ISSN 0036-0244, Russian Journal of Physical Chemistry A, 2020, Vol. 94, No. 1, pp. 30–40. © Pleiades Publishing, Ltd., 2020. CHEMICAL THERMODYNAMICS AND THERMOCHEMISTRY Hyperbolic Correlation Between the Viscosity Arrhenius Parameters at Liquid Phase of Some Pure Newtonian Fluids and their Normal Boiling Temperature E. Mlikia,b, T. K. Srinivasac,*, A. Messaâdid, N. O. Alzamela,b, Z. H. A. Alsunaidia, and N. Ouerfellia,b aDepartments of Mathematics and Chemistry, College of Science, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, 31441 Saudi Arabia b Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, 31441 Saudi Arabia cDepartment of Physics, A.S.N Women’s Engineering College Tenali, Andhra Pradesh, 522 201 India dUniversité de Tunis El Manar, Laboratoire de Biophysique et Technologies Médicales, LR13ES07, Institut Supérieur des Technologies Médicales de Tunis, Tunis, 1006 Tunisia *e-mail: [email protected] Received April 29, 2019; revised April 29, 2019; accepted May 14, 2019 Abstract—Viscosity is the most important characteristic of hydraulic fluid and which is influenced by pressure and temperature. Using statistical methods for correlation analysis and regression, eventual causal relationship between parameters of the Arrhenius-type equation and principally the boiling point of some classical Newtonian liquids is attempted. Empirical validation utilizing data on viscositabty of pure Newtonian liquids studied at atmospheric pressure and at different ranges of temperature gives excellent statistical results. Indeed, we found a significant strong correlation between the Arrhenius activation energy (Ea), the Arrhenius temperature (TA) and the boiling point (Tb). Consequently, an original hyperbole-type equation modeling this relationship is proposed which allows the prediction of the boiling temperature and the type of isobaric liquid-vapor diagram through information on viscosity Arrhenius parameters values, only in liquid phase, and will be thus very useful in the handling of engineering data especially for the study of hydraulic components and systems efficiency. Keywords: Viscosity, Arrhenius behavior, Boiling temperature, Newtonian liquids, Correlation DOI: 10.1134/S0036024420010239 Effectively, this will be very gainful for hydraulic fluid properties simulation in the design and optimization of several industrial processes, such as in Food Industry, Chemical Industry, Pharmaceuticals, Cosmetics and Hydraulic-Mechanics, etc. INTRODUCTION Liquids viscosity is one key transport property embroiled in chemical engineering, process design and development [1–6]. Numerous empirical and semi-theoretical equations of liquid viscosity have been suggested based generally on three principal theories: the theory of reaction rate [7–9], the molecular dynamic approach [10] and the distribution function theory [11]. Several semi-empirical expressions have been proposed in literature [12–17] and investigated in previous works to discuss the viscosity-temperature dependence [18–22]. The present study focuses on analyzing the existence of any eventual causal association between the Arrhenius-type equation parameters and boiling points of some classical Newtonian liquids. This relationship may allow estimating these phase change temperatures through the study of variation of viscosity versus temperature of a pure solvent only in its liquid phase. LITERATURE REVIEW ON THE LIQUID VISCOSITY-TEMPERATURE DEPENDENCE Explicit Dependence and Newtonian Behavior As the temperature increases, the thermal agitation increase and consequently the mean free path decreases. Hence, numerous expressions, generally classified under three types: two-, three- and multiconstant equations, have been suggested in the literature for describing the liquid shear viscosity (η) versus temperature (T) through available experimental data of literature for interpolation methods. 30 HYPERBOLIC CORRELATION BETWEEN The simplest form of representation of Newtonian fluid viscosity versus temperature is a relationship with 1 two parameters where the neperian logarithm form becomes the most popular and it is the so called Arrhenius type-equation which may be linearly formulated as follows [18–22]: () Ea 1 (1) , R T where R is the perfect gas constant, As and Ea, the preexponential factor and the Arrhenius activation energy for the pure fluid system respectively. ln η = ln As + For the Newtonian fluids not obeying to the viscosity Arrhenius behavior, the most popular nonlinear equation is suggested firstly by Vogel and lately known as Vogel-Fulcher-Tammann-type model [18–22]: B , T −C where A, B and C are constants. ln η = A − (2) Notice that in this work we focus on the Arrhenius type-equation for studying any eventual relationship between its parameters and boiling points (Tb) of some classical Newtonian liquids in the hope to predict its value through information on viscosity Arrhenius parameters only in liquid phase. Implicit Dependence and Derived Temperatures From thermodynamic quantities and similar amounts, certain specific temperatures can be determined by manipulating certain derivatives as well as those of the Maxwell equations. Noting that there are certain common points with the definition of the enthalpy of vaporization and the enthalpy of activation viscous flow (ΔH*) as well as the viscosity activation energy (Ea) [23–37]. This leads us to think that the specific temperatures derived from the previous quantities are causally correlated with the boiling temperature of the fluid that we studied its dynamic viscosity. It is in this context that we will present certain correlations and try to reduce a way of predicting the boiling temperature. Among the derived temperatures we cite the Arrhenius temperature (TA) of a pure liquid (Eqs.1 and 3), the Arrhenius mean temperature (TAm) of a liquid binary mixture (Eq. 4), the current temperature of Arrhenius (TAc) deduced from partial molar magnitudes (Eq. 5) and the thermodynamic temperature (T#) deduced from Maxwell’s equations (Eq. 11). TA = − Ea . R ln As (3) quasi-linear behavior where the slop gives the Arrhenius mean temperature (TAm) expressed as follows: Ea (4) =TAm ln As + B. R Where B is the intercept of the ordinate when ln(As) is mathematically null and it is thoroughly related to the shear viscosity of the binary fluid system at boiling point; we remark that the Tam value is close to the boiling temperature Tb of one of the two pure constituting components of the binary mixture [23–37]. Nonetheless, if we eliminate the hypothesis of Eq. (4) that the Arrhenius mean temperature (TAm) nothing is more constant over the whole domain of molar composition, we can review it as variable Arrhenius’ current temperatures (TAci ) related to the pure component (i). It can be determinate from the following partial derivatives at selected molar composition (x1): − ⎛ ∂(Eai ) ⎞ (5) TAci = ⎜ ⎟ . ⎝ ∂(−R ln( Asi )) ⎠P Whereas in several cases of previously studied binary liquid mixtures, we have observed a strong correlation existing between the limiting Arrhenius’ current temperatures (TAci ) and the vaporization temperature of the isobaric liquid vapor equilibrium of the corresponding binary system [23–37]. The following equation asserts all these observations, ⎛ ∂(Eai ) ⎞ ⎜ ⎟ ∂xi ⎟ (6) lim ⎜ ≈ −RΔTbi , xi →1 ⎜ ∂ ( ln A ) ⎟ si ⎜ ∂x ⎟ ⎝ ⎠ i and from which we have estimated in previous works with a good approximation the boiling point (Tbi) of some pure liquid constituents (i). In addition, the free energy (ΔG*) of activation of viscous flow are related as follows [7–9, 38]: ⎛ η( x ,T , P )V ( x1,T , P ) ⎞ ΔG *( x1,T , P ) = RT ln ⎜ 1 ⎟ , (7) hN A ⎝ ⎠ where R, η, h, NA and V are the perfect gas constant, dynamic viscosity of binary fluid system, Plank’s constant, Avogadro’s number and molar volume of fluid system at molar composition (x1), absolute temperature T and pressure P respectively, and: (8) ΔG * = ΔH * − T ΔS *. Where ΔH* and ΔS* are the enthalpy and the entropy of activation of viscous flow of binary fluid system at molar composition (x1) and can be calculated in the general case as follows: ⎛ ∂(ΔG */T ) ⎞ ΔH * = ⎜ ⎟ , ⎝ ∂(1/T ) ⎠P Generally for binary mixtures the plot (Ea) as a function of (ln As) at different compositions exhibit a RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A 31 Vol. 94 No. 1 2020 (9) 32 MLIKI et al. 100 1000 T* Tm TA Ti/K Tb 100 1000 Fig. 1. (Color online) Classification of different mean temperatures (Ti ) used in this statistical investigation. Great vertical bar (|): Average value and the small vertical bar (l): delimitation of the Confidence Interval (CI). ⎛ ∂(ΔG *) ⎞ ΔS * = ⎜ ⎟ . ⎝ ∂(T ) ⎠P (10) Moreover, considering the partial derivatives functions of Maxwell equations and the Gibbs free energy expression (Eq. (7)), we can determinate the thermodynamic temperature (T#) at constant pressure by the partial derivative (ΔH*) with respect to (ΔS*) as follows: ⎛ ∂(ΔH *) ⎞ # ⎜ ⎟ =T . ⎝ ∂(ΔS *) ⎠P (11) RESULTS AND DISCUSSION Arrhenius and Temperature Parameters A sample of 103 experimental data provided previous works and from the literature [14–19, 23–37] on viscosity of pure Newtonian liquids studied at atmospheric pressure and at different temperature ranges is used in this paper in order to analyze the existence of any eventual correlation between parameters of the Arrhenius type-equation, such as the activation energy (Ea) and the entropic factor (ln As), and the melting temperature (Tm) and boiling temperature (Tb). Table 1 presents the experimental values of these parameters. Also, the table provides experimental values of additional temperature parameters which are the Arrhenius temperature (TA) and the Arrhenius activation temperature (T*= Ea/R) as introduced by Ouerfelli et al. [23–35]. We note that the existence of some samples’ repetitions, that’s because the data are provided from different references of the literature and it are realized at different ranges of temperature, which it’s statistically considered as different and independent observations and it will well enrich the investigated set of samples, and lead to reliable statistical dispersion. Table 2 reports the principle descriptive statistics of temperature parameters for the 103 studied observations, such as the Arithmetic mean (Ti ), the Confidence Interval (CI) of the mean and the coefficient of variation (CV) from which we can admit the following classification, also show by Fig. 1 which confirms that there is no net intersection between any consecutive (CI). TA < Tm < Tb < T *. Moreover, according to the (CV) values, the (T*)parameter is the most dispersed variable, inversely to the boiling temperature (Tb) which is the most 2 homogenous. Analysis of Correlation Firstly, we investigate direct mutual correlation between the three parameters of Arrhenius (TA, ln As and Ea) and, the melting temperature Tm and boiling temperature Tb. For that, in order to measure and test any eventual association between the target parameters, we have applied the correlations tests of Spearman’s rank [39], where the null hypothesis supposes the independence of the variables. Table 3 presents the result of the test for all possible pairs. Also, we present in Fig. 2 the pairwise scatter plots, which is a good graphical method to show any possible correlation. Based on the graphs of Fig. 2, we deduce that there is no clear direct relationship between any of the pairwise analysis. In addition, the results of the Spearman’s rank correlations tests show that there is no relationship between any of the target parameters. Thus, we deduce that there is no direct correlation between the Arrhenius parameters (TA, ln As and Ea) and the temperatures Tm and Tb. Nevertheless, instead of that, we think that a complicated association may exist. For that, we have tried several possible complicated relationships between two or more transformed parameters from which we found the most interesting result indicating a strong nonlinear correlation (Fig. 3) may exist between Ea and (1/TA – 1/Tb). Regarding results given in Figs. 1 and 2b, Tables 2 and 3, and the finite values of the two parameters of Arrhenius straight line (Eq. 1) whether T tends to zero or to infinity, we can ascertain that the curvature of Fig. 3 mathematically admits two asymptotes, one vertical very near of zero and a horizontal one at infinity. RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 94 No. 1 2020 HYPERBOLIC CORRELATION BETWEEN 33 Table 1. Arrhenius parameters of some pure solvents studied at previous works and from the literature [14–19, 23–37], Arrhenius activation energy Ea (kJ mol–1), the logarithm of the entropic factor ln(As/Pa s), the melting point (Tm/K), the boiling point (Tb/K), the Arrhenius temperature (TA/K) and Arrhenius activation temperature (T*/K) No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Tm Tb lnAs Ea TA T* K K – kJ mol–1 K K 257.15 273.15 288.35 191.15 165.15 183.15 158.15 150.05 260.65 279.62 178.15 182.60 278.65 228.15 178.15 248.65 182.55 143.45 180.15 225.35 250.23 189.55 156.85 289.75 178.45 183.35 266.85 257.15 214.15 293.15 159.15 293.15 293.15 159.15 159.15 214.15 212.15 275.65 253.15 468.15 373.15 478.45 289.75 339.15 390.85 372.15 351.65 475.75 353.89 341.88 371.15 353.15 405.15 409.15 417.15 371.15 309.25 383.75 412.25 349.87 350.15 307.75 391.15 329.20 390.85 457.28 468.15 461.35 503.15 465.15 503.15 503.15 465.15 465.15 461.35 425.00 483.15 438.55 –14.945 24.955 200.83 3001.4 –13.414 –16.486 –15.048 –10.474 –13.925 –15.860 –10.699 –12.404 –12.831 –11.798 –12.367 –11.812 –10.695 –11.027 –11.145 –11.302 –10.886 –11.135 –10.975 –11.152 –11.728 –11.446 –11.308 –11.097 –13.689 –13.564 –14.945 –22.128 –16.210 –20.681 –16.438 –16.485 –21.510 –21.857 –18.266 –10.780 –12.442 –10.914 15.835 26.777 20.025 6.9072 19.742 24.852 7.2626 15.142 14.461 9.1501 11.292 10.940 8.7094 9.1110 9.8360 8.6196 6.0998 9.0229 8.7485 10.329 9.9183 7.5203 11.213 7.4406 19.114 19.997 24.955 47.765 33.359 43.910 33.904 34.033 45.933 46.763 37.551 9.0530 16.410 9.7590 141.98 195.34 160.05 79.310 170.52 188.46 81.639 146.82 135.56 93.281 98.023 111.39 97.939 99.375 106.14 91.723 67.393 97.461 95.872 111.39 101.72 79.021 119.26 80.643 167.94 177.32 200.83 259.62 247.51 255.36 248.07 248.31 256.84 257.32 247.25 101.00 158.63 107.54 1904.5 3220.5 2408.5 830.74 2374.4 2989.0 873.49 1821.2 1739.3 1100.5 1358.1 1315.8 1047.5 1095.8 1183.0 1036.7 733.64 1085.2 1052.2 1242.3 1192.9 904.48 1348.6 894.9 2298.9 2405.1 3001.4 5744.8 4012.2 5281.1 4077.7 4093.3 5524.5 5624.3 4516.3 1088.8 1973.7 1173.7 Pure Solvent n-octanol Water Benzyl alcohol Ethylamine Tetrahydrofuran 1-butanol 2-butanol 1-chlorobutane Methyl benzoate Cyclohexane n-hexane Heptane Benzene Chlorobenzene ethylbenzene o-xylene n-heptane n-pentane Toluene m-xylene Carbone tetrachloride Ethylacetate Diethylether Aceticacid Acetone Butyl Alcohol Aniline n-octanol Propylene Glycol Butane-1,4-diol Butane-1,2-diol 1,4-Butanediol 1,4-Butanediol 1,2-Butanediol 1,2-Butanediol Propylene Glycol N,N-dimethylformamide Formamide N,N-dimethylacetamide RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 94 No. 1 2020 34 MLIKI et al. Table 1. (Contd.) No. Tm Tb lnAs Ea TA T* K K – kJ mol–1 K K Pure Solvent 40 41 42 2-Methoxyethanol Water N,N-dimethylacetamide 188.15 273.15 253.15 397.65 373.15 438.55 –12.602 –13.284 –10.896 15.185 15.510 9.4260 144.93 140.42 104.05 1826.3 1865.4 1133.7 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 2-Ethoxyethanol N,N-dimethylacetamide 1,4-dioxane Water Isobutyric acid Water Ethanol Water Methanol Water p-xylene Dimethylsulfoxide o-xylene Dimethylsulfoxide Ethylene glycol 1,4-dioxane Water Triethyl amine Water Glycerol TEGMME* n-Heptane Propargylalcohol Allylalcohol t-butanol 2-propanol 1-propanol Bromobenzene Chlorobenzene Ethylbenzene Benzene Dimethylsulfoxide 3-amino-1-propanol Isoamylalcohol 2-propanol Ethanol 1,4-dioxane N-methylacetamide 183.15 253.15 284.15 273.15 226.15 273.15 159.15 273.15 175.55 273.15 285.65 290.65 248.65 290.65 260.15 284.15 273.15 158.15 273.15 293.15 229.15 182.15 220.15 144.15 298.84 183.65 149.15 242.15 228.15 178.15 278.65 290.65 284.15 156.15 183.65 159.15 284.15 300.15 408.15 438.55 374.15 373.15 426.65 373.15 351.15 373.15 337.75 373.15 411.15 462.15 417.15 462.15 470.15 374.15 373.15 361.95 373.15 455.15 395.15 371.15 387.65 370.15 355.55 355.15 370.15 429.15 405.15 409.15 353.15 462.15 458.65 403.15 355.15 351.15 374.15 478.15 –12.682 –10.934 –11.853 –13.443 –11.200 –13.383 –12.166 –13.232 –11.528 –13.334 –10.761 –11.872 –11.044 –12.002 –16.146 –11.430 –13.742 –11.248 –13.389 –24.143 –13.638 –12.613 –12.607 –12.866 –18.476 –15.032 –13.415 –13.450 –10.406 –10.807 –13.254 –10.975 –18.036 –14.322 –16.403 –12.997 –11.607 –13.155 15.803 9.7973 12.660 15.920 11.126 15.749 13.204 15.433 9.9340 15.640 8.3920 14.058 9.5730 14.333 29.941 11.669 16.684 8.2120 15.786 59.608 21.245 14.344 15.105 15.305 32.029 21.950 17.792 16.771 7.9331 8.5016 14.955 11.721 36.059 21.640 25.430 15.680 12.074 19.128 149.87 107.77 128.47 142.43 119.48 141.54 130.50 140.28 103.64 141.07 93.800 142.42 104.26 143.63 223.03 122.78 146.02 87.810 141.80 296.95 187.36 136.78 144.10 143.07 208.50 175.63 159.51 149.98 91.687 94.617 135.71 128.44 240.46 181.72 186.46 145.10 125.11 174.88 1900.7 1178.3 1522.6 1914.7 1338.1 1894.2 1588.1 1856.2 1194.8 1881.1 1009.3 1690.8 1151.4 1723.9 3601.1 1403.5 2006.6 987.68 1898.6 7169.2 2555.2 1725.2 1816.7 1840.8 3852.2 2640.0 2139.9 2017.1 954.13 1022.5 1798.7 1409.7 4336.9 2602.7 3058.5 1885.9 1452.2 2300.6 RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 94 No. 1 2020 HYPERBOLIC CORRELATION BETWEEN 35 Table 1. (Contd.) No. Tm Tb lnAs Ea TA T* K K – kJ mol–1 K K Pure Solvent 81 82 83 84 85 2-Methoxyethanol Water Propylene carbonate 1,2-diethoxyethane Acetonitrile 188.15 273.15 218.15 199.15 222.15 397.65 373.15 513.15 394.15 354.65 –12.662 –13.307 –11.729 –10.819 –10.793 14.504 15.568 14.192 7.5690 6.9895 137.77 140.70 145.53 84.142 77.885 1744.4 1872.4 1706.9 910.34 840.64 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 Tetrahydrofuran Methanol Octane Pyridine Benzene Water Propylene Glycol N, N-dimethylacetamide Water 2-propanol 2-mehoxyethanol Isoamyl alcohol Glycerol 1,4-Dimethylbezene n-decane n-eicosane Benzene n-tetracosane 165.15 175.55 216.15 232.00 278.65 273.15 214.15 253.15 273.15 183.65 397.65 156.00 563.00 286.30 243.30 310.00 278.65 324.00 339.15 337.75 398.75 388.55 353.15 373.15 461.35 438.55 373.15 355.15 397.65 403.15 455.15 411.15 447.25 616.25 353.15 664.55 –10.393 –11.629 –12.886 –12.635 –9.7233 –11.900 –18.266 –10.914 –13.443 –15.030 –12.822 –14.320 –24.140 –10.760 –11.355 –12.675 –13.250 –12.253 6.7382 10.198 13.275 14.925 5.5473 12.043 37.551 9.7590 15.920 21.950 15.704 21.640 59.610 8.3920 10.633 18.630 14.960 18.962 77.977 105.47 123.90 142.08 68.618 121.71 247.25 107.54 142.43 175.63 147.31 181.72 296.95 93.800 112.63 176.78 1135.7 186.13 810.42 1226.5 1596.6 1795.1 667.20 1448.4 4516.3 1173.7 1914.7 2640.0 1888.8 2602.7 7169.4 1009.3 1278.9 2240.7 1799.3 2280.6 * TEGMME is an abbreviation of triethylene glycol monomethyl ether. Thus, using statistical modeling techniques to find the expression which best fits this relationship, we suggest the following general equations relying Ea and (1/TA – 1/Tb) for both directions which may hold well. a1 Ea = + a3, ⎛1 ⎞ 1 ⎜T − T ⎟ − a2 ⎝ A b⎠ (12) Table 2. Descriptive statistics of temperatures` parameters: Arithmetic mean (T1), Confidence Interval (CI) and coefficient of variation (CV) Parameters T1/K CI (mean) CV (%) Tm 229.16 217.72–240.60 22 Tb 403.23 391.87–414.58 12 TA 146.19 133.89–158.49 37 T* 2093.55 1784.59–2402.52 65 RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A ⎛1 b1 1⎞ ⎜T − T ⎟ = E − b + b3, ⎝ A b⎠ a 2 (13) where ai (I = 1,2,3) and bj (j = 1,2,3) are the models’ parameters. Indeed, according to the Table 4, which presents the result of statistical estimations of suggested models, only a3 is statistically non-significant and tend thus to zero, all the other estimated parameters are statistically significant, the R-squared tends to one and the value of Fisher statistics are very high for both models. This result allows as expecting a very good predictive power and quality of approximation during their practical use. Note that their relationship is not necessarily bijective i.e., the equations resuming their relationship in the two directions are not necessarily inverse functions (Ea = f(1/TA – 1/Tb) or 1/TA – 1/Tb = g(Ea)). For that Vol. 94 No. 1 2020 36 MLIKI et al. we have presented a best model for each direction independently. To confirming the good quality of approximation of the suggested equations, Eq. (12) and Eq. (13), we have estimated Ea by replacing the experimental values of (1/TA – 1/Tb) in the Eq. (12) and we have estimated (1/TA – 1/Tb) by replacing the experimental values of Ea in Eq. (13). Table 5 provides descriptive statistics of estimated versus experimental values for the two variables, in addition to the Average Percentage Deviation (APD) which is an excellent indicator of the approximation quality. The descriptive statistics indicates distinctly that the experimental data values are close to the corresponding ones estimated from Eqs. (12) and (13). Also, the values of the Average Percentage Deviations (APD) are very low, 3.7% and 3.2% for Ea and (1/TA – 1/Tb) respectively, showing the small discrepancy between the estimated and the experimental values (Table 5). Nevertheless, in order to affirm the obtained results, we have to apply some statistical test of populations’ comparison. So, we have compared the estimated values with the experimental ones of the two variables using the Wilcoxon signed-rank test [40], which is an equality tests on matched data. Note that the null hypothesis supposes that the distributions of estimated and experimental values are the same. Result of the test is presented in the Table 6. The test of signed-rank of Wilcoxon leads to agree the null hypothesis for all variables showing that the estimated and the experimental distributions of both variables are significantly and statistically the same, which justify the excellent predictive power of the Eqs. (12) and (13). Table 3. Spearman rank correlation tests Parameters Ea lnAs TA Prob > |t | Tm 0.25 0.03 Tb 0.41 0.00 Tm –0.16 0.18 Tb –0.25 0.03 Tm 0.24 0.06 Tb 0.49 0.00 These results can also be revealed graphically. In fact, we display in Figs. 4 and 5 the experimental observations of one variable on the x-axis versus simultaneously the estimated and experimental observations of the other variable on the y-axis. Thus, we can see frankly that the gap between estimated and experimental values is showing a small discrepancy. For providing some physical meaning to the suggested empirical equations (Eq. (12) and Eq. (13)), fruitful for Chemical Engineering, we can propose some variable transformations to provide new semiempirical equations interesting for theoreticians as follows: Ea = Rα1 + E01, ⎛1 1⎞− 1 − ⎜T T ⎟ T ⎝ A b⎠ 01 (14) Rβ1 ⎛1 1⎞ 1 (15) ⎜T − T ⎟ = E − E + T , ⎝ A b⎠ a 02 02 where R is the ideal gas constant, α1 and β1 are dimensionless constants, E01 and E02 are the minimal values 70 600 Tm, K Tb, K (a) Spearman rho 60 Ea, kJ mol−1 (b) - R.lnAs/3 Ea and - R.lnAs/3 (Tm and Tb)/K 500 400 300 200 100 50 40 30 20 10 0 10 20 30 40 50 60 0 50 100 150 200 250 300 TA/K Fig. 2. Scatter plots of the direct mutual correlation between the Arrhenius parameters (TA, ln As and Ea) and the melting temperature Tm and boiling temperature Tb of some pure liquids. (Ea/kJ mol–1; –Rln(As/Pa s)/3 in J K–1 mol–1). RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 94 No. 1 2020 HYPERBOLIC CORRELATION BETWEEN 37 0.015 12 1/TA − 1/Tb (1/TA − 1/Tb)exp 0.010 8 (1/TA − 1/Tb)est A (1/TA − 1/Tb)/kK−1 10 0.005 6 4 0 0 20 2 10 20 30 40 50 60 Ea, kJ mol−1 Fig. 3. Scatter plots for correlations between Ea and (1/TA – 1/Tb). that the activation energy can theoretically take for the set of studied viscous liquids, and T01 and T02 are the minimal values that the difference between the reciprocal of Arrhenius and boiling temperatures (1/TA – 1/Tb) in Eqs. (14) and (15) can theoretically be taken 3 for the set of studied liquids group and which it is schematized by the dashed line in Fig. 3. For practical use of the Eq. (14) and the Eq. (15), we report in the Table 7 the estimated values of the new parameters. Thus, the practical form of the proposed equations, Eq. (14) and Eq. (15), can be defined as follows: Ea = 0.0661 , ⎛1 ⎞ −4 1 ⎜T − T ⎟ − 1.36 × 10 ⎝ A b⎠ Ea, kJ mol−1 60 Fig. 4. Graphical comparison between the experimental and the estimated values of (1/TA – 1/Tb) as function of the experimental values (Ea)exp. 0 0 40 (16) We add that mathematically, the relationship between the two equations (14) and (15) requires that there is equality by couple of similar parameters, i.e. (α1 = β1), (E01 = E02) and (T01 = T02). Nevertheless, the small and negligible difference between estimated values of each of these couples of parameters is due to several reasons such as the measurement and estimation errors, the heterogeneity of the studied temperature range and the implicit dependence of these parameters to specific properties of the studied solvents. Also, though the quality of the estimates is very good, the values are very small for which the error of the estimates does not make it possible to obtain them with such precision. In addition, there is a distinct difference between the theoretical and experimental values of {(1/TA – 1/Tb)th and (1/TA – 1/Tb)exp}, where these last ones are due principally to statistical distribution of observations, errors of calculation and measurements, and to the nature of the studied group of fluids for which its specific features can play an substantial role in this fact. Causal Correlation Origin ⎛1 −4 1⎞ 0.0596 ⎜T − T ⎟ = E − 0.8521 + 2.6 × 10 , ⎝ A b⎠ a (17) where Ea is in (kJ mol–1) and Ti in K. As a preliminary theoretical explanation of the causal correlation between the boiling point (Tb/K) and the Arrhenius temperature (TA/K) we can give the Table 4. Result of econometric estimations of the models (12) and (13) Equation (3) (4) Parameter a1/kJ K–1 mol–1 a2/K–1 a3/kJ mol–1 0.0661 (53.21) 0.000136 (3.99) tends to 0 b1/J K–1 mol–1 0.0596 (30.220) b2/kJ mol–1 0.8521 (4.84) b3/K–1 0.00026 (3.2) Values between parentheses are t-statistics RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 94 No. 1 2020 R-squared F-statistics 0.994 7467 0.995 7471 38 MLIKI et al. Table 5. Descriptive statistics on experimental versus estimated values Mean σ Experimental 17.41 11.24 6.10 59.61 Estimated 17.43 11.10 5.76 63.89 APD (%) 3.7 3.3 0.2 16.5 Experimental 0.0052 0.0024 0.0012 0.0116 Estimated 0.0052 0.0025 0.0013 0.0116 APD (%) 3.2 2.7 0.1 Variables Ea/kJ mol–1 (1/TA – 1/Tb) following points: (i). The vaporization phenomenon is a transition between two thermodynamic states where molecules escape from the free surface layer of liquid towards the free space [12, 13]. (ii). The activation energy (Ea) value is necessary to the transfer between two transition states where molecules move from one layer to another adjacent layer in the frame of shear flow of Newtonian fluid [14–17]. (iii). Similar causal correlation is observed in previous works suggesting an estimation of the boiling temperature of some liquid components forming some binary fluid mixtures through the viscosity-temperature and viscosity-composition dependences [23–37] and using some mathematical derivations of partial molar activation energies. (iv). We precise that the novel notion of partial molar activation energies is justified, as a first approximation, by the observation of similar compositiondependence between the variation of the viscosity activation energy (Ea) and the enthalpy (ΔH*) of activation of viscous flow deduced from the temperatureGibbs free energy dependence for each mole fraction of fifteen studied binary mixtures in previous works [23–37]. Table 6. Results of Wilcoxon signed-rank test: experimental versus estimated values Variable z Ea (1/TA – 1/Tb) Prob > |z| –0.22 0.84 0.90 0.37 Table 7. The suggested parameters’ optimal values of the Eq. (14) and Eq. (15) Equation Eq. 14 Eq. 15 Parameter α1 T01/K 0.00795 7353 β1 T02/K E02/kJ mol–1 0.00717 3846 0.8521 E01/kJ mol–1 0 Min Max 13.4 CONCLUSION Statistical and econometric techniques are used in this paper in order to investigate the existence of any eventual causal association between parameters of the Arrhenius-type equation, the melting and boiling points of some classical Newtonian liquids. For empirical analysis, 103 data set of viscosity of pure Newtonian fluids studied at atmospheric pressure and at different temperatures from the literature is used. Several analyses of mutual and complex possible correlations are made allowing as finding original and interesting result. A significant strong nonlinear correlation between the Arrhenius activation energy (Ea), the Arrhenius temperature (TA) and the boiling point (Tb) is seen. Consequently, several statistical and econometric estimations have been made to find the model which best fit this association. Finally we have suggested original equations modeling this relationship, Eqs. (12) and (13). In order to validate the proposed equation, we have made several statistical analyses for comparing the experimental values of parameters with the corresponding estimated values obtained using the Eqs. (12) and (13). Indeed, a descriptive statistics indicates clearly that the experimental data are close to the corresponding values estimated from Eqs. (12) and (13). The values of the Average Percentage Deviations (APD) are very low showing the small discrepancy between the estimated and the experimental values. The application of Wilcoxon signed-rank test confirmed that the experimental and the estimated distributions of the two variables are significantly and statistically the same, which justify the good predictive power of the suggested equations. For giving some physical significance to the proposed empirical equations to be useful for chemical engineering, we have made some variables transformations to obtain new semi-empirical equations, Eqs. (14) and (15), which may be very interesting for the theoreticians. Also, for practical use, we have proposed the Eqs. (14) and (15) which allow the predic- RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 94 No. 1 2020 HYPERBOLIC CORRELATION BETWEEN Ea, kJ mol 80 60 (Ea)exp (Ea)est 40 20 0 0 0.005 0.010 0.015 (1/TA 1/Tb)exp Fig. 5. Graphical comparison between the experimental and the estimated values of (Ea) as function of the experimental values (1/TA – 1/Tb)exp. tion of the boiling temperature through information on viscosity Arrhenius parameters of liquid phase and may be thus very useful in the handling of engineering data especially for the study of hydraulic components and systems efficiency. Finally, we hope that this present work open a window for theoreticians to give some theoretical justifications for the understanding of the liquid state and some meaningful insight into molecular interactions. ACKNOWLEDGMENTS Authors thank Dr. R.H. Kacem (FSEGN, University of Carthage, Tunisia) for helpful discussions and for some econometric and statistical calculations. DISCLOSURE STATEMENT Authors declare that there is no conflict of interest. REFERENCES 1. N. Ouerfelli, M. Bouaziz, and J. V. Herráez, Phys. Chem. Liq. 51, 55 (2013). 2. J. V. Herráez, R. Belda, O. Diez, and M. Herráez, J. 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