For mixtures, the corresponding parameters are calculated as follows: V* ) x1V*1 + x2V*2 T* )

(2)

φ1P*1 + φ2P*2 - φ1θ2X12 φ2P*2 φ1P*1 + T*1 T*2

(3)

P* ) φ1P*1 + φ2P*2 - φ1θ2X12

siηii

(5)

2

2V*s

as X12 )

s1∆η

(6)

2V*s 2

where ∆η ) η11 + η22 - 2η12. In eqs 5 and 6, V*s (reduction volume for segment) and ηij are changed from molecular units E is determined from to molar units per segments. Hm HEm )

(

)

(

x1V*θ 1 1 1 2X12 1 1 + x1V*P + x2V*P 1 * 1 2 * 2 j j j j j V V1 V V2 V

which can be also written as

j 0 ) φ 1V j 2. The term that depends directly on j 1 + φ 2V where V X12 in eq 8 is usually named the interaction contribution32 to E Hm . The other terms are the so-called equation of state E j , in . The reduced volume of the mixture, V contribution32 to Hm eqs 7 and 8 is obtained from the equation of state. Therefore, the molar excess volume can be also calculated: j - φ1V j 1 - φ2V j 2) VEm ) (x1V*1 + x2V*)(V 2

(9)

2.2. Kirkwood-Buff Formalism: Theory. The theory17,18 describes thermodynamic properties of solutions in an exact manner in the whole concentration range using the KirkwoodBuff integrals: Gij )

∫

∞

0

(gij - 1)4πr2 dr

(10)

The radial distribution function, gij, denotes the probability of finding a molecule of species i in a volume element at the distance r of the center of a molecule of species j. So, this function provides information about the solution structure on the microscopic level. The product FjGij (Fj is the number density of molecules of species j) represents the average excess (or deficiency) number of molecules j in the whole space around a molecule i with respect to the bulk average. The Gij values can be obtained from FjGij by a process of normalization with respect to concentration and can be interpreted as follows: Gij > 0 represents the excess of molecules of the i type in the space around a given molecule of species j. This means attractive interactions between molecules of i and j. Gij < 0 means that interactions of i-i and j-j are preferred to mutual interactions.17,36 The Kirkwood-Buff integrals can be derived from experimental data of thermodynamic properties as chemical potential; partial molar volumes and isothermal compressibility factor. The resulting equations are17,37

(4)

In eqs 3 and 4, φi ) xiV*/Σx i iV* i is the segment fraction and θ2 is alternatively calculated as θ2 ) φ2/(φ2 + S12φ1). S12 is the so-called geometrical parameter of the mixture, which, assuming that the molecules are spherical, is calculated as S12 ) (V*1 / V*2 )-1/3. The energetic parameter, X12, also present in eqs 3 and 4, is defined by similarity with P*i )

E j + x1V*1 φ2((V j1 - V j 2)/V j o)(P*2 /V j 2 - P*1 /V j 1) + Hm ) x1V*1 θ2X12 /V E j 2 Vm /(Vo) (φ1P*1 + φ2P*2 ) (8)

)

(7)

G11 ) RTκT +

j 22 x 2V V x1VD x1

(11)

G22 ) RTκT +

j 12 x 1V V x2VD x2

(12)

j 1V j2 V VD

(13)

G12 ) G21 ) RTκT -

j i are the mole fraction and where R is the gas constant, xi and V the partial molar volume of component i, respectively (i ) 1, 2), V is the molar volume of the solution, and κT is the isothermal compressibility of the mixture. D is defined as D)1+

( )

x1x2 ∂2GEm RT ∂x2 1

(14) P,T

E In this expression, Gm is the excess molar Gibbs energy. Using the Gij quantities, it is possible to estimate the so-called linear coefficients of preferential solvation:38,39

δ011 ) x1x2(G11 - G12)

(15)


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Table 1. Properties of Pure Compounds at T ) 298.15 K Needed for the Application of the Flory Theory or the Kirkwood-Buff Formalism a

compundb

Vi/cm3 mol-1

Rp/10-3 K-1

κT/10-12 Pa-1

3 V*/cm mol-1 i

P*/J cm-3 i

1O1 2O2 3O3 4O4 5O5 8O8 1O4 1O2O1 2O2O2 1O2O4 1O3O3 1O4O2 1O5O1 2O2O3 2O3O2 1O1O1 2O1O2 1O2O2O1 3O2O2O3 1O2O2O2O1 1O2O2O2O2O1 THF THP 1,3-dioxolane 1,3-dioxane 1,4-dioxane hexane heptane octane decane dodecane tetradecane hexadecane

69.67c 104.74e 137.68f 170.45g 203.40h 302.09i 118.75k 104.34c 141.33m 157.01n 157.86n 157.11n 154.98n 158.69n 159.03n 89.49o 126.38o 142.93l 248.64p 181.72l 221.02l 81.76r 98.19c 69.98u 85.64V 85.71c 131.57w 147.45x 163.52x 195.9x 228.47x 261.09x 294.04x

1.996c 1.654e 1.261f 1.1336g 1.027h 0.68j 1.36k 1.268l 1.225m 1.17n 1.12n 1.12n 1.04n 1.13n 1.12n 1.495o 1.281o 1.060l 1.275q 0.965l 0.921l 1.2265r 1.156s 1.164u 1.05V 1.115c 1.387w 1.256x 1.164x 1.051x 0.960x 0.886y 0.883w

2514.6d 1967e 1440f 1205.9g 1067h 580j 1497.4k 1114.5l 1140.5m 1053n 1069.5n 1036.6n 939.7n 1093.2n 1107.4n 1485o 1411o 821.6l 958q 707.1l 589.6l 962.3r 990t 758.3u 733V 738c 1794w 1461x 1302.4x 1110x 988x 872y 862w

49.02 76.57 106 133.74 162.02 236.59 90.16 80.25 109.38 122.51 124.12 123.53 123.41 124.58 123.53 66.3 97.0 113.44 191.04 146.49 179.51 63.26 76.78 54.65 68.08 67.44 99.48 113.6 127.71 155.71 184.33 213.33 240.42

478.1 469.1 440.7 455.2 450.8 485 469.8 573.4 534.6 544.1 505 521 520.4 500 535.4 574.2 465.3 610.6 666.6 626.1 706.1 634.7 569.4 750.3 675.7 727.5 424.2 431.9 436.8 447 445.2 453.7 457

a reduction parameter for volume; P*, reduction Vi, molar volume; Rp, isobaric thermal expansion coefficient; κT, isothermal compressibility; V*, i i parameter for pressure in the Flory model. b The figures represent the number of aliphatic groups attached to the O atoms, e.g., 1O1 is dimethyl ether and 1O2O2O1 is 2,5,8-trioxanonane; THF, tetrahydrofuran; THP, tetrahydropyran. c Reference 93. d Reference 94. e Reference 95. f Reference 96. g Reference 97. h Reference 21. i Reference 98. j Estimated value. k Reference 99. l Reference 100. m Reference 82. n Reference 101. o Reference 23. p Reference 102. q Reference 103. r Reference 104. s Reference 105. t Reference 106. u Reference 107. V Reference 108. w Reference 19. x Reference 3. y Reference 109.

δ012 ) x1x2(G12 - G22) δ021 ) x1x2(G12 - G11) δ022 ) x1x2(G22 - G12) which are useful quantities to determine the local mole fractions of the i species around the central j molecule:37-39 xij ) xi +

δ0ij Vc

influence on the final calculations of the Kirkwood-Buff E data at the same temperature integrals.49 In the absence of Vm E data at 298.15 K have been that the VLE measurements, Vm used. This introduces a small error to calculate the partial molar volumes of the components, magnitudes, which are of secondary importance when Gij values are determined.50 To evaluate Vi and κTi values at T * 298.15 K, we have applied the well-known equations for the density, F, Rp, and γ () Rp/κT):51 F ) Fo exp(-Ro∆T)

(17)

R ) Ro + Ro2(7 + 4RoT)∆T/3

(18)

γ ) γo - γo(1 + 2RoT)∆T/T

(19)

(16)

where Vc is the volume for solvation sphere. This value may be roughly estimated38 as the volume of a sphere of radius Rc ) 3r, where r is the radius of the central molecule. This leads to a value of Vc equal to approximately (33 - 1)Vo ) 26Vo, being Vo the molar volume of the solvated component. 2.2.1. Source of Data. D values were obtained using E Redlich-Kister type expressions for Gm determined from vapor-liquid equilibrium (VLE) data at the temperature T E data required for the available in the literature.40-44 The Vm calculations were taken from refs 41 and 45-48. Table 1 collects the isothermal compressibilities of pure compounds, κTi, as well as their molar volumes, Vi. For the mixtures, their isothermal compressibilities were calculated as κT ) Φ1κT1 + Φ2κT2, where Φi is the volume fraction of the component i of the system. That is, when the compressibility of the system is calculated, the solution is assumed to be ideal. This assumption does not

where ∆T ) T - 298.15 K 3. Estimation of the Flory Energetic Parameter E measurement at a given X12 can be determined from a Hm composition from the equation

( )

x1P*V 1 * 1 1 X12 )

( )

j1 j2 T T + x2P*V 2 * 2 1 j j T T x1V*θ 1 2

(20)

For the use of this expression, it must be taken into account E jT j is a function of Hm : that V

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E Table 2. Molar Excess Enthalpies, Hm , at 298.15 K and Equimolar Composition for Ether + n-Alkane Systemsa

systemc 1O1 + n-C 2O2 + n-C7 2O2 + n-C10 2O2 + n-C12 2O2 + n-C16 3O3 + n-C7 3O3 + n-C8 3O3 + n-C10 3O3 + n-C12 3O3 + n-C16 4O4 + n-C7 4O4 + n-C8 4O4 + n-C10 4O4 + n-C12 4O4 + n-C16 5O5 + n-C7 5O5 + n-C8 5O5 + n-C10 5O5 + n-C12 5O5 + n-C16 8O8 + n-C8 8O8 + n-C10 8O8 + n-C16 1O4 + n-C7 1O4 + n-C8 1O4 + n-C10 1O4 + n-C12 1O4 + n-C16 1O2O1 + n-C7 2O2O2 + n-C7 1O2O4 + n-C8 1O3O3 + n-C8 1O4O2 + n-C8 1O5O1 + n-C8 2O2O3 + n-C8 2O3O2 + n-C8 1O1O1 + n-C7 2O1O2 + n-C7 1O2O2O1 + n-C7 1O2O2O1 + n-C8 1O2O2O1 + n-C10 1O2O2O1 + n-C12 3O2O2O3 + n-C12 c 10

E Hm /J mol-1

X12/J cm-3

E Hm,int /J mol-1

E Hm,EOS /J mol-1

E c σr(Hm )

ref

643 42.33 530 113 0.044 40 362 16.47 268 94 0.053 20 437 19.81 361 76 0.021 20 483 21.89 461 62 0.043 20 616 26.75 556 60 0.063 20 204 7.10 148 56 0.021 60 242 8.21 179 63 0.046 19 293 9.65 226 67 0.032 19 352 11.44 284 68 0.167 19 482 14.97 403 79 0.040 19 119 3.66 90 29 0.020 110 142 4.11 106 36 0.022 111 186 5.11 142 44 0.020 110 110 239 6.44 190 49 0.022 354 9.10 292 62 0.039 110 94 2.81 79 15 0.059 21 109 2.90 85 24 0.071 21 143 3.43 109 34 0.036 21 185 4.26 144 41 0.047 21 286 6.24 231 55 0.065 21 47 0.95 35 12 58 59 1.17 47 12 58 68 1.46 63 5 58 307 11.96 222 85 0.060 99 337 12.91 250 87 0.041 99 382 14.33 297 85 0.042 99 425 15.82 347 78 0.022 99 552 19.72 468 84 0.043 99 1285 54.07 922 363 0.055 112 889 30.31 644 245 0.065 66 735 22.47 542 193 0.048 67 587 17.84 437 150 0.029 67 674 20.55 501 173 0.025 67 763 23.45 575 188 0.034 67 701 21.23 520 181 0.035 67 610 18.54 449 161 0.028 67 1064 51.31 755 309 0.058 113 605 22.34 437 168 0.039 113 1621 55.10 1203 418 0.065 68 1680 54.37 1253 427 0.066 69 1827 54.59 1376 451 0.092 114 1936 56.54 1479 457 0.095 114 971 20.42 753 218 0.082 102 861c 18.24 654 207 0.085 102 1O2O2O2O1 + n-C7 1704 49.23 1301 403 0.055 115 1O2O2O2O1 + n-C8 1877 51.74 1430 447 0.095 115 1O2O2O2O1 + n-C10 2110 54.25 1612 498 0.091 115 1O2O2O2O1 + n-C12 2214 54.03 1705 509 0.109 115 1O2O2O2O2O1 + n-C7 1897 48.64 1474 423 0.106 114 1O2O2O2O2O1 + n-C10 1785 40.34 1383 402 0.194 114 THF + n-C7 791 39.58 573 218 0.062 78 815 40.78 0.064 116 THF + n-C8 856 41.73 628 228 0.057 116 THF + n-C10 907 42.61 681 226 0.075 78 THF + n-C14 1000 45.19 793 207 0.032 78 THP + n-C7 598 26.18 439 159 0.028 78 607 26.57 0.018 116 THP + n-C8 648 27.45 479 169 0.020 116 THP + n-C10 666 27 502 164 0.028 78 THP + n-C14 759 29.42 603 156 0.110 78 1,3-dioxolane + n-C7 1937 108.04 1397 540 0.094 75 1,3-dioxolane + n-C8 2029 110.28 1479 550 0.099 75 1,3-dioxolane + n-C10 2156 112.75 1604 552 0.111 75 1,3-dioxolane + n-C12 2260 113.42 1715 545 0.118 75 1,3-dioxolane + n-C14 2344 116.77 1816 528 0.127 75 1,3-dioxane + n-C7 1561 74.83 1152 409 0.159 78 1,3-dioxane + n-C10 1562 69.16 1177 385 0.186 78 1,4-dioxane + n-C7 1783 85.61 1301 482 0.071 48 1,4-dioxane + n-C8 1861 86.66 1369 492 0.069 48 1,4-dioxane + n-C10 2036 90.62 1521 515 0.073 48 1,4-dioxane + n-C12 2158 92.98 1642 516 0.073 48 1,4-dioxane + n-C14 2260 95.07 1752 508 0.082 48 1,4-dioxane + n-C16 2361 97 1835 526 0.096 48 a E E E at equimolar compostion and the interactional and equation of state contributions, Hm,int and Hm,EOS , The interaction parameters calculated from Hm respectively, are also included. c For symbols, see Table 1 . c T ) 323.15 K.


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x1P*V x2P*V 2 * 2 1 * 1 1 j + x2P*V j (x P*V*T + + 2 *T 2 2) j j j j 1 1 1 1 V1 V2 VT (21) j j j and that, from the equation of state, V ) V(T). For normal applications, it is possible to use the so-called pj ≈ 0 approximation of the equation of state, which is a good approximation at atmospheric pressure. More details have been given elsewhere.14,15 Equation 20 generalizes that previously given to calculate X12 from HEm at x1 ) 0.5.52 Properties of the pure compounds, molar volumes, RPi and κTi, and the corresponding reduction paramV*i (i ) 1, 2), needed for calculations are listed in eters, P*and i E data at Table 1. X12 values determined from experimental Hm x1 ) 0.5 are collected in Table 2. HEm )

4. Results E E and Vm obtained from the Flory model using Results on Hm X12 values at x1 ) 0.5 are listed in Tables 2 and 3, respectively. E Table 3. Molar Excess Volumes, Vm , at 298.15 K and Equimolar Composition for Ether + n-Alkane Systemsa E Vm /cm3 mol-1

systemb

exp

Floryc

ref

2O2 + n-C7 2O2 + n-C16 3O3 + n-C7 3O3 + n-C8 3O3 + n-C10 3O3 + n-C12 3O3 + n-C16 4O4 + n-C7 4O4 + n-C8 1O2O1 + n-C7 2O2O2 + n-C7 1O2O4 + n-C8 1O3O3 + n-C8 1O4O2 + n-C8 1O5O1 + n-C8 2O2O3 + n-C8 2O3O2 + n-C8 1O1O1 + n-C7 2O1O2 + n-C7 1O2O2O1 + n-C7 1O2O2O1 + n-C8 1O2O2O1 + n-C10 1O2O2O1 + n-C12 3O2O2O3 + n-C12 1O2O2O2O1 + n-C7 1O2O2O2O1 + n-C8 1O2O2O2O1 + n-C10 1O2O2O2O1 + n-C12 1O2O2O2O2O1 + n-C7 1O2O2O2O2O1 + n-C10 THF + n-C7 THF + n-C8 THF + n-C10 THF + n-C14 THP + n-C7 THP + n-C8 THP + n-C10 THP + n-C14 1,3-dioxolane + n-C7 1,3-dioxolane + n-C8 1,3-dioxolane + n-C10 1,3-dioxolane + n-C14 1,4-dioxane + n-C7 1,4-dioxane + n-C8 1,4-dioxane + n-C10 1,4-dioxane + n-C14

0.375

0.419 -0.147 0.217 0.236 0.186 0.130 0.0004 0.056 0.123 1.275 0.818 0.656 0.481 0.541 0.483 0.603 0.591 1.313 0.649 0.914 1.122 1.415 1.633 1.676 0.456 0.800 1.232 1.510 -0.058 0.543 0.685 0.854 0.995 1.154 0.415 0.561 0.665 0.793 1.511 1.724 1.959 2.234 1.164 1.395 1.717 2.095

117

0.2558 0.2679 0.2312 0.1847 0.0971 0.086 0.1371 1.092 0.7426 0.5914 0.5608 0.5814 0.5099 0.6637 0.6327 1.217 0.699 0.902 1.099 1.340 1.601 0.712 0.7486 0.970 1.281 1.477 0.6023 1.185 0.3202 0.4212 0.555 0.739 0.2491 0.3924 0.480 0.776 0.7475 0.917 1.127 1.410 0.728 0.904 1.154 1.435

81 59 59 59 59 118 119 100 82 101 101 101 101 101 101 23 23 83 69 64 62 102 63 63 63 45 63 63 46 46 120 120 46 46 120 120 75 75 75 120 48 48 48 48

a Comparison of experimental (exp) results with Flory calculations. For symbols, see Table 1. c Values obtained using interaction E parameters determined from Hm at equimolar composition (Table 2).

b

E Figure 1. Hm for linear monoether (1) + heptane (2) mixtures at 298.15 K. Points, experimental results: (b) diethyl ether;20 (9) dipropyl ether;60 (2) dipentyl ether.21 Solid lines, Flory calculations using interaction parameters from Table 2.

E Figure 2. Hm for linear polyether (1) + heptane (2) mixtures at 298.15 K. Points, experimental results: (b) 2,5-dioxahexane;112 (9) 2,5,8-trioxanonane;68 (2) 2,5,8,11-tetraoxadodecane;115 (1) 2,5,8,11,14-pentaoxapentadecane.114 Solid lines, Flory calculations using interaction parameters from Table 2.

A comparison between experimental and theoretical values for E E Hm and Vm are shown graphically in Figures 1-3. For the sake of clarity, Table 2 also includes the relative standard deviations E defined as for Hm σr(HEm)

[ ∑(

1 ) N

HEm,exp - HEm,calc HEm,exp

)]

2 1/2

(22)

E where N () 19) is the number of data points and Hm,exp E represents smoothed Hm values calculated at ∆x1 ) 0.05 in the composition range [0.05, 0.95] from polynomial expansions given in the original works. To obtain detailed information on the concentration dependence of X12, this magnitude has been E values at ∆x1 determined using eq 20 and the mentioned Hm,exp ) 0.05. The X12(x1) variation is estimated from the equation:

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Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 E Table 4. Variations of the X12 Values, ∆i, Obtained from Hm Data at 298.15 of Ether + n-Alkane Mixtures in the Concentration Ranges [0.05, 0.5] (i ) 1) and [0.5, 0.95] (i ) 2) Calculated According to Eq 23

systema 1O1 + n-C 2O2 + n-C7 2O2 + n-C10 2O2 + n-C12 2O2 + n-C16 3O3 + n-C7 3O3 + n-C8 3O3 + n-C10 3O3 + n-C12 3O3 + n-C16 4O4 + n-C7 4O4 + n-C8 4O4 + n-C10 4O4 + n-C12 4O4 + n-C16 5O5 + n-C7 5O5 + n-C8 5O5 + n-C10 5O5 + n-C12 5O5 + n-C16 1O4 + n-C7 1O4 + n-C8 1O4 + n-C10 1O4 + n-C12 1O4 + n-C16 1O2O1 + n-C7 2O2O2 + n-C7 1O2O4 + n-C8 1O3O3 + n-C8 1O4O2 + n-C8 1O5O1 + n-C8 2O2O3 + n-C8 2O3O2 + n-C8 1O1O1 + n-C7 2O1O2 + n-C7 1O2O2O1 + n-C7 1O2O2O1 + n-C8 1O2O2O1 + n-C10 1O2O2O1 + n-C12 3O2O2O3 + n-C12 b 10

E Figure 3. Vm for the oxaalkane (1) + n-alkane (2) mixtures at 298.15 K. Points, experimental results: (2) dipropyl ether + heptane;81 (b) 2,5,8trioxanonane + heptane;83 (9) 2,5,8,11-tetraoxadodecane + dodecane.45 Solid lines, Flory calculations using interaction parameters from Table 2.

∆i )

|∆X12 | max i X12(x1 ) 0.5)

(23)

where |∆X12|max is the maximum absolute value of the X12(x1) i - X12(x1 ) 0.5) difference in the ranges [0.05, 0.45] (i ) 1) and [0.55, 0.95] (i ) 2). The corresponding values are listed in Table 4 (see also Figures 4 and 5). Values of the Kirkwood-Buff integrals and of the local mole fractions are collected in Table 5. Figures 6 and 7 show Gij results for some selected systems. 5. Discussion Thermodynamic properties of mixtures can be examined by taking into account differences in molecular size and shape, anisotropy, dispersion forces, orientational effects, and so forth. To investigate the impact of polarity on bulk properties, the effective dipole moment, µ j , can be used.53-56 To compare the relative changes in intermolecular forces of homomorphic compounds upon replacing a CH2 group by a given X group (e.g., × ) -O-), the standard enthalpy of vaporization, ∆Hvap, is a useful magnitude.29,56 So, if ∆∆Hvap ) ∆Hvap(oxaalkane) - ∆Hvap(isomeric alkane) > 0, then the interactions in the ether are stronger than in the homomorphic alkane. Table 6 lists µ j and ∆∆vapH for some ethers. In the following, we are referring to values of the excess functions at equimolar composition and 298.15 K; n stands for the number of C atoms in the alkane. 5.1. Linear Monoether Systems. HEm values of CH3(CH2)u-1O(CH2)u-1CH3 + fixed alkane mixtures decrease when u is increased (Table 2), because the ether-ether interactions become weaker, as the corresponding decrease of µ j and ∆(∆Hvap) reveal (Table 6). In addition, it should be taken into account that the number of those interactions, available to be broken up during the mixing process, also decreases for molecules characterized by large u values, as the -O- atom is then more sterically E increases in solutions with a hindered. On the other hand, Hm given ether when n is increased. This may be due to (i) longer alkanes being more active breakers of the ether-ether interactions and (ii) changes in the orientational order of long chain alkanes produced by di-n-alkyl ethers29 (Patterson’s effect57).

1O2O2O2O1 + n-C7 1O2O2O2O1+ n-C8 1O2O2O2O1 + n-C10 1O2O2O2O1 + n-C12 1O2O2O2O2O1 + n-C7 1O2O2O2O2O1 + n-C10 THF + n-C7 THF THF THF THP

+ + + +

n-C8 n-C10 n-C14 n-C7

THP + n-C8 THP + n-C10 THP + n-C14 1,3-dioxolane + n-C7 1,3-dioxolane + n-C8 1,3-dioxolane + n-C10 1,3-dioxolane + n-C12 1,3-dioxolane + n-C14 1,3-dioxane + n-C7 1,3-dioxane + n-C10 1,4-dioxane + n-C7 1,4-dioxane + n-C8 1,4-dioxane + n-C10 1,4-dioxane + n-C12 1,4-dioxane + n-C14 1,4-dioxane + n-C16 a

∆1

∆2

ref

0.031 0.139 0.035 0.076 0.133 0.053 0.082 0.043 0.046 0.036 0.049 0.058 0.043 0.039 0.060 0.139 0.399 0.079 0.124 0.168 0.085 0.073 0.105 0.034 0.123 0.160 0.192 0.117 0.075 0.064 0.090 0.089 0.074 0.149 0.068 0.180 0.164 0.229 0.277 0.203 0.242b 0.076 0.354 0.273 0.277 0.352 0.735 0.172 0.145 0.109 0.242 0.032 0.049 0.018 0.011 0.023 0.228 0.219 0.233 0.248 0.238 0.164 0.621 0.802 0.186 0.150 0.185 0.142 0.133 0.148

0.069 0.073 0.014 0.022 0.038 0.017 0.078 0.063 0.048 0.050 0.024 0.022 0.016 0.026 0.038 0.028 0.114 0.044 0.045 0.058 0.125 0.067 0.034 0.023 0.025 0.017 0.020 0.039 0.011 0.010 0.015 0.017 0.015 0.012 0.063 0.046 0.056 0.185 0.125 0.073 0.063b 0.117 0.052 0.078 0.181 0.161 0.466 0.036 0.079 0.080 0.046 0.037 0.037 0.039 0.047 0.078 0.282 0.069 0.083 0.102 0.121 0.160 0.159 0.168 0.036 0.045 0.054 0.068 0.094 0.146

40 20 20 20 20 60 19 19 19 19 110 111 110 110 110 21 21 21 21 21 99 99 99 99 99 112 66 67 67 67 67 67 67 113 113 68 69 114 114 102 102 115 115 115 115 114 114 78 116 116 78 78 78 116 116 78 78 75 75 75 75 75 78 78 48 48 48 48 48 48

For symbols, see Table 1. b T ) 323.15 K.

Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 E Vm

Figure 4. Flory interaction parameters, X12, for linear oxaalkane (1) + E heptane (2) mixtures at 298.15 K. Points, values determined from Hm at ∆x1 ) 0.05: (b) dipropyl ether;60 (9) dibutyl ether;110 (2) dipentyl ether;21 (1) 2,5-dioxahexane;112 ([) 2,5,8,11-tetraoxadodecane.115 Solid lines, X12 E values calculated from Hm at x1 ) 0.5 (Table 2).

Figure 5. Flory interaction parameters, X12, for cyclic ether (1) + heptane E (2) mixtures at 298.15 K. Points, values determined from Hm at ∆x1 ) 0.05: (2) tetrahydrofuran;116 (b) 1,4-dioxane;48 (9) 1,3-dioxolane.75 Solid E lines, X12 values calculated from Hm at x1 ) 0.5.

The latter is supported, e.g, by discrepancies between experimental HEm values and those predicted by the group contribution model DISQUAC for solutions with longer alkanes.29 The existence of an orientational order in linear monoethers of long chains has been pointed out in the case of dioctyl ether by E values of systems containing linear or branched comparing Hm E isomeric alkanes58 (those with n-alkanes have the higher Hm E values). It may be interesting here to show the ratios Hm (ether E (ether + heptane) for a number of mixtures: + dodecane)/Hm 1.33 (diethyl ether); 1.38 (butyl methyl ether); 1.73 (dipropyl ether); 2.01 (dibutyl ether); 1.97 (dipentyl ether); 1.19 (2,5,8trioxanonane); 1.21 (1,4-dioxane). On the other hand, µ j and ∆(∆Hvap) values of butyl methyl ether are higher than those of E is higher for systems dibutyl ether (Table 6), and therefore, Hm with the asymmetric ether (Table 2).

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values depend on the relative size of the mixture The components, which indicates the existence of structural effects. So, for dipropyl ether systems (all values in cm3 mol-1):59 E E E Vm (hexane) ) 0.1939 < Vm (octane) ) 0.2679 > Vm (decane) ) E 0.2312 > Vm(hexadecane) ) 0.097. These positive values E is the interactional indicate that the main contribution to Vm E becomes one. Structural effects may be predominant and Vm negative, as in the case of the dibutyl ether + hexane mixture41 (-0.0443 cm3 mol-1). Finally, it should be mentioned that the E ’s of these solutions are negative and of low absolute value Cp,m E (dipropyl ether + heptane)60 ) -0.5955 J mol-1 K-1), (Cp,m which is characteristic of systems with weakly polar interactions. E is larger than for 5.2. Linear Polyether Systems. Here, Hm solutions including linear monoethers (Table 2), accordingly to the higher µ j and ∆(∆Hvap) values of the CH3(CH2)u-1O(CH2CH2O)V(CH2)u-1CH3 compounds (Table 6). Therefore, interactions between oxaalkane molecules are stronger in these types of systems. In mixtures with a fixed alkane, HEm decreases when V is constant and u increases, which may partially be due to the oxygen atoms becoming more sterically hindered, while E increases with V in systems involving ethers characterized Hm by constant u values. This is newly consistent with the relative variation of µ j and ∆(∆Hvap) with u and V (Table 6). Thus, interactions between 2,5,8,11-tetraoxadodecane or 2,5,8,11,14pentaoxapentadecane molecules are stronger than those between 2,5-dioxahexane molecules. In fact, at 298.15 K, the systems formed by the tetraether and dodecane or the pentaether and decane are close to their upper critical solution temperatures (280.81 and 291.98 K, respectively61). Similarly, the µ j and ∆(∆Hvap) values of acetals and diethers point out that proximity E (acetal) < effects lead to, in mixtures with a given alkane, Hm E Hm(diether) (Table 2). E E Both Hm and Vm increase with n in mixtures with a fixed polyether (Tables 2 and 3). The positive VEm values suggest that, in such a case, the interactional contribution to this excess E function is dominant. However, the opposite variation of Hm E and Vm , observed in systems with a given alkane when V is increased (u constant), reveals that structural effects are also present. In fact, the data suggest that the increment of the positive contribution to VEm from the disruption of the strengthened ether-ether interactions is overcompensated by the negative increment from the contribution related to the difference E decrease. in size of the mixture compounds, leading to a Vm This magnitude may be even negative for solutions with E (2,5,8,11,14-pentaoxapcomponents of very different size (Vm entadecane + pentane)62 ) -0.39 cm3 mol-1). Moreover, the E curves of systems such as 2,5,8,11-tetraoxadodecane or Vm 2,5,8,11,14-pentaoxapentadecane + hexane or heptane are shifted to higher mole fractions of the smaller compound,63,64 and this points out the existence of free volume effects. CEp,m values of these systems strongly depend on the involved ether. For mixtures including diethers (2,5-dioxahexane, 3,6dioxaoctane, or dioxanonane isomers) and heptane or octane, E is negative over all the composition range.65-67 For Cp,m solutions including ethers characterized by u ) 1 and V ) 2, 3, E shows a concentration 4 and different alkanes,64,68-70 Cp,m dependence of the W-shape type: two minima at the extremes of the concentration range separated by a central maximum. This behavior is characteristic of systems at temperatures close to the UCST and is connected with dipolar interactions and with nonrandomness in the solution.71-74 5.3. Cyclic Ether Systems. The cyclization effect leads to increased excess enthalpies as the µ j and ∆(∆Hvap) values of the cyclic oxaalkanes are higher than those of the linear

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Table 5. Kirkwood-Buff Integrals, Gij, and Local Mole Fractions, xij, at Temperature T and Mole Fraction x1 for Ether + n-Alkane Mixtures systema 1O1 + n-C

f

b 10

T/K

x1

G11/cm3 mol-1

G22/cm3 mol-1

G12/cm3 mol-1

x11

x12

323.15

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

44.6 32.2 13.9 -14.5 -177.7 -180 -177.5 -173 70.8 62.6 22.9 -37.5 29.4 185.7 73.7 -144.3 443.9 495.4 404.6 78.7 420.7 555.7 373.4 6.2

-207.6 -221.4 -240.9 -270.6 -116.1 -104.1 -91.9 -81.8 -180.2 -149.2 -129.4 -89.5 -238.5 -147.5 117.6 -180.8 -129 -89.7 76.3 293.7 -142.1 -72 144.1 181.6

-79.4 -92.9 -112 -141.4 -162.5 -155.1 -146.6 -137.2 -100.1 -131.3 -163.3 -170.6 -248.2 -411.5 -534.9 -375.9 -129 -249.9 -410.1 -345.8 -159.2 -332.4 -479.8 -289.6

0.210 0.416 0.616 0.811 0.199 0.399 0.598 0.799 0.213 0.422 0.621 0.810 0.208 0.426 0.627 0.807 0.253 0.503 0.712 0.839 0.242 0.496 0.692 0.821

0.204 0.406 0.606 0.804 0.198 0.396 0.596 0.797 0.202 0.401 0.598 0.797 0.199 0.390 0.576 0.786 0.200 0.389 0.566 0.770 0.199 0.384 0.561 0.780

4O4 + n-C6c

298.15

THF + n-C7d

298.15

1O2O2O2O1 + n-C12e

435.26

1,3-dioxolane + n-C6f

308.15

1,4-dioxane + n-C7g

298.15

a E ) 0. For symbols, see Table 1. b Reference VLE;40 Vm E 47 g E 48 . . Reference VLE;44 ref Vm Reference VLE;43 ref Vm

c

E 41 . Reference VLE and Vm

d

E 46 . Reference VLE;43 ref Vm

e

E 45 . Reference VLE;42 ref Vm

Figure 6. Kirkwood-Buff integrals, Gij, for the dibutyl ether (1) + hexane (2) system at 298.15 K. Dashed lines, Gijsize (see text).

Figure 7. Kirkwood-Buff integrals, Gij, for cyclic ether (1) + heptane (2) systems at 298.15 K. Solid lines, 1,4-dioxane; dashed lines, tetrahydrofuran.

homomorphic ethers (Table 6; µ j of 1,4-dioxane is an exception). Moreover, the oxygen atoms in cyclic rings are less sterically hindered and ether-ether interactions are more easily formed. This is consistent with the dense packing of 1,3-dioxolane, 1,3dioxane, and 1,4-dioxane (densities: 1058.66, 1028.69, and 1027.93 kg m-3, respectively48,75,76) and may explain the high ∆(∆Hvap) value of 1,4-dioxane, which contrasts with its low µ j. It has been suggested that the interactions between 1,4-dioxane molecules are mainly dispersive and of quadrupolar type.77 Thus, the significant order in cyclic diethers is modified upon mixing E E and Vm are obtained (Tables 2 and and large and positive Hm 3). For mixtures with a given alkane, these excess functions change in the same sequence: tetrahydropyran < tetrahydrofuran < 1,4-dioxane < 1,3-dioxolane, in agreement with the relative E for the variation of ∆(∆Hvap) (Table 6). The close values of Hm systems 1,3-dioxane or 1,4-dioxane + cyclohexane (1488 and 1558 J mol-1, respectively76) suggest that proximity effects are

not very important. For solutions containing a fixed cyclic ether, E E Hm and Vm increase with n (Tables 2 and 3). The independence E of the maximum Hm value upon n, derived from the measurements by Inglese et al.,78 is not supported by the data obtained by Calvo et al.48 and Brocos et al.75 Discrepancies may be due to the increase of viscosity of the alkane when passing from n ) 7 to n ) 14, which reduces the efficiency of the Picker calorimeter used by Inglese et al.48 Mixtures with 1,3-dioxolane, 1,3-dioxane, or 1,4-dioxane also show W-shaped CEp,m curves.48,75,76 E The Cp,m values for the heptane solutions are 1.91 J mol-1 K-1 (1,3-dioxolane) and -0.1304 J mol-1 K-1 (1,4-dioxane). That is, nonrandomness is weaker in the 1,4-dioxane system (UCST ) 269 K79). 5.4. Excess Functions at Constant Volume. HEm is the result not only of interactional effects but also of structural effects. The former are more properly considered using UVEm, the excess

Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 a

Table 6. Physical Constants of Some Pure Compounds Considered in This Work oxaalkaneb 1O1 1O4 2O2 3O3 4O4 5O5 1O2O1 2O2O2 1O1O1 2O1O2 1O2O2O1 1O2O2O2O1 1O2O2O2O2O1 THF THP 1,3-dioxolane 1,3-dioxane 1,4-dioxane

TC/K d

400 512.8d 466.7d 530.8d 580d 626e 536e 565e 477.4e 524.1e 583.5e 636.4e 702.8e 540.1d 572.2d 554.1e 594.4e 587d

PC/bar d

52.4 33.7d 36.4d 30.3d 25.3d 20.7e 38.7a 30e 42.2e 33.3e 29.4e 23.7e 19.5e 51.9d 47.7d 58.2e 52.3e 52.1d

µ/D

∆(∆Hvap)c/kJ mol-1

µ j

3.05 0.87 0.71 -0.87 -1.43 -1.93 4.74 1.74

0.596 0.456 0.486 0.396 0.352 0.322 0.641 0.518 0.271 0.317 0.631 0.636 0.636 0.719 0.618 0.672 0.881 0.165

d

1.3 1.3d 1.3d 1.2d 1.2d 1.2f 1.71f 1.61g 0.67g 0.93g 1.97f 2.24g 2.47g 1.7d 1.6d 1.47g 2.13g 0.4d

-0.92 4.8 5.2 3.44 1.66 7.08h 6.07 5.62

a TC, critical temperature; PC, critical pressure; µ, dipole moment in gas phase; µ j , effective dipole moment calculated as µ j (effective dipole moment53) ) [µ2NA/(4πεoVkBT)]1/2 where, NA is the Avogadro’s number, εo is the permittivity of the vacuum, and kB is the Boltzmann constant. ∆(∆Hvap), difference between molar enthalpies of vaporization at 298.15 K of the ether and of the isomeric alkane. b For symbols, see Table 1. c Calculated using data from ref 121. d Reference 122. e Estimated from Lyndersen’s method.122 f Reference 93. g Reference 123. h Calculated using ∆Hvap from ref 93 for 1,3-dioxolane.

internal energy at constant volume. Neglecting terms of higher E , UVEm is displayed to a good approximation to54,80 order in Vm UEVm ) HEm -

Rp E TV κT m

(24)

where (Rp/κT)TVEm is termed the equation of state (eos) contribuE and Rp is the isobaric thermal expansion coefficient tion to Hm of the mixture. Due to the lack of experimental data, Rp and κT have been calculated from the F ) Φ1F1 + Φ2F2 equation (Fi is the property of the pure compound i). This is a reasonable approximation for the studied mixtures in view of the low values of the excess compressibilities available in the literature.69,81-83 E Values of (Rp/κT)TVm and UVEm are listed in Table 7. E (eq 24). The variation of UVEm strongly depends on that of Vm E For dipropyl ether mixtures, the UVm values increase with n more E ones (Figure 8), in agreement with the steeply than the Hm E , which decrease of the equation of state contribution to Hm E E decrease. In contrast, Vm and the mentioned comes from the Vm eos contribution increase with n for systems including linear polyethers or cyclic oxaalkanes, and UVEm varies with n more E smoothly than Hm does (Figure 9). It seems that the members of homologous series including such ethers mainly differ by structural effects. The more important UEVm variation for solutions with 1,3-dioxolane or 1,4-dioxane in comparison to those of cyclic monooxaalkanes merely indicates the difference between these ethers.84 Wu and Sandler developed a carefully study of the ability of the classical UNIFAC model to treat cyclic ethers85,86 and showed, on the basis of principles of unchanging geometry and approximate group electroneutrality, that new groups must be introduced for the correct description of systems with cyclic ethers. From the study of 1-alkanol + cyclic ether mixtures in terms of DISQUAC, it was also concluded that it is convenient to distinguish between cyclic monoethers and cyclic diethers (1,3-dioxolane behaves differently), for a better prediction of the thermodynamic properties of such solutions.87 In mixtures with a linear polyether and a fixed alkane, when V is increased (u ) 1), UVEm changes with n more rapidly than

9519

E of the Equation of State Term, rPVm T/KT, Energies at Constant Volume, UVEm, at

Table 7. Contribution Molar Excess Internal Equimolar Composition at 298.15 K and the Flory Interaction Parameter, X12, Determined from These Data (for References, see Tables 2 and 3) systema

E (RPVm T/κT)/J mol-1 UVEm/J mol-1 X12/J cm-3

2O2 + n-C7 3O3 + n-C7 3O3 + n-C8 3O3 + n-C10 3O3 + n-C12 3O3 + n-C16 4O4 + n-C7 4O4 + n-C8 1O2O1 + n-C7 2O2O2 + n-C7 1O2O4 + n-C8 1O3O3 + n-C8 1O4O2 + n-C8 1O5O1 + n-C8 2O2O3 + n-C8 2O3O2 + n-C8 1O1O1 + n-C7 2O1O2 + n-C7 1O2O2O1 + n-C7 1O2O2O1 + n-C8 1O2O2O1 + n-C10 1O2O2O1 + n-C12 3O2O2O3 + n-C12 1O2O2O2O1 + n-C7 1O2O2O2O1 + n-C8 1O2O2O2O1 + n-C10 1O2O2O2O1 + n-C12 1O2O2O2O2O1 + n-C7 1O2O2O2O2O1 + n-C10 THF + n-C7 THF + n-C8 THF + n-C10 THF + n-C14 THP + n-C7 THP + n-C8 THP + n-C10 THP + n-C14 1,3-dioxolane + n-C7 1,3-dioxolane + n-C8 1,3-dioxolane + n-C10 1,3-dioxolane + n-C14 1,3-dioxane + n-C7 1,4-dioxane + n-C7 1,4-dioxane + n-C8 1,4-dioxane + n-C10 1,4-dioxane + n-C14 a

95 66 71 63 51 28 23 37 312 211 204 161 169 149 189 179 332 184 272 354 426 516 245 234 310 421 491 202 416 93 125 171 238 71 115 145 244 221 279 357 469 219 369 279 370 481

267 138 171 230 301 454 96 105 973 678 531 426 505 614 512 431 732 421 1349 1326 1406 1420 726 1470 1567 1689 1723 1695 1369 698 731 736 762 527 533 521 515 1716 1750 1799 1875 1342 1564 1582 1667 1779

12.38 5.15 5.82 7.65 9.88 14.17 2.98 3.04 40.97 23.13 16.24 12.95 15.40 18.90 15.52 13.10 35.42 15.59 45.49 42.95 42.90 41.51 15.49 42.56 43.26 43.45 42.06 43.56 30.99 34.93 35.65 34.64 34.74 23.08 22.58 21.14 20.21 95.76 95.16 94.11 93.64 64.41 75.11 73.7 74.23 75

For symbols, see Table 1.

HEm (Table 7), which reflects the decrease of the eos contribution E also decreases). (note that Vm The question if the whole UVEm(x1) curve should be used for a more exact discussion has been also briefly examined. It seems E to UVEm, the shape of the curves that, when passing from Hm remains unchanged for the studied systems. This occurs for the solutions tested: dipropyl ether + hexadecane, 2,5,8-trioxanonane or 1,3-dioxolane + octane, and 2,5,8,11-tetraoxadodecane + dodecane (see below). E and CVEm curves of The same trend is observed for the Cp,m the systems: 2,5,8-trioxanonane + heptane or + octane and 1,4dioxane + heptane (CVEm calculated according to the equations proposed by Benson and Kiyohara88). However, the W shape E of the 1,3-dioxane + cyclohexane mixture vanishes of the Cp,m in the CVEm function, probably due to the low absolute values of the CEp,m in the middle of the concentration range76 (CEp,m ) 0.197 J mol-1 K-1). This merely indicates that, in such cases, accurate Rp and κT data are needed for a correct evaluation of CVEm.76

9520


Figure 8. F ) M(dipropyl ether + n-alkane)/M(dipropyl ether + heptane) E E at 298.15 K: solid line and (b), M ) Hm ; dashed line and (b), M ) UVm ; E ; dashed line and (O), M solid line and (O), M ) X12 determined from Hm E ) X12 determined from UVm .

Figure 9. F ) M(ether + n-alkane)/M(ether + heptane) at 298.15 K. (b) E 1,3-Dioxolane systems: solid line, M ) Hm ; dashed line, M ) UVEm. (9) E 2,5,8-Trioxanonane systems: solid line, M ) Hm ; dashed line, M ) UVEm. (O) 1,3-Dioxolane mixtures: solid line, M ) X12 determined from HEm; dashed line, M ) X12 determined from UEVm. (0) 2,5,8-Trioxanonane mixtures: solid E line, M ) X12 determined from Hm ; dashed line, M ) X12 determined from UVEm.

5.5. Flory Results. A part of the observed discrepancies E values may be due to between measured and theoretical Hm experimental inaccuracies related to miscibility problems45,48 E when the mixtures are prepared (e.g, Hm of the 2,5,8,11,14pentaoxapentadecane + decane or of 1,3-dioxane solutions seem E values to be too low (Table 2)), and to the consideration of Hm at x1 ) 0.05, 0.95, obtained from measurements over the central region of the concentration range. In spite of this, some general trends concerning the results obtained from the model can be E ) values that are lower than stated. The model provides σr(Hm 0.10 for most of studied systems. As a matter of fact, the average E E ) () Σσr(Hm deviations, σ j r(Hm )/NS; NS, number of systems) are 0.046, 0.042, 0.095, 0.049, and 0.104 for mixtures including linear monoethers, diethers, or polyethers, cyclic monooxaal-

E ) values kanes, and cyclic diethers, respectively. Large σr(Hm are encountered for the systems 2,5,8,11-tetraoxadodecane + heptane or + dodecane, probably because, at 298.15 K, it is close to the UCST (see above), and nonrandomness effects are E ) expected. The model describes reasonably well (0.07 < σr(Hm E < 0.10) Hm of systems characterized by rather high values of this magnitude (see results, e.g., for 2,5,8-trioxanone or 1,4dioxane solutions). As expected, the model fails when describing E curves. For the 2,5,8,11-tetraoxadodecane + octane the Cp,m E ) 0.88 J mol-1 K-1 (experimental value, 2.5764 mixture, Cp,m in the same units). Regarding the X12 variation with x1 we note that, usually, X12(x1) > X12(0.5) at [0.05, 0.45] and X12(x1) < X12(0.5) at [0.55, 0.95] and that ∆1 is much larger than ∆2 (Table 4, Figures 4 and 5). Thus, orientational effects are more relevant in the region [0.05, 0.45], where the theory underestimates the interactions between oxaalkane molecules as X12(x1) > X12(0.5). It is remarkable that for some systems X12(x1) > X12(0.5) also when x1 ∈ [0.55, 0.95]. In particular, this occurs for the 2,5,8,11tetraoxadodecane mixtures (Figure 2) and means that the E curves are not as flat as the experimental ones, theoretical Hm a typical behavior observed for mixtures that are close to the critical temperature.29,89-91 It should be noted that the X12 variation with x1 substantially decreases when the range of concentrations is limited to [0.2, 0.8], which is in agreement E . with the good results obtained for Hm Tables 2 and 7 show that for the most of the systems UVEm < E E . Consequently, the interaction parameter obtained from Hm Hm E measurements is then higher than that calculated from UVm data (Tables 2 and 7), and interactions between ether molecules are overestimated when determined from the former. This should be taken into account when, e.g., the X12 variation with n is examined (see below). However, as already mentioned, the E and UVEm curves are very similar and the shapes of the Hm conclusions concerned with the concentration dependence of X12 remain unchanged. In addition, the σr(UEVm) values determined for the systems dipropyl ether + hexadecane (0.025), 2,5,8trioxanonane + octane (0.068), 2,5,8,11-tetraoxadodecane + dodecane (0.097), or 1,3-dioxolane + octane (0.095) are in E ) values (Table practice the same as the corresponding σr(Hm 2). In mixtures with a fixed ether, the dependence of X12 with n E E and UVm values used for its reflects the variations of the Hm determination. Solutions containing 2,5,8,11-tetraoxadodecane or cyclic ethers show X12 values that remain nearly constant with n (Tables 2 and 7, Figure 9). This supports the previous conclusion that the homologous mixtures that include such oxaalkanes are similar from an interactional point of view. If the alkane is fixed and V increases (u ) 1), X12 varies more or less erratically (Tables 2 and 7). This shows that size effects may be important when the interaction parameter is calculated, as X12V*1 increases at the mentioned conditions. 5.5.1. Molar Excess Volumes. The model correctly predicts E some important features regarding the relative variation of Vm (Table 3): (i) for dipropyl ether systems, the theoretical values also decrease from decane, indicating the increase of structural effects; (ii) for mixtures including 2,5,8-trioxanonane, 2,5,8,11E increases with n; (iii) tetraoxadodecane, or cyclic ethers, Vm E decreases when a fixed alkane is mixed with ethers of Vm increasing V values and u ) 1, which reveals an increment of E at this condition. the structural contribution to Vm The observed discrepancies between experimental and calculated values could be rather easily analyzed in terms of the Prigogine-Flory version of the theory. This has been shown in detail previously62 and here only a few general trends will

Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 E Vm

be given. In this version of the model, the curvature term of j1 - V j 2)2, and the so-called P* term is is proportional to -(V j 2) difference j 2). The (V j1 - V j1 - V proportional to (P*1 - P*2 )(V depends on the (R1 - R2) value, in such way that if R1 < R2, j 2. In the case of dipropyl ether mixtures, (V j1 - V j 2) j1 < V then V increases in absolute value for larger n values, as R2 decreases. The large decrease observed for VEm may be due to very negative contributions from the curvature term. The contribution from the P* is expected to be small in view of the close values of P*1 and P*2 . For solutions including 2,5,8,11-tetraoxadodecane, the mentioned contributions decrease when n is increased as then R2 is close to R1. In addition, P*1 is high and this makes that the P* contribution is important for the shorter alkanes. For systems formed by, e.g., heptane, and linear polyethers with u ) 1, the curvature and P* terms decrease when V is increased, E values. It is remarkable that the and this leads to decreased Vm E E E /Hm,int ratio contributions to Hm vary differently, as the Hm,eos decrease at this condition (Table 2). E data and theoretical Discrepancies between experimental Hm results for systems with linear polyethers have been attributed to a weak self-association of these oxaalkanes.62 The mentioned W-shaped CEp,m and the small dVEm/dT values70,92 have been also E merely invoked to support such a conclusion.62 However, Cp,m indicates the existence of dipolar interactions, and the corresponding concentration fluctuations close to the UCST, and the latter may be ascribed to structural effects, which already have E /dT is -0.0022 cm3 mol-1 K-1 been pointed out. In fact, dVm for the system 2,5,8,11,14-pentaoxpentadecane + heptane92 and -0.0025 cm3 mol-1 K-1 for the dibutyl ether + hexane mixture.41 The dominant role of dipolar interactions is supported E E E () Hm - Gm ) of these solutions. by the large and positive TSm E E The TSm magnitude has been calculated at 298.15 K using Hm E data listed in Table 2 and Gm values determined from the DISQUAC model using the interaction parameters available in E (in J mol-1) is the literature.29 For mixtures with heptane, TSm 632, 734, and 804 for the systems including 2,5-dioxahexane, 2,5,8-trioxanonane, and 2,5,8,11,14-pentaoxapentadecane, respectively. In view of these results, the random mixing hypothesis is a reasonable approximation for mixtures including linear or cyclic monoethers or linear diethers. Orientational effects become stronger in solutions with V ) 2, 3, 4 and u ) 1. In the case of 1,3-dioxolane mixtures, this type of effects are more relevant than in systems with 1,4-dioxane. 5.6. Results from the Kirkwood-Buff Formalism. For systems including linear or cyclic monoethers, the local mole fractions are similar to the bulk ones (Table 5). Therefore, it may be concluded that the mixture structure is close to that of random mixing. This is in agreement with the rather low σr(HEm) values obtained from the Flory model, which reveals the existence of weak orientational effects in such solutions. On the other hand, size effects on the Kirkwood-Buff integrals E 38 can be examined using the Flory-Huggins equation for Gm E and assuming Vm ) 0 to calculate Gsize . The values determined ij under these assumptions are compared with the corresponding Gij integrals for the dibutyl ether + hexane mixture in Figure 6. We note that the Gij values are close to those of Gsize ij , which underlines the importance of structural effects in this system. A similar conclusion is valid for the dimethyl ether + decane 3 mixture at 323.15 K, in view of its Gijand Gsize ij values (in cm size -1 size mol ): G11 ) -26.9; G11 ) 24; G22 ) -237; G22 ) -230.2; size ) -83; G12 ) -101.5. G12 In the case of mixtures containing cyclic diethers, we note that x11 > x1 (Table 6). This indicates that the ether-ether

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interactions are prevalent, particularly at x1 < 0.5, In fact, the corresponding G11 curves show a maximum of ≈500 J cm-3 at x1 ≈ 0.4 (Figure 7). Such values contrast with those characteristic of solutions with strong orientational effects, as the mixture methanol + dipropyl ether. For this system, the G11 curve a maximum of ≈1200 J cm-3 occurs at x1 ≈ 0.2.15 Finally, temperature effects have been examined calculating the Gij integrals for the 2,5,8,11-tetraoxadodecane + dodecane mixture at 435.26 K. This temperature is far from the UCST, and, as a consequence, a decrease of the orientational effects should be expected. This is confirmed by the local mole fractions obtained, which are close to those of systems with monoethers (Table 6). 6. Conclusions E E and Vm data of linear or cyclic ether + n-alkane systems Hm have been discussed in terms of µ j and ∆∆Hvap values of the ether, the number and relative positions of the oxygen atoms, the shape of the ether, and the relative size of the mixture compounds. The mentioned solutions have also been studied using the Flory model and the Kirkwood-Buff formalism. Both theories provide consistent results. At 298.15 K, the random mixing hypothesis is a good approximation for mixtures including linear or cyclic monoethers or linear diethers. Orientational effects become stronger in solutions with 2,5,8trioxanonane, 2,5,8,11-tetraoxadodecane, or 2,5,8,11,14-pentaoxapentadecane. In the case of 1,3-dioxolane mixtures, this type of effect is more relevant than in systems with 1,4-dioxane. E curves, large variations of This is supported by W-shaped Cp,m X12(x1), and x11 values that are higher than the bulk ones, particularly at lower x1 values. The last means that orientational effects are more important at this condition, as is confirmed by large X12(x1) variations in the region of low x1 values. X12(x1 ) 0.5) values of solutions with 2,5,8,11-tetraoxadodecane or cyclic ethers remain nearly constant with n, which reveals that systems with such oxaalkanes are similar from an interactional point of view. From UVEm values, it is shown that interactions between like molecules are usually overestimated. The general trends E or UVEm data considered. observed are independent of the Hm Structural effects are present in mixtures with components which differ largely in size.

Acknowledgment The author gratefully acknowledges the financial support received from the Consejerıá de Educacioń y Cultura of Junta de Castilla y Leoń, under the Project VA052A09 and from the Ministerio de Educacioń y Ciencia, under the Projects FIS200761833 and FIS2010-16957. Literature Cited (1) Flory, P. J. Statistical thermodynamics of liquid mixtures. J. Am. Chem. Soc. 1965, 87, 1833. (2) Aicart, E.; Menduinã, C.; Arenosa, R. L.; Tardajos, G. Correlation of the Prigogine-Flory theory with isothermal compressibility and excess enthalpy data for benzene + n-alkane mixtures. J. Solution Chem. 1983, 12, 703. (3) Riesco, N.; Gonza´lez, J. A.; Villa, S.; Garcıá de la Fuente, I.; Cobos, J. C. Thermodynamics of organic mixtures containing amines. III. Molar excess volumes at 298.15 K for tripropylamine + n-alkane systems. Application of the Flory theory to N,N,N-trialkylamine + n-alkane systems. Phys. Chem. Liq. 2003, 41, 309. (4) Wang, L.; Benson, G. C.; Lu, B. C.-Y. Excess molar enthalpies of methyl tert-butyl ether + n-hexane + (n-decane or n-dodecane) ternary mixtures at 298.15 K. Thermochim. Acta 1993, 213, 83.

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Thermodynamics of Mixtures Containing Oxaalkanes. 4. Random

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