1 Introduction

Currently as a potential candidate for sustained generation of CO2 free electrical energy, nuclear power generation with a closed fuel cycle is an important step toward net-zero carbon emission goals. The extraction power of tri-n-butyl phosphate (TBP) was reported in 1949 by Warf [1] and since the end of 1955, it has remained work-horse of PUREX (Plutonium Uranium Extraction) process, used for recycling the unused nuclear fuel to enhance the energy recovery [2]. TBP has two severe limitations- appreciable aqueous solubility (typically 200 mg·L−1 [3]) leading to runaway reactions at elevated temperatures as well as limited loading capacity for tetravalent metal ions via solvates [3, 4]. Due to these reasons, alternate solvent for PUREX process was desired. As compared to TBP, Tri-iso-amyl phosphate (TiAP) has better attributes on both the counts. Thus it is often referred as alternate solvent for the PUREX process [5,6,7,8].

The density of the organic and aqueous phases used in solvent extraction operations is an important process parameters for nuclear fuel cycle and nuclear recycle plants as direct access is often not possible and remote means are employed for process operation and control. In a typical three-purge-probe sensor fitted in a process tank, controlled amount of air is passed through the probes as shown in Fig. 1. From the first pair of measurements to get the differential air pressure, density of the liquid inside tank is estimated. From the other pair of measurements for differential air pressure, employing the estimated density value, the level in the close tank is estimated remotely. As high level of ambient radiation doses is quite harmful to conventional electronics, even in concurrent hot cells for radiochemical processing, the three-purge probes are employed to get an estimate of density and level of the liquid inside tank as these differential pressure measurement system are located at quite distance at the outside of the radiation shielded area. In addition, phase disengagement of dispersed phase from the continuous phase is solely dependent upon the density of the respective phases. Depending upon the phase flow ratio, the phase with lower flow rate is dispersed into the phase with the higher flow rate. For modeling of drop formation and coalescence inside the solvent extractor like pulse-column or centrifugal extractors, knowledge of density is utmost important. Precise measurements of density of organic solvent or aqueous process solutions are required so that the installed storage tanks can be calibrated precisely during cold commissioning.

Fig. 1
figure 1

Remote measurement of density and level inside a process tank in shielded radiochemical plant

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Physical properties of n-dodecane and n-dodecane/TiAP binary solutions were reported by few researchers [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. Most of the published work just reports the density of TiAP and its 1.1 M solutions at few temperatures [13,14,15] and this information is not just sufficient for precise calibration works. Singh [16] reported TiAP density data, correlations and derived Redlich–Kister parameters in the limited range of 288.15 K to 338.15 K only. Since temperatures higher than 338.15 K may be required for distillation based recovery of solvent from used extractant for recycle, density data at additional temperature range of 338.15 to 363.15 are required.

In this work, extensive experimental density data for TiAP-dodecane binary mixtures in well controlled conditions, in the range of 283.15 K to 363.15 K, were reported. Related Redlich–Kister parameters were estimated and reported. In addition, extension of Kirkwood–Buff theory to estimate partial molar volume of TiAP in the n-dodecane was also discussed.

2 Materials and Methods

Heavy Water Plant, Tuticorin, India synthesized and provided purified tri-iso-amyl phosphate (TiAP) with the stated purity of 99.6 %. The supplied TiAP was passed through a bed of indicating silica gel. The collected TiAP was further filtered through a 0.2-µm pore size 47 mm dia PTFE membrane filter. n-Dodecane (Sigma-Aldrich AR Grade) was used as such without any purification. Table 1 lists the details of chemicals used in this study. Important purity descriptors for both the components were taken as experimentally determined density values at 298.15 K and refractive index measured for sodium D line at 298.15 K. Thermophysical descriptors for pure fluids are listed in Table 2 and compared against the values reported in the literature.

Table 1 Chemical sample table
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Table 2 Physical properties of pure components at 298.15 K and 0.1 MPa and comparison with literature data
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Eight binary solutions of TiAP/n-dodecane were made on gravimetric basis. Required approximate amounts of TiAP and n-dodecane were weighed on SHIMADZU 220D balance (80/220 g, 0.01/0.1 mg resolution) in capped condition and mixed in a factory-certified glassware. Two pure fluids (TiAP and n-dodecane) were also taken as experimental solutions (to have x1 = 1 and x2 = 1). These solutions were stored in air-tight, certified and factory calibrated glass bottles.

The density of binary solutions, of TiAP and n-dodecane, was measured by a vibrating-tube density meter (VTDM Anton Paar DMA-5000) as per ASTM-D4052. The DMA-5000 was connected to a 30-position sample changer SP-3m and refractometer RXA-156 and fitted with automatic filling, rinsing and drying system. ASTM type-I demineralized water and AR grade ethanol were used as rinsing agents for cleaning glass U-tube of the density meter. Dried and filtered air was used for drying the thermostatted U-tube. The experimentally determined density data for binary solutions of TiAP and n-dodecane at several temperature levels are listed in Table 3. Estimated uncertainties u(x1), u(T) and u(ρ) are 10−4, 0.01 and 3.0 × 10−2 respectively.

Table 3 Density of TiAP/n-dodecane solutions for different composition and temperatures at 0.1 MPa
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3 Theory

3.1 Density of the Binary Solutions

The density of binary solutions is an engaging excellent research topic. The popular empirical correlations, from the published literature [25,26,27,28,29,30,31,32], are listed in Table 4. It may be observed that many of them had empirical background and choice of the equation was purely subjective.

Table 4 Correlations for densities of mixed solutions reported in literature
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3.1.1 Equation Proposed by Kumar and Koganti [33]

Based on the extension of their work on the density of mixed aqueous solutions, Kumar and Koganti [33] proposed the following equation for binary organic solutions containing tri-n-butyl phosphate and dodecane.

$$\left( {\frac{{\rho_{m} }}{{\rho_{o} }}} \right) = 1 + \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{2} {C_{i}^{(j + 1)/2} (b_{ij}^{o} + b_{ij}^{\prime } (T - 298.15) + b_{ij}^{\prime \prime } (T - 298.15)^{2} )} }$$
(1)

C is the concentration of solute in molar concentration scale. This equation is dimensionally inconsistent and needs correction.

For the solvent density ρo, at the required temperature T, they assumed a quadratic variation with temperature. A reference density value at 298.15 K was also required.

$$\left( {\frac{{\rho_{o} \left| {_{T} } \right.}}{{\rho_{o} \left| {_{298.15} } \right.}}} \right) = 1 - \alpha \left( {\frac{T - 298.15}{K}} \right) - \beta \left( {\frac{T - 298.15}{K}} \right)^{2}$$
(2)

In this work, a modified equation is proposed for estimation for the density of binary solutions. Equation 3 is a dimensionally correct form of Eq. 1, proposed earlier by Kumar and Koganti [33]-

$$\left( {\frac{\rho }{{\rho_{o} }}} \right) - 1 = \sum\limits_{j = 1}^{2} {\left( {\frac{{C_{1}^{j} }}{{C^{0} }}} \right)\left( {a_{oj} + a_{1j} \left( {\frac{T}{298.15} - 1} \right) + a_{2j} \left( {\frac{T}{298.15} - 1} \right)^{2} } \right)}.$$
(3)

Here, C0 is the dimensional correction factor and has a numerical value of unity. Corresponding equation in mole fraction terms will be as follows-

$$\left( {\frac{\rho }{{\rho_{o} }}} \right) - 1 = \sum\limits_{j = 1}^{2} {x_{1}^{j} \left( {b_{oj} + b_{1j} \left( {\frac{T}{298.15} - 1} \right) + b_{2j} \left( {\frac{T}{298.15} - 1} \right)^{2} } \right)}$$
(4)

Equation 4 may be obtained from the correlation proposed by Gonzalez et al. [31]. His expanded correlation is as follows-

$$\rho_{m} = A_{o} + A_{1} x + A_{2} x^{2}$$
(5)

It may be written as

$$\left( {\frac{{\rho_{m} }}{{A_{o} }}} \right) = 1 + \left( {\frac{{A_{1} }}{{A_{o} }}} \right)x + \left( {\frac{{A_{2} }}{{A_{o} }}} \right)x^{2}$$
(6)

Further

$$\left( {\frac{{\rho_{m} }}{{A_{o} }}} \right) = 1 + B_{1} x + B_{2} x^{2}$$
(7)

If adjustable coefficients Bjs are assumed to a quadratic dependence on the parametric temperature τ as

$$\begin{aligned} & B_{j} = B_{oj} + B_{1j} \tau + B_{2j} \tau^{2} \\ & {\text{where\;t}} = \left( {\frac{T}{298.15} - 1} \right). \end{aligned}$$
(8)

Ao can be the density of the solvent (component 1) at the required temperature and may be denoted by ρo. Now after rearranging Eqs. 4–7, one gets

$$\left( {\frac{{\rho_{m} }}{{\rho_{o} }}} \right) - 1 = \left( {B_{o1} + B_{11} \tau + B_{21} \tau^{2} } \right)x + \left( {B_{o2} + B_{12} \tau + B_{22} \tau^{2} } \right)x^{2}$$
(9)

Equation 9 is identical to Eq. 4.

It may be observed that to cover 17 temperature levels from 283.15 K to 363.15 K in steps of 5 K, 17 values of density of TiAP as well as n-dodecane at each level and 3 coefficients at each level are required for Jouyban–Acree correlation. However, for Eq. 4, density of dodecane at 298.15 K, 2 parameters for temperature related variation of density of n-dodecane and 6 parameters of right-hand side (R.H.S.) of Eq. 3 are only required to model the experimental density values for any experimental dataset of any size consisting of multiple levels of concentration and temperature.

4 Results and Discussions

4.1 The Excess Molar Volume for Binary Solutions of TiAP-n-Dodecane in the Range of 283.15–363.15 K

Shapes of both the molecules- extractant TiAP and diluent n-dodecane, are quite different as shown in Fig. 2a and b. Therefore, upon mixing both the fluids, the intermolecular as well as intramolecular interactions do happen. If intramolecular interactions (i–i and j–j) prevail upon the intermolecular interactions (i–j), the observed excess molar volumes will be negative. In the case of mixing of ethyl alcohol and water, this phenomenon is observed and total volume, after mixing, contracts or shrinks over the volume of individual unmixed fluids. If intermolecular interactions (i–j) prevail upon intramolecular interactions (i–i and j–j), the final volume after mixing will exceed the total volume before mixing and the observed excess molar volumes will be positive. In the case of mixing of tri-n-butyl phosphate with a non-polar diluent (n-dodecane), the final volume of mixed solution typically swells by 0.4 % due to stronger intermolecular interactions (i–j) [34]. Therefore, in case of mixing TBP (or TiAP) with the non-polar n-dodecane, the excess molar volume is predicted to be positive. The excess molar volume was estimated from Eq. 10.

$$V^{E} = \frac{1}{{\rho_{m} }}\sum\limits_{i = 1}^{2} {x_{i} M_{i} } - \left[ {\sum\limits_{i}^{2} {\frac{{x_{i} M_{i} }}{{\rho_{i} }}} } \right]$$
(10)
Fig. 2
figure 2

(a) Structure of n-dodecane molecule. (b) Structure of tri-iso amyl phosphate molecule

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The calculated excess molar volumes were found positive for the binary solution of TiAP/n-dodecane and could be correlated with a Redlich–Kister expression [35] with two parameters-

$$V^{E} = x_{1} x_{2} \sum\limits_{i = 0}^{2} {A_{i} \left( {x_{1} - x_{2} } \right)^{i} }$$
(11)

The estimated excess molar volume values of TiAP/n-dodecane solutions were plotted against mole fractions in Fig. 3a and b for different temperature levels. Parameters of Eq. 11 are listed in Table 5 for each temperature level. If desired, density of mixed solutions may be estimated with the linear density addition of constituent liquids with the addition of contribution from the excess molar volume.

$$\rho_{m} = \sum\limits_{i = 1}^{2} {x_{i} \rho_{i} } + \frac{T}{{x_{1} x_{2} \sum\limits_{i = 0}^{n} {A_{i} \left( {x_{1} - x_{2} } \right)^{i} } }}$$
(12)
Fig. 3
figure 3

(a) Variation of excess molar volume (VE) in TiAP-n-dodecane binary solutions. Points depict experimental values multiplied by a factor (listed in legends) to avoid clutter. Lines indicate a 2nd order Redlich–Kister representation multiplied by the same factor. (b) Variation of excess molar volume (VE) in TiAP-n-dodecane binary solutions. Points depict experimental values multiplied by a factor (listed in legends) to avoid clutter. Lines indicate a 2nd order Redlich–Kister representation multiplied by the same factor

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Table 5 Redlich–Kister coefficients for excess molar volume of TiAP in the binary solutions of TiAP and n-dodecane at 283.15–363.15 K and 0.1 MPa
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Jouyban and Acree’s equation [32] listed in Table 3 has a similar but not identical form.

The experimental excess molar volume of TiAP values, generated in this work at 298.15 K, were compared with the experimental values reported by Singh [16] as well as values reported by Das and Ali [36], obtained by molecular simulation, as shown in Fig. 4. It was observed that values reported in this work are the lowest as compared to the values reported by Singh [15] as well as Das and Ali [36]. Das and Ali [36] reported simulated density of TiAP at 298 K as 950 kg·m−3 against experimentally observed value of ~ 948 kg·m−3 as listed in Table 2. MD/MS simulated [37] and experimental [17] excess molar volumes of TBP/n-dodecane system were also compared. The values of excess molar volumes reported by Singh [16] were about six times higher than that observed in the experiments in this work. Reason for the disparity could not be ascertained.

Fig. 4
figure 4

Comparison of excess molar volume (VE) in TiAP-n-dodecane binary solutions at 298.15 K and 0.1 MPa. Values for TBP are plotted for the sake of comparison

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4.2 The Correlation for Density of Binary Solutions of TiAP-n-Dodecane in the Range of 283.15–363.15 K

4.2.1 Molar Concentration Basis

In the flowsheet calculations, usually molar concentrations are employed. The experimental density data listed in Table 3 and converted to molar scale for concentration could be represented by Eq. 3 with a R.2 value of 0.99. ρo, the density of n-dodecane at temperature T K, was estimated from Eq. 13

$$\begin{aligned} \left( {\frac{{\rho_{o} \left| {_{T} } \right.}}{{\rho_{o} \left| {_{298.15} } \right.}}} \right) &= 1 - \left( {{ - 2}{{.9070 \times 10}}^{{ - 1}}\pm {7}{{.4769 \times 10}}^{{ - 5}} } \right)\left( {\frac{T}{298.15} - 1} \right) \\ & \quad - \left({{9}{{.6380 \times 10}}^{{ - 3 }} \pm { 1}{{.1284 \times 10}}^{{ - 3}} } \right)\left( {\frac{T}{298.15} - 1} \right)^{2} \end{aligned}$$
(13)

Average absolute percentage deviation (AAPD) was 5.3 × 10−4 % for Eq. 13. ρo|298.15 was taken as 745.30 kg·m−3, being the experimental density of n-dodecane at 298.15 K, as listed in Table 2. A plot of residuals between predicted values from Eq. 3 and the experimental data is shown in Fig. 5. An excellent agreement was observed. The parameters of the Eq. 3 are listed in Table 6. AAPD and standard deviations for Eq. 3 were observed as 1.20 × 10−1 % and 8.95 × 10−2 % respectively. Figure 6 shows a histogram plot mapped to normal probability distribution.

Fig. 5
figure 5

A plot of residuals (molar basis) for varying concentrations of TiAP. Legends indicate molar concentrations

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Table 6 Estimated coefficients of the density equations reported in this work
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Fig. 6
figure 6

A histogram plot of deviations (molar basis)

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4.2.2 Mole Fraction Basis

The density data of the binary solution, listed in Table 3, could be represented by Eq. 4 involving mole-fraction with a R2 value of 0.99. The parameters of the Eq. 4 are listed in Table 6. The AAPD and standard deviations were 5.88 × 10−2 % and 5.59 × 10−2 %, respectively. A parity plot of predicted values from Eq. 4 and the experimental data is shown in Fig. 7. An excellent agreement was observed. Figure 8 shows a histogram plot mapped to normal probability distribution.

Fig. 7
figure 7

A plot of residuals for varying concentrations of TiAP. Legends indicate mole fraction

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Fig. 8
figure 8

A histogram plot of deviations (mole fraction basis)

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It may be noted that for similar but not identical data in a limited range of 288.15 K to 338.15 K, Singh [16] claimed AAPD of 1.4341 % for their model II, 2.4749 % for their model I, 2.3052 % for Redlich–Kister model, and 6.8182 % for Jouyban–Acree model. The values of AAPD observed for Eq. 4 are much smaller to those reported by Singh [16] for all the listed models.

4.3 Kirkwood–Buff Integrals Based Estimation of Partial Molar Volume

For understanding of the phenomenon involved in a real-life mixing of two types of molecules with different sizes and structures, Flory and Huggins [38, 39] provided a statistical thermodynamics-based insight. Gibbs energy change for mixing could be defined in their terms as

$$\Delta G_{mix} = RT\left( {\sum\limits_{i = 1}^{2} {n_{i} \ln \varphi_{i} } + n_{1} \varphi_{2} \chi_{12} } \right)$$
(14)

where χ12 is the interaction/mixing parameter for the free energy. Towing a slightly different line, Kirkwood and Buff [40] provided a statistical mechanical theory of solutions without assumption of pair-wise additivity of the total interaction energy. The Kirkwood–Buff theory was valid for any shape of the molecule. Kirkwood and Buff [40] demonstrated that the derivatives of the chemical potentials and osmotic pressures with respect to concentrations, the compressibilities and the partial molar volume of solute as well as solvent could be represented in the form of integrals of radial distribution functions of molecular pairs existed in the solution. Zolkiewski [41] noted Kirkwood–Buff relations as generalizations of the compressibility equation-

$$\int_{0}^{\infty } {\left[ {g_{ij} (r) - 1} \right]4\pi r^{2} dr = RT\kappa_{T} - \frac{1}{\rho }} ,$$
(15)

where gij(r), known as pair correlation function for the pair (i,j) [42, 43], is the radial distribution function involving species i and j. The Kirkwood–Buff integrals represent a measure of affinity of the species i to the species j as-

$$G_{ij} = \int_{0}^{\infty } {\left[ {g_{ij} (r) - 1} \right]4\pi r^{2} dr}$$
(16)

Inversion of Kirkwood–Buff integrals has been discussed well in the literature [44,45,46]. Based upon inversion, for a binary solution, the three integrals were obtained [47] as-

$$\begin{aligned} G_{ii} &= k_{B} T\kappa_{T} - \frac{1}{{\rho_{i} }} + \frac{{\rho_{j} \overline{{V_{j}^{2} }} (\rho_{i} + \rho_{j} )}}{{\rho_{i} D}} \\ G_{jj} &= k_{B} T\kappa_{T} - \frac{1}{{\rho_{j} }} + \frac{{\rho_{i} \overline{{V_{i}^{2} }} (\rho_{i} + \rho_{j} )}}{{\rho_{j} D}} \\ G_{ij} &= k_{B} T\kappa_{T} - \frac{{\overline{{V_{i} }} \overline{{V_{j} }} (\rho_{i} + \rho_{j} )}}{D} \end{aligned}$$
(17)

where the parameter D is given by the expression

$$D = \frac{{x_{i} }}{{k_{B} T}}\left( {\frac{{\partial \mu_{i} }}{{\partial x_{i} }}} \right)_{T,P} = 1 + \frac{{x_{1} x_{2} }}{RT}\left( {\frac{{\partial^{2} G^{E} }}{{\partial x_{i}^{2} }}} \right)_{T,P}$$
(18)

From Eq. 18, it is apparent that at terminal concentrations (xi = 1 or xj = 1), the parameter D will be equal to unity. In this case, limiting Gii will be given as

$$\mathop {\lim }\limits_{{x_{i} \to 1}} G_{ii} = RT\kappa_{T} - V_{i}^{o}$$
(19)

This condition has to be satisfied at the terminal concentrations (xi = 1 or xj = 1). At temperatures different from than 298.15 K, κT can be estimated as

$$\begin{aligned} \alpha_{p}& = \alpha_{298.15K} + \alpha_{298.15K}^{2} \left( {7 + 4\alpha_{298.15K} T} \right)\left( {\frac{\Delta T}{3}} \right) \\ \gamma & = \gamma_{298.15K} - \gamma_{298.15K} \left( {1 + 2\alpha_{298.15K} T} \right)\left( {\frac{\Delta T}{T}} \right) \\ \kappa_{T} &= \frac{{\alpha_{p} }}{\gamma },\quad \Delta T = T - 298.15\end{aligned}$$
(20)

Individual κT values were taken from supplementary information of Singh et al. [13]. For binary mixture, κT was estimated as

$$\left( {\kappa_{T} } \right)_{mix} = \sum\limits_{j = 1}^{2} {x_{i} } \left( {\kappa_{T} } \right)_{i}$$
(21)

Estimated KBIs, for TiAP/n-dodecane binary solution at 298.15 K, are depicted in Fig. 9. For a binary mixture, Simon et al. [48] proposed the following set of parametric equations-

$$\eta = \rho_{1} + \rho_{2} + \rho_{1} \rho_{2} \left( {G_{11} + G_{22} - 2G_{12} } \right)$$
(22)
Fig. 9
figure 9

Kirkwood–Buff integrals at 298.15 K for TiAP/n-dodecane binary mixture

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Now partial molal volumes of components 1 and 2 can be estimated as

$$\begin{aligned} V_{1} & = \frac{{\left( {1 + \rho_{2} \left( {G_{22} - G_{12} } \right)} \right)}}{\eta } \\ V_{2} & = \frac{{\left( {1 + \rho_{1} \left( {G_{11} - G_{12} } \right)} \right)}}{\eta } \\ \end{aligned}$$
(23)

These estimated volumes have been depicted and compared with experimental values in Figs. 10 and 11. A good agreement was observed.

Fig. 10
figure 10

Comparison of experimental partial molar volumes (TiAP) with those estimated from Kirkwood–Buff integrals at 298.15 K for TiAP/n-dodecane binary mixture

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Fig. 11
figure 11

Comparison of experimental partial molar volumes (n-Dodecane) with those estimated from Kirkwood–Buff integrals at 298.15 K for TiAP/n-Dodecane binary mixture

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The theoretical concept of local composition of solvent molecules around a solute molecule in form of preferential solvation deals with the solute–solvent interactions, solubility and structural stability of the solute. The molecular level preferential solute–solvent interactions can be modeled through the Kirkwood–Buff integrals linked with the experimental data. The same was discussed in detail by several researchers [49,50,51,52]. Linear coefficients of preferential solvation \(\delta_{ij(i \ne j)}^{o}\) could be written as

$$\begin{aligned} \delta_{11}^{o} & = x_{1} x_{2} \left( {G_{11} - G_{12} } \right) \\ \delta_{12}^{o} & = x_{1} x_{2} \left( {G_{12} - G_{22} } \right) \\ \delta_{21}^{o} & = x_{1} x_{2} \left( {G_{12} - G_{11} } \right) \\ \delta_{22}^{o} & = x_{1} x_{2} \left( {G_{22} - G_{12} } \right) \\ \end{aligned}$$
(24)

The variation of estimated preferential solvation terms \(\delta_{12}^{o}\) and \(\delta_{21}^{o}\) with respective mole fractions is depicted in Fig. 12. Now in terms of preferential solvation terms, a local composition of ith type molecule around jth type molecule may be defined as follows-

$$x_{ij} = x_{i} + \frac{{\delta_{ij}^{o} }}{{26V_{o} }}$$
(25)
Fig. 12
figure 12

Linear coefficients of preferential solvation \(\delta_{ij(i \ne j)}^{o}\) at 298.15 K for TiAP/n-dodecane binary mixture

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Vo is the molar volume of the component being solvated. Thus the variation of local mole fractions x12 and x21 with the mole fraction of TiAP (x1) is shown in Table 7. It is interesting to note that except at terminal concentrations (xi = 1), the sum of local mol-fractions is not equal to unity. In the region xi ≠ 1, the local mole fractions were slightly less than the actual mole fractions for both the components—TiAP as well as n-dodecane. This deficiency indicates the nonideality prevailing in the concerned binary system.

Table 7 Comparison of actual and local mole fraction compositions for TiAP/n-dodecane binary solutions at 298.15 K
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For molar volume of the solution, a response surface model could be written in terms of linear, bilinear, and quadratic variables as-

$$V = a_{0} + \sum\limits_{j = 1} {a_{j} x_{j}} + \sum\limits_{j = 1} {a_{jj} x_{j}^{2} } + \sum\limits_{j \ne k} {a_{jk} x_{j} x_{k}}$$
(26)

It should be noted that the response surface space is severely constrained as for binary system,

$$x_{1} + x_{2} = 1$$
(27)

However, in the present case, the molar volume of the solution could be represented by following simplification of Eq. 26 with a R2 value of 0.99 as

$$V = \left( {{228}{\text{.61 }} \pm { 0}{\text{.11}}} \right) + \left( {98.89 \pm 0.72} \right)x_{1} - \left( {2.57 \mp 0.73} \right)x_{1}^{2}$$
(28)

Predicted values from Eq. 28 were compared with the experimental values in Fig. 13. A good agreement was observed.

Fig. 13
figure 13

Comparison of experimental and predicted molar volume of TiAP/n-Dodecane binary mixture at 298.15 K

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Rahbari et al. [53] presented a novel method based on Monte-Carlo (MC) simulations for computation of partial molal properties. For selected case studies, they reported excellent agreement between their results and results from PC-SAFT equation of state (EOS). However, the results were different from those from Peng-Robinson EOS. For denser binary liquid systems, these MC simulations have to be explored extensively.

5 Conclusions

Based on the experimental data on the density of binary solutions of TiAP/n-dodecane at 283.15–363.15 K temperature range, density equations on molar as well as mole fraction basis were was reported. Analysis of data yielded derivative properties like excess molar volume, molar volume, and linear coefficients of preferential solvation as well as local mole fractions. Kirkwood–Buff integrals were also estimated. Based on KBI, partial molar volumes of TiAP and n-dodecane were estimated at 298.15 K and a geed agreement was observed between the experimental values and values estimated from KBI.