J Solution Chem (2012) 41:1631–1648 DOI 10.1007/s10953-012-9892-6
Volumetric Properties of the Ternary System Dimethyl Carbonate + Butyl Methacrylate + Allyl Methacrylate and Its Binary Butyl Methacrylate + Allyl Methacrylate at 293.15 K and p = 101.325 kPa Jaime Wisniak · Gladis Cortez · René D. Peralta · Ramiro Infante · Luis E. Elizalde · Tláloc A. Amaro · Omar García · Homero Soto Received: 2 October 2011 / Accepted: 27 November 2011 / Published online: 14 September 2012 © Springer Science+Business Media, LLC 2012
Abstract Densities of the ternary system dimethyl carbonate + butyl methacrylate + allyl methacrylate and its binary subsystem butyl methacrylate + allyl methacrylate have been measured in the whole composition range, at 293.15 K and atmospheric pressure, using an Anton Paar DMA 5000 oscillating U-tube densimeter. The calculated excess molar volumes of the binary system are positive and were correlated with the Redlich–Kister equation and with a series of Legendre polynomials. Several models were used to correlate ternary behavior from the excess molar volume data of their constituent binaries and found to fit the data equally well. The best fit was based on a direct approach, without information on the component binary systems. Keywords Densities · Excess molar volumes · Dimethyl carbonate · Methacrylates · Ternary systems Glossary of Symbols 123 , cm3 ·mol−1 ρ, g·cm−3 φV , cm3 ·mol−1 A, . . . , G, cm3 ·mol−1 Ai , cm3 ·mol−1 Aij , Bij , Cij , cm3 ·mol−1 ai , cm3 ·mol−1
contribution of the ternary effect, Eqs. 18–22 densities apparent molar volume, Eq. 5 parameters of the Redlich–Kister equation for a ternary system, Eq. 20 parameters of the Redlich–Kister equation for a binary system, Eq. 8 parameters of Eq. 22 parameters of the Legendre equation, Eq. 10
J. Wisniak () Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel e-mail:
[email protected] G. Cortez · R.D. Peralta · R. Infante · L.E. Elizalde Centro de Investigación en Química Aplicada, Saltillo 25253, Coahuila, Mexico T.A. Amaro · O. García · H. Soto Facultad de Ciencias Químicas, Universidad Autónoma de Coahuila, Saltillo 25280, Coahuila, Mexico
1632
Bi , atm−1 A, B, C, cm3 ·mol−1 d, Hz R, atm·cm−3 ·mol−1 ·K−1 s, cm3 ·mol−1 V , cm3 ·mol−1 Vm0i , cm3 ·mol−1 V¯i∞ , cm3 ·mol−1 , cm3 ·mol−1 VmE,∞ i E V , cm3 ·mol−1 V¯i , cm3 ·mol−1 xi
J Solution Chem (2012) 41:1631–1648
parameters of the Nagata model, Eq. 21 parameters for the ternary effect, Eq. 19 Debye parameter universal gas constant, Eq. 22 standard deviation, Eq. 12 molar volume molar volume of pure component partial molar volumes at infinite dilution partial excess molar volume at infinite dilution excess molar volume partial molar volume mole fraction
1 Introduction The thermodynamic properties of multicomponent liquid mixtures and their analysis in terms of models are important for the design of industrial processes and the search of models capable of correlating the molecular structure and macroscopic properties of liquids. The mixing of different compounds gives rise to solutions that generally do not behave ideally. The deviation from ideality is expressed by many thermodynamic functions, particularly by excess or residual properties. Excess thermodynamic properties of mixtures correspond to the difference between the actual property and the property if the system behaves ideally. In particular, they reflect the interactions that take place between solute–solute, solute–solvent, and solvent–solvent species. Binary and ternary mixtures are important classes of systems and the behavior of some of their properties is still not clear. This work is part of our program to provide data for the eventual characterization of the molecular interactions between solvents and commercially important monomers, in particular, the influence of the chemical structure of the solute in the systems under consideration. So far, we have studied the volumetric behavior of several monomers with cyclic hydrocarbons [1], aromatic solvents [2–4], and aliphatic and cyclic ethers [5–7]. Dimethyl carbonate (DMC) is an eco-friendly powerful solvent, having a low dipole moment (μ = 0.91 D), low solubility in water, and is useful in both extraction and reaction processes such as polymerization, methylation, and carbonylation. It is used in making low-boiling temperature solvents, cleaning agents, propellants and solvents for special paints and sprays, synthetic lubricants, and electrolytes for high-performance batteries [8]. The monomers considered in this study are important industrial chemicals used in the large-scale preparation of useful polymers; acrylic acids and their esters are among the most used monomers for improving the performance characteristics of a large number of polymer formulations. These monomers are also interesting for structure studies because they contain simultaneously one or more double bonds and an ester group. Here, we report experimental values for the excess molar volumes for the ternary system of dimethyl carbonate (1)–butyl methacrylate (2)–allyl methacrylate (3), and the binary mixture butyl methacrylate (2) + allyl methacrylate (3). Wisniak et al. [9] measured the excess molar volumes for the binary mixtures dimethyl carbonate + butyl methacrylate and dimethyl carbonate + allyl methacrylate at (293.15, 303.15, and 313.15) K and found that they are positive for all concentrations and temperatures. To the best of our knowledge no literature data are available for the excess molar volumes of the systems reported here.
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Table 1 Purity of the components used in this work and their experimental densities ρ along with literature densities ρ(lit) at T = 293.15 K Component
Purity/mass per cent
ρ / g·cm−3
ρ(lit) / g·cm−3
Dimethyl carbonate
99.95
1.06977
1.07002 [18] 1.06969 [19] 1.06995 [20] 1.06990 [21] 1.0700 [22]
Butyl methacrylate
99.5
0.89521
0.89526 [23]
Allyl methacrylate
99.3
0.93302
0.93365 [24]
Determination of the thermodynamic properties of mixtures becomes more complicated and time consuming as the number of components increases. In this work we have evaluated several empirical equations allowing prediction of the excess molar volumes of ternary systems from the corresponding excess molar volumes of their constituent binary subsystems.
2 Experimental 2.1 Experimental The materials used in this work were obtained from Aldrich with the following minimum stated purity (in mass percent): Dimethyl carbonate 99.95 and contained 0.023 mass percent water, butyl methacrylate (BM) 99.5, and allyl methacrylate (AM) 99.3. Prior to use, butyl methacrylate and allyl methacrylate were vacuum distilled to eliminate the stabilizer (about 0.002 mass percent hydroquinone monomethyl ether). The supplier certified the purity of all the reagents by GC analysis. The purity of the solvents was further ascertained by comparing their densities at T = 293.15 K, listed in Table 1, with the values reported in the literature. 2.2 Equipment and Process The densities of the samples were measured with an Anton Paar model DMA 5000 oscillating U-tube densimeter, provided with automatic viscosity correction, and two integrated Pt 100 platinum thermometers (DKD traceable), with a stated accuracy of 5 × 10−6 g·cm−3 . The temperature of the densimeter was regulated to ±0.001 K with a solid-state thermostat. The densimeter was calibrated daily with both dry air and bi-distilled degassed water. All liquids were boiled or heated to remove dissolved air. Solutions of different compositions were prepared by mass in a 10 cm3 rubber-stoppered vial to prevent evaporation, using a Mettler AG 204 balance accurate to ±10−4 g. To minimize the errors in composition, the heavier component was charged first and the sample kept in ice water. Total uncertainties (ISO 9001) in the mole fractions for the binary and ternary systems were estimated to be better than ±0.0001, precision of the density (duplicate) measurement ±2 × 10−6 g·cm−3 , and of the temperature ±0.002 K. Total uncertainty in the density measurement, as reported by the equipment manufacturer, was better than 1 × 10−5 g·cm−3 . Proper safety measures were taken when handling all the materials. The 13 C NMR spectra were determined using a JEOL apparatus (model Eclipse 300, operating at 7.05 Tesla), at a frequency of 75.44 Hz with a pulse of 90°, 3 second relaxation
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J Solution Chem (2012) 41:1631–1648
Fig. 1 Map of the experimental measurements for the ternary system
time and 25 scans. Samples were introduced into a 5 mm diameter tube containing 1.5 mL of the pure compound or of a mixture and a sealed ampoule that contains a deuterium oxide.
3 Results and Discussion Twenty-one density measurements were performed (with repetition) for the binary system in the full mole fraction range (0 < x < 1). Several series of measurements were carried out for the ternary compositions resulting from adding allyl methacrylate to a binary mixture composed of {(x1 ) dimethyl carbonate + (x3 ) butyl methacrylate} (the apostrophe denotes the initial binary composition). A total of ninety-eight points were thus determined over the full concentration range (0 < x < 1). The composition of the regions close to the pure components was studied in more detail. The extent of coverage of the ternary diagram by experimental measurements is shown in Fig. 1. The excess molar volumes, V E , of the solutions of molar composition x were calculated from the densities of the pure liquids and their mixtures according to the following equations [10]: V E = [x1 M1 + x2 M2 ]/ρ − [x1 M1 /ρ1 + x2 M2 /ρ2 ]
(binary system)
(1)
V = [x1 M1 + x2 M2 + x3 M3 ]/ρ E
− [x1 M1 /ρ1 + x2 M2 /ρ2 + x3 M3 /ρ3 ]
(ternary system)
(2)
where ρ, and ρi are the densities of the solution and pure component i, and Mi the molar mass of pure component i. The corresponding values of ρ and V E are reported, respectively, in Tables 2 and 3 and in Figs. 2 and 3. The first term in Eqs. 1 and 2 represents the actual molar volume of the solution and the second, the molar volume that it would occupy if the mixture behaved ideally. In general, while these two molar volumes are similar in size (usually larger than 100 cm3 ·mol−1 ), their difference is usually smaller by two to three orders of magnitude and thus its calculation may carry a significantly larger percentage error.
J Solution Chem (2012) 41:1631–1648
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Fig. 2 Excess molar volumes at 293.15 K for the system butyl methacrylate (2) + allyl methacrylate (3)
Table 2 Experimental densities ρ, molar volumes V , calculated excess molar volumes V E , partial molar volumes V¯i and apparent molar volumes (φV )i for the system {x1 butyl methacrylate + (1 − x1 ) allyl methacrylate} obtained from Eq. 1, at T = 293.15 K and mole fraction x1 x1
ρ g·cm−3
V cm3 ·mol−1
V E × 103 cm3 ·mol−1
V¯1 cm3 ·mol−1
V¯2 cm3 ·mol−1
φV a MB + MA
φV a MA + MB
0
0.93302
135.21
0
158.95
135.21
0.0248
0.93190
135.80
3.5263
158.94
135.21
135.21
158.98
0.0496
0.93080
136.39
5.2145
158.94
135.21
135.21
158.95
0.0999
0.92859
137.58
10.190
158.94
135.21
135.22
158.94
0.1498
0.92644
138.76
14.479
158.93
135.21
135.23
158.94 158.94
0.1997
0.92431
139.95
20.867
158.93
135.21
135.24
0.2501
0.92221
141.14
25.702
158.93
135.21
135.24
158.94
0.2997
0.92017
142.32
30.752
158.92
135.21
135.25
158.94
0.3499
0.91816
143.51
33.664
158.92
135.21
135.26
158.94
0.3998
0.91618
144.69
38.281
158.92
135.22
135.27
158.94
0.4498
0.91424
145.88
40.678
158.91
135.22
135.28
158.93
0.4996
0.91233
147.06
43.404
158.91
135.23
135.30
158.93
0.5502
0.91044
148.26
46.233
158.90
135.23
135.31
158.92
0.5998
0.90862
149.43
46.759
158.89
135.24
135.33
158.92
0.6496
0.90682
150.61
47.653
158.88
135.26
135.35
158.91
0.7000
0.90504
151.80
46.358
158.88
135.27
135.36
158.91
0.7495
0.90333
152.96
43.031
158.87
135.30
135.38
158.90
0.8000
0.90161
154.15
40.262
158.86
135.32
135.41
158.89
0.8499
0.89996
155.33
32.954
158.85
135.36
135.43
158.88
0.8998
0.89834
156.50
24.972
158.85
135.40
135.46
158.87
12.036
0.9498
0.89676
157.67
158.84
135.45
135.45
158.85
0.9749
0.89598
158.25
5.3785
158.84
135.48
135.42
158.85
1
0.89521
158.84
0
158.84
135.51
a The first component is the solvent and the second the solute
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J Solution Chem (2012) 41:1631–1648
Table 3 Experimental densities ρ, molar volumes V , calculated excess molar volumes V E for the system dimethyl carbonate + butyl methacrylate + allyl methacrylate, obtained from Eq. 2 at T = 293.15 K and mole fraction x1 , x2 x1
x2
ρ g·cm−3
V E × 103 cm3 ·mol−1
0.0580
0.3272
0.92342
49.646
0.0878
0.4962
0.91860
93.146
0.1198
0.6699
0.91393
0.0570
0.1417
0.93128
134.98 43.689
0.1143
0.2838
0.92951
95.615
0.1716
0.4265
0.92777
141.36
0.2289
0.5699
0.92610
174.09
0.0846
0.1223
0.93443
54.192
0.1675
0.2427
0.93589
97.394
0.2492
0.3609
0.93740
136.39
0.3292
0.4775
0.93855
219.78
0.1112
0.1038
0.93754
0.3218
0.2998
0.94669
194.30
63.633 287.07
0.4211
0.3931
0.95103
0.1376
0.0853
0.94079
0.2672
0.1660
0.94866
122.68 252.59
0.5064
0.3146
0.96468
0.1630
0.0676
0.94393
0.3140
0.1301
0.95518
63.378
73.100 130.47
0.4543
0.1882
0.96670
183.40
0.5850
0.2424
0.97849
233.42
0.1878
0.0502
0.94718
0.3588
0.0957
0.96184
69.818 126.69
0.5148
0.1371
0.97682
194.73
0.6580
0.1752
0.99268
213.93
0.2122
0.0331
0.95042
0.4019
0.0624
0.96868
70.438 112.79
0.5721
0.0889
0.98768
144.82
0.7258
0.1128
1.00784
133.72
0.2361
0.0165
0.95358
0.4432
0.0305
0.97553
105.27
83.403 126.92
0.6262
0.0433
0.99845
0.7891
0.0546
1.02299
0.0145
0.0803
0.93059
17.512
0.0426
0.0617
0.93373
25.905
95.643
0.0698
0.0433
0.93682
42.563
0.0963
0.0258
0.94004
37.966
0.1220
0.0085
0.94331
26.333
0.0164
0.8845
0.89987
27.283
0.0715
0.8705
0.90357
0.1018
0.8629
0.90566
83.133 116.26
J Solution Chem (2012) 41:1631–1648
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Table 3 (continued) x1
x2
ρ g·cm−3
V E × 103 cm3 ·mol−1 145.43
0.1333
0.8544
0.90793
0.9272
0.0068
1.05330
39.824
0.9290
0.0232
1.05176
61.318
0.9309
0.0367
1.05064
73.948
0.9321
0.0492
1.04943
92.767
0.9335
0.0605
1.04884
0.2811
0.6985
0.9242
72.305 271.66
0.4003
0.5804
0.93947
314.41
0.5081
0.4736
0.95529
338.57
0.6059
0.3764
0.97201
311.60
0.6952
0.2881
0.98937
272.99
0.7768
0.2069
1.00772
194.56
0.8520
0.1325
1.02640
150.90
0.0103
0.0025
0.93378
1.3690
0.0042
0.0331
0.93191
0.48317
0.0086
0.0304
0.93238
3.1084
0.0128
0.0274
0.93287
3.4877
0.0179
0.0241
0.93345
0.0407
0.0093
0.93601
3.3942 14.800
0.0086
0.9304
0.89793
12.621
0.0623
0.9161
0.90144
76.381 91.508
0.0817
0.9106
0.90280
0.0140
0.8847
0.89972
25.534
0.0425
0.8774
0.90163
51.730
0.0718
0.8701
0.90362
80.800
0.0505
0.8759
0.90208
71.662
0.1332
0.8548
0.90787
0.0141
0.8857
0.89973
19.083
0.1019
0.8626
0.90581
93.426
0.0199
0.9495
0.89805
0.0256
0.9479
0.89835
151.57
4.3441 22.927
0.0317
0.9465
0.89876
26.225
0.0457
0.9424
0.89968
43.221
0.9607
0.0353
1.05667
81.827
0.9599
0.0281
1.05738
71.004
0.9589
0.0206
1.05815
56.021
0.9580
0.0127
1.05908
31.617
0.9569
0.0044
1.05982
21.402
0.9740
0.0239
1.06107
44.184
0.9737
0.0220
1.06125
41.241
0.9735
0.0199
1.06155
30.416
0.9733
0.0176
1.06170
35.033
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Table 3 (continued) x1
ρ g·cm−3
x2
V E × 103 cm3 ·mol−1
0.9729
0.0153
1.06187
34.370
0.0290
0.1612
0.92829
17.913
0.3900
0.2424
0.95657
192.12
0.2184
0.2035
0.94223
111.93
0.0422
0.8778
0.90153
0.2810
0.6988
0.92425
0.9608
0.0353
1.05681
0.0817
0.9108
0.90274
63.759 264.29 72.037 100.46
0.0623
0.9161
0.90144
76.815
0.0086
0.9301
0.89790
20.451 65.489
0.0475
0.9203
0.90041
0.0373
0.9227
0.89972
59.448
0.0259
0.9259
0.89894
51.055 23.179
0.9725
0.0127
1.06220
0.9713
0.0037
1.06307
8.1815
Fig. 3 Three dimensional representation of the ternary system
Partial molar volumes of the binary system were calculated using the relations [10]: V¯m1 = Vm + x 2 (dVm /dx1 )
(3)
V¯m2 = Vm − x 1 (dVm /dx1 )
(4)
The values of the derivatives were calculated using Eq. 7 and the values of the parameters are given in Table 5. The pertinent values are reported in Table 2 and are necessarily consistent. Partial molar volumes can also be calculated using the concept of apparent molar volume, (φV )i defined for the solute i as: (φV )i =
V − nj Vj Mi (ρi − ρ) = − ni ni mi ρi ρ
(5)
J Solution Chem (2012) 41:1631–1648
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Table 4 Molar volume of pure component Vm0 i , partial molar volume at infinite dilution V¯i∞ calculated from the apparent molar volume, and excess partial molar volume at infinite dilution. All values in cm3 ·mol−1 Vm0 i
Component
V¯i∞
(V¯mE,∞ )a i
(V¯mE,∞ )b i
Butyl methacrylate (2)
158.84
158.93
0.093249
0.10703
Allyl methacrylate (3)
135.21
135.46
0.25427
0.30066
a From apparent volumes b From Eqs. 12–15
where nj and Vj represent the number of moles and molar liquid volume of component j (the solvent in this case), Mi and ni the molar mass and the number of moles of the solute, and mi the molality (moles of solute per kg of solvent) of the solution, respectively [11]. Since, in the case of the binary system reported here, each component may be considered as solvent or solute, we have applied Eq. 5 to both situations. The pertinent values of φV are reported in Table 2. The partial molar volume of the solute can be calculated from Eq. 5 using the relation V¯i = (∂V /∂ni )p,T ,nj yielding [11]: V¯i = ni (∂φV i /∂ni )nj + (φV )i = mdφV i /dmi + (φV )i
(6)
The left-hand side of Eq. 6 results from V¯i being calculated at nj constant so that mi = nj . If the apparent molar volume, φV , is determined at various molalities, then the partial molar volume can be calculated, at any composition, from the slope of the plot of φV against nj or against mi . An important characteristic of this plot, for the binary system BM–AM studied here, is that at molalities below 0.015 mol·kg−1 it becomes a straight line (containing at least the last seven experimental points), a fact that can be used to calculate very easily the partial molar volumes at infinite dilution, V¯i∞ . Values of (V¯i∞ ) for the binary system BM (2) + AM (3) are reported in Table 4. Once again, attention should be paid to the fact that V¯i∞ is calculated as the difference between two numbers that are necessarily of the same magnitude. Hence, the result is prone to carry more error than each of the terms. The calculated VmE values of the binary system were correlated with composition using two procedures: (a) The Redlich–Kister expression [12]: VmE = x1 x2
n
Ak (x1 − x2 )k
(7)
k=0
where the Ak ’s are the adjustable parameters of the empirical equation. The Redlich–Kister equation, developed originally to correlate the excess Gibbs function and activity coefficients, has proven to be such a very powerful and versatile regression tool that its use has been extended to the description of the concentration functionality of a variety of properties of mixtures, among them, excess molar volumes, excess enthalpies of mixing, excess viscosities, excess isobaric heat capacities, and excess refractive indexes. Nevertheless, it suffers from the important drawback that the values of its adjustable parameters change as the number of terms in the series is increased, so that no physical interpretation can be attached to them.
1640 Table 5 Parameters (cm3 ·mol−1 ) of Eqs. 7, 8 and standard deviation s (cm3 ·mol−1 )
J Solution Chem (2012) 41:1631–1648 Redlich–Kister, Eq. 7
Legendre, Eq. 8
Dimethyl carbonate (1) + butyl methacrylate (2) A0 = 1.450
a0 = 1.480
A1 = 0.1215
a1 = 0.1205
A2 = 0.09012
a2 = 0.06002
s = 0.0047
s = 0.0048
Dimethyl carbonate (1) + allyl methacrylate (3) A0 = 0.3803
a0 = 0.3864
A1 = 0.01387
a1 = 0.01387
A2 = 0.01831
a2 = 0.01220
s = 0.0013
s = 0.0013
Butyl methacrylate (2) + allyl methacrylate (3) A0 = 0.17629
a0 = 0.18548
A1 = 0.096813
a1 = 0.096813
A2 = 0.027549
a2 = 0.018365
s = 0.00080
s = 0.00083
(b) A series of Legendre polynomials Lk (x1 ): VmE = x1 x2
n
ak Lk (x1 )
(8)
k=0
which for the four first terms (k = 0, 1, 2, 3) becomes: VmE = x1 x2 a0 + a1 (2x1 − 1) + a2 (6x12 − 6x1 + 1) + a3 20x13 − 30x12 + 12x1 − 1
(9)
Legendre polynomials belong to the category of orthogonal functions such as Fourier, Bessel, and Chebyshev, which have the valuable property that, for a continuous series of observations (infinite), the values of the coefficients do not change as the number of terms in the series is increased. This is an important characteristic because if a physical explanation can be attached to one of its coefficients, its value remains constant and independent of the number of terms taken into account in the series. A more detailed description of this method appears in a previous publication [9]. An important fact is that the first four terms of the Redlich–Kister expansion have the same algebraic structure as the first four terms of the Legendre ones. For higher terms the two series diverge. Equations 7 and 8 were fitted using a least-squares optimization procedure, with all points weighted equally and minimizing the following objective function, OF: OF =
N
2 VmEi ,expt − VmEi ,calc /N
(10)
1
where N is the number of observations. The values of the different adjustable parameters, Ak of Eq. 7 and ak , of Eq. 8 are reported in Table 5 for different values of k, together with the pertinent statistics. The standard deviation s was calculated from: N 1/2 2 (11) VmEi ,expt − VmEi ,calc (N − k) s= 1
J Solution Chem (2012) 41:1631–1648
1641
Fig. 4 Residual distribution plot for the system butyl methacrylate (2) + allyl methacrylate (3) according to the fit given in Table 5
where k is the number of adjustable parameters. The statistical significance of adding one or more terms after the third was examined using a χ 2 based test, with the simultaneous requirement that the residues (defined as the difference between the calculated and experimental value of the molar excess volume) be randomly distributed, as suggested by Wisniak and Polishuk [13]. Randomness of the residues was tested using the Durbin–Watson statistic. It was not deemed necessary to perform a step-wise regression. Figure 4 shows the residuals distribution of the Redlich–Kister fit for the (butyl methacrylate + allyl methacrylate) binary system at 293.15 K, which is random as shown by the Durbin–Watson statistic. The variation of VmE /x1 x2 with composition was used to test the quality of the binary data; this function is extremely sensitive to experimental errors, particularly in the dilute ranges and helps in detecting outliers. In addition, its values at infinite dilution represent [10] (also the partial the values of the partial excess molar volume at infinite dilution, V¯mE,∞ i ∞ ¯ molar volume of mixing at infinite dilution, Vmi ) which can be also calculated from the adjustable parameters using the relations: = A0 − A1 + A2 − · · · = V¯m∞1 − Vm01 = V¯m∞1 V¯mE,∞ 1 = A0 + A1 + A2 + · · · = V¯m∞2 − Vm02 = V¯m∞2 V¯mE,∞ 2
(12) (13)
for the Redlich–Kister expression and = a0 − a1 + a2 − · · · = V¯m∞1 − Vm01 = V¯m∞1 V¯mE,∞ 1 V¯mE,∞ = a0 + a1 + a2 + · · · = V¯m∞2 − Vm02 = V¯m∞2 2 Vm0i
(14) (15)
is the molar volume of pure component i. for the Legendre polynomials. In Eqs. 12–15 In addition, it should be realized that, in the absence of self-association, the value of the partial excess molar volume at infinite dilution reflects the true solute–solvent interaction. . The values of this property Equations 12 and 14 or 13 and 15 yield the same values of V¯mE,∞ i for the butyl methacrylate + allyl methacrylate binary system are reported in Table 4 and compare well with the ones calculated using the apparent volumes. It should be realized that the values of the property at infinite dilution are probably less accurate because the data have been fitted with a technique that assigns equal statistical weight to all the points. Inspection of the results of Table 2 and Fig. 2 indicates that the excess molar volumes for the binary butyl methacrylate + allyl methacrylate are positive for the whole composition
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range, an expected result since both methacrylates have the same basic structure, differing only in the length of and branching of the alkyl chain. The increase in the molar volume of the mixture and positive contribution to the excess volume is due to a disruption of the pure component liquid structure. The magnitude and sign of V E is a reflection of the types of interactions taking place in the mixture. Table 3 shows that, as expected from the behavior of the binary mixtures, the ternary system shows positive values of V E at all compositions. Several methods for estimating the excess molar volume of the ternary system from binary data were tested in this work. The simplest one is to assume that there are no ternary effects; the only important clusters are the binary ones, that is: E E E E = V12 + V13 + V23 V123
(16)
Other methods, which are based on the existence of a ternary effect (123 ), follow a relation of the general form: E E E E = V12 + V13 + V23 + 123 V123
(17)
In Eqs. 16 and 17 the first three terms on the right hand side represent the contribution of the pertinent binary, and in Eq. 17 123 is the contribution of the ternary effect. Some expressions for the ternary effect are as follows: Cibulka [14] 123 = x1 x2 x3 (C1 + C2 x1 + C3 x2 )
(18)
123 = x1 x2 x3 A + Bx1 (x2 − x3 ) + cx12 (x3 − x2 )
(19)
123 = x1 x2 x3 A + B(x1 − x2 ) + C(x2 − x3 ) + D(x3 − x1 ) + E(x1 − x2 )2 + F (x2 − x3 )2 + G(x3 − x1 )2 + · · ·
(20)
Singh [15]
Redlich–Kister [12]
and Nagata [16] 123 = RT[B0 − B1 x1 − B2 x2 − B3 x12 − B4 x22 − B5 x1 x2 − B6 x13 − B7 x23 + B8 x12 x2 ]
(21)
where R is the gas constant and T the absolute temperature. The data for the binary systems dimethyl carbonate + butyl methacrylate and dimethyl carbonate + allyl methacrylate were taken from Wisniak et al. [9]. A third possibility is the direct fit of the data, along the same type of approach used in correlating the boiling points of a ternary mixture without use of binary data [9, 17]. The pertinent expression is 123 = x1 x2 A12 + B12 (x1 − x2 ) + C12 (x1 − x2 )2 + · · · + x1 x3 A13 + B13 (x1 − x3 ) + C13 (x1 − x3 )2 + · · · + x2 x3 A23 + B23 (x2 − x3 ) + C23 (x2 − x3 )2 + · · · (22) The coefficients of the different models were determined using the Excel Solver® optimization algorithm, minimizing the mean standard deviation given by Eq. 11. Table 6 lists the values of the adjustable parameters and the mean standard deviation for each model tested. Inspection of this table shows that consideration of a ternary effect results in a reduction of at least 20 % in the mean standard deviation compared to when the mixture is
J Solution Chem (2012) 41:1631–1648
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Table 6 Parameters (cm3 ·mol−1 ) of Eqs. 18–22 and standard deviation s (cm3 ·mol−1 ) Dimethyl carbonate (1) + butyl methacrylate (2) + allyl methacrylate (3) Predictive, Eq. 16 s = 0.019
Redlich–Kister, Eq. 20
Cibulka, Eq. 18
Singh, Eq. 19
C1 = 0.49444
C1 = −0.94801
C2 = −0.85477
C2 = −6.3558
C3 = −3.5431
C3 = −2.7481
s = 0.015
s = 0.015
Nagata, Eq. 21 (atm−1 )
A = −0.69169
B0 = −0.0000036611
B6 = 0.0020205
B = 1.9863
B1 = 0.00033530
B7 = 0.0012278
C = 0.053018
B2 = 0.00013601
s = 0.013
D = 0.96067
B3 = −0.0018439
E = −3.8811
B4 = −0.00093545
s = 0.014
B5 = 0.00051611
Direct fit, Eq. 22 A12 = 1.3390
A13 = 0.38162
A23 = 0.074012
B12 = 0.058630
B13 = 0.055487
B23 = −0.11778
C12 = −0.031989
C13 = 0.029011
C23 = −0.055626 s = 0.013
Direct fit, Eq. 22 A12 = 1.3400
A13 = 0.39602
A23 = 0.049797
B12 = −0.093110
B13 = 0.032128
B23 = −0.082784
C12 = 0.0011592
C13 = 0.0030188
C23 = 0.10578
D12 = 0.39094
D13 = 0.0034059
D23 = 0.14272 s = 0.012
considered ideal. In addition, there is no significant difference in the abilities of the different models tested here to correlate the data. In the case of the Redlich–Kister model, an increase in the number of ternary constants from two to seven decreases the mean standard deviation from 0.016 to 0.014. The direct fit procedure is clearly superior to all the other methods tested, because it yields a standard deviation more than 30 % lower than the ideal case and needs only nine constants. As shown in Table 6, an increase of the parameters from nine to twelve does not improve the correlation ability. The only disadvantage of this model is that its structure does not allow prediction for the binary sub-systems. Excess molar volume isoclines for the ternary system calculated using the Cibulka model are presented in Fig. 5, and show positive values of V E at all compositions. 3.1 Spectroscopic Analysis of Binary and Ternary Mixtures of BM, AM and DMC 13 C NMR spectroscopy is a technique that allows establishment of the local magnetic environment of a carbon atom that is determined by the delocalized bonding electrons (π ) and non-bonding electrons (n). The location of these electrons in a chemical structure defines the uniqueness of the molecule by itself and in solution and can be evaluated by nuclear magnetic resonance spectroscopy. In mixtures of solvents, the interaction between molecules
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Fig. 5 Excess molar volumes isoclines according to the fit given by Eq. 18
Table 7 Chemical shift for the carbon atom in the carbonyl group of butyl methacrylate, allyl methacrylate, and dimethyl carbonate Compound
Carbonyl carbon electrostatic charge*
Oxygen sp2 electrostatic charge*
Dipole moment (Debye)*
δ (Hz)
Molecular volume* (cm3 ·mol−1 )
Butyl methacrylate
0.599
−0.509
2.03
12596.20
169.79
Allyl methacrylate
0.541
−0.494
1.66
12576.53
147.25
Dimethyl carbonate
0.800
−0.501
4.96
11855.534
90.50
* Calculated by the theoretical density function method using Spartan software (Spartan 04, version 1.01,
Wavefunction Inc., Irvine, CA)
due to either the effect of van der Waals forces or steric effects, change the magnetic environment and thus, the chemical shifts of the atoms that participate in the mixture. In order to get a better understanding of the effect of the components in mixtures of allyl methacrylate, butyl methacrylate and dimethyl carbonate, it was decided to study the chemical shift of the carbon atom present in the carbonyl group of the three compounds. As stated above, the associations and interactions among molecules making up the particular mixture affect the magnetic characteristics in this carbon atom and are reflected in the chemical shift. In the analysis of the difference in the chemical shift of the three carbonyl groups it is important to consider that these differences originate from the magnetic environment characteristic of each molecule. To understand these differences, the electronic parameters of the
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Fig. 6 (a) Dipole–dipole interactions in BM molecules between the carbonyl carbon atom and the oxygen atom with sp2 hybridization. (b) Dipole–dipole interactions in BM and DMC molecules between the BM carbonyl carbon atom and the DMC oxygen atom with sp2 hybridization
molecules were calculated theoretically using the density function method. In an attempt to explain the differences in the chemical shifts, the electrostatic charges of the carbon atoms in the carbonyl group of the three compounds (two esters and one carbonate) and of the bound oxygen atom (with sp2 hybridization) were calculated. The results are shown in Table 7 and indicate that the difference in electrostatic charge between these two atoms results in the formation of a permanent dipole. Thus, in the liquid state, dipole–dipole interactions are present; the proximity of the oxygen atom of the neighboring molecule changes the magnetic environment of the carbonyl carbon atom. Figure 6(a) exemplifies this effect for butyl methacrylate. The observed differences in chemical shifts are due to the magnetic environment and dipole–dipole interactions; any change in the dipole–dipole interactions is reflected in differences in the chemical shifts. The magnitude of the latter will be a function of the type of atom and electrostatic charge; the stronger the dipole–dipole interactions the stronger is the difference in chemical shift. Table 8 shows the chemical shift of the carbon atom in the carbonyl group of BM in mixtures with AM and DMC. For the BM–DMC binary system the chemical shift changes from 12596.2 Hz (Table 7) to 12614.02 Hz (Table 8). This change
1646 Table 8 Chemical shift for the carbon atom in the carbonyl group of BM in mixtures with AM and DMC
J Solution Chem (2012) 41:1631–1648 Mixture
Chemical shift (Hz)
BM–AM (0.60)
12605.50
BM–AM (0.65)
12607.70
BM–AM (0.70)
12605.50
BM–DMC (0.5)
12614.02
BM–AM–DMC (0.62, 0.32, 0.06)
12698.89
Fig. 7 13 C spectrum for the BM carbonyl carbon atom and in mixtures with AM and DMC
is a result of the formation of a dipole–dipole interaction with DMC: the proximity of the oxygen atom of the DMC alters the magnetic environment, increasing the electronic current on the carbon atom in the carbonyl group of BM, and thereby inducing deprotection diamagnetic anisotropy. This interaction is favored by the small molar volume of DMC which diffuses easily among the larger BM molecules. An additional consequence of this interaction is an increase in the excess molar volume of the BM–DMC mixture since the presence of DMC between BM molecules, and without any other stabilizing interaction between these two molecules, will provoke a volume increase as shown in Fig. 6(b). An analysis of binary mixtures of AM and BM has to consider that these two molecules have similar electrostatic charges and magnetic environments and, thus, their chemical shifts are very similar. Nonetheless, it is interesting to analyze the sample with molar fraction x2 = 0.65 where a maximum in V E occurs. 13 C NMR analysis of mixtures around this maximum, (x2 = 0.60 and x2 = 0.70), indicates that a slight increase in chemical shift (Fig. 7) occurs when V E increases.
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The increase in the chemical shift of the carbon atom in the carbonyl group of BM can be attributed to a dipole–dipole interaction between the BM and AM molecules; this interaction results in an approach of the oxygen atom with sp2 hybridization to this carbon atom. Since no other stabilizing interactions take place, the net result is an increase in V E due to the intercalation of the smaller volume AM molecules into BM. In the ternary mixture dipole–dipole intermolecular interactions are also present, as demonstrated by the change in the chemical shift of the carbon atom in the carbonyl group of BM. However, the combined effect among the three molecules gives rise to a smaller change than in the case of the binary mixture BM–DMC, which also shows an increase in V E .
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