Correlations among Composition, Temperature, and Density

Article pubs.acs.org/IECR

Correlations among Composition, Temperature, and Density, Viscosity, or Derived Thermodynamic Properties of Binary Mixtures of Tri‑n‑butyl Phosphate with n‑Hexane or n‑Dodecane Mani Lal Singh,†,‡ Subhash C. Tripathi,*,§ P. P. K. Venkata,∥ and Vilas G. Gaikar‡ †

Nuclear Recycle Board, §Fuel Reprocessing Division, and ∥Computer Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400 085, India ‡ Department of Chemical Engineering, Institute of Chemical Technology, Matunga, Mumbai 400 019, India S Supporting Information *

ABSTRACT: Correlations proposed in this work predict density and viscosity with better accuracy in comparison with existing correlations. Dimensionless specific volume and logarithmic viscosity of binary mixtures are nth order (1 ≤ n ≤ 4) polynomials of the mole fraction of one of the components at the system temperature, and parametric coefficients of such polynomials are mth order (1 ≤ m ≤ 4) polynomials of system temperatures. Parametric coefficients of fitted fourth order Redlich−Kister type equations (RK-eq) are linear functions of temperature. Excess properties and derived thermodynamic properties of both mixtures have been discussed in terms of intramolecular and intermolecular interactions of molecules. Like virial equations of state (EOS), dimensionless RK-eq quantifies to calculate virial coefficients and hence to compute interactions among component 1−1, component 1−2, and component 2−2. Parametric coefficients of the Vogel−Fulcher−Tammann equation of viscosity are polynomials of mole fractions of one of the components.

1. INTRODUCTION Tri-n-butyl phosphate (TBP) is a universally accepted extractant for organic compounds, inorganic acids, and metal ions in the chemical, nuclear, and hydrometallurgical industries.1−7 In the past six decades, the nuclear industry has been using TBP, diluted with heavy normal paraffins comprising n-alkanes (C6−C16), mostly n-dodecane, for reprocessing of irradiated nuclear fuels, due to its excellent resistance to thermal, radiation, and chemical degradation, low vapor pressure, and good selectivity for U and Pu with fast kinetics and physical properties. Density and viscosity are very important physical properties required for optimization of the design of process equipment as well as processes, and also for computations in thermodynamics, and multiphase, multicomponent transport phenomena with chemical reactions for research and development.8,9 The densities and viscosities of individual chemicals such as TBP, n-hexane, and n-dodecane have been studied by many researchers.10−19 The density and viscosity values of TBP reported differ from each other.10,11,18,19 Reported binary mixture densities of TBP + hexane between the temperatures 298.15 and 323.15 K also do not match.18,19 Effects of changes in the composition on the density and viscosity of binary mixtures of TBP + hexane and TBP + dodecane have been reported between the temperatures 298.15 and 328.15 K.18 The operating temperature varies between the temperatures 288.15 and 338.15 K in the PUREX (Plutonium and URanium EXtraction) process, for optimization of extraction, scrubbing, and stripping. In a liquid−liquid mass transfer contactor, when 1g·L−1 Pu(IV) and 225 g·L−1 U(VI) are loaded in 30% (v/v) TBP + 70% dodecane, excess molar volume increases by approximately 3−5%, which makes material accounting © 2014 American Chemical Society

challenging in the organic stream. Most of the reported physical and transport properties were experimental to meet the industrial requirements of process plants, which need to be carried out scientifically. No thermodynamic analyses of both mixtures have been reported in the literature. Hence, it is required to measure density as well as viscosity of the pure components and both binary mixtures for the entire compositions at temperatures between 288.15 and 338.15 K. It is also required to derive correlations pertaining to volumetric and thermodynamic properties, and the equation of state (EOS) from the experimental density and viscosity of both mixtures.23−25 The excess molar volumes and deviations of the viscosity and thermal expansion coefficient of the binary liquid mixtures provide quantifiable information pertaining to understanding intermolecular as well as intramolecular interactions.20 The EOS and derived thermodynamic properties will be significant in understanding the mechanism of solvent extraction.21,22 Parametric coefficients of Redlich−Kister type equations (RK-eq)26 for excess molar volumes, viscosity deviations, and excess isobaric thermal expansion coefficients of both binary mixtures are linear functions of the system temperature. Based on the data,27−37 the dimensionless specific volume and logarithmic viscosity of binary mixtures are nth order (1 ≤ n ≤ 4) polynomials of the mole fraction of one of the components, at constant temperature, and parametric coefReceived: Revised: Accepted: Published: 3795

November 1, 2013 January 22, 2014 January 22, 2014 January 22, 2014 dx.doi.org/10.1021/ie4036912 | Ind. Eng. Chem. Res. 2014, 53, 3795−3804

Industrial & Engineering Chemistry Research

Article

Table 1. Purity Grades, Experimental Densities, and Viscosities for Pure Compounds with Corresponding Literature Values at 298.15 K ρ/(kg m−3)

η/(mPa·s)

component

purity/mass fr

T/K

expt

lit.18

expt

lit.18

TBP n-hexane n-dodecane

0.992 0.994 0.995

298.15 298.15 298.15

0.9707 0.6566 0.7453

0.970 83 0.656 33 0.745 79

3.38 0.299 1.347

3.399 0.285 1.336

ficients of such polynomials are mth order (1 ≤ m ≤ 4) polynomials of system temperatures. Activation free enthalpy, excess activation free enthalpy, activation enthalpy, excess activation enthalpy, activation entropy, and excess activation entropy for viscous flow of the binary mixtures have been derived from experimental viscosities of the systems. Various correlations have been used to relate densities, viscosities, and derived thermodynamic properties of the binary mixtures. Proposed correlations have been compared with the existing ones to prove the superiority in prediction. Excess properties of both mixtures have been discussed in terms of intramolecular and intermolecular interactions of molecules.38−41 Human ingestion of about 0.6 μg of Pu is a lethal dose of intake. Red-oil formation and a subsequent event or explosion can take place when the temperature of TBP and nitrates of any one or all of them (H, U, Pu, Th, etc.) is raised beyond 408 K.

tion.43,44 The Supporting Information contains comprehensive experimental data, derived excess properties, plots of parity, and residuals for correlations, among others. 3.1. Density of Binary Mixtures. Excess properties were quantified for a binary liquid mixture using the following equation: i=2

ΔX mE = X m −

∑ xiXi

(1)

i=1

Excess molar volumes VmE =

i=2 ∑i = 1 xiMi

ρm

VEm

i=2

−

∑ i=1

were evaluated applying eq 1:

xiMi ρi

(2)

where M, ρ, and are x are the molecular mass, density, and mole fraction, respectively. Subscripts “1”, “2”, and “m” represent component 1 (TBP), component 2 (hexane or dodecane, one at a time), and mixture, respectively. The computed excess molar volumes of the binary mixtures were correlated with the following Redlich−Kister type polynomial equation:26

2. EXPERIMENTAL SECTION 2.1. Materials. AR grade TBP, n-hexane, and n-dodecane were supplied by E. Merck, Germany, s. d. Fine Chemicals Ltd., Mumbai, and Spectrochem Pvt. Ltd., Mumbai, India, and were verified for their compositions. The compositions of the reagents were determined by gas chromatography using a Shimadzu gas chromatograph (GC) (Model 2014) equipped with a flame ionization detector using “GC solution” software for data processing. The purity of the organic solvents was greater than 0.99 mass fraction, which was further checked in Table 1, by measuring and comparing the densities and viscosities with their literature values.10−19 All molar quantities were based on the IUPAC relative atomic mass table. 2.2. Apparatus and Procedure. The densities (ρ) of the pure liquids and binary mixtures were measured as per ASTM D4052, using an Anton Paar (Austria) DMA 5000 digital vibrating U-tube densitometer (with automatic viscosity correction) having a stated accuracy of ±5 × 10−6 g·cm−3. The kinematic viscosities (ν) of the pure liquids and binary mixtures were measured with a digital Stabinger viscometer SVM 3000/G2 from Anton Paar as per ASTM D7042-11. Data of the kinematic viscosities of pure liquids and mixtures were multiplied by the corresponding values of density obtained in DMA 5000, to enhance the quality of dynamic viscosity data. Details of the experimentation could be referred from the literature.42 The stated reproducibility of measurement of dynamic viscosity is 0.0035 times the experimental value and that of density is 5 × 10−4 g·cm−3 in the temperature interval 288.15−338.15 K. The uncertainty in the values of density, viscosity, and excess molar volume are about ±5 × 10−6 g·cm−3, ±2 × 10−3 mPa·s, and ±3 × 10−3 cm3 mol−1 respectively.

i=n

VmE = x1x 2∑ Ai (x1 − x 2)i i=0

(3)

where Ai are the fitting parameters obtained by the unweighted least-squares method.27 Equation 3 was extended in order to represent the excess molar volume as a function of temperature by considering the coefficients Ai to be polynomial of the temperature T (K):24 j=m

Ai =

∑ aij(T ) j j=0

(4)

In eq 4, aij are the temperature-independent parameters for these binary mixtures and T is the absolute temperature. aij were obtained by the unweighted least-squares method.27 Experimental as well as predicted data were ascertained in the Supporting Information by using the standard deviation σ/ standard uncertainty u/combined uncertainty uc/combined expanded uncertainty Uc = kuc.45 The level of confidence is 0.9545, at a coverage factor k = 2. Density based thermodynamic properties [excess molar volumes VEm, partial molar volumes V̅ m,i, isobaric thermal expansion coefficients αi, excess isobaric thermal expansion coefficient αEm, and isothermal coefficient of pressure excess molar enthalpy (∂HEm/∂P)T,x] for TBP + hexane and TBP + dodecane were computed based on experimental values given in Table S2 in the Supporting Information and plotted against the mole fraction of TBP at different temperatures, as shown in Figures 1 and 2, respectively.24 Derived thermodynamic properties are given in Tables S3 and S4 in the Supporting Information. Excess molar volumes are negative for TBP + hexane over the entire composition range and increase with temperature, as

3. RESULTS AND DISCUSSION Volumetric (inclusive of EOS), viscometric, and derived thermodynamic properties are computed, tabulated, and depicted elaborately in the associated Supporting Informa3796

dx.doi.org/10.1021/ie4036912 | Ind. Eng. Chem. Res. 2014, 53, 3795−3804


Article

Figure 2. Plots of (a) density, (b) excess molar volume, (c) partial molar volume of TBP, (d) partial molar volume of n-dodecane, (e) thermal expansion coefficient, (f) excess thermal expansion coefficient, and (g) isothermal coefficient of pressure excess molar enthalpy against mole fraction for [TBP(1)+n-dodecane(2)] at temperatures 288.15 (□), 298.15 (○), 308.15 (△), 318.15 (▽), 328.15 (◇), and 338.15 K (∗). Solid curves represent values calculated from equations reported in this work with coefficients from Table S4 in the Supporting Information.

Figure 1. Plots of (a) density, (b) excess molar volume, (c) partial molar volume of TBP, (d) partial molar volume of n-hexane, (e) thermal expansion coefficient, (f) excess thermal expansion coefficient, and (g) isothermal coefficient of pressure excess molar enthalpy against mole fraction for [TBP(1)+n-hexane(2)] at temperatures 288.15 (□), 298.15 (○), 308.15 (△), 318.15 (▽), 328.15 (◇), and 338.15 K (∗). Solid curves represent values calculated from equations reported in this work with coefficients from Table S3 in the Supporting Information.

one component into the other’s structure due to the differences in free volume, shape, and size of the components. The first factor appears to be poor for both mixtures. Nonpolar molecules of hexane and dodecane are cylindrical in shape having 6 and 12 linear carbon−carbon chains, respectively. The carbon chain length of hexane is close to that of butyl. Therefore, the hexane molecule can easily enter the space between the three butyls of TBP and approaches butyl nicely. On the one hand, this makes the intermolecular distance between the hexane molecule and butyl smaller; on the other hand, this also decreases the total volume of TBP and hexane. Hence VEm becomes negative for the mixture of TBP + hexane. In the mixture of TBP + dodecane, dodecane has a longer carbon chain than butyl, so dodecane cannot easily enter the space between three butyls of TBP. Because of differences in the free volume, shape, and size of the molecules of TBP and the dodecane, they cannot fully approach. Therefore, the total volume of TBP and dodecane increases and hence the values of VEm for the mixture of TBP + dodecane are positive. Because of the dominance of intermolecular interaction based associative forces, effective volumes of unlike molecules shrink and hence excess molar volumes of the mixture will be negative. Applying this concept to Figures 1b and 2b, the following possibilities are derived: (i) The interactive force between TBP and hexane is dominant over the forces among TBP−TBP and hexane− hexane. This dominance is of such an extent that TBP−hexane volume shrinkage increases with increase in temperature. As the

shown in Figure 1b. The effects of composition and temperature on excess molar volumes of TBP + dodecane are opposite those for TBP + hexane, as shown in Figure 2b. Excess molar volumes of both mixtures matched well with those reported by Tian,18 between the temperatures 298.15 and 328.15 K. When intermolecular attractions between molecules are stronger than intramolecular attractions, excess molar volumes are negative and vice versa.18,38 The derived excess molar volumes indicate that the TBP−hexane interaction is stronger than the interactions of TBP−TBP and hexane−hexane, while the TBP−dodecane interaction is weaker than the interactions of TBP−TBP and dodecane−dodecane. The following analysis could analyze these results. Values of excess molar volumes VEm are affected by three factors. The first factor comprises the specific forces between molecules, such as hydrogen bonds and charge-transfer complexes. Breaking of hydrogen bonds gives rise to positive excess molar volumes and vice versa. The second factor comprises physical intermolecular forces, including electrostatic forces between permanent dipoles, induction forces between a permanent dipole and an induced dipole, and dispersion forces between induced dipoles and repulsion between nonpolar groups. Often, the dispersion force is the main contributor to the physical intermolecular force, which is inversely proportional to the sixth power of the distance between two molecules. Physical intermolecular forces are usually weak. The third factor is the structural characteristics of the components, arising from the geometrical fitting of 3797



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system temperature increases, the interactions between TBP− TBP and hexane−hexane become weaker in comparison to the TBP−hexane interaction. Therefore, VEm values increase with temperature as shown in Figure 1b. (ii) The interactive force between TBP and dodecane is weaker than the forces among TBP−TBP and dodecane− dodecane. As a result, positive values of excess molar volumes are shown by binary mixtures of TBP + dodecane, as shown in Figure 2b. (iii) TBP−hexane interactions are stronger than TBP− dodecane interactions. The ranges of molar volumes of hexane, dodecane, and TBP are 129.4−139.2, 226.2−237.8, and 271.9−284.5 cm3 mol−1 respectively, between the temperatures of 288.15 and 338.15 K. Polar molecules of TBP are in a tetrahedral pyramidal shape with a PO43− moiety and tend to self-associate. Due to mismatch in the shapes of TBP and dodecane, there is a poor molecular orientation effect among them. This tends to demix TBP + dodecane as compared to that in TBP + hexane where hexane, though similar in shape to dodecane, is smaller by six methylene units. The molar volume of hexane is almost half that of TBP, so possibly TBP can accommodate the smaller chain hexane inside the tetrahedral PO43− moiety by establishing a dipole−induced dipole interaction with the PO part of TBP.19 Large molecules such as dodecane fail to do so. The steric effect plays a vital role in the observed difference in the volumetric properties for the two alkanes. Increased temperature decreases the self-association of TBP which causes better accommodation of hexane into TBP (TBP-rich mixtures) and greater extent of TBP−hexane interactions due to increasing availability of TBP monomers (hexane-rich mixtures). With an increase in temperature, thermal motion rises, which causes better mixing by a further decrease in interspacing among components like, hexane and TBP, whereas molecular interactions become weak with increase in temperature, as in case of dodecane and TBP and excess molar volume increases. The Redlich−Kister type equation (3) was solved for VEm of mixtures of TBP + hexane and TBP + dodecane, and corresponding parametric coefficients of the RK-eq are reported in Table S5 in the Supporting Information, between the temperatures 288.15 and 338.15 K. It also contains parametric coefficients for the viscosity deviation Δηm and the excess isobaric thermal expansion coefficient αEm of both binary mixtures. Parametric coefficients Ai of each RK-eq for the three excess properties are linearly dependent on temperature. These linear equations have been solved numerically using the multiparametric least-squares method27 to get slopes and intercepts, which are given in Table S6 in the Supporting Information. The partial molar volumes V̅ m,1 and V̅ m,2 of the components in binary mixtures can be determined from excess volume by deriving (∂VEm/∂P)xj≠i,p,T from eq 3. V0i is the volume of pure component i at the system temperature. The values of V̅ m,1 and V̅ m,2 can differ in sign, size, and slope.

Vm,2 ̅ =

VmE

+

V 20

+ x1 (∑ Ai [(2x1 − 1)i 2

i=0 i−1

− 2ix1(2x1 − 1)

])

(6)

Setting x1 = 0, eq 5 leads to partial molar volumes of component 1, at infinite dilution, V̅ ∞ m,1. i=n ∞ = V10 + V̅m,1

∑ Ai(−1)i i=0

(7)

For x1 = 1, eq 6 simplifies to partial molar volumes of component 2, at infinite dilution, V̅ ∞ m,2. i=n ∞ = V 20 + V̅m,2

∑ Ai i=0

(8)

Partial molar volumes of TBP are almost constant with respect to composition at a given temperature, but increase with temperature at a given composition as per Figures 1c and 2c. Similar observations were noted for hexane as well as dodecane. Increase in temperature increases the amplitude of vibration of molecules about their mean position, so the partial molar volume increases. Partial molar volumes for both components could also be obtained from25 V̅ = (ρ−1 + (1 − x1) ∂(ρ−1)/∂xi)Mi

(9)

The apparent molar volumes of TBP in n-alkanes and n-alkanes in TBP, denoted as VΦ,1 and VΦ,2, respectively, are calculated from experimental data using eqs 10 and 11:19 VΦ,1 = V10 + VmEx1−1

(10)

VΦ,2 = V 20 + VmEx 2−1

(11)

At a given temperature, dimensionless specific volumes are nth order (1 ≤ n ≤ 4) polynomials of the mole fraction of one of the components of binary liquid mixture and parametric coefficients of such polynomials are mth order (1 ≤ m ≤ 4) polynomials of system temperatures. This proposition has also been tested with this work and various data.14−19,23−25,28−37 Estimation of density using a fourth order RK-eq involves very lengthy steps, so correlation 1 (eqs 12 and 13) of dimensionless specific volume is introduced based on the above proposition, to evaluate densities of pure liquids and their binary mixtures, which could be represented by the following nth order equation of the mole fraction of component 1: i≤n≤4

ρi0 ρ−1 =

∑

bi(x1)i

i=0

(12)

−3

where ρi0 = 1.0 g·cm and the parametric coefficients bi of eq 12 are an mth order polynomial of temperature, as follows: i≤m≤4

bi =

∑ j=0

cij(T ) j (13)

which have been optimized based on the experimental density data. n = 2 and m = 1 have been found to be appropriately representing dimensionless specific volumes of mixtures of TBP + hexane (or dodecane). Dimensionless specific volumes of eq 12 have been solved numerically by a least-squares method, using density data, and

E 0 2 Vm,1 ̅ = Vm + V1 + (1 − x1) i=n

× (∑ Ai [(2x1 − 1)i + 2i(1 − x1)(2x1 − 1)i − 1]) i=0

(5) 3798



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0 Table 2. Values of Pure Molar Volume V0i /(cm3 mol−1) and Predicted Partial Molar Volumes at Infinite Dilution, Vi,RK‑eq , by Redlich−Kister eq 3 and This Work (V0i,eq 12) for TBP (1) in (n-Hexane or n-Dodecane), between the Temperatures of 288.15 and 338.15 K

T/K V0i /(cm3

a

−1

mol )

288.15

V01,expt V01,expt,Basua 0 V1,RK‑eq 0 Vi,eq 12

271.92

V01,expt 0 V1,RK‑eq 0 Vi,eq 12

271.91 271.91 272.15

V02,expt V02,expt,Basua 0 V2,RK‑eq 0 V2,eq 12

129.40

V02,expt 0 V2,RK‑eq V02,eq 12

226.20 226.20 225.91

272.16 276.07

129.40 127.33

298.15

308.15

Solute TBP (1) + n-Hexane (2) 274.34 276.80 273.47 275.97 274.58 277.04 278.62 281.23 Solute TBP (1) + n-Dodecane (2) 274.34 276.80 274.32 276.78 274.56 277.03 TBP (1) + Solute n-Hexane (2) 131.25 133.15 131.53 133.40 131.25 133.15 129.11 130.94 TBP (1) + Solute n-Dodecane (2) 228.42 230.69 228.42 230.69 228.13 230.39

318.15

328.15

338.15

279.31 278.52 279.54 283.88

281.86

284.45

282.09 286.59

284.69 289.34

279.31 279.29 279.53

281.86 281.84 282.08

284.45 284.44 284.67

135.11 135.36 135.11 132.82

137.12

139.20

137.12 134.76

139.20 136.76

233.00 233.00 232.70

235.36 235.36 235.05

237.77 237.77 237.46

Reference 19.

Isobaric thermal expansion coefficients αi, of pure components and binary mixtures, were calculated using eqs 15 and 16. Values of the thermal expansion coefficient αm are shown in Figures 1e and 2e for mixtures of TBP + hexane and TBP + dodecane, respectively. The dominance of the interaction of TBP−hexane is higher than the interactions of both TBP−TBP and hexane−hexane. As the content of TBP increases in the binary mixture, it goes on accommodating most of the hexane molecules in its tetrahedral structure, which is evident from Figure 1e: curves of the thermal expansion coefficient of mixture, αm, fall as a power function, which flattens later on because of the nonavailability of TBP. Thermal expansion coefficients of the mixture of TBP + dodecane αm are linear with respect to the composition of TBP at a given temperature, which also varies linearly with temperature at constant composition. Therefore, αm will form a three-dimensional plane with variables x1 and T in Cartesian coordinate system. The basic expression relating molar volumes of mixtures and excess molar volume is in accordance with eq 1:

their parametric coefficients are given in Table S7 in the Supporting Information. Parametric coefficients of eq 13 have also been solved by the least-squares method to get the parametric constants listed in Table S8 in the Supporting Information for both systems. The temperature-dependent density of pure components has been fitted to eq 14 of correlation 2 (eqs 14 and 15): j=2

ln[ρi (ρi0 )−1] =

∑ djT j j=0

(14)

where ρi0 = 1.0 g·cm−3 and the parametric coefficients are a first order polynomial of the mole fraction of component 1, as follows: k=1

dj =

∑ f jk (x1)k k=0

(15)

The dimensionless logarithmic density of eq 14 has been solved by the least-squares method, using density data, and their parametric coefficients are given in Table S9 in the Supporting Information. Parametric coefficients of eq 15 have also been solved by the least-squares method to get the parametric constants listed in Table S10 in the Supporting Information for both mixtures. The density could be predicted using correlation 2 (eqs 14 and 15) not only for pure liquids, but also for binary liquid mixtures. Pure component molar volumes using Redlich−Kister based equations, eqs 5, 6, and 12, along with experimental values, are listed in Table 2. Values of the molar volume of pure hexane matched well with the scarcely reported values.19 The thermal expansion coefficient αi of pure component i can be obtained by analytical differentiation of the density fitting eq 16: αi = −(ρi )−1(∂(ρ−1)/∂T )P

i=2

Vm = VmE +

∑ xiVi

(17)

i=1

where Vi and xi are the molar volume and mole fraction of component i, respectively. By differentiating eq 17 with respect to temperature T and dividing by the molar volume of the mixture Vm, we obtain24 ⎡⎛ E ⎞ ∂V Vm = (Vm)−1⎢⎢⎜ m ⎟ + ∂T ⎠ ⎣⎝ P ,x i

⎤ α ( x V ) ∑ i i i ⎥⎥ i=1 ⎦ i=2

(18)

where αm is the thermal expansion coefficient of the binary mixture. The values of excess thermal expansion coefficient αEm could be calculated as

(16) 3799



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i=2

αmE = αm −

∑ Φiαi i=1

3.2. Viscosities of Binary Mixtures. The viscosity deviations for a binary mixture of components, in accordance with eq 1, were calculated from the following correlation:

(19)

where Φi represents the volume fraction component i:

i=2

Δηm = ηm −

i=2

Φi = (xiVi )(∑ xiVi ) i=1

(22)

where ηm is the viscosity of the mixtures and η1 and η2 are the viscosities of components 1 and 2, respectively. For viscous flow of mixtures, the experimental viscosity ηm, viscosity deviation Δηm, and derived thermodynamic properties based on viscosity [activation free enthalpy ΔG#m, excess # activation free enthalpy ΔG#E m , activation enthalpy ΔHm, excess #E # activation enthalpy ΔHm , activation entropy ΔSm and excess activation entropy ΔS#E m ] were calculated based on experimental data provided in Table S2 in the Supporting Information, and trends are shown in Figures 3 and 4, respectively, for the mixtures TBP + hexane and TBP + dodecane.25 Derived thermodynamic properties are given in Tables S11 and S12 in the Supporting Information. The viscosity of the mixture strongly depends on the entropy of the mixture, which is related to the structure and enthalpy of liquids, as shown in Figures 3 and 4.42 Therefore, viscosity is also related to the shape, size, and molecular interactions of components of the mixture.

(20)

Values of the excess thermal expansion coefficient αmE calculated based on eqs 18−20 are shown in Figures 1f and 2f for TBP + n-hexane and TBP + n-dodecane, respectively. Liquid expansion is affected by the extent of volume compaction occurring in the liquid mixture. Higher volume compaction is indicative of higher cohesion which lowers the volume of the liquid thereby decreasing the extent of thermal expansion. Excess thermal expansion coefficient values describe the packing of the molecules and their orientation in the mixtures. The positive values of αEm describe the self-association of molecules in the mixtures, whereas the negative values imply the presence of hydrogen bonds within the molecules of components present in the mixture.41 The values of αEm for TBP + hexane mixtures are negative. The effect of hydrogen bonding between unlike molecules of TBP and n-hexane plays a significant role and decreases αEm, which corresponds to a power function of x1. αEm becomes positive due to an increase in the number and size of the microphases. The αEm values for the TBP + dodecane mixtures showed a nonlinear trend over the entire range of temperature and concentration. The graphical display represents nonlinear behavior of αEm attributed to the presence of stronger intermolecular interactions between the unlike molecules than between the like ones, in the mixture of TBP + hexane. At initial regions of the curves of the mixtures of TBP + dodecane, the effect of self-association dominates in contributing molecules, which makes αmE values positive. The increasingly positive TBP + dodecane αEm values with the increase in temperature clearly indicate the reduced molecular interaction between TBP + dodecane molecules. The isothermal coefficients of the excess molar enthalpy− pressure derivative (∂HEm/∂P)T,x can be derived accurately from volumetric measurements by application of the following expression: (∂HmE/∂P)T , x = VmE − T (∂VmE/∂T )P , x

∑ xiηi i=1

−1

(21)

This quantity represents the dependence of the excess molar enthalpy of mixing at constant composition and temperature. The isothermal coefficients of the excess molar enthalpy− pressure derivative are shown in Figures 1g and 2g for TBP + nhexane and TBP + n-dodecane, respectively. (∂HEm/∂P)T,x of TBP + n-hexane mixtures is positive, which smoothly increases from zero to a maximum at x1 ≈ 0.4 and reduces smoothly to zero. At a constant composition, (∂HEm/∂P)T,x increases with an increase in temperature for mixtures of TBP + hexane,while the (∂HEm/∂P)T,x decreases for mixtures of TBP + dodecane. Sigmoidal profiles have been observed for TBP + dodecane mixtures, but the maximum is at x1 ≈ 0.1 and minimum is at x1 ≈ 0.6. (∂HEm/∂P)T,x becomes less positive when the temperature increases for both binary mixtures under study. The effect of temperature on (∂HEm/∂P)T,x is more prominent in TBP + hexane mixtures, in comparison to TBP + dodecane.

Figure 3. Plots of (a) experimental viscosity, (b) viscosity deviation, (c) activation free enthalpy for viscous flow of mixtures, (d) excess activation free enthalpy for viscous flow of mixtures, (e) activation enthalpy, (f) excess activation enthalpy, (g) activation entropy, and (h) excess activation entropy against mole fraction for TBP (1) + n-hexane (2) at temperatures, 288.15 (□), 298.15 (○), 308.15 (△), 318.15 (▽), 328.15 (◇), and 338.15 K (left-pointing open triangle). Solid curves represent values calculated from equations reported in this work with coefficients from Table S5 and S6 in the Supporting Information. 3800



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The Grunberg−Nissan43 equation was also used to determine the molecular interactions responsible for viscosity changes with the single binary interaction parameter G12 as well as to estimate the dynamic viscosity of binary liquid mixtures. The standard uncertainties of G12 were rather poor, so the Grunberg−Nissan equation was not explored further (Supporting Information, Table S13). The Vogel−Fulcher−Tammann (VFT)47−49 equation suitably correlates the viscosities of pure components and both binary mixtures as a function of temperature for the entire range of compositions. ln(η) = A + [B(T − T0)−1]

(26)

where A, B, and T0 are parametric coefficients of eq 26, which have been found to be an nth order polynomial of the mole fraction of component 1, as follows: i≤n≤4

A or B or T0 =

∑

ui(x1)i

i=0

The VFT equation, eq 26, has been solved numerically to obtain the values of A, B, and T0, as listed in Table S14 in the Supporting Information. These parametric coefficients of eq 27 have also been computed one by one by the least-squares method and are given in Table S15 in the Supporting Information. With this, the scope of the VFT equation, eq 26, has been enhanced, and hereafter it is called the enhanced VFT equation. At a given temperature, the logarithmic viscosity of binary mixtures is an nth order (1≤ n ≤ 4) polynomial of the mole fraction of one of the components and parametric coefficients of such polynomials are mth order (1≤ m ≤4) polynomials of system temperatures. This proposition has also been tested with various data.14−19,25,28−37 When ln(η) was plotted against x1, a linear relationship was observed, at different temperatures, for the mixtures of TBP + n-hexane (or n-dodecane). Therefore, correlation 3 (eqs 28 and 29) is introduced in this work, based on the above propositions, to relate viscosities of the pure components as well as the binary mixtures as a function of temperature for whole range of mole fractions of TBP.

Figure 4. Plots of (a) viscosity, (b) viscosity deviation, (c) activation free enthalpy for viscous flow of mixtures, (d) excess activation free enthalpy for viscous flow of mixtures, (e) activation enthalpy, (f) excess activation enthalpy, (g) activation entropy, and (h) excess activation entropy against mole fraction for TBP (1) + n-dodecane (2) at temperatures 288.15 (□), 298.15 (○), 308.15 (△), 318.15 (▽), 328.15 (◇), and 338.15 K (left-pointing open triangle). Solid curves represent values calculated from equations reported in this work with coefficients from Table S5 and S6 in the Supporting Information.

The viscosity deviation of both mixtures matched well with those reported, between the temperatures of 298.15 and 328.15 K.18 The viscosity deviation of mixtures of TBP + hexane is less negative than that of TBP + dodecane, and it reduces with an increase in temperature for both mixtures. TBP−hexane interactions are so dominant that, even at elevated temperatures, the viscosity deviation of TBP + hexane is lower than that of TBP + dodecane. Breaking of the self-association and weak TBP−dodecane interactions are the reasons for the negative viscosity deviation of TBP + dodecane. In terms of the absolute rate theory, the viscosity of a liquid is given by25

i≤n≤4

ln(η) =

= [NA h(Vm) ]

exp[ΔHm# (RT )−1]

(28)

j≤m≤4

pi =

exp[−ΔSm# (R )−1]

∑ j=0

vij(T ) j (29)

Equation 28 has been solved numerically to obtain the values of pi, as listed in Table S16 in the Supporting Information. n = 1 and m = 2 have been found to be appropriately representing viscosities of TBP + hexane (or dodecane). The parametric coefficients of eq 29 have also been computed one by one numerically and are given in Table S17 in the Supporting Information. The Teja−Rice50,51 correlation is reported to replicate viscosities of binary mixtures better25 than the correlations given by Katti−Choudhri52 and Grunberg−Nissan.46 Therefore, viscosities have been calculated based on the Teja−Rice correlation, the enhanced VFT equations (eqs 26 and 27), and

where h is Planck’s constant, NA is Avogadro’s number, and Vm is the molar volume. Thus, activation free enthalpies for viscous flow of the mixtures could be obtained from (24)

The activation enthalpy of the viscous flow has been obtained by plotting ln(ηmVm) as a function of T−1 and the activation entropy calculated from ΔGm# = ΔHm# − T ΔSm#

ln(pi ) (x1)i

where parametric coefficients pi are a polynomial function of the temperature, of the following form:

(23)

ΔGm# = RT ln[ηVmh−1(NA )−1]

∑ i=0

ηm = hNA(Vm)−1 exp[ΔGm# (RT )−1] −1

(27)

(25) 3801



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Z = 1 + B(Vm)−1 + C(Vm)−2 + D(Vm)−3 + E(Vm)−4

correlation 3 (eqs 28 and 29), which have been compared with the experimental viscosity data. Correlation 3 has been found to be better than the other two correlations (enhanced VFT equations and Teja−Rice equations), which is evident from the residual and parity plots of Figures S2 and S3 (Supporting Information) for both mixtures. The standard deviations of correlation 3, the enhanced VFT equations, and Teja−Rice equations are 0.1250, 0.3653, and 0.3361 mPa·s. respectively, for the viscosity data of the TBP + n-hexane system. For TBP + n-dodecane binary mixtures, the standard deviations of correlation 3, the enhanced VFT equations, and the Teja− Rice equations are 0.0369, 0.7328, and 0.2708 mPa·s, respectively. The ten parametric correlation of RK-eq (eqs 3 and 4) and the six parametric correlation 2 (eqs 14 and 15) of this work have been compared with the six parametric correlation 1 (eqs 12 and 13) of this work in Table 3, for prediction of densities of

(30)

The EOS has been applied to both binary mixtures at each temperature to obtain the values of the coefficients given in Table S18 in the Supporting Information. Virial coefficients are straight lines of temperatures for both mixtures; intercepts along with slopes are tabulated in Table S19 in the Supporting Information. The virial EOS has been optimally regressed to get virial coefficients for both mixtures, as listed in Table 4. Numerically, it is also possible to represent the EOS or the compressibility factor for both mixtures as well as for pure liquids, at temperatures between 288.15 and 338.15 K, using the following proposed EOS: Z = (7.81478 × 10−5 − 1.24913 × 10−7T )Vm

with a standard deviation of ±1.221 88 × 10 . Authors present the dimensionless Redlich−Kister polynomial equation, as follows:

Table 3. Comparison of Proposed Correlations of This Work with the Existing Correlations for Prediction of Densities and Viscosities of Mixtures of Tri-n-butyl Phosphate (1) + n-Hexane (or n-Dodecane) (2), between the Temperatures of 288.15 and 338.15 K

i=n

VmE(x1x 2A 0)−1 = 1 +

no. params

TBP + hexane

RK-eqs 2−426 eqs 14 and 15 (correln 2) eqs 12 and 13(correln 1)

10 6 6

viscosity eqs

no. params

TBP + hexane

TBP + dodecane

eqs 26 and 2747−49 Teja−Rice eq50,51 eqs 28 and 29(correln 3)

15 8 6

0.3653 0.3361 0.125

0.7328 0.2708 0.0369

∑ [Ai(A 0)−1(x1 − x2)i ]

(32)

i=1

which could also be written as

±σ(ρ)/(g·cm−3) density eqs

(31)

−7

Z RK = 1 + (A1(A 0)−1)[(x1)2 − (x 2)2 ] + (A 2 (A 0)−1)

TBP + dodecane

× (x1 − x 2)2 + (A3(A 0)−1)(x1 − x 2)3

0.0508 0.001 0.0283 0.0084 0.0127 0.0079 ±σ(η)/(mPa·s)

+ (A4 (A 0)−1)(x1 − x 2)4

(33)

where ZRK is analogous to the compressibility factor Z, the deviation of which from unity is also a measure of the nonideality of mixtures of fluids. The value of the mixture molar volume is not a must to use the Redlich−Kister EOS eq 33, whereas the virial EOS eq 30 needs not only the mole fraction of one constituent, but also the mixture molar volume. The second coefficient of the Redlich−Kister EOS eq 33 contains both intramolecular interactions, without containing the intermolecular term, while the third coefficient of eq 33 contains intramolecular and intermolecular interactions. In line with the virial EOS eq 30, intramolecular and intermolecular interactions could be quantified from the dimensionless Redlich−Kister EOS eq 33 by calculating virial coefficients.20

both mixtures. It also incorporates a comparison of the proposed correlation 3 with correlations given by enhanced Vogel−Fulcher−Tammann and Teja−Rice equations. Correlation 1 of this work has been found to be better than RK-eq and correlation 2 for the prediction of densities of mixtures. Residual and parity plots of correlation 1 are shown in Figure S1 in the Supporting Information for both binary mixtures. The standard deviations of correlation 2, RK-eq, and correlation 1 are 0.0508, 0.0283, and 0.0127 g·cm−3, respectively, for the density data of TBP + hexane mixtures. For binary mixtures of TBP + dodecane, the standard deviations of correlation 2, RKeq, and correlation 1 are 0.001, 0.0084, and 0.0079 g·cm−3, respectively. 3.3. Equation of State of the Binary Mixtures. TBP becomes associated easily with organic nonpolar hexane and dodecane. Hence, the following fourth order virial equation of state (EOS) was also used to get the compressibility factor, based on volumetric data, by quantifying the second to fifth virial coefficients for both liquid binary mixtures:20

4. CONCLUSION Excess molar volumes, partial molar volumes of components, isobaric thermal expansion coefficients, excess isobaric thermal expansion coefficients, and isothermal coefficients of pressure excess molar enthalpy have been derived from the observed densities of the mixtures of tri-n-butyl phosphate + n-hexane (or n-dodecane), between 288.15 and 338.15 K, at atmospheric pressure. Six parametric correlations introduced in this work predict the densities of both binary mixtures better than the 10 parametric Redlich−Kister type equation (RK-eq). For viscous flow of mixtures, the viscosity deviation, activation free enthalpy, excess activation free enthalpy, activation enthalpy,

Table 4. Virial Coefficients of EOS eq 30 for Mixtures of Tri-n-Butyl Phosphate (1) + n-Hexane (or n-Dodecane) (2), between the Temperatures of 288.15 and 338.15 K mixture

B

C

D

E

TiAP + hexane TiAP + dodecane

−750.3577 −1156.0977

208 964 498 418

−2.5372 −9.412 × 107

1.1323 × 109 6.58035 × 109

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Article

present investigation. This article forms a part of of the doctoral thesis of M.L.S.

excess activation enthalpy, activation entropy, and excess activation entropy have also been evaluated based on experimental viscosities of the mixtures. Dimensionless specific volume and logarithmic viscosity of binary mixtures are nth order (1≤ n ≤ 4) polynomials of the mole fraction of one of the components, at the system temperature, and parametric coefficients of such polynomials are mth order (1≤ m ≤ 4) polynomials of system temperatures. The six parametric correlation proposed in this work predicts viscosities of both binary mixtures with better accuracy in comparison to the 15 parametric enhanced Vogel−Fulcher−Tammann correlation and the multiparametric Teja−Rice correlation. Parametric coefficients of RK-eq are polynomials of the mole fraction of component 1. Excess molar volumes of TBP + n-hexane are negative because of the dominance of TBP−hexane interaction, and those of TBP + n-dodecane are positive due to poor TBP− dodecane interaction. Viscosity deviations are negative for both mixtures. Fourth order RK-eqs have been fitted for the excess molar volumes, viscosity deviations, and excess isobaric thermal expansion coefficients of the binary mixtures. Parametric coefficients of all fitted RK-eqs are linear functions of temperature. A fourth order virial equation of state (EOS) along with a proposed EOS has also been presented for both mixtures. The presented dimensionless RK-eq is analogous to the virial EOS and provides calculation of the second to fifth virial coefficients of the virial EOS, thereby enabling us to compute interactions among different mixture components.

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(1) Siddall, T. H., III. Trialkyl phosphates and dialkyl alkyl phosphonates in Uranium and Thorium extraction. Ind. Eng. Chem. 1959, 51, 41. (2) Siddall, T. H., III. Thermodynamics for the Extraction of Uranyl Nitrate and Nitric Acid by Esters of the Types (RO)3P(UNK)O and RO2RP[UNK]O. J. Am. Chem. Soc. 1959, 81, 4176. (3) Science and Technology of Tri Butyl Phosphate; Schulz, W. W., Burger, L. L., Novratil, J. D., Bender, K. P., Eds.; CRC Press: Boca Raton, FL, 1984. (4) Sood, D. D.; Patil, S. K. Chemistry of nuclear fuel reprocessing: Current Status. Radioanal. Nucl. Chem. 1996, 203, 547. (5) Benedict, M.; Pigford, T. H.; Levi, H. W. Nuclear Chemical Engineering; McGraw Hill:New York,1981. (6) Siddall, T. H., III. The effects of altering alkyl substituent in trialkyl phosphates on the extraction of actinides. J. Inorg. Nucl. Chem. 1960, 13, 151. (7) Wright, A.; Paviet-Hartmann, P. Review of physical and chemical properties of tributyl phosphate/diluent/nitric acid systems. Sep. Sci. Technol. 2010, 45, 1753. (8) Zhou, Z.; Liang, S.; Qin, W.; Fei, W. Extraction Equilibria of Lithium with Tributyl Phosphate, Diisobutyl Ketone, Acetophenone, Methyl Isobutyl Ketone, and 2-Heptanone in Kerosene and FeCl3. Ind. Eng. Chem. Res. 2013, 52, 7912. (9) Kumar, A.; Hartland, S. Computational Strategies for Sizing Liquid−Liquid Extractors. Ind. Eng. Chem. Res. 1999, 38, 1040. (10) Kannan, S.; Kishore, K. Absolute viscosity and Density of Trisubstituted Phosphoric Esters. J. Chem. Eng. Data 1999, 44, 649. (11) De Lorenzi, L.; Fermeglia, M.; Torriano, G. Density, refractive index, and kinematic viscosity of diesters and triesters. J. Chem. Eng. Data 1997, 42, 919. (12) Singh, M. L.; Tripathi, S. C.; Lokhande, M.; Gandhi, P. M.; Gaikar, V. G. Density, Viscosity, and Interfacial Tension of Binary Mixture of Tri-iso-Amyl Phosphate (TiAP) and n-Dodecane: Effect of Compositions and Gamma Absorbed Doses. J. Chem. Eng. Data 2014, DOI: http://dx.doi.org/10.1021/je400493x. (13) Bolotnikov, M. F.; Neruchev, Y. A. Viscosities and Densities of Binary Mixtures of Hexane with 1-Chlorohexane between 293.15 and 333.15 K. J. Chem. Eng. Data 2003, 48, 739. (14) Aminabhavi, T. M.; Patil, V. B. Density, refractive index, viscosity, and speed of sound in binary mixtures of ethenylbenzene with hexane, heptane, octane, nonane, decane, and dodecane. J. Chem. Eng. Data 1997, 42, 641. (15) Aralaguppi, M. I.; Jadar, C. V.; Aminabhavi, T. M. Density, refractive index, viscosity, and speed of sound in binary mixtures of cyclohexanone with hexane, heptane, octane, nonane, decane, dodecane, and 2,2,4-trimethylpentane. J. Chem. Eng. Data 1999, 44, 435. (16) Nayak, J. N.; Aralaguppi, M. I.; Aminabhavi, T. M. Density, viscosity, refractive index, and speed of sound for the binary mixtures of ethyl chloroacetate with n-alkanes (C6 to C12) at (298.15, 303.15, and 308.15) K. J. Chem. Eng. Data 2001, 46, 891. (17) Trenzado, J. L.; Matos, J. S.; Segade, L.; Carballo, E. Densities, Viscosities and Related Properties of Some (Methyl Ester + Alkane) Binary Mixtures in the Temperature Range from 283.15 to 313.15 K. J. Chem. Eng. Data 2001, 46, 974. (18) Tian, Q.; Liu, H. Densities and viscosities of binary mixtures tributyl phosphate with hexane and dodecane from (298.15 to 328.15) K. J. Chem. Eng. Data 2007, 52, 892. (19) (Ali) Basu, M.; Samanta, T.; Das, D. Volumetric and acoustic properties of binary mixtures of tri-n-Butyl phosphate with n-hexane, cyclohexane, and n-heptane from T= (298.15 to 323.15) K. J. Chem. Thermodyn. 2013, 57, 335. (20) Prausnitz, J. M.; Litchtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid Phase Equilibria; Prentice Hall: Upper Saddle River, NJ, 1999.

ASSOCIATED CONTENT

S Supporting Information *

Tables S1 (data from the literature), S2 (experimental density and viscosity), S3 and S4 (derived thermodynamic properties based on experimental density), S5 and S6 (parametric coefficients of Redlich−Kister equation, eq 3, and coefficients of eq 4), S7 and S8 (parametric coefficients of dimensionless specific volume, eq 12 and coefficients of eq 13), S9 and S10 (parametric coefficients of logarithmic dimensionless density, eq 14 and coefficients of eq 15), S11 and S12 (derived thermodynamic properties based on experimental viscosity), S13 (coefficients of temperature-dependent Grunberg−Nissan interaction parameter), S14 and S15 (parametric coefficients of VFT equation, eq 26, and coefficients of eq 27), S16 and S17 (parametric coefficients of eq 28 and coefficients of eq 29), S18 and S19 (parametric coefficients of virial EOS eq 30 and coefficients of temperature-dependent parametric coefficients of virial EOS). Figures S1 (residual and parity plot of density), S2 and S3 (residual and parity plot of viscosities of TBP + hexane (or dodecane). This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*Tel.: +91-22-25591201. Fax: +91-22-25505340 or +91-2225505118. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors sincerely thank Mr. P. K. Wattal, Director, NRG, Mr. K. Agarwal, General Manager, NRB, and Mr. S. Basu, Chief Executive, NRB, and Director, BARC, Mumbai, India, for their keen interest and encouragement during the course of the 3803



Article

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Correlations among Composition, Temperature, and Density

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