1. Presented By: Dr. Debashis Das Department of Chemistry
THE RELATIVE REDUCED REDLICH-KISTER, ARREHENIUS ACTIVATION ENERGY AND HERRÁEZ EQUATIONS FOR CORRELATING VISCOSITIES OF N,N-DIMETHYLACETAMIDE WITH FORMAMIDE AND N,N-DIMETHYLFORMAMIDE, MIXTURES AT TEMPERATURES FROM K TO K.
Subject:
Presented By: Dr. Debashis Das
Department of Chemistry
Dinhata College
Dinhata, Cooch Behar,

2. INTRODUCTION
Among the physico-chemical properties of liquid mixtures, density and viscosity is a valuable tool for developing new theoretical models to understand the liquid state. On the other hand, excess thermodynamic functions and deviations of analogous non-thermodynamic functions of binary liquid mixtures are fundamental for understanding different types of intermolecular interactions in these mixtures. This paper is a continuation of our earlier work that includes the study of the binary liquid mixture of N,N-DMA with FA and DMF [1]. In this paper the measured density and viscosity values were used to calculate the excess properties and then fitted to the Redlich-Kister equation. Here, density and viscosity deviations are fitted to the reduced Redlich-Kister (R-K) and Herráez equations [2-4] and are interpreted in terms of molecular interactions and structural effects. Comparison between the two models at different temperatures and the effect of using different numbers of
parameters will be discussed.

3. The excess viscosity has been calculated using the following equation
Table 1 Experimental density (ρ ), absolute viscosity (η), for the binary mixtures of N, N-dimethylacetamide with formamide at , and K.
Fig. 1 Variation of viscosity deviations with the mole fraction of DMA in binary mixtures of N, N-dimethylacetamide with formamide at K(●), K(■) and K(▲).
x1(DMA)
r g cm-3
h mPas
T = K
T = K
T = K
0.0000
1.1254
2.953
1.1118
2.384
1.1072
1.951
0.0350
1.1120
3.217
1.1045
2.532
1.0944
2.077
0.0700
1.0997
3.457
1.0923
2.677
1.0826
2.180
0.1110
1.0866
3.656
1.0791
2.847
1.0700
2.286
0.1500
1.0752
3.822
1.0677
2.950
1.0590
2.357
0.2053
1.0606
3.966
1.0522
3.037
1.0446
2.416
0.2414
1.0518
3.971
1.0442
3.057
1.0361
2.420
0.3009
1.0387
3.932
1.0307
3.027
1.0228
2.389
0.3563
1.0275
3.789
1.0195
2.920
1.0117
2.310
0.4177
1.0162
3.537
1.0080
2.741
1.0001
2.160
0.4580
1.0092
3.338
1.0011
2.594
0.9930
2.076
0.5011
1.0021
3.094
0.9940
2.430
0.9859
1.953
0.5474
0.9948
2.826
0.9867
2.235
0.9785
1.810
0.5972
0.9873
2.530
0.9791
2.028
0.9709
1.650
0.6500
0.9797
2.251
0.9715
1.812
0.9632
1.505
0.7092
0.9717
1.916
1.579
0.9550
1.316
0.7500
0.9663
1.720
0.9578
1.441
0.9495
1.205
0.7948
0.9607
1.540
0.9522
1.289
0.9437
1.096
0.8910
0.9492
1.210
0.9406
1.035
0.9318
0.901
1.0000
0.9371
0.933
0.9283
0.819
0.9193
0.726 X1
The excess viscosity has been calculated using the following equation
η = η – (η1x1 + η 2x2 ) (1)
where η is the respective solution properties and η1 , η2 are the viscosities of the pure components. Graphical representations of  η as functions of mole fraction of DMA(x1) are given in Fig.1

4. The excess viscosity has been calculated using the following equation
Table 2 Experimental density (ρ), absolute viscosity (η),
for the binary mixtures of N, N-dimethylacetamide with
N,N-dimethylformamide at , and K.
Fig. 2 Variation of viscosity deviations with the mole fraction of DMA in binary mixtures of N, N-dimethylacetamide with N,N-dimethylformamide at K(●), K(■) and K(▲).
x1(DMA)
r g cm-3
h mPas
T = K
T = K
T = K
0.0000
0.9436
0.802
0.9348
0.713
0.9256
0.638
0.0500
0.9433
0.809
0.9345
0.718
0.9253
0.642
0.1062
0.9429
0.816
0.9341
0.724
0.9249
0.647
0.1500
0.9426
0.822
0.9338
0.729
0.9246
0.652
0.2171
0.9421
0.831
0.9332
0.737
0.9242
0.658
0.2600
0.9418
0.837
0.9330
0.742
0.9239
0.662
0.3094
0.9414
0.843
0.9326
0.747
0.9235
0.666
0.3500
0.9411
0.849
0.9323
0.752
0.9233
0.670
0.4050
0.9407
0.856
0.9319
0.758
0.9229
0.675
0.4550
0.9404
0.863
0.9316
0.763
0.9226
0.679
0.5042
0.9400
0.870
0.9313
0.769
0.9223
0.684
0.5500
0.9397
0.876
0.9309
0.774
0.9219
0.688
0.6071
0.9393
0.883
0.9305
0.781
0.9216
0.694
0.6550
0.9389
0.890
0.9302
0.786
0.9213
0.698
0.7140
0.9385
0.898
0.9298
0.793
0.9209
0.704
0.7500
0.9383
0.903
0.9295
0.797
0.9206
0.707
0.7968
0.9379
0.909
0.9292
0.803
0.9203
0.712
0.8500
0.9375
0.916
0.9288
0.9199
0.717
0.9112
0.9370
0.925
0.9284
0.9196
0.723
0.9500
0.9368
0.930
0.9281
0.820
0.9193
0.727 ∆η
mPas
The excess viscosity has been calculated using the following equation
η = η – (η1x1 + η 2x2 ) (1)
where η is the respective solution properties and η1 , η2 are the viscosities of the pure components. Graphical representations of  η as functions of mole fraction of DMA(x1) are given in Fig. 2.

5. REDLICH-KISTER EQUATIONS
The excess viscosity η were fitted to the Redlich- Kister equations [3].
YE = x1(1 – x1)  A j (1 – 2x1)j (2)
Where A0, A1, A2 are adjustable parameters. These parameters were evaluated by fitting YE/x1(1 – x1) to equation 2 by the method of least squares. The values of these parameters along with the standard deviation σ(YE) of YE as defined by the equation.
σ (YE) = [  (YEobs - YEcal )2 / (N – M) ] (3)
In equation 3, N is the total no. of points and M is the number of parameters. 5

6. Reduced Redlich-Kister Equations
In our earlier paper [1] the measured density and viscosity values of N,N-DMA with FA and DMF binary mixtures were used to calculate the viscosity deviations of this system and then fitted to the Redlich-Kister equation [3]. In this context, Desnoyers et al. [4] stated that
“While there is a general consensus among chemists on the use of the Redlich-Kister equation to analyze thermodynamic data on mixtures of liquids, we would like to suggest, in agreement with the original statements of Redlich and Kister, (3) that it is better to plot YEx/x2(1 - x2) for this purpose. This function is directly related to the apparent molar quantities of both components and therefore has a thermodynamic significance. A rapid examination of the trends of the YEx dependence on x2 suggests that many of these systems are similar but that the differences in interactions are mostly significant in the solution rich in component 2. Excess thermodynamic quantities have the advantage of illustrating
the sign and magnitude of the nonideality but YEx/x2(1 - x2)
gives a much better handle on the origin of the nonideality”.

7. From the treatment of excess thermodynamic quantities for liquid mixtures proposed by Desnoyers et al. [4], we can conclude that the excess Quantity ∆η gives an overall view of the origin of the nonideality in the mixtures but still can be quite misleading, especially for systems that show strong interactions at high dilution. In agreement with the original statements of Redlich and Kister [3], it is better to use the ratio ∆η / x1(1 – x1) for this purpose.
This ratio is so called the experimental reduced Redlich-Kister excess properties QY,exp,T(x1) which is expressed by Eq. 3,
QY,exp,T(x1)= YjE/(x1(1- x1) (4)
where YjE denotes Dh.
We note that the reduced Redlich-Kister excess property is more sensitive than the direct excess property YE or ∆η to interactions that occurs at low concentrations [4,5]. We note that Redlich and Kister treated the data with a power series, putting all the weight on data near 0.5 mole fraction [4]. As Desnoyers et al. have shown, this is not always the best approach for mixtures having specific interactions such as association at low concentration. Therefore, elimination of the factor [x1(1 – x1)] in the reduced Redlich-Kister excess function QY,exp,T(x1) (Eq. 3) removes this effect and gives a specific reduced function QY,exp,T(x1) characterizing the viscosity or other property and also gives evidence to the existence of important interactions.

8. Fig. 4: DMA (1) + DMF (2) Fig. 3: DMA (1) + FA (2)
Reduced R-K viscosity deviation Qη,T(x1) ), Eq. 4, vs x1 of DMA is plotted in Figs. 3 and 4. at temperatures: (●): K ; (○): K; (▲): K.
Fig. 4: DMA (1) + DMF (2)
Fig. 3: DMA (1) + FA (2)
N,N-dimethylacetamide is a dipolar aprotic solvent having dielectric constant (ε = at K) [6]. Formamide is a protic solvent with a high dielectric constant (ε = at K) [6] and are strongly self-associated through extensive networks of hydrogen bonds. N,N-dimethylformamide is an aprotic, protophylic solvent with a moderate dielectric constant (ε = 37.5 at K) and a weak hydrogen bonded interaction existed between DMF molecules. The changes in curvature found in Fig.1 indicates more hydrogen bonded interactions existed between DMA+FA than DMF. As the temperature increases, the magnitude of the viscosity deviation sharply decreases due to the breaking up of the hydrogen bonds in both the systems. Large separation of curves near x1 ≈ 0 and their closeness near
x1 ≈ 1 (Fig. 3)suggests that the solute-solvent interaction parameter is much larger for N,N-DMA in FA than for the reverse case but it is same for DMA in DMF and for the reverse case (Fig. 4).

9. relative reduced Redlich-Kister function, QY,rel,T(x1) = QY,T(x1)/Y.
Like the relative deviation ∆η /η, the Qη,rel,T(x1) versus mole fraction x1 of DMA is shown below:
Fig. 5: DMA (1) + FA (2)
Fig. 6: DMA (1) + DMF (2)
Comparing with earlier figure it seems that Qη,rel,T(x1) depend very slightly on temperature. Fig. 5 shows that the nearness and then the larger separation of curves in Fig. 3 at the two limits of infinite dilution do not depend only on the value of the solute-solvent interaction parameter but are also affected by temperature and further it affects the FA rich-region more than the DMA one. Comparing Figs. 4 and 6 it is noticed that temperature has no significant effect in
the viscosity of binary mixtures of the DMA + DMF system. We remark that the
differences between their Arrhenius activation energy of viscosity Ea can play an important role in this effect.

10. Viscosity temperature-dependence Arrhenius equation
ln h = lnAs + Ea/RT (18)
The plot of ln η against (1/T) is practically linear (Fig. 7 & 8) and the Arrhenius parameters Ea and As are thus independent of temperature over the studied temperature range ( to ) K. Using both graphical and least-squares fitting methods, the slope of the straight line is equal to Ea/R and the intercept is equal to ln(As).
Fig. 7: DMA (1) + FA (2)
Fig. 8: DMA (1) + DMF (2)
at some fixed mole fractions. (●): x1 = 0.0 ;
(○): x1 = ; (▲): x1 = ; (∆): x1 = ; (■): x1 = ; (□): x1 = (♦): x1 =
and (◊): x1 = 1.0.
at some fixed mole fractions x1. (●): x1 = ;
(○): x1 = ; (▲): x1 = ; (∆): x1 = ;
(■): x1 = ; (□): x1 = ; (♦): x1 = ;
(◊): x1 = and (▼): x1 =

11. For system DMA (1) + DMF (2) For system DMA (1) + FA (2)
ARRHENIUS ACTIVATION ENERGY EA / (kJ·mol-1), ENTROPIC FACTOR OF ARRHENIUS As / (10-6 Pa·s) AS A FUNCTION OF THE MOLE FRACTION OF N,N-DMA (x1) OVER THE TEMPERATURE RANGE ( to ) K.
For system DMA (1) + DMF (2)
For system DMA (1) + FA (2) x1 Ea As
kJ·mol-1
10-6 Pa·s
0.0000
9.053
20.815
9.097
20.614
9.147
20.392
9.183
20.240
9.237
20.020
9.272
19.872
9.309
19.736
9.339
19.627
9.379
19.475
9.413
19.363
9.445
19.266
9.475
19.167
9.512
19.049
9.543
18.948
9.579
18.850
9.600
18.786
9.624
18.739
9.653
18.669
9.685
18.599
9.704
18.565
1.0000
9.726
18.538 x1
Ea/ kJ·mol−1
As /10−6 Pa·s
0.0000
16.41
3.95
0.0350
17.25
3.05
0.0700
17.97
2.44
0.1110
18.57
2.04
0.1500
19.07
1.74
0.2053
19.47
1.53
0.2414
19.62
1.45
0.3009
19.65
1.42
0.3563
19.52
1.44
0.4177
19.09
1.60
0.4580
18.73
0.5011
18.15
2.05
0.5474
17.57
2.35
0.5972
16.81
2.87
0.6500
15.85
3.73
0.7092
14.81
4.87
0.7500
14.03
6.00
0.7948
13.26
7.29
0.8910
11.70
10.82
1.0000
9.884
17.30
Since, the As-values are closely related to the viscosity of the system in vapor-phase [7], we
observe that for DMA+FA system, high values of viscosity in the DMA-rich region indicates that the DMA molecules bounding are more correlated and ordered in vapor state and for another
system it happens in the DMF-rich region. We can predict that in vapor-liquid equilibrium study, the values of the mixing enthalpy must be high positives values in this region.

12. absolute reaction rate theory
In case of liquid phase Eyring et al. [8] and Ali et al. [9] relates kinematic viscosity with the free energy (DG*) of activation of viscous flow:
(19)
where h, h, R, NA, r and M are the dynamic viscosity of binary mixture, Plank’s constant, universal gas constant, Avogadro’s number, density and molar mass of mixture (Eq. 21), respectively, and:
DG* = DH* - T DS* (20)
M = x1(M1 – M2) + M (21)
where M1 and M2 are the molar mass of the pure component 1 (DMA )and component 2
( FA or DMF) respectively.
By assuming that the activation parameters ∆H* and ∆S* [9,10] are independent of temperature, we obtained, for each composition of mixture (x1,x2) these parameters when ln(h·V/(h·NA)) is plotted against 1/T. Using both graphical and least-square fit method, the slope is equal to ∆H*/R and the intercept on the ordinate is equal to – ∆S*/R. We can use then Eq. 20 to determine the activation parameters ∆H* and ∆S*. In fact, the plot of ln(η·V/(h·NA)) against the reciprocal of absolute temperature (1/T) is practically linear and the enthalpy ∆H* and the entropy ∆S* of activation of viscous flow become independent of temperature in the studied range of temperature ( to ) K. Values of the
enthalpy of activation of viscous flow ∆H* are presented in Table

13. For system DMA (1) + DMF (2) For system DMA (1) + FA (2)
ENTHALPY OF ACTIVATION OF VISCOUS FLOW ∆H* / (kJ·mol-1) FOR THE TWO STUDIED MIXTURES AS A FUNCTION OF THE MOLE FRACTION OF N,N-DIMETHYLACETAMIDE (x1) OVER THE TEMPERATURE RANGE ( TO ) K.
For system DMA (1) + DMF (2)
For system DMA (1) + FA (2) x1 Ea As
DH*
kJ·mol-1
10-6 Pa·s
0.0000
9.053
20.815
8.292
9.097
20.614
8.336
9.147
20.392
8.386
9.183
20.240
8.423
9.237
20.020
8.478
9.272
19.872
8.515
9.309
19.736
8.552
9.339
19.627
8.583
9.379
19.475
8.625
9.413
19.363
8.659
9.445
19.266
8.692
9.475
19.167
8.722
9.512
19.049
8.761
9.543
18.948
8.793
9.579
18.850
8.829
9.600
18.786
8.851
9.624
18.739
8.876
9.653
18.669
8.906
9.685
18.599
8.939
9.704
18.565
8.959
1.0000
9.726
18.538
8.982 x1
Ea/ kJ·mol−1
As /10−6 Pa·s
DH*/ kJ·mol−1
0.0000
16.41
3.95
15.769
0.0350
17.25
3.05
16.622
0.0700
17.97
2.44
17.349
0.1110
18.57
2.04
17.958
0.1500
19.07
1.74
18.470
0.2053
19.47
1.53
18.870
0.2414
19.62
1.45
19.027
0.3009
19.65
1.42
19.044
0.3563
19.52
1.44
18.906
0.4177
19.09
1.60
18.458
0.4580
18.73
18.095
0.5011
18.15
2.05
17.504
0.5474
17.57
2.35
16.921
0.5972
16.81
2.87
16.148
0.6500
15.85
3.73
15.182
0.7092
14.81
4.87
14.129
0.7500
14.03
6.00
13.334
0.7948
13.26
7.29
12.554
0.8910
11.70
10.82
10.974
1.0000
9.884
17.30
9.1447

14. Fig.9 DMA (1) + FA (2) Fig. 10 DMA (1) + DMF (2)
VARIATION OF ARRHENIUS ACTIVATION ENERGY Ea AND ENTHALPY OF ACTIVATION OF VISCOUS FLOW ∆H* FOR THE TWO MIXTURES VERSUS MOLE FRACTION x1 OF N,N-DIMETHYLACETAMIDE IN THE TEMPERATURE RANGE ( to K). (●): Ea / kJ·mol-1; (○): ∆H* / kJ·mol-1.
Fig.9 DMA (1) + FA (2)
Fig. 10 DMA (1) + DMF (2)
Both the curves shows that the Ea and ∆H* values are very closely related. Variation of the Arrhenius activation energy Ea versus molar fraction (x1) can give evidence of eventual change in the complex structure or cluster formed in binary liquid mixtures under temperature effect and/or mixture composition [11-14]. Fig. 9 indicates strong interactions between the formamide and N,N-Dimethylacetamide molecules in the vicinity of x1 = There is no observable change of curvature found in Fig. 10. Monotonous variation of the activation energy Ea, suggests that there is a progressively change of solvent’s structure when one component is introduced into other in the mixture.

15. Correlation between the Arrhenius activation energy Ea (kJ·mol-1) of viscosity and the logarithm of the Arrhenius entropic factor (lnAs) for N,N-dimethylacetamide with formamide and dimethylformamide mixtures in the temperature range ( to K).
Fig.11 DMA (1) + FA (2)
Fig.12 DMA (1) + DMF (2)
The behaviors of the studied systems are clearly shown when the correlation between disorder and order is plotted in above figures (enthalpy factor against the corresponding entropy one). Fig. 11 indicates the formation of association type structures by a composition around mole fraction x1 ≈ 0.3 of N,N-Dimethylacetamide, but Fig.12 does not give such type of indication.

16. THE PROPOSED HERRÁEZ EQUATION
In recent paper [2], Herráez et al propose a new empirical correlation equation (Eqs. H12-a and H21-b) which introduces a correcting polynomial (Eq. H12-c) as an exponential-acting upon the molar fraction of one of mixture components. It is observed that, the viscosity deviations calculated with this model generally yield satisfactory results for many studied mixtures showing monotonous variation in viscosity values with molar fraction, but records inferior performance when distribution exhibits a maximum or minimum [15].
η(x1) = η 2 + (η 1 – η2)·x1P12,T(x1) (H12-a)
where P12,T(x1) is a power polynomial with order (n) and (n+1) adjustable parameters Bp,T:
P12,T(x1) =
(H12-b)
Hence, the Herráez Pn,T(x1) polynomials of Eq. H12-b can be inspected experimentally by using Eq. H12-c.
P12,exp,T(x1) =
(H12-c)
where η1 and η2 are the dynamic viscosity of pure components (1) and (2)
respectively and ηexp,T(x1) the dynamic viscosity of mixture at molar fraction x1 and temperature T for [x1 =0,1].

17. Fig.13 DMA (1) + FA (2) Fig.14 DMA (1) + DMF (2)
VARIATION OF THE EXPERIMENTAL HERRÁEZ EXPONENT P12,EXP,T (x1) (Eq. H12-c) FOR DMA (1) WITH FA AND DMF (2) MIXTURES AGAINST MOLE FRACTION x1 IN DMA AT THE TEMPERATURES, (●): K; (○): K; (▲): K.
Fig.13 DMA (1) + FA (2)
Fig.14 DMA (1) + DMF (2)

18. η(x2) = η1 + (η2 – η1)·x2P21,T(x1) (H21-a)
On the other face of the model and when it mathematically possible we can write:
η(x2) = η1 + (η2 – η1)·x2P21,T(x1) (H21-a)
where P21,T(x2) is a power polynomial with order (n) and (n+1) adjustable parameters Bp,T:
P21,T(x1) =
(H21-b)
Hence, the Herráez Pn,T(x2) polynomials of Eq. H21-b can be inspected experimentally by using Eq. H21-c.
(H21-c)
P21,exp,T(x2) =
where η1 and η2 are the dynamic viscosity of pure components (1) and (2) respectively and ηexp,T(x2) the dynamic viscosity of mixture at molar
fraction x2 and temperature T for [x2 =0,1].

19. Fig.15 DMA (1) + FA (2) Fig.16 DMA (1) + DMF (2)
VARIATION OF THE EXPERIMENTAL HERRÁEZ EXPONENT P21,EXP,T (X2) (Eq. H21-c) FOR DMA (1) WITH FA AND DMF (2) MIXTURES AGAINST MOLE FRACTION X2 IN FA AND DMF RESPECTIVELY AT THE TEMPERATURES, (●): K; (○): K; (▲): K.
Fig.15 DMA (1) + FA (2)
Fig.16 DMA (1) + DMF (2)
However, at infinite dilution (xi → 0+), the Pexp,T(xi) values converge to a surprising single point (Pexp,T(0) = 1.0) independent of temperature (Figs ) showing a fixed value of B0,T or B’0,T constant corresponding to the first monomial of Pij,T(xi) indicates that the ion-ion or solute-solute interaction is absent at very high dilution of DMA in the two regions. Note that the same remark is observed in our previous works [12,16-19] investigating viscosity in isobutyric acid + water and 1,4-dioxanne + water mixtures. We have concluded that B0 is considered as a universal exponent characterizing the type of preponderant interaction at infinite dilution.

20. CONCLUSION:
Correlation between the two Arrhenius parameters for N,N-DMA+FA mixtures can give evidence of the existence of distinct composition regions with different behaviors. Therefore, there will be a significant degree of H-bonding leading to strong correlation between the molecules.
Correlation between the two Arrhenius parameters for DMA+DMF mixtures can give no observable change in curvature. Therefore, there will be a weak dipole-dipole interaction and a slight structure breaking effect observed in the DMA+DMF mixture when we introduce one component into other in the mixture.
The reduced R-K function and calculation of the Arrhenius activation energy is also required along with the direct excess property to get information about the different interactions played in the binary systems.
In addition, the relative reduced R-K equation has been introduced to reduce the temperature effect and can also be good tools, like the reduced R-K function for interpreting different types of interactions.

21. In the case of viscosity data having a maximum or minimum as a function of mole fraction, this divergence is more pronounced in the Redlich–Kister model than in the Herráez model.
The Redlich–Kister polynomial is more sensitive to the number of experimental points whereas the Herráez model offers a good smoothed interpolation without any oscillation between data points, even in the case with few experimental measurements.
It is observed that, the viscosity excesses calculated with the Herráez model generally yield satisfactory results for many studied mixtures showing monotonous variation in viscosity values with molar fraction, but records inferior performance when
Distribution exhibits a maximum or minimum. In the case of some nonelectrolyte mixtures, the net absence of solute-solute interactions leads to a fixed value of the Herráez constant of 1.0 or more.

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23. ACKNOWLEDGEMENT:
The author is grateful to Dr. N. Ouerfelli (Laboratoire de Chimie Analytique et Electrochimie, Département de Chimie, Faculté des Sciences de Tunis, Campus Universitaire, 2092 El Manar Tunisia) for all the calculations made by their computer programmes and helpful suggestions.

24. Perspectives
The study of other physicochemical properties to apply other methods or models for estimating excess functions to highlight the existing interactions along with some more information's inside the mixtures in solution in collaboration with thermodynamics and organic chemists.

25. Thank You all