ARTICLE pubs.acs.org/IECR
Thermodynamics of Partitioning System with Dimerization Taking Place in Both Phases Bingwen Long*,†,‡ and Yiheng Luo† † ‡
College of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta T6G 2 V4, Canada ABSTRACT: The partition coefficients of benzoic acid in partially miscible two phase mixtures of (methanol þ cyclohexane) were measured at the temperature of 293.15, 303.15, and 313.15 K. The experimental results show that the benzoic acid is more soluble in methanol than in cyclohexane and thus it makes the partition coefficients, defined as the ratio of the total molar concentration of benzoic acid in methanol phase to cyclohexane phase, be greater than unity. In addition, the partition coefficient at each temperature is not a constant but increases with the increase of benzoic acid concentration in each phase even at very low concentration. This seeming deviation to the Nernst law can be explained by chemical theory of strong dimerization reactions taking place in both phases. Accordingly, a new generalized equilibrium model was proposed to describe the partition system with solute selfdimerization in both phases. The new model predictions agree well with the experimental data and the equilibrium constants for all the proposed association and partition reactions are obtained simultaneously by fitting the experimental data with nonlinear leastsquares method. The results show that there is much stronger dimerization reaction for benzoic acid in methanol phase than in cycolhexane phase, which results in that the dimerization in cycolhexane phases can be totally neglected. In the methanol phase alone, the calculation results further show that dimerization of benzoic acid is so strong that almost all the solute molecules form dimers within the concentration investigated. The standard thermodynamics functions of Gibbs energy, enthalpy and entropy changes for the proposed reactions were estimated with the obtained parameters at different temperature.
1. INTRODUCTION The partition behavior of a solute in two phases has been an important subject for both experimental and theoretical study for many years. When dissolving and partitioning in a two immiscible solvent mixture, the solute will generally show very different concentration in each phase. This phenomenon implies the ability of the solute to transfer from one phase to another at certain conditions. Partition coefficient, which is defined as the ratio of the solute concentration in each phase, is usually experimentally determined and thus used as a measure of such ability.1 Theoretically, the partitioning is also a thermodynamic equilibrium process of a solute transferring from one phase solvent to another, so the partition coefficient, like many other equilibrium functions, is also strongly affected by many factors such as temperature, mutual phase saturation, phase miscibility, solute concentration, solute and solvent purity, and solute stability, etc.1,2 On the other hand, partition coefficient measurements can provide information on solution thermodynamics at a molecular level as well, which helps us increase our understanding of the interactions between the solute and solvent molecules in each phase. The thermodynamic properties derived from a partition coefficient can also be used for the predictions of extraction, adsorption, and membrane permeability.3 In theory, Nernst made the first landmark contribution to this subject by his famous Nernst law,4 the partition coefficient would be constant only if a single molecular species were being considered as partitioned between the two phases. However, many measurements on partition coefficients evidently show considerable deviations to this law.1 These deviations by no means imply that Nernst is wrong but are due to the complexity of the real partitioning system. In many cases, the solute molecules can exist in different forms in the two r 2011 American Chemical Society
phases and the equilibrium concentrations we measured are often the apparent total concentrations of the solute in each phase, which cannot explicitly show the exact nature of the molecular species undergoing partitioning.5 To describe a true partition coefficient, one must consider every species in both phases at the same time. Many such partition behaviors were successfully explained by the chemical theory which tries to explain the nonideal behavior of the solutions by describing both the chemical and phase equilibrium relationships among the “real” species formed in the solution instead of the “apparent” ones.5,6 A good example is the partition behavior studies of carboxylic acid partitioning between an aqueous phase and an organic phase. The carboxylic acid will form polymers in the organic phase, while being ionized in the aqueous phase. So for the carboxylic acid, there are at least four species in the two phases, monomer, dimer, ionized, and un-ionized molecule, which makes the measured apparent partition coefficients show a strong linear dependency on the solute concentration. When the reliable ionization constants become available through other measurements, the observed apparent partition coefficients can be used to estimate the association equilibrium constant of the carboxylic acid.7-10 If such measurements can be carried out at different temperatures, the standard thermodynamic functions like enthalpy change ΔH, Gibbs energy change ΔG, and entropy change ΔS for the association reaction can also be estimated.11-13 Such a method has been used to treat the partition coefficient data of benzoic acid,7,8,10 halo fatty acids,14,15 and Received: November 14, 2010 Accepted: February 24, 2011 Revised: February 14, 2011 Published: March 09, 2011 4752
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Table 1. Comparison of Measured Densities and Refractive Indices of the Pure Components with Literature Values at 293.15 K F (g 3 cm-3)
a
a
nD
purity (mole %)
this work
lit
this work
lita
methanol
99.47
0.7917
0.7914
1.3291
1.3288
cyclohexane
99.74
0.7779
0.7786
1.4247
1.4255
Literature data was taken from ref 26.
substituted benzoic acids16 in two-phase systems of water plus organic solvents like benzene, toluene, trichloromethane, and xylene.17,18 The obtained association constants were summarized and compared by Leo.1 The ionization and self-association behavior in the aqueous phase and the organic phase, respectively, was also discussed in detail. Other types of partition behavior like solutes tending to form micelles in water were also investigated,19,20 and there is a large collection of data on partition coefficient especially for the (n-octanol þ water) system.21 Despite many similar investigations being performed, the two-phase solvent systems constitute mainly the completely immiscible solvent systems (e.g., water and alkanes), and few partitioning studies were carried out in partially miscible solvent systems which exhibit more departures from ideality and thus are more difficult to be treated in a rigorous thermodynamics way.1,3 Cyclohexane is a typical nonpolar solvent widely used in chemical industries, and it has been found to be able to present very good correlations with the partitioning of solutes in biological membranes.22 Methanol is a benign polar solvent for many organic solids and liquids and it has approximately equivalent mutual solubility with cyclohexane.23 Benzoic acid is a model solute for many partition studies with a tendency to form polymers, mainly dimers, in the organic phase.7,8,10 It also shows remarkably diverse solubility for different solvents, like methanol and cyclohexane.24,25 Therefore the objective of this study is to quantitatively investigate the partition behaviors of benzoic acid in partially miscible two phase-systems of methanol þ cyclohexane in an equilibrium thermodynamic way. The partition coefficients were measured at different benzoic acid concentrations and three temperatures of 293.15, 303.15, and 313.15 K. Furthermore, a generalized equilibrium model is proposed to describe the partition behaviors by simultaneously considering dimerization reactions taking place in both phases. The equilibrium constants for the proposed reactions are obtained by nonlinear regression from the experimental data, and some meaningful thermodynamic functions are also derived.
2. EXPERIMENTAL SECTION 2.1. Materials. Analytical reagent benzoic acid, sodium hydroxide (NaOH), methanol, and cyclohexane were purchased from Beijing Chemical Reagent Co., and the stated minimum purity was 99.5% (mass). The melting point range of benzoic acid samples was determined with a digital melting point system (type WRS-1A1B, HangZhou Kebo Co.) as between 394.95 and 396.00 K and a value of 395.52 K was reported in the handbook.26 The chemicals were stored over A3 molecular sieves and were analyzed for purity by gasliquid chromatography. The measured purity (molar percentage) values are listed in Table 1. Densities and refractive indices of pure methanol and cyclohexane at 293.15 K were also measured for comparison with the literature values. Densities were measured using 5 cm3 pycnometers and refractive indices were measured with an Abbe-3 L refractometer (type WYA-2W, Shanghai Optical
Figure 1. Equilibrium time tests for the binary system of methanol þ cyclohexane (red square) and the ternary system of benzoic acid þ methanol þ cyclohexane (green circle). nD is the refractive index of the cyclohexane layer sample.
Instrument Co.). The devices were carefully precalibrated and temperature controlled. The results are listed in Table 1 together with literature data.26 Aqueous NaOH solutions of a concentration of about 0.0050 mol 3 L-1 were prepared for titration analysis, and the phenolphthalein was used as the indictor. Before each analysis, the exact concentrations of the NaOH solutions were calibrated with potassium acid phthalate which was put into a temperature controlled oven at 420 K for overnight before each calibration. An analytical balance (type Adventurer AR2140, OHAUS Co.) with precision of (0.0001 g was used for all the mass measurements. 2.2. Partition Coefficient Measurements. A detailed description of the measuring procedure has been presented elsewhere,1,2,10 so only a brief explanation is given here. The experiments were carried out in a magnetically stirred, jacketed equilibrium cell with a working volume of 200 mL, and the cell was sealed to prevent the evaporation of solvent as described in our previous work.27-30 The temperature of the equilibrium cell was controlled by circulating water from a thermostat (type DTY-15A, Beijing DeTianYou Co.) through the jacket of the cell, which is capable of maintaining the temperature within (0.05 K. The solution temperature was measured using a glass thermometer with 0.1 K resolutions. The liquid-liquid equilibrium was first achieved by continuous stirring of the methanol and cyclohexane mixture with known volume at constant temperature. The stirring speed was carefully adjusted to avoid possible emulsification. Small amounts of samples of upper phase were periodically taken out for refractive index measurements to determine a suitable equilibrium time. Figure 1 shows the change of the measured refractive index data with stirring time. It was found that 4 h was enough for mutual saturation. Then the concentration of each phase was analyzed by gas-liquid chromatography and compared with literature liquid-liquid equilibrium data of methanol þ cyclohexane systems23 to ensure that equilibrium was reached. Good agreements were observed between them at different temperatures. Those data are listed in Table 2. After the methanol phase and cyclohexane phase reached mutual saturation, a small amount of preweighed benzoic acid (around 0.02-0.05 g) was added into the solutions by a special long pipe without opening the cell. Then stirring was restarted, and samples of the upper phase were periodically taken out for refractive index measurements again. The change of the refractive index is also shown in Figure 2. It can be seen that the refractive index became stable very quickly, but it may also be because the amount of benzoic 4753
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Table 2. Binary Liquid-Liquid Equilibrium Measurements (x, Molar Fraction) and Literature Data of Methanol (1) þ Cyclohexane (2)
a
cyclohexane-rich phase
methanol-rich phase
T (K)
xexp 1
xlita 1
xexp 1
xlita 1
293.15
0.1053
0.108
0.8443
0.842
303.15
0.1569
0.159
0.8116
0.810
313.15
0.2408
0.242
0.7366
0.739
Literature data was taken from ref 23.
Figure 2. Experimental partition coefficient of benzoic acid in methanol þ cyclohexane two-phase system at 303.15 K and its modeling. (Green C circle) cM T ; (blue triangle) cT ; (lines) calculated by proposed model.
acid added was too small to cause a detectable change of the refractive index. Therefore the mixture was allowed to be stirred continuously for another 4 h before the gravitational phase separation. After enough time of mixing and gravitational settling, the phase separation occurred. Then the solutions were kept standing at the equilibrium temperature for another 3 h to reach complete phase separation. Several samples of about 2 mL of both the upper cyclohexane layer and lower methanol layer were withdrawn with two graduated syringes respectively. The resolution of the graduation is 0.1 mL. Then a small amount of preweighed benzoic acid was added into the system to run the next partition equilibrium, and the sample solutions were titrated with the standardized aqueous NaOH solution for the methanol layer and the cyclohexane layer (several drops of nonionic surfactant solution were added during the titration for cyclohexane layer samples), respectively. Blank titration experiments were also performed as corrections of the possible reaction of CO2 with the NaOH solutions. The concentration of benzoic acid in each phase was calculated then. An average value was taken from at least three independent measurements for the same composition of solutions. The estimated uncertainty of the concentration values based on error analysis and repeated observations was less than 1%.
3. RESULTS AND DISCUSSION The measured partition equilibrium concentrations of benzoic acid in two layer of (methanol þ cyclohexane) two phase system at (283.15, 293.15, 303.15, and 313.15 K) are listed in Table 3.
The apparent partition coefficient KP is calculated using the measured equilibrium concentration data by the following definition: KP ¼
cM T cCT
ð1Þ
C where cM T and cT are the measured total apparent molar concentration of benzoic acid in the methanol layer and cyclohexane layer, respectively, mol 3 L-1. The calculated partition coefficients KP by eq 1 are also listed in Table 3. It can been seen that within all the concentrations and temperature ranges investigated KP values are greater than 1, which means that benzoic acid is more soluble in methanol than in cyclohexane and the more benzoic acid dissolves, the bigger the KP. The observations agree with the solubility results from Long24 and Thati,25 but do not simply follow the Nernst partitioning law, which gives a constant KP prediction. Similar deviation was also observed from the studies of the partition behavior of benzoic acid in (benzene þ water)7,8 and in (n-dodecane þ water).10 In their studies, the deviation to the Nernst law was well explained by the chemical theory as new species of benzoic acid dimers and ionized benzoic acid molecules are formed in the hydrocarbon phase and aqueous phase, respectively, by the proposed reactions. The dimerization equilibrium constants were also quantitatively estimated from the measured partition coefficient data. On the basis of their work, we first assume the benzoic acid molecules form dimers in the cyclohexane phase while remaining as monomers in the methanol phase, and thus there are two equilibrium reactions in the systems. (1) Partition equilibrium of benzoic acid monomer:
½BACM T ½BAM M C where [BA]M M and [BA]M stand for the benzoic acid monomers in methanol phase and cyclohexane phase, respectively. In this work, the superscript of all symbols and variables refers to the phase, C for cycolhexane phase and M for methanol phase, and the subscript represents the form of the molecule, M and D for the monomer and dimer, respectively. So the real partition equilibrium constant K1 for benzoic acid monomers could be expressed as
K1 ¼
cM M cCM
ð2Þ
C where cM M and cM are the molar concentration of benzoic acid monomer in methanol phase and cyclohexane phase, respectively, mol 3 L-1. In this assumption of only dimerization in the M C C cyclohexane phase, cM M = cT and cM < cT . (2) Dimerization equilibrium of benzoic acid in cyclohexane phase:
2½BACM T ½BACD where [BA]CD stands for the benzoic acid dimers in cyclohexane phase and the equilibrium constant K2C (L 3 mol-1) is given as K2C ¼
cCD ðcCM Þ2
ð3Þ
where cCD is the molar concentration of dimeric benzoic acid in cyclohexane phase, mol 3 L-1. The total mass balance of benzoic 4754
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Table 3. Concentrations of the Benzoic Acid in the Two Phases and Calculated Apparent Partition Coefficients methanol phase
cyclohexane phase
calculated concentration for each specie (mol 3 L-1) -1 T (K) 103 cM T (mol 3 L )
293.15
303.15
313.15
calculated concentration for each specie (mol 3 L-1)
103 cM T
103 cM M
103 cM D
103 cCT (mol 3 L-1)
103 cCT
103 cCM
103 cCD
KP
1.683
1.555
0.912
0.322
1.322
1.222
1.213
0.005
1.273
2.493
2.479
1.257
0.611
1.698
1.689
1.672
0.009
1.468
3.309
3.050
1.442
0.804
2.106
1.940
1.917
0.012
1.572
3.466
3.508
1.579
0.964
2.102
2.127
2.100
0.014
1.649
4.853 4.281
4.288 4.654
1.796 1.890
1.247 1.382
2.742 2.349
2.423 2.553
2.387 2.514
0.018 0.020
1.770 1.823
6.843
7.375
2.509
2.434
3.159
3.405
3.336
0.035
2.166
8.735
9.978
3.003
3.488
3.583
4.093
3.993
0.050
2.438
10.583
10.767
3.140
3.814
4.211
4.284
4.176
0.055
2.513
12.593
12.230
3.383
4.425
4.761
4.624
4.497
0.063
2.645
15.550
14.871
3.786
5.544
5.429
5.192
5.034
0.080
2.864
18.071
17.054
4.094
6.482
5.964
5.629
5.443
0.093
3.030
21.028
21.189
4.628
8.282
6.342
6.390
6.153
0.119
3.316
22.930
23.372
4.889
9.243
6.636
6.765
6.501
0.133
3.455
25.994
26.044
5.193
10.428
7.189
7.203
6.904
0.150
3.616
1.391
1.484
0.651
0.416
1.251
1.334
1.333
0.000
1.112
1.983
1.819
0.741
0.539
1.655
1.519
1.517
0.001
1.198
2.137
1.881
0.756
0.562
1.762
1.551
1.549
0.001
1.213
3.699
3.429
1.091
1.169
2.413
2.237
2.234
0.001
1.533
3.556
3.485
1.101
1.191
2.305
2.258
2.255
0.001
1.543
4.913 5.500
5.405 5.607
1.423 1.454
1.990 2.076
2.655 2.926
2.920 2.982
2.915 2.977
0.002 0.002
1.851 1.880
5.992
6.109
1.527
2.291
3.073
3.133
3.127
0.003
1.950
7.296
6.840
1.628
2.605
3.564
3.342
3.335
0.003
2.047
6.708
7.122
1.666
2.727
3.220
3.419
3.412
0.003
2.083
7.780
8.545
1.846
3.349
3.450
3.789
3.781
0.004
2.255
9.033
8.956
1.895
3.530
3.924
3.891
3.882
0.004
2.302
11.489
11.254
2.152
4.550
4.511
4.419
4.407
0.005
2.547
14.036 15.286
14.066 16.000
2.433 2.610
5.815 6.693
4.987 5.122
4.997 5.362
4.983 5.346
0.007 0.008
2.815 2.984
17.112
16.249
2.632
6.807
5.695
5.407
5.391
0.008
3.005
18.393
18.335
2.811
7.761
5.793
5.775
5.756
0.009
3.175
19.880
19.647
2.918
8.363
6.066
5.995
5.975
0.010
3.277
21.605
20.856
3.013
8.920
6.415
6.192
6.171
0.010
3.368
27.325
28.065
3.533
12.264
7.074
7.265
7.236
0.014
3.863
31.144
31.480
3.756
13.860
7.643
7.725
7.692
0.016
4.075
35.905 39.324
35.338 39.179
3.994 4.218
15.669 17.478
8.349 8.712
8.216 8.679
8.179 8.638
0.018 0.020
4.301 4.514
44.180
44.987
4.537
20.222
9.172
9.339
9.291
0.023
4.817
47.065
47.074
4.647
21.210
9.564
9.566
9.516
0.024
4.921
50.070
49.583
4.775
22.400
9.928
9.832
9.779
0.026
5.043
2.117
2.753
0.903
0.925
1.911
2.485
2.484
0.000
1.108
4.717
4.563
1.215
1.673
3.455
3.343
3.341
0.000
1.365
4.595 5.131
4.778 5.090
1.248 1.294
1.765 1.898
3.301 3.587
3.433 3.560
3.431 3.558
0.000 0.000
1.392 1.430
5.999
6.002
1.421
2.290
3.908
3.910
3.908
0.000
1.535
7.682
7.686
1.633
3.025
4.491
4.495
4.492
0.001
1.710
8.691
8.524
1.731
3.396
4.856
4.762
4.759
0.001
1.790
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Table 3. Continued methanol phase
cyclohexane phase
calculated concentration for each specie (mol 3 L-1) -1 T (K) 103 cM T (mol 3 L )
calculated concentration for each specie (mol 3 L-1)
103 cM T
103 cM M
103 cM D
103 cCT (mol 3 L-1)
103 cCT
103 cCM
103 cCD
KP
10.098 11.557
9.977 11.517
1.888 2.044
4.043 4.735
5.259 5.643
5.196 5.624
5.193 5.621
0.001 0.001
1.920 2.048
13.554
13.503
2.229
5.635
6.157
6.135
6.131
0.001
2.201
15.653
15.674
2.417
6.626
6.642
6.653
6.649
0.001
2.356
19.583
19.466
2.717
8.372
7.524
7.478
7.473
0.001
2.603
23.051
22.870
2.962
9.951
8.219
8.153
8.148
0.002
2.805
26.368
26.208
3.186
11.508
8.823
8.768
8.762
0.002
2.989
29.454
29.058
3.366
12.843
9.388
9.263
9.256
0.002
3.137
37.565
38.192
3.889
17.147
10.529
10.704
10.695
0.003
3.568
acid in cyclohexane phase leads to cCT ¼ cCM þ 2cCD
ð4Þ
Substituting eq 2 and eq 3 into eq 4 yields c CT 1 1 2K2C M ¼ ¼ þ 2 cT K K cM K1 P 1 T
ð5Þ cM T
Equation 5 indicates that the dependence of 1/KP on should be linear and the intercept of the straight line gives the value of K1, with which the slope may give the value of K2C. However, our experimental data plots in Figure 2 do not support such a conclusion of eq 5. Meanwhile, the results from the partitioning,31 solubility,32 and IR spectroscopic33,34 measurements all show that the self-association of benzoic acid in organic solvent is seldom beyond the dimer stage. This gives us the idea that the assumption of only benzoic acid monomers in methanol phase may not be reasonable. So we further consider adding an addition dimer reaction taking place in the methanol phase to describe the partition behavior. (3) Dimerization equilibrium of benzoic acid in methanol phase: ½BAM M
T
(eq 8) and the seeming deviation comes from the dimerization reactions (eqs 3 and 6) taking place in both phases. Therefore we need to further extract such equilibrium constants from the experimental apparent partition coefficient data. Next the total mass balance of benzoic acid in methanol phase is M M cM T ¼ c M þ 2c D
Substituting eqs 2, 3, and 6 into eq 4 and eq 9 to remove the dimer concentration, we get C 2 C 2 cM T ¼ K1 cM þ 2K2M K1 ðcM Þ
cCT ¼ cCM þ 2K2C ðcCM Þ2 Dividing eq 10 with eq 11, we can get the expression for cCM ¼
cM T ¼
where stands for the benzoic acid dimers in methanol phase and the equilibrium constant K2M (L 3 mol-1) is given as ð6Þ
where cM D is the molar concentration of dimeric benzoic acid in M the methanol phase, mol 3 L-1. In this assumption case, cM M < cT C C and cM < cT. Then the real partition equilibrium constant for the dimers in the two phases K2 can be defined as K2 ¼
cM D cCD
ð7Þ
K2M 2 K K2C 1
ð12Þ
ð13Þ
K1 KP ðK1 - KP ÞðK2C - K2M K1 Þ 2ðK2C KP - K2M K12 Þ2
ð14Þ
Equation 13 and eq 14 are the general equilibrium equations for the partition system with solute dimerization in both phases. If there is no dimerization reaction taking place in methanol phase (K2M = 0), either equation will be reduced into eq 5. Now we are able to obtain the partition equilibrium constant K1 and dimer equilibrium constants K2C and K2M simultaneously by fitting the experimental data. Since the concentrations of both phases are measured independently, they should be fitted together. So the following objective function is adopted: NP
min f2 ðK1 , K2C , K2M Þ ¼
Substituting eqs 2, 3, and 6 into eq 7, we get K2 ¼
ð11Þ cCM:
K1 ðK1 - KP ÞðK2C - K2M K1 Þ 2ðK2C KP - K2M K12 Þ2
cCT ¼
½BAM D
cM D 2 ðcM MÞ
c CT K1 - c M T 2 c 2ðK2C c M T - K2M K1 cT Þ
ð10Þ
Substituting eq 12 into eq 10 or eq 11, we arrive at
[BA]M D
K2M ¼
ð9Þ
ð15Þ
ð8Þ
It can be seen here that the Nernst law is still applicable to the partitioning for both benzoic acid monomers (eq 2) and dimers
∑ ½ðcMT,,iexp - cMT,,ical Þ2 þ ðcCT,, iexp - cTC,, ical Þ2 i¼1
where the superscripts exp and cal stand for the experimental and the calculated total benzoic acid concentration, respectively. Np is the number of data points for each temperature. The 4756
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Table 4. Regression Results for Equation 15: Equilibrium Constants and Calculated Standard Deviation K2
K2C
K2M
104 σM
104 σC
T (K)
K1
293.15
0.7521
69.70
3.1389
386.71
5.95
2.44
303.15 313.15
0.4883 0.3636
870.31 5763.78
0.2691 0.0260
982.39 1133.81
4.77 2.95
1.73 1.81
Figure 4. Scheme of strong dimerization of benzoic acid in methanol 2 phase: experimental cM T /KP data at different benzoic acid concentrations: (red square) 293.15; (black diamond) 303.15; (green triangle) 313.15 K; dashed lines are reference lines of 1/(2K2MK12) at respective temperature.
Figure 3. Experimental partition coefficient of benzoic acid in methanol þ cyclohexane two-phase system at 313.15 K and its modeling by eq 17 (dashed line).
Table 5. Partition Coefficient Regression Results Using eq 17 and eq 19, Respectively: Equilibrium Constants and the Squared Correlation Coefficient by eq 17
cM T ¼
by eq 19 2
2
T (K)
K1
K2M
R
K1
K2M
R
293.15
0.8391
275.27
0.9953
0.8280
283.70
0.9858
303.15
0.5065
899.67
0.9991
0.5015
918.48
0.9960
313.15
0.3536
0.9993
0.4108
866.04
0.9958
1202.1
become available, the partition equilibrium constant K2 can be calculated from eq 8 and the concentrations of benzoic acid monomer and dimer in both phases can be calculated from eqs 12, 2, 3, and 6, respectively. Such calculated concentrations are also summarized in Table 3, and the calculated K2 values are listed in Table 4. It should be noted that in Table 3, the concentrations of benzoic acid dimer in cycolhexane phase cCD are very small in comparison with those of other species and in Table 4; the obtained K2M values are much bigger than K2C, which indicate that there is much stronger dimerization in the methanol phase than in the cyclohexane phase. Therefore we further simplified eq 14 by setting K2C = 0, which leads to
Levenberg-Marquardt method was used as the optimization algorithm for minimizing the above objective function.35 The optimized equilibrium constants are summarized in Table 4, and the calculated total apparent concentration of benzoic acid in both phases are listed in Table 3 for comparison. The goodness of the fit of the models to the experimental data can be expressed by the standard deviations σ, which is calculated according to the following definition: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u NP u u ðcexp - ccal Þ2 t ð16Þ σ ¼ i¼1 NP ðNP - nÞ
∑
where n is the number of parameters and is 3 for this work. The calculated standard deviations σ for the concentrations in both phases are also listed in Table 4. The calculated the 1/KP-cM T and 1/KP-cCT curves at 303.15 K are also shown in Figure 2. As can been seen from the figure, the proposed model perfectly reproduces the experimental data. In addition, when K1, K2C, and K2M
KP2 KP 2 2K2M K1 2K2M K1
ð17Þ
eq 17 indicates that the dependence of cM T on KP should be a quadratic equation without intercept and K1 can be obtained by dividing the two multinomial coefficients of the regressed quadratic equation, with which K2M can be straightforwardly obtained from either multinomial coefficient. Figure 3 shows the experimental data and the smoothed quadratic curves calculated by eq 17 at 313.15 K. The obtained K1 and K2M values are listed in Table 5 along with the squared correlation coefficient R2. All the R2 values are greater than 0.99 and the equilibrium constants obtained from eq 17 are very close to those from eq 15, so it can be concluded that the simplified equation of eq 17 can reasonably represent the experimental results. Furthermore, if KP/K1 . 1, which means most benzoic acid molecules exist as dimers instead of monomers in the methanol phase (in this assumption case, M M M KP/K1 = cM T /cM = 1 þ 2cD /cM), the last term in eq 17 can be ignored and we get the following expression: cM 1 T 2 KP 2K2M K1 2
ð18Þ
Equation 18 indicates that if all the solute molecules form dimers 2 completely, the value of cM T /KP should be a composition free M 2 constant. The data of cT /KP at different benzoic acid concentrations and at different temperatures were plotted in Figure 4. The 4757
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Table 6. Standard Thermodynamic Functions of the Association Reactions of Benzoic Acid in Benzoic Acid in Methanol þ Cyclohexane Two-Phase System T (K) ΔH (KJ 3 mol-1) ΔG (KJ 3 mol-1) ΔS (J 3 mol-1 3 K-1) ξΔH ξTΔS Partition Reaction for Monomers between the Two Phases 293.15
-27.79
0.69
-97.32
0.493 0.507
303.15
-27.79
1.80
-97.32
0.485 0.515
313.15
-27.79
2.63
-97.32
0.477 0.523
Partition Reaction for Dimers between the Two Phases 293.15 303.15
168.7 168.7
-10.34 -17.06
611.40 611.40
0.485 0.515 0.476 0.524
313.15
168.7
-22.55
611.40
0.468 0.532
Dimerization Reaction in Methanol Phase
Figure 5. Experimental partition coefficient of benzoic acid in methanol þ cyclohexane two-phase system at 303.15 K and its modeling by eq 19 (line).
293.15
41.36
-14.52
191.65
0.424 0.576
303.15
41.36
-17.37
191.65
0.416 0.584
313.15
41.36
-18.31
191.65
0.408 0.592
293.15
-182.91
-2.79
-614.39
0.504 0.496
303.15 313.15
-182.91 -182.91
3.31 9.50
-614.39 -614.39
0.495 0.505 0.487 0.513
Dimerization Reaction in Cyclohexane Phase
of K2M in Table 5 indicated that there is a very strong dimerization of benzoic acid in the methanol phase. Similarly, if K2M is large enough to cause 2K21K2McCT . K1, then eq 19 may be reduced to cM T 2K2M K1 2 ðcCT Þ2
ð20Þ
and this is the same equation as eq 18. In addition, the standard thermodynamic functions of the association reactions can be estimated based on the following expressions:
cM T cCT
¼ KP ¼ K1 þ 2K2M K1 2 cCT
ð19Þ
Equation 19 indicates that the dependence of KP on cCT should be linear and the equilibrium constant could be obtained from the intercept and the slope. The experimental data of KP against cCT at 303.15 K was plotted in Figure 5. It is clear that the experimental results agree well with the predictions from eq 19. The obtained equilibrium constants are also listed in Table 5, and they are very close to those obtained from eq 17. The large values
ð21Þ
TΔS ¼ ΔH - ΔG
ð22Þ
0 1 ΔG D B ΔHðTÞ B RT C C B C ¼ @ DT A RT 2
Figure 6. Temperature effect on equilibrium constants of benzoic acid in methanol þ cyclohexane two-phase system. (red diamone) K1; (blue square) K2; (green triangle) K2M; (purple circle) K2C.
data for each temperature was scattered in the figure but approach the constants as eq 18 predicts. On the other hand, if we directly assume that there is negligible dimerization in the cycolhexane phase, we can get the following equation in the same way as we derived eq 5:
ΔG ¼ - RT ln K
ð23Þ
p
Substituting eq 21 into eq 23 and assuming ΔH is temperature independent within the measured temperature range, we get ln K ¼ -
ΔH ΔS þ RT R
ð24Þ
So a straight line could be expected by plotting ln K against 1/T and ΔH and ΔS can be estimated then from the slope and intercept, respectively. Such plots are shown in Figure 6. The equilibrium constants used for the plots and the following thermodynamic calculations are those listed in Table 4. The Gibbs free energy change ΔG can be calculated straightforward from the reaction equilibrium constant by eq 21 or from the ΔH and ΔS by eq 22. The calculated standard thermodynamic functions for each reaction at different temperatures are listed 4758
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in Table 6. The enthalpy and entropy changes imply the energetic requirements and the molecular randomness involved in the reactions, respectively. The respective contributions by enthalpy and entropy change toward the Gibbs energy change can be evaluated by eq 25 and 26, with which we can identify whether the energy changes or molecular organization changes take the dominant effect.36,37 ξΔH ¼
jΔHj jΔHj þ jTΔSj
ð25Þ
ξTΔS ¼
jTΔSj jΔHj þ jTΔSj
ð26Þ
The calculated respective contributions for all the reactions are also presented in Table 6. From the data in that table, it can be said that the enthalpy change and entropy change, respectively, take almost equally half the contributions to the Gibbs energy change for all the reactions, but the entropy change will gradually become the dominant driving force as the temperature rise.
4. CONCLUSION The partition coefficients of benzoic acid in methanol þ cyclohexane two-phase mixtures were measured at the temperatures of 293.15, 303.15 and 313.15 K. The partition behavior cannot be simply explained by the Nernst law because nonconstant partition coefficients were observed at different benzoic acid concentrations. A new general equilibrium model was proposed to describe the partition system, taking the dimerization of benzoic acid in both phases into account. The dimerization equilibrium constants for the proposed reactions were obtained by fitting the experimental data. The results show that benzoic acid has much stronger dimerization reactions in the methanol phase than in cyclohexane phase. So a simplified model was applied by neglecting the dimerization in the cyclohexane phase. The simplified model fits the experimental results quite well. The standard thermodynamics functions of Gibbs energy, enthalpy, and entropy changes of each proposed reaction were estimated with the obtained parameters at different temperature. The analysis shows that the enthalpy and entropy changes almost make equivalent contributions to the Gibbs energy change. ’ AUTHOR INFORMATION Corresponding Author
*Tel.: 780-492-5963. Fax: 780-492-2881. E-mail: bingwen_
[email protected].
’ ACKNOWLEDGMENT John M. Shaw (University of Alberta) is greatly acknowledged for his helpful comments on this paper as well as his hospitality and encouragements. B. Long would also like to thank Lyenna Wood for our excellent collaboration in writing this Paper. ’ REFERENCES (1) Leo, A.; Hansch, C.; Elkins, D. Partition Coefficients and Their Uses. Chem. Rev. 1971, 71, 525–616. (2) Dearden, J. C.; Bresnen, G. M. The Measurement of Partition Coefficients. Quant. Struct., Act. Relat. 1988, 7, 133–144.
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