Analysis of the aqueous solubility of trialkyl phosphates, dialkyl alkylphosphonates, dialkylphosphites and alkyl dialkylphosphinates

Linear free energy relationship (LFER) analysis based on the molar intrinsic volume (Vx), McGowan hydrophile‒lipophile balance (HLB) and cumulative Kabachnik constants (Σσϕ) of considered solutes has been performed for the solubility of trialkyl phosphates, dialkyl alkylphosphonates, dialkyl phosphites and alkyl dialkylphosphonates in pure water (called also as their aqueous solubility). The descriptors mentioned above are known for all considered solutes and are strongly correlated with their aqueous solubility. From the correlation found for trialkyl phosphates, dialkyl alkylphosphonates and dialkyl phosphites it follows that the mean contribution of each methylene group in their structure (Δlog Sw) is a constant for these homologous series and equal to − 0.41. It means that within of homologous series the aqueous solubility of succeeding solutes decreases regularly. The established increment has been further used as a tool for the preliminary selection of solutes whose aqueous solubility at 25 °C is in agreement with this condition. The adequate model (further denoted as AR model) based on the accessible data of temperature effects upon the aqueous solubility of tributyl phosphate, dibutyl butylphosphonate and butyl dibutylphosphinate has been formulated and then tested with positive results for the other solutes whose solubility is known at a selected constant temperature, e.g. at 25 °C. It should noticed, however, that in this case there are some serious restrictions. First, the interpretation and formulation of conclusions for the solutes which aqueous solubility is known at one selected temperature (e.g. 25 °C) or if it has been determined at a few temperatures only, should be made with care. Second, the obtained results with AR model cannot be used for prediction of temperature effects upon the aqueous solubility of such solutes. Simultaneously, the results obtained with AR model have been compared with those following from the modified Abraham model. Such comparison has been possible due to a significant progress in the different application of Abraham model for several organophosphorus derivatives as monobasic dialkyl- and diarylphosphoric acids, dialkyl- and diphenylphosphinic acids, trialkyl- and triaryl phosphates, dimethyl methylphosphonate, dialkylphosphites and some of trialkyl- and triarylphosphine oxides, respectively.


Introduction
Alkyl and aryl derivatives of organophosphorus based acids play a crucial role in the different areas of human activity. For instance, they are widely applied as extractants or synergists in metal extraction systems, high temperature lubricants and flame retardands [1][2][3][4]. The physicochemical properties of these compounds affect their importance for practical purposes and determine their toxicity towards the natural environment and biosphere. From the last point of view, the extremely toxic are isopropyl methylphosphonofluoridate (sarin) and pinacolyl methylphosphonofluoridate (soman) which are considered as a chemical weapons of mass destruction [5]. The global application of organophosphorus compounds is very large and has an evident effect on the pollution of atmosphere, aquatic systems (mainly of the surface waters) and finally of soils. The aqueous solubility of organophosphorus compounds makes many difficulties in the purification of polluted water prior to its utilization for bath, washing, laundering, cooking, as well as drinking water for the human population, cattle and animals. For example, the problems existing in this field in Sweden have been adequately presented and discussed by Marklund and coworkers [6,7].
Physicochemical properties of trialkyl phosphates, dialkyl phosphites (dialkyl hydrogen-phosphonates), dialkyl alkylphosphonates, alkyl dialkylphosphinates, and trialkyl-as well as of triarylphosphine oxides could be correlated with their molar intrinsic volumes (V x ) of McGowan, hydrophile-lipophile balances (HLB) in the McGowan scale and cumulative Kabachnik constants (Σσ ϕ ) of alkyl and alkoxy substituents [8][9][10][11]. Nevertheless, both two first descriptors are not sufficient to describe quantitatively the differences between the isomers [8,9]. However, the cumulative Kabachnik constant Σσ ϕ , which is equal to the sum of individual constants σ ϕ of substituents on the atom of phosphorus, is specific for a given organophosphorus compound and differentiates it from its isomers [10,11]. The descriptors mentioned above were used with some success in the previous papers of Apostoluk and coworkers [12,13]. The present paper relates to the accessible data of aqueous solubility of trialkyl phosphates, dialkyl alkylphosphonates, dialkyl phosphites and alkyl dialkylphosphinates. This property of considered solutes has been analyzed in terms of the linear free energy relationships (LFER) method involving three descriptors, the same as used previously [13]. Their physical and chemical meaning has been indicated and discussed in Sect. 3. The same has been done for five solute descriptors involved in the model of Abraham [14,15].

Aqueous solubility of trialkyl phosphates, dialkylphosphites, dialkyl alkylphosphonates and alkyl dialkylphosphinates at 25 °C
The aqueous solubilities of considered compounds, treated as their concentrations in the saturated aqueous solutions, are presented in Table 1. From the most comprehensive experimental study performed at 1958 by Burger and Wagner it follows that at 25 °C the solubility of considered compounds in water increases in the following order: trialkyl phosphates < dialkylphosphites and dialkyl alkylphosphonates < alkyl dialkylphosphinates < trialkyl phosphine oxides, which indicates the increasing polarity of P-O bonds in each mentioned homologous series [16]. It should be noted, however, that these authors studied only the solutes of low or a moderate aqueous solubility and concluded that the lowest members of each homologous series containing only methyl or ethyl groups attached to phosphorus atom through the C-P or C-O-P bonds are miscible with water [16]. However, this opinion is fallacious since the hydrophilicity of the corresponding lower members of trialkyl phosphates, dialkyl phosphites and dialkyl alkylphosphonates, expressed here in the scale of McGowan (HLB > 7), proves that they are mutually miscible with water and consequently, their aqueous solubility at 25 °C should be treated as a very high or as relatively high [1,4]. On the other hand, the reported values of aqueous solubility of trimethyl phosphate (HLB = 9.73) may differ significantly. Indeed, in this case for instance the following values, i.e. 1000, 500 and 300 g/dm 3 , respectively, have been reported [1,4,17]. The highest aqueous solubility of trimethyl phosphate, equal to 1000 g/dm 3 , seems to be reasonable since the similar solubility of dimethylphosphite (HLB = 8.90) has been reported [4] and it could be also accepted. It is also surprising that Nikolotova and Kartashova [1] as well as Yalkowski et al. [17] reported in their monographs the same aqueous solubility of trimethyl and triethyl phosphates, both equal to 1000 or 500 g/ dm 3 , respectively, even this quantity should decrease with the increasing molecular weight or with the decreasing hydrophilicity of solutes of the same homologous series. Therefore, one should find the appropriate tool necessary for the proper selection of available data and then for the formulation of the best correlation describing quantitatively the dependence of solubility of all considered compounds in water on their characteristic descriptors in AR [8,[11][12][13] and Abraham [14,15] models, respectively.

Temperature effect on the aqueous solubility trialkyl phosphates, dialkyl alkylphosphonates, and alkyl dialkylphosphinates
Burger and Wagner [16] indicated that the solubility of some trialkyl phosphates passes through the minimum at about of 70 °C. For instance, the solubility of diethyl i-butyl phosphate [16] at 75 °C is about 15% lower than that at 25 °C. However, the detailed studies were performed only in the case of tributyl phosphate whose solubility decreases with temperature, reaches the minimum at about 65 °C and then rises with temperature [16,18]. Burger in the monograph [19] compared and critically evaluated the latter results obtained by different authors,  [16] however, the study carried out by Nikolayev et al. [20] had been omitted. These authors found that their results are in a good agreement with those reported for tributyl phosphate by Higgins et al. [18] and demonstrated that the aqueous solubility of dibutyl butylphosphonate and butyl dibutylphosphinate changes with temperature in a similar manner as that of tributyl phosphate. A main drawback of the work [20] lies in that the experimental results presented graphically were expressed in the scale of weight percent solubility of considered solutes in water while the densities of corresponding aqueous solutions were not indicated. In this case, however, there are two possible ways of calculation of the molar concentrations of solutes. First, the weight percent of a solute could be converted to its molar concentration assuming that the density of an aqueous solution is equal to the density of pure water at temperature expressed in °C. The effect of temperature within the range from 0 to 100 °C upon the density of water has been found from the experimental data reported in the tables of its physical properties [25]. Similarly as in the work of Jones and Harris [26] the polynomial quartic equation in temperature has been applied: Second, the weight percent of a solute can be easily recalculated to its molality, i.e. concentration in mole per one kilogram of pure water. In a sufficiently diluted solution the molality of a solute is practically equal to its molar concentration. The results obtained in both ways are identical for tributyl phosphate, and practically the same for dibutyl butylphosphonate. They are slightly different for butyl dibutylphosphinate, however, the differences are small and approximately equal to ± 0.01 of logarithmic unit. The results obtained in both ways of calculation are indicated in Table 1 by an asterisk.
Note that the aqueous solubility of the other considered solutes were usually determined in the molar scale at 25 °C and sporadically at a slightly higher temperatures [17,22]. The set of analyzed data is presented in Table 1 where the data written in italics have been omitted in further calculations.

AR model
At a constant temperature the set of solubility in water (S w ) of organophosphorus compounds collected in Table 1 has been analyzed according to Eq. (2): where S w stand for the molar concentration of solutes in the saturated aqueous solutions and V x , HLB and Σσ ϕ are their characteristic descriptors collected in Table 2. The values of V x , and HLB have been calculated from the composition and structure of solutes, i.e. according to the formulae: where n i and V x,i stand for number of ith atoms and their molar volume, n B denotes the total number of bonds, irrespective of their nature (single, double or triple) [8].
Since the studied esters belong to the different homologous series, therefore, their hydrophile-lipophile balances in the scale of McGowan have been calculated from the known values of V x and the number of oxygen atoms (n O ) attached to the phosphorus atom in their structure and acting as the hydrogen bond acceptors from water molecules [9]. The cumulative Kabachnik constants, Σσ ϕ , have been calculated from the individual σ ϕ constants of alkyl and alkyloxy substituents attached to the phosphorus atom. The individual σ ϕ constants of several alkyl and alkyloxy substituents can be found elsewhere [10,11]. They can be also calculated from their structure according to our approach [13]: where N C stands for a number of carbon atoms in the chain of substituent; Z is equal to 0 and 1 for alkyl and alkyl ester groups, respectively; n is equal to 1, 2 or 3 for primary, secondary and tertiary carbon atom in C-P and/or C-O-P bonds, respectively; m is a total number of branches of the carbon chain. The found correlation of excellent quality (R 2 = 0.9983, SD = 0.018, F = 3326, P = 0.0000) reproduces nicely the values of individual σ ϕ constants of 30 substituents and indicates that σ ϕ values of ester groups are always by (0.84 ± 0.01) unit higher than those of corresponding alkyl substituents with the same length and structure of carbon chain [13]. The contribution of a molar volume V x of solutes is always negative since it reflects an endoergic effect of cavity formation in the structure of solvent, in this case of water. Within each homologous series V x regularly increases while HLB decreases, which means that the aqueous solubility of solutes regularly decreases due to the increase of their hydrophobicity. However, the values of V x and HLB, being strongly correlated from definition (4), are the same for the isomers of a corresponding solutes, e.g. for tributyl, tri-i-butyl and tri-s-butyl phosphates with Σσ ϕ values equal to − 1.11, − 1.38 and − 1.32, respectively. Thus, the different values of Σσ ϕ distinguish them as independent solutes and indicate also the effect of branching of carbon chain on the formation of hydrogen bonds with water molecules. The deeper insight into a nature of this descriptor gives its splitting on the appropriate parts in accordance with the correlation (6) found for twenty five neutral esters with the straight and branched carbon chains of alkyl-or alkyl ester substituents, respectively, and four dialkylphosphites which are the potential hydrogen bond donors: where n HBA stands for the number of oxygen atoms in the structure of all esters and n HBD denotes the number of H-P bonds and is equal to 0 for all neutral alkyl esters and to 1 for dialkylphosphites, respectively. Both positive effects of the hydrogen bond acceptors and donors, respectively, compensate the negative effect of V x on hydrophilicity of homologous series of esters, but do not distinguish the differences between their isomers. The isomers of neutral esters may deviate from correlation (6) more than ± 3 SD, for instance tri-i-propyl phosphate with the straight carbon chain of substituents and involving secondary C-O-P bonds is only one outlier with the deviation equal to − 3.44 SD. The observed deviations of isomers from Eq. (6) seem to be clear in terms of the approach (5) and our calculation performed previously [13]. Both positive effects of the hydrogen bond accepting and donating abilities of all esters are reasonable and justified in terms of their physicochemical properties and from the statistical criteria.
The effect of temperature on the aqueous solubility of an organophosphorus based ester can been studied in accordance with the model: where t stands for temperature expressed in the scale of Celsius. The temperature at which the particular solute attains minimum of its solubility in pure water has been log S w = f t, t 2 calculated from the derivative d log S w /dt of an equation based on the model (7). If the different solutes are compared it is obvious that their descriptors, V x , HLB and Σσ ϕ , affect the temperature of their minimal solubility in water. Consequently, the derivative (d log S w /dt) should also depend upon the descriptors of solutes: Combination of the simple models (2), (7) and relation (8) leads to the following form of AR model: which previously has never been used.

Abraham model
The model of Abraham [14,15] for such phenomenon as the aqueous solubility (S w ) of different solutes at a constant temperature is as follows: where S w is the molar aqueous solubility of a solute. The symbols c, e, s, a, b and v stand for the regression coefficients, whereas E, S, A, B and V x are the solute descriptors. E is the solute excess molar refractivity, S is the solute dipolarity/polarizability, A and B denote its hydrogen bond acidity and basicity, respectively, and V x is identical with the McGowan molar intrinsic volume of solute used in AR model.
If the solute is liquid, E can be easily calculated [27] in units (cm 3 mol −1 )/10 from its refractive index ( n 20 D ) determined at 20 °C. V x in Eq. (10) is calculated from the composition and structure of a solute with the known McGowan and Abraham [8,14] algorithms and then is expressed in units (cm 3 mol −1 ])/100. The unknown values of S, A and B of solute descriptors could be estimated from their properties and behavior in several processes and phenomena, e.g. partition in the two phase liquid systems, partition between gas phase and liquids and solubility in organic solvents [15]. Except of dialkyl phosphites, which are a relatively week hydrogen bond donors with A equal to 0.10, the neutral alkyl esters of phosphoric, alkylphosphonic and dialkylphosphinic acids, respectively, and trialkyl phosphines are only the hydrogen bond acceptors [15] with A equal to 0. The reported values of Abraham solutes descriptors [15] for five trialkyl phosphates, five dialkyl phosphites, dimethyl methylphosphonate and three trialkylphosphine oxides, respectively are collected in Table 3. Note that for the other considered solutes the list of descriptors is incomplete. As has been stated above, the calculation of V x is very simple, but the determination of the excess molar refractivity (E) of solutes is not so trivial [27]. The refractive indices, n 20 D , necessary for its calculation are not always known [1,15,29,30]. The calculated values of E descriptor are also indicated in Table 3. The estimation of dipolarity/polarizability (S) and hydrogen bond basicity (B) of solutes is laborious, time consuming and requires the advanced techniques of calculations [15]. In such case for instance, the approximate hydrogen bond basicity of solutes (except of dimethyl methylphosphonate and tribubutylphosphine oxide which have been excluded) can be evaluated from the simple correlation: valid for trimethyl-, triethyl-, tripropyl-, tributyl-and triamyl phosphates, dimethyl-, diethyl-, dibutyl-, dihexyland dioctylphosphites, as well for trioctyl-and tri(2-ethylhexyl)phosphine oxides, respectively. Hence, it is clear that hydrogen atom attached directly to phosphorus in dialkylphosphites decreases their hydrogen bond basicity. Equation (11) is also in a good accordance with the  Table 3 the values of B estimated from Eq. (11) are written in italics. Unfortunately, there is no of any reasonable rule for the rough estimation of dipolarity/polarizability of solutes, i.e. their S descriptors.

Direct comparison of both models
Having in mind that both models differ in the number of solute descriptors we have been selected the set of four trialkyl phosphates and three dialkyl phosphites for which the aqueous solubility at room temperature and appropriate solute descriptors are known. The direct comparison of models should indicate these descriptors of Abraham model which affect the aqueous solubility of solutes in question. The results of preliminary calculations show that both classes of solutes in Abraham model are correlated  Table 3). Instead of Eq. (13) one can consider the correlation (14): with one deviation equal to 3.76 SD for dibutyl butylphosphinate. Consequently, the aqueous solubility of trialkyl phosphates, alkyl dialkylphosphonates and alkyl dialkylphosphinates, which are not the hydrogen bond donors, depends in the Abraham model on the molar intrinsic volume, excess molar refractivity or basicity of solutes. Therefore, the effect of temperature on the aqueous solubility of tributyl phosphate, butyl dibutylphosphonate and butyl dibutylphosphinate should be considered according to the modified form of Abraham model: which is evidently simpler than that presented by Eq. (9).
The statistical assessment of all derived correlations has been characterized by means of their determination coefficient (R 2 ), standard deviation (SD) of explained variable (log S w ), standard deviations of the regression coefficients of explanatory variables in the model used, the number of experimental points (N) and the level (P) of their statistical significance.
The predictive power of Eqs. (23) and (24) resulting from both models have been characterized with their internal cross validation coefficient (Q 2 ) calculated by means of the leave-one-out (LOO) procedure [31,32] and corresponding sets of data [18,20]. It should be also added that the number of diverse solutes and limited sets of experimental data [18,20] are rather not enough for validation of derived correlations through the many-leave-out (MLO) method or through the comparison of training and test sets of data, respectively (Fig. 1).

Effect of methylene groups on the aqueous solubility of trialkyl phosphates at 25 °C
The set of aqueous solubility of diethyl butyl, diethyl i-butyl, diethyl amyl, dibutyl methyl, dibutyl ethyl, tributyl and tri(2-chloroethyl) phosphates, determined at 25 °C by Burger and Wagner [16], has been completed with those of trimethyl phosphate [1] and tripropyl phosphate [7] and then used for the calculation of logarithmic increment of methylene groups involved in the structure of mentioned phosphates. Application of the model (2) leads to the correlation of excellent quality presented in Fig. 2, where the absolute values of deviations are not higher than 2SD: (16) log S w = (3.652 ± 0.117)−(2.969 ± 0.076) V x 100 −(0.236 ± 0.069) Correlation (16) can be used for prediction of some unknown or verification of uncertain and/or misleading aqueous solubility of trialkyl phosphates. As an example the tri(i-butyl) phosphate can be used. Its aqueous solubility at 25 °C reported in the review [4] is equal to 0.264 g/ dm 3 . However, the aqueous solubility of tri(i-butyl) phosphate at 25 °C calculated from Eq. (15) is equal to 0.583 g/ dm 3 and does not differ significantly from the value of 0.617 g/dm 3 determined experimentally [22] at (26 ± 1) °C. Logarithmic increment of the aqueous solubility of trialkyl phosphates is calculated per one methylene group in their structure. The molar intrinsic volume V x of methylene group [7] is equal to 14.09 cm 3 and according to Eq. (16) its contribution to the aqueous solubility of trialkyl phosphates is a constant: In Eq. (16) the negative contribution of McGowan molar volume of a particular trialkyl phosphate predominates over the positive contribution of its cumulative Kabachnik constant. Therefore, the calculated increment (17) could be used for the rough estimation of unknown values of the aqueous solubility of trialkyl phosphates. For instance, the estimated solubility of triamyl phosphate is equal to 0.0205 g/dm 3 and comparable with the value of 0.019 g/ dm 3 determined experimentally [22] at (26 ± 1) °C. From the reasons indicated in Sect. 2.1, all reported values of the aqueous solubility of triethyl phosphate [1,7] are erroneous. The next correlation (18) has been obtained involving the aqueous solubility of tri(i-butyl) and triamyl and tri(2methylbutyl) phosphates determined experimentally [22] at (26 ± 1) °C: As has been expected the absolute deviations from Eq. (18) are lower than 2 SD. In this case, however, the value of methylene group increment, Δlog S w , is a slightly lower and equal to − 0.44. As a result, Eq. (18) relates to the aqueous solubility of trialkyl phosphates at room temperature.

Aqueous solubility of trialkyl phosphates, dialkyl alkylphosponates, dialkylphosphites and alkyl dialkylphosphinates at 25 °C
The set of aqueous solubility of trialkyl phosphates used for the formulation of correlation (16), has been completed with the solubility of dibutyl ethyl-, dibutyl butyl-, dibutyl hexyl-and di-i-amyl methylphosphonates, dimethyl, diethyl, and dibutyl phosphites and butyl dibutylphosphinate, respectively. Further calculations have been performed in order to explain the behavior of three dialkyl phosphites, which can act simultaneously as a hydrogen bond acceptors and donors, respectively, in comparison with the neutral trialkyl phosphates, dialkyl alkylphosphonates and alkyl dialkylphosphinates. From this reason, apart from V x and Σσ ϕ descriptors, two additional explanatory variables have been introduced, namely n HBA and n HBD , the same as those in Eq. (6). Application of the step wise selection of explanatory variables leads to the very interesting result: with two absolute deviations of 3.33 SD for di-i-amyl methylphosphonate and equal exactly to 3 SD for butyl dibutyl phosphinate. Correlation (19) neglects entirely the hydrogen bond acceptance of all considered esters and indicates the negative contribution of hydrogen bond donating ability of dialkyl phosphites. These facts are clear from the following reasons: (1) in the homologous series the molar volume of analogous solutes decreases regularly by a constant value (6.87 cm 3 mol −1 ) per one -CH 2 O-P bond in their structure; (2) the value of σ ϕ of hydrogen atom bonded directly to phosphorus is equal from definition to 0, thereby the aqueous solubility of dialkyl phosphites are always greater than those of corresponding trialkyl phosphates and dialkyl alkylphosphonates. As a result, the mean increment (Δ log S w ) calculated from Eq. (19) per one methylene group is equal to − 0.41, irrespective of the homologous series of considered esters. It is important and useful conclusion. For instance the aqueous solubility at 25 °C of dimethyl methylphosphonate should be higher than the reported value [4] of 322 g/dm 3 (log S w = 0.41). The solubility in water of diamyl hexylphosphonate, equal to 0.090 g/dm 3 at non specified temperature (log S w = − 3.53), is probably overestimated [28] assuming that it has been determined at 25 °C, similarly as that for dibutyl hexylphosphonate [23].

Effect of temperature on the aqueous solubility of tributyl phosphate, dibutyl butylphosphonate and butyl dibutylphosphinate and its analysis in terms of AR and Abraham models
The effect of temperature on the solubility of tributyl phosphate in water was studied in a sufficiently large range of temperature [18,20]. Both sets of data are in very good agreement which is demonstrated in Fig. 3. Except of the points of lowered [18] solubility at 25 °C and elevated [18] at 50 °C, the parabolic shape of this dependence is confirmed by the following correlation, with only one absolute deviation exceeding 2 SD. From the derivative of Eq. (20) versus temperature it follows that the minimum aqueous solubility of tributyl phosphate lies at about of 60 °C. Aqueous solubility of dibutyl butylphosphonate reported [16,23]. at 25 °C are lower than that predicted from Eq. (19). This conclusion is also confirmed by the analysis of Russian workers data [20] : (20) log S w = − (2.324 ± 0.008)−(0.0245 ± 0.0005)t + (0.000205 ± 0.000006)t 2 R 2 = 0.9961, SD = 0.015, log S w = −(1.887 ± 0.019)−(0.0224 ± 0.00089)t + (0.000150 ± 0.000073)t 2 R 2 = 0.9940, SD = 0.019, F = 497, N = 9, P = 0.0000, Fig. 3 Effect of temperature on the aqueous solubility of tributyl phosphate according to Eq. (20). The experimental data have been taken from the appropriate references: ○- [17]; •- [19]; ∆- [18]; ×- [15,17] with only one absolute deviation equal to 2.01 SD. According to Eq. (21), the minimum of dibutyl butylphosphonate solubility in pure water is achieved at about of 75 °C.
All reported values of butyl dibutylphosphinate solubility in pure water [16,20] fulfill the following dependence upon temperature: The quality of correlation (22)  The analysis of temperature upon the aqueous solubility of tributyl phosphate, dibutyl butylphosphonate and butyl dibutylphosphinate is possible with AR model if the next solute is involved. According to Burger and Wagner [16], the aqueous solubility of diethyl-i-butyl phosphate at 25 and 75 °C, respectively, has been used as a reference assuming a priori that its dependence on temperature passes through the minimum at about of 70 °C. Consequently, the correlation of high statistical importance has been obtained:  The obtained result is presented in Fig. 4. The minima of solubility of tributyl phosphate, dibutyl butylphosphonate and butyl dibutylphosphinate calculated from the appropriate derivatives dlog S w /dt of Eq. (24) are attained at 61.5, 72 and 73 °C, respectively, and are identical with those determined from Eq. (23). Note that these results are in fairly good accordance with those estimated above from Eqs. (20), (21) and (22), respectively. The statistical quality of correlation (24) is excellent since it predicts nicely the aqueous solubility of diethyl i-butyl phosphate [16] at 25 and 75 °C, respectively. However, we fill that our AR model predicting the temperature effects upon the aqueous solubility of four solutes mentioned above, it is also able to predict at room temperature the solubility in water of a solute with the known values of its characteristic descriptors. Hence, it is interesting to check the predictive power of further correlation based on AR model against the aqueous solubility of all solutes taken from Table 1, except of those indicated in italics. Tri(2-chloroethyl) phosphate and di(i-amyl) methylphosphonate have been excluded as the outliers from the following correlation: with only one absolute deviation greater than 2 SD, namely 2.51 SD for tri(2-methylbutyl)phosphate at 26 °C. It should pointed out that correlation (25) is fulfilled by eighteen solutes, involving ten trialkyl phosphates, three dialkyl alkylphosphonates, butyl dibutylphosphinate and three dialkyl phosphites, respectively. The obtained results are illustrated in Fig. 5.

Final remarks
The determined increment of the aqueous solubility per one methylene group added in the structure of trialkyl phosphates, dialkyl alkylphosphonates, dialkyl phosphites and alkyl dialkylphosphinates is a constant for all considered solutes and it can be used for the estimation (25) log S w = −(11.68 ± 0.33) + (0.488 ± 0.018)t + (0.1590 ± 0.0167) t 2 1000 + (1.186 ± 0.042)HLB −(2.53 ± 0.08) −(0.101 ± 0.003) t ⋅ V x 100 − (0.038 ± 0.002)t ⋅ HLB + (0.068 ± 0.003)t ⋅ R 2 = 0.9932, Q 2 = 0.9865, SD = 0.086, F = 932, N = 53, P = 0.0000, of their approximate solubility in water. Such a increments are also constant for Kovats retention indices (I) of analogous trialkyl phosphates and dialkyl alkylphosphonates [33] which, according to our knowledge, are strongly correlated with the aqueous solubility of mentioned solutes. The recent work of Maksimov and Kovalenko [34] and references therein do not affect on the results of our calculations of tributyl phosphate concentration in the aqueous solutions. Stenzel et al. [35] used the model of Abraham for determination of polyparameter linear free energy relationship for established and alternative flame retardants, but considered only three solutes, namely triethyl phosphate, tri(2-chloroethyl) phosphate and diethyl ethylphosphonate, which are interesting for us, but we have been unable to check the utility of their reported descriptors.

Conclusions
The accessible literature data on the solubility data of neutral esters of phosphoric, phosphonic and phosphinic acids are rather limited, mainly for dialkyl alkylphosphonates, dialkyl phosphites and particularly for alkyl dialkylphosphinates. The same relates to the solutes of high and low hydrophobicity in terms of their HLB in the McGowan scale. In general, the values of the aqueous solubility of trialkyl phosphates, dialkyl alkylphosphonates, dialkyl phosphites and alkyl dialkylphosphinates could be estimated applying the calculated mean contribution of each methylene group in their molecular structure or can be verified by means of the appropriate correlations established in the present work.
The AR model (9) has been proposed for interpretations of temperature effect upon the aqueous solubility of tributyl phosphate, butyl dibutylphosphonate and butyl dibutylphosphinate and it has been found that after addition of existing data for the solubility of diethyl i-butyl phosphate at 25 and 75 °C, respectively, the correlation (24) of excellent quality reproduces all considered experimental data. Based on the model (9) and correlation (24), the derived relationship (25) of a good statistical quality is fulfilled by eighteen solutes: (1) eleven trialkyl phosphates; (2) three dialkyl alkylphosphonates and three dialkylphosphites; (3) butyl dibutylphosphinate; (4) tri(2-chloroethyl) phosphate and di(i-amyl) methylphosphonate are the outliers. The appropriate modification of Abraham model (15) have been also proposed and it has found as a more convenient for the interpretations of temperature effect upon the aqueous solubility of tributyl phosphate, butyl dibutylphosphonate and butyl dibutylphosphinate as the analogous solutes. Both models predict the same temperatures of minimal solubility of solute in question. On the other hand, they can be also used for the prediction of aqueous solubility of a solute at temperature at which the experimental value is not known. However, there are some serious restrictions in the interpretation and formulation of conclusions for the solute whose aqueous solubility is known at one selected temperature (e.g. 25 °C) or if it has been determined at a few temperatures only. In such situation it is obvious that there are some serious restriction in application of the correlation (25) for the estimation of dependence log S w = f(t) for solutes for which the aqueous solubility is known at only one temperature. When the aqueous solubility of a solute is known at two different temperatures one can consider the following cases: (1) diethyl i-butyl phosphate for which the difference of temperature of experimental solubility is equal to 50 °C ; (2) tripropyl phosphate for which this difference is equal to 5 °C , respectively. As been mentioned in Sect. 4.3, the aqueous solubility diethyl i-butyl phosphate has been an exception, whereas the other solutes cannot be treated in such a way, i.e. the estimation of log S w = f(t) dependences are not permissible.

Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of interests.

Human and animal rights
The performed LFER studies of aqueous solubility of titled compound have not involved any experiments with of Human Participants and/or Animals.
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