Prediction of Solvatochromic Polarity Parameters for Aqueous Mixed-Solvent Systems

Abstract

Solvent polarity is important data being used in solvent selections for preliminary engineering design of chemical processes. In this work, a predictive model is proposed for estimating the solvatochromic polarity of electronic transition energy (E_T) of Reichardt indicator for aqueous mixtures. To validate the model, the E_T values of eighteen aqueous mixtures collected from the literature were used. The predictive model provided a good estimation of E_T values with an overall deviation of 2.1%, compared with an ideal model (5.1%) from the mole fraction average. The linear relationship of the contribution factor of hydrogen bond donor interactions (CF^HBD) in the predictive model with Kamlet–Taft acidity was newly proposed in order to extend the model for other aqueous mixtures. The predictive model is applicable to many aqueous mixtures and simply requires three properties of pure components as: (i) E_T values, (ii) gas-phase dipole moment and (iii) Kamlet–Taft acidity.

1. Introduction

The use of aqueous mixtures is preferable in many chemical processes (e.g., biomass conversion [1,2], separation and fractionation [3] and processing of active pharmaceutical ingredients [4,5,6]) because of the benefit of a safe solvent. Solvent polarity is an informative data being used in solvent selections [7,8,9,10] for preliminary engineering design and understanding the solvent effects on the chemical processes [11]. Polarity [12] is generally referred to as a solvent’s capability for solute dissolution and can be quantified with many physical properties of solvents (e.g., electronic transition energy, solubility parameters and dielectric constant). Among the physical properties of solvents, the electronic transition energy (E_T) is commonly used to quantify an empirical solvent scale (also known as Reichardt’s polarity) because the parameter requires a simple measurement using a solvatochromic technique with an indicator [13,14].

The E_T values of aqueous mixtures [15] generally tend to have a negative deviation from the ideality line (Figure 1b) due to preferential interactions of an indicator with cosolvents. There are the correlative models (preferential solvation model [16,17,18] and Jouyban–Acree model [19]) for representing a preferential trend in the aqueous mixtures, while there is no predictive model for estimating E_T values of the mixture. It is supposed that an ideal model (Equation (1), Section 2.1) is most likely applied for estimations due to a simple calculation from a mole fraction average. However, the ideal model (dashed line, Figure 1b) caused a large deviation from experimental data (symbol, Figure 1b).

According to our previous works, the predictive models for estimating E_T values [20] and Kamlet–Taft dipolarity/polarizability (KT-π*) [21,22] of nonpolar-polar mixtures have been proposed and the models provided a good predictive result as shown by the example for E_T values in Figure 1a (blue solid line). Since both nonpolar-polar mixtures (Figure 1a) and aqueous mixtures (Figure 1b) show a similar negative deviation from ideality, the predictive model (Equation (2)) is expected to be applicable to the aqueous mixtures.

With preliminary assessment, the predictive model (Equation (2)) is applicable to the aqueous mixtures (blue solid line, Figure 1b). The objective of this work is to predict the polarity parameters (E_T values) of aqueous systems using the predictive model (Equation (2)). The predictive model was validated by experimental E_T data of Reichardt indicator for eighteen aqueous mixtures that were collected from the literature [15,23,24,25].

2. Models and Methods

2.1. Ideal Model

The ideal model is used to compare the deviation of experimental data from ideality and is defined by Equation (1):

$$E_{\mathrm{T}}(\mathrm{ideal})=x_1{E}_{\mathrm{T},1}^0+x_2{E}_{\mathrm{T},2}^0$$

(1)

where $E_{\mathrm{T},\mathrm{i}}^0$ is the electronic transition energy of pure component i. Component 1 denotes water and component 2 denotes hydrogen bond acceptor (HBA) cosolvent or hydrogen bond donor (HBD) cosolvent.

2.2. Predictive Model

A predictive model was originally proposed for estimating Kamlet–Taft dipolarity/polarizability (KT-π*) [22] of binary nonpolar-polar mixtures with an assumption that the gas-phase dipole moment (µ) of the polar component can quantify a trend of mixture KT-π* values at a fixed mole composition. Due to a linear relationship between KT-π* and E_T values (homomorphism line, Figure 2), the predictive model for KT-π* values can directly transform into a function form for estimating E_T values for nonpolar-polar mixtures reported in our previous work [20], as shown by Equation (2). In this work, the model (Equation (2)) was used to predict the E_T values of aqueous mixtures due to similar negative deviation trends in both aqueous mixtures and nonpolar-polar mixtures as mentioned in the introduction (Figure 1).

$$\Delta E_{\mathrm{T},\mathrm{m}\mathrm{i}x}^{\mathrm{N}}={\mu }_2\left(-x_1\ln (x_1+1.981x_2)-x_2\ln (x_2+0.181x_1)\right)\times C{F}^{\mathrm{HBD}}$$

(2)

where µ₂ is the gas-phase dipole moment of component 2. The 1.981 and 0.181 values are the universal Wilson constant parameters (Λ₁₂ and Λ₂₁) for predictions and were evaluated by correlating experimental data with Wilson thermodynamic excess function [22]. The $\Delta E_{\mathrm{T},\mathrm{m}\mathrm{i}x}^{\mathrm{N}}$ is relative normalized electronic transition energy and defined by Equations (3)–(5).

$$\Delta E_{\mathrm{T},\mathrm{m}\mathrm{i}x}^{\mathrm{N},}=E_{\mathrm{T},\mathrm{m}\mathrm{i}x}^{\mathrm{N}}-(x_1{E}_{\mathrm{T},1}^{\mathrm{N}}+x_2{E}_{\mathrm{T},2}^{\mathrm{N}})=E_{\mathrm{T},\mathrm{m}\mathrm{i}x}^{\mathrm{N}}-x_2$$

(3)

$$E_{\mathrm{T},\mathrm{m}\mathrm{i}x}^{\mathrm{N}}=\frac{E_{\mathrm{T},\mathrm{m}\mathrm{i}x}-E_{\mathrm{T},1}^0}{E_{\mathrm{T},2}^0-E_{\mathrm{T},1}^0}$$

(4)

$$E_{\mathrm{T}}(\mathrm{kcal}\cdot {\mathrm{mol}}^{-1})=28591/{\lambda }_{\max }(\mathrm{nm})$$

(5)

where λ_max (nm) is the maximum absorption of the wavelength of a solvatochromic indicator (Reichardt indicator) obtained from UV-Vis spectroscopy. The $E_{\mathrm{T},\mathrm{m}\mathrm{i}x}$ and $E_{\mathrm{T},\mathrm{m}\mathrm{i}x}^{\mathrm{N}}$ are the electronic transition energy and the normalized electronic transition energy of the binary mixtures. To apply the predictive model (Equation (2)) to aqueous systems studied in this work, component 1 refers to water and component 2 refers to cosolvent (HBA or HBD). According to normalization (Equation (4)), the $E_{\mathrm{T},1}^{\mathrm{N}}$ and $E_{\mathrm{T},2}^{\mathrm{N}}$ of pure components 1 and 2 in Equation (3) are set equal to zero and unity, respectively.

The CF^HBD parameter in Equation (2) is the HBD contribution factor and the parameter is only applied to an aqueous mixture of HBD cosolvents due to specific interaction of HBD solvent with indicator [20] that causes a deviation in the pure $E_{\mathrm{T},2}^0$ value (HBD solvent) from the homomorphism line (Figure 2). The CF^HBD values in Equation (6) can be estimated by a deviation of the actual E_T value of pure HBD cosolvent ($E_{\mathrm{T},2}^0$) from the homomorphism line in Figure 2 as shown in Equation (6).

$$C{F}^{\mathrm{HBD}}=E_T^{0,\mathrm{Non}}-E_{\mathrm{T},2}^0$$

(6)

where $E_T^{0,\mathrm{Non}}$ is a non-HBD bonding E_T value of pure HBD cosolvents defined as a linear function of dipolarity/polarizability (KT-π*), that is the homomorphism linear line in Figure 2. The linear relationships of $E_T^{0,\mathrm{Non}}$ were given in detail in our previous work [20], and the CF^HBD values of nine HBD cosolvents studied in this work are given in

Table 1. Pure properties of hydrogen bond donor (HBD) at 25 °C showing Kamlet–Taft acidity (KT-α) [27,28] and HBD contribution factor (CF^HBD) evaluated from three indicators (Ind.) ^a.

Entry	HBD Solvents	KT-α	Actual CFHBD (-) b				Calculated CFHBD (-) c
(Symbol)	(Abbreviation)	(-)	Ind. 1	Ind. 2	Ind. 3	Avg	(Equation (7))
1 ( )	Methanol (MeOH)	1.00	3.65	2.32	2.34	2.77	2.75
2 ( )	Ethanol (EtOH)	0.89	3.05	2.21	2.21	2.49	2.45
3 ( )	1-Propanol (PrOH)	0.84	2.79	1.96	2.51	2.42	2.31
4 ( )	2-Propanol (iPrOH)	0.76	2.44	1.72	2.17	2.11	2.09
5 ( )	1-Butanol (BuOH)	0.84	2.65	2.04	3.04	2.58	2.31
6 ( )	Tert-Butanol (T-BuOH)	-	2.65	-	-	2.65	-
7 ( )	Ethylene glycol (ETG)	0.90	3.01	-	1.59	2.30	2.48
8 ( )	Acetic acid (AcOH)	1.12	4.34	1.83	2.78	2.98	3.09
9 ( )	Formamide (FA)	0.63	2.09	1.27	1.32	1.56	1.73

^a Indicators: Ind. 1 = 2,6-diphenyl-4-(2,4,6-triphenyl-1-pyridinio) phenolate indicator; Ind. 2 = phenol blue and Ind. 3 = Nile red. ^b Actual CF^HBD values are evaluated from Figure 2. ^c Calculated CF^HBD values are obtained from Equation (7).

along with their Kamlet–Taft acidity (α).

An average CF^HBD value from three indicators (Table 1) was used in predictions (Equation (2)) for nine aqueous mixtures of HBD cosolvents. To extend to other HBD cosolvents, a linear relationship of CF^HBD with Kamlet–Taft acidity (α) was proposed in Equation (7) and Figure 3, because the α values of HBD cosolvents are widely available in the literature [27,28].

$$C{F}^{\mathrm{HBD}}=2.766\alpha -0.0115,({\mathrm{R}}^2=0.89)$$

(7)

Figure 4 shows a flow chart developed in this work for predicting E_T values of aqueous mixtures that is divided into five steps. In step 1, pure properties of electronic transition energy ($E_{\mathrm{T},\mathrm{i}}^0$) and gas-phase dipole moment (µ₂) of components 1 and 2 are compiled from the literature. In step 2, solvent characteristics of component 2 (cosolvent) are determined to check HBD ability by considering their molecular structures and KT-acidity. For example, HBD cosolvents are able to donate a proton so that their KT-acidity values are relatively high (Table 1). On the other hand, HBA cosolvents lack proton donor groups and, thus, the CF^HBD value of HBA cosolvent is equal to unity. In step 3, CF^HBD values of HBD cosolvent are calculated by either Figure 2 or Equation (7). In steps 4 and 5, the predictive model (Equation (2)) was used to estimate the E_T values of aqueous mixtures and was validated, respectively. To validate the predictive model in step 5, nine aqueous mixtures of HBA cosolvents and nine aqueous mixtures of HBD cosolvents (

Table 2. Comparison between electronic transition (E_T, kcal·mol⁻¹) values calculated with the predictive model (Equation (2)) and ideal model (Equation (1)) for nine aqueous mixtures of hydrogen bond acceptor (HBA) cosolvents and nine aqueous mixtures of hydrogen bond donor (HBD) cosolvents at 25 °C that were obtained with Reichardt indicator. Average relative deviations (ARD) were calculated with Equation (8) along with dipole moment of component 2 (µ₂), HBD contribution factor (CF^HBD), Hunter basicity (β^H) of component 2, pure properties of electronic transition energy ($E_{\mathrm{T},\mathrm{i}}^0$ ) and the number of data used (N). Gray-shaded rows indicate the ARD results estimated from the calculated CF^HBD (Equation (7)).

Entry	Component 2	µ2	CFHBD	βH	${\mathit{E}}_{\mathbf{T}}^0$		ARD (%)		N	Ref.
(Symbol)		D	(-)	(-)	(1)	(2)	Predict	Ideal
Water (1)—HBA cosolvent (2)
1 (■)	ACN	3.92	1.00	4.7	63.10	46.00	6.64	3.36	11	[15]
2 (■)	THF	1.63	1.00	5.3	63.10	37.50	4.36	4.97	11	[15]
3 (●)	GBL	3.82	1.00	5.3	63.00	44.62	3.45	4.42	12	[23]
4 (●)	GVL	5.30	1.00	5.3	63.00	47.85	2.61	6.27	12	[23]
5 (●)	PYR	3.10	1.00	8.3	63.00	47.90	1.49	5.24	11	[24]
6 (●)	NMP	4.09	1.00	8.3	63.00	42.20	1.85	9.85	10	[24]
7 (●)	DMF	3.86	1.00	8.3	63.10	43.80	1.37	4.77	11	[15]
8 (■)	DMSO	3.96	1.00	8.9	63.10	45.00	0.75	5.29	11	[15]
9 (■)	Pyridine	2.19	1.00	7.0	63.10	40.30	1.29	4.77	11	[15]
		Overall (aqueous HBA mixtures)						2.65	5.44
Water (1)—HBD cosolvent (2)
10 ( )	MeOH	1.70	2.77	5.8	63.10	55.70	0.23	2.40	11	[15]
			2.75				0.23
11 ( )	EtOH	1.69	2.49	5.8	63.10	51.70	0.34	3.92	11	[15]
			2.45				0.38
12 ( )	PrOH	1.66	2.42	5.8	63.10	50.60	1.42	5.14	11	[15]
			2.31				1.53
13 ( )	iPrOH	1.66	2.11	5.8	63.10	48.70	2.64	6.73	11	[15]
			2.09				2.67
14 ( )	BuOH	1.66	2.58	5.8	63.10	43.90	2.58	4.28	10	[15]
			2.31				2.19
15 ( )	T-BuOH	1.67	2.65	5.8	63.10	43.30	4.67	12.54	11	[25]
16 ( )	ETG	2.31	2.30	-	63.10	56.24	0.50	2.44	10	[15]
			2.48				0.69
17 ( )	AcOH	1.74	2.98	5.3	63.10	55.00	1.03	2.10	11	[15]
			3.09				1.12
18 ( )	FA	3.73	1.56	5.8	63.10	53.74	0.97	3.22	11	[15]
			1.73				1.32
		Overall (aqueous HBD mixtures)						1.60	4.75
	Overall (both HBA and HBD mixtures)							2.12	5.10	197

The µ₂ values for all solvents are taken from handbook [29], except for GVL [32]. The overall average relative deviation (ARD) values are calculated without considering the values in the gray-shaded rows. Solvent abbreviations for HBD cosolvent are given in Table 1. Solvent abbreviations for HBA cosolvent: ACN = acetonitrile; THF = tetrahydrofuran; GBL = gamma butyrolactone; GVL = gamma valerolactone; PYR = 2-pyrrolidinone; NMP = N-methyl-2-pyrrolidone; DMF = dimethylformamide and DMSO = dimethyl sulfoxide.

) were used and discussed in Section 3. Table 2 tabulates the $E_{\mathrm{T},\mathrm{i}}^0$, µ₂ [29], Hunter basicity (β^H) of component 2 [30,31] and CF^HBD values of solvents used in the predictions.

2.3. Evaluation of the Frameworks

Average relative deviation (ARD) was used to evaluate a deviation between experimental ($E_{\mathrm{T}}^{\mathrm{Exp}}$) and calculated ($E_{\mathrm{T}}^{\mathrm{Cal}}$) data as shown in Equation (8):

$$\mathrm{ARD}(\%)=(1/N){\displaystyle \sum \left|(E_{\mathrm{T}}^{\mathrm{Cal}}-E_{\mathrm{T}}^{\mathrm{Exp}})/E_{\mathrm{T}}^{\mathrm{Exp}}\right|}\times 100$$

(8)

3. Results and Discussion

3.1. Prediction for Aqueous Mixtures of HBA Cosolvents

Figure 5 shows the E_T values of nine aqueous mixtures of water (1)—HBA cosolvent (2) as a function of component 2 (HBA cosolvent). The E_T values of all aqueous mixtures (Figure 5) exhibited a negative deviation from the ideality (dashed lines, Figure 5), except for the aqueous mixtures of acetonitrile (ACN), tetrahydrofuran (THF) and gamma-butyrolactone (GBL) that showed a sigmoid function (Figure 5a–c). Blue solid lines (Figure 5) show predicted E_T values of the aqueous mixture using Equation (2) without CF^HBD value (CF^HBD = 1) that generally tended toward the experimental data (symbols, Figure 5).

Table 2 shows a comparison of ARD values obtained between the ideal model (Equation (1)) and predictive model (Equation (2)) for the aqueous mixtures of HBA cosolvents (entries 1–9, Table 2) and HBD cosolvent (entries 10–18, Table 2). The predictive model (Equation (2)) generally provided a lower ARD value (2.7%, entries 1–9, Table 2) than that estimated from the ideal model (5.4%). However, the model gave a relatively high ARD value (entries 1–3, Table 2) for the aqueous mixtures that showed the sigmoid functions (Figure 5a–c). These results inferred a limitation of the predictive model and were discussed later in Section 3.4. The predictive results of aqueous mixtures of HBD cosolvents were discussed in the following section.

3.2. Prediction for Aqueous Mixtures of HBD Cosolvents

Figure 6 shows the E_T values of nine aqueous mixtures of water (1)—HBD solvent (2) as a function of component 2 (HBD solvent) that also exhibited a negative deviation from the ideality. A comparison in predictive model (Equation (2)) with considering CF^HBD value (Table 1) and without considering CF^HBD value (CF^HBD = 1). Red solid lines in Figure 6 show predictions with considering CF^HBD value, while the blue ones are the predictions without considering CF^HBD value.

The predictive model with the addition of CF^HBD values (red solid lines, Figure 6) could provide a better result in the calculated E_T value that followed the experimental data than the result without the CF^HBD value (blue solid lines, Figure 6). The ARD values obtained from the predictive model (Equation (2)) with considering CF^HBD value (1.6%, entries 10–18, Table 2) are lower than those estimated from the ideal model (4.8%). Figure 7 shows parity plots of calculated E_T values that are estimated from the predictive model (Figure 7a) and the ideal model (Figure 7b). The estimated E_T values from the predictive model (R² = 0.91, Figure 7a) are less scattered than those obtained from the ideal model (R² = 0.81, Figure 7b).

3.3. Evaluation of CF^HBD Methods for Predictions

In Section 3.2, the predicted E_T values of aqueous mixtures of HBD cosolvents were estimated based on the actual CF^HBD values evaluated from Figure 2 and Table 1. In the section, a comparison in evaluated E_T values between actual data (Table 1) and calculations with KT-acidity (Equation (7)) was made for predictions. Red solid lines (Figure 6) show predicted E_T values of the mixtures with actual CF^HBD values, while the green solid lines (Figure 6) show predictions using calculated CF^HBD values from the correlation (Equation (7)). The predictive model with the calculated CF^HBD values provided a similar trend in predictions with the actual CF^HBD values (Figure 6), in which the ARD values from the calculated CF^HBD (gray-shaded rows, Table 2) are comparable to those estimated from the actual ones (entries 10–18, Table 2).

3.4. Limitation of the Predictive Model

According to predictive results in Section 3.1, the model (Equation (2)) provided a high ARD value for aqueous mixtures having a sigmoid function (entries 1–3, Table 2). Marcus reported [33,34] that the sigmoid trends were found in aqueous mixtures when microheterogeneity occurred. It is expected that the microheterogeneity phenomenon caused a high error in predictions because this phenomenon was not considered in the development of the model as mentioned in Section 2.2. Hunter basicity of HBA solvents (β^H, Table 2) can quantify the hydration shell strength in aqueous mixtures [35]. It was found that the sigmoid behaviors occurred in the aqueous mixtures that have low β^H values (β^H ≤ 5.3, entries 1–3, Table 2). Thus, the model (Equation (2)) is effective for predicting the aqueous mixtures having high β^H values (β^H ≥ 5.3).

4. Conclusions

In this work, a predictive model is proposed for estimating the solvatochromic polarity of electronic transition energy (E_T) for aqueous mixtures. The function form for E_T values for aqueous mixtures can be adopted from the previous model for nonpolar-polar mixtures due to similar interactions and trends in the mixture E_T values in both systems. The predictive model was validated by eighteen aqueous mixtures and was found to give a reliable E_T value with an overall deviation of 2.1%. Three properties of pure components are basically needed for predictions: (i) E_T values, (ii) gas-phase dipole moment and (iii) Kamlet–Taft acidity.