Solubility and Dissolution Thermodynamic Properties of 3,3′,4,4′-Oxydiphthalic Anhydride in Binary Aqueous Solutions of Acetic Acid and Propionic Acid from (278.15 to 313.15) K

Abstract

The solubility of 3,3′,4,4′-oxydiphthalic anhydride in binary solvents of (acetic acid + water) and (propionic acid + water) at temperatures ranging from 278.15 to 313.15 K were measured using a gravimetric method. The measured solubility data of 3,3′,4,4′-oxydiphthalic anhydride in (acetic acid + water) and (propionic acid + water) mixed solvents exhibit monotonic behavior under the studied conditions. The solubilities increased with increasing of both temperature and mass fraction of acetic acid or propionic acid. The solubility values were correlated by the three parameters equation, the $\lambda h$ equation, the NRTL model and the binary solvent models, the Ma model and the Sun model. The calculated values by the five models agreed well with the experimental data for the solubility of 3,3′,4,4′-oxydiphthalic anhydride in (acetic acid + water) and (propionic acid + water) solutions. The standard thermodynamic properties of dissolution of 3,3′,4,4′-oxydiphthalic anhydride, including the dissolution Gibbs energy change ($\Delta _{\text{sol}} G_{{}}^{\text{o}}$),molar dissolution enthalpy change ($\Delta _{\text{sol}} H_{{}}^{\text{o}}$) and molar dissolution entropy change ($\Delta _{\text{sol}} S_{{}}^{\text{o}}$) were calculated from the experimentally measured solubility data. The values of the change of standard molar enthalpy are positive, which shows that the dissolution process of 3,3′,4,4′-oxydiphthalic anhydride in the binary system of (acetic acid + water) and (propionic acid + water) is endothermic.

1 Introduction

3,3′,4,4′-Oxydiphthalic anhydride (CAS Reg. No. 1823-59-2) is an important chemical for preparing its corresponding dicarboxylic acids and their derivatives. It is a key monomer used to prepare high temperature polyetherimides by polymerization with an appropriate diamine [1–3]. The polymerization process generally requires monomers with high purity in order to provide good reaction kinetics and to effectively construct polymers of appropriate molar mass [4–6].

Various methods have been described for the preparation of 3,3′,4,4′-oxydiphthalic anhydride in the literature. It is often synthesized from 4-chlorophthalic anhydride by combining two molecules of 4-chlorophthalic anhydride to form crude 3,3′,4,4′-oxydiphthalic anhydride. The crude product usually contains some unreacted raw material, catalyst, and various colored impurities [7–12]. Several methods have been proposed to purify the crude 3,3′,4,4′-oxydiphthalic anhydride [5, 13–18]. A valuable way with low cost is crystallization from some suitable solvent such as acetic acid, propionic acid or aqueous solutions of them [16–18]. During the purification of 3,3′,4,4′-oxydiphthalic anhydride via solvent crystallization, the solubility has a direct effect on the quality of the product. As a result, the solubility data and the thermodynamic properties of dissolution process are important factors in the design and optimization of the crystallization process. However, to the best of the authors’ present knowledge, no solubility data for 3,3′,4,4′-oxydiphthalic anhydride in solvents have been reported. The aim of this work is to (1) determine the solubility of 3,3′,4,4′-oxydiphthalic anhydride in binary solvents of (acetic acid + water) and (propionic acid + water) at temperatures ranging from 278.15 to 313.15 K using the gravimetric method, (2) fit the solubility data with the three parameter equation, the $\lambda h$ equation, the NRTL model and two mixed solvents models, the Ma model and the Sun model, and (3) evaluate the thermodynamic properties (Gibbs energy, enthalpy and entropy) from solubility data for the dissolution process of 3,3′,4,4′-oxydiphthalic anhydride in solvent mixtures of (acetic acid + water) and (propionic acid + water) by the three parameter equation and Gibbs equation.

2 Experimental Section

2.1 Materials

3,3′,4,4′-Oxydiphthalic anhydride (purity more than 0.98 in mass fraction) was supplied by TCI Shanghai. It was recrystallized three times from aqueous acetic acid solution. The final purity of 3,3′,4,4′-oxydiphthalic anhydride was measured using a Shimadzu-6A high-performance liquid phase chromatograph (HPLC), which was 0.9921 in mass fraction. Anhydrous acetic acid with a purity of 0.9953 in mass fraction, and propionic acid with a mass fraction of 0.9941, were both purchased from Grace Chemical Technology Co., LTD. The compositions of the two acids were further verified by a gas chromatograph equipped with a flame ionization detector. The properties of chemicals used in this work and their sources are provided in Table 1.

Table 1 Provenance and purity of the materials used

2.2 DSC Measurement

Although the melting temperature of 3,3′,4,4′-oxydiphthalic anhydride was reported in the literature [19], the fusion enthalpy was not found. In the present work, the melting point T _m and fusion enthalpy ∆_fus H of 3,3′,4,4′-oxydiphthalic anhydride were measured by differential scanning calorimetry (type: NETZSCH 204 F1) under a nitrogen atmosphere, the flow rate of which was 5 mL·min⁻¹. The heat flow and temperature of the instrument were precalibrated with indium as the reference material before experiment. The weight of the sample was about 1.9 mg, and the heating rate 5 K·min⁻¹. The measurement uncertainty was evaluated to be less than ±2 %.

2.3 Solubility Measurements

The gravimetric method is often used for determination of solubility [20–23]. In this work the method was employed to measure the solubility of 3,3′,4,4′-oxydiphthalic anhydride in aqueous solutions of acetic acid and propionic acid. The binary solvent was prepared by using a BSA224S analytical balance with a precision of ±0.0001 g and then was added to a glass bottle, which had a volume of 50 mL. The temperature of the system was controlled by a thermostatic water bath with a precision of 0.01 K, and the temperature was calibrated in advance using a mercury thermometer (Arno Amavell, 6983 Kreuzwerthelm with precision of ±0.01 K).

During the solubility measurements, about 25 mL of solvent was introduced to a 50 mL glass bottle. Excess solute was added to the solvent and the system was stirred continuously with a magnetic stirrer. The glass bottle was equipped with a condenser to avoid solvent evaporation. Each experiment was carried out at a constant temperature controlled by the thermostatic water bath (model: DZKW-4), which was produced by Ningbo Scientz Biotechnology Co., Ltd. In order to check the equilibrium, the liquid phase was extracted using a syringe equipped with a filter having a 0.45 μm pore size and the composition was determined using HPLC. If the concentration of 3,3′,4,4′-oxydiphthalic anhydride in the liquid phase did not vary, then it was believed that the system had arrived at equilibrium. To confirm the equilibrium, the composition of 3,3′,4,4′-oxydiphthalic anhydride was measured again after 2 additional days. Results indicated that it took about 13 h to reach equilibrium. When the system was at equilibrium, the magnetic stirring was ceased for 1 h before sampling. When all solid had precipitated out of the aqueous solution, 2–3 mL (precision: ±0.01 mL) of the equilibrium liquid phase was withdrawn with a syringe preheated at the same temperature for 5 min, and immediately transferred into a preweighed flask of 50 mL. Then the flask was covered quickly with a stopper to avoid solvent evaporation. The total mass of the flask was reweighed on the analytical balance. Subsequently deionized water was added to the flask and the sample was diluted to 50 mL; 20 μL of solution was taken from the flask to measure the solute concentration by HPLC.

2.4 Analysis

The content of 3,3′,4,4′-oxydiphthalic anhydride in the mixed solvents of (acetic acid + water) and (propionic acid + water) was determined using a Shimadzu-6A high-performance liquid phase chromatograph (HPLC) with a single wavelength spectrophotometric detector. The chromatographic system was equipped with a STI UV501 detector and a column of Unimicro Kromasil C18, 5 $\upmu{\text{m}}$ (250 mm × 4.6 mm). The wavelength of the single spectrophotometric detector was set to 260 nm, and the column temperature was maintained at 303 K. The composition of mobile phase was 5 mL acetic acid: 450 mL methanol and 550 mL water, the flow rate of which was 1 mL·min⁻¹; 20 μL samples were injected into the column.

The HPLC internal standard method was adopted in the solubility determination of 3,3′,4,4′-oxydiphthalic anhydride in mixed solvents. Before the experiments, a series of mixtures with known concentrations of 3,3′,4,4′-oxydiphthalic anhydride were prepared, and 20 μL aliquots of these were used to calibrate the HPLC. The dependence of peak area on the concentration of 3,3′,4,4′-oxydiphthalic anhydride was determined and was used to calculate the concentrations of the samples. Each analysis was performed three times, and the average value was employed to evaluate the mole fraction of 3,3′,4,4′-oxydiphthalic anhydride in the mixed solvents.

The solubility of 3,3′,4,4′-oxydiphthalic anhydride in mole fraction (x) in aqueous solutions of acetic acid and propionic acid can be evaluated from Eq. 1, and the initial concentrations (w) of mixed solvents in mass fraction were calculated with Eq. 2:

$$ x{ = }\frac{{{{m_{1} } \mathord{\left/ {\vphantom {{m_{1} } {M_{1} }}} \right. \kern-0pt} {M_{1} }}}}{{{{m_{1} } \mathord{\left/ {\vphantom {{m_{1} } {M_{1} { + }{{m_{2} } \mathord{\left/ {\vphantom {{m_{2} } {M_{2} { + }{{m_{3} } \mathord{\left/ {\vphantom {{m_{3} } {M_{3} }}} \right. \kern-0pt} {M_{3} }}}}} \right. \kern-0pt} {M_{2} { + }{{m_{3} } \mathord{\left/ {\vphantom {{m_{3} } {M_{3} }}} \right. \kern-0pt} {M_{3} }}}}}}} \right. \kern-0pt} {M_{1} { + }{{m_{2} } \mathord{\left/ {\vphantom {{m_{2} } {M_{2} { + }{{m_{3} } \mathord{\left/ {\vphantom {{m_{3} } {M_{3} }}} \right. \kern-0pt} {M_{3} }}}}} \right. \kern-0pt} {M_{2} { + }{{m_{3} } \mathord{\left/ {\vphantom {{m_{3} } {M_{3} }}} \right. \kern-0pt} {M_{3} }}}}}}}} $$

(1)

$$ w{ = }\frac{{m_{2} }}{{m_{2}\,+\,m_{ 3} }} $$

(2)

where m ₁, m ₂ and m ₃ stand for the mass of 3,3′,4,4′-oxydiphthalic anhydride, acetic acid or propionic acid, and water, respectively. M ₁, M ₂ and M ₃ are the corresponding molar masses.

3 Results and Discussion

3.1 Pure Component Properties

Figure 1 shows the differential scanning calorimetry (DSC) curve of 3,3′,4,4′-oxydiphthalic anhydride. The melting point (T _m) and fusion enthalpy (∆_fus H) of 3,3′,4,4′-oxydiphthalic anhydride are 501.8 K and 35.77 kJ·mol⁻¹, respectively. According to the DSC data, the fusion entropy (∆_fus S) of 3,3′,4,4′-oxydiphthalic anhydride was evaluated with Eq. 3:

$$ \Delta_{\text{ful}} S = \frac{{\Delta_{\text{ful}} H}}{{T_{\text{m}} }} $$

(3)

Fig. 1

DSC curve of 3,3′,4,4′-oxydiphthalic anhydride

The value of $\Delta_{ful} S$ was calculated to be 71.28 J·mol⁻¹·K⁻¹ by solving Eq. 3.

3.2 Solubility of 3,3′,4,4′-Oxydiphthalic Anhydride

The solubilities of 3,3′,4,4′-oxydiphthalic anhydride in the mixed solvents of (acetic acid + water) and (propionic acid + water) are given in Tables 2 and 3, respectively. In order to compare the experimental values, plots of the solubility date against temperature are presented in Figs. 2 and 3.

Table 2 Mole fraction solubility ($x$) of 3,3′,4,4′-oxydiphthalic anhydride in acetic acid (w) + water (1 − w) with the temperature range from T = (278.15 to 313.15) K under atmospheric pressure

Table 3 Mole fraction solubility (x) of 3,3′,4,4′-oxydiphthalic anhydride in propionic acid (w) + water (1 − w) with the temperature range from T = 278.15 to 313.15 K under atmospheric pressure

Fig. 2

Mole fraction solubility x of 3,3′,4,4′-oxydiphthalic anhydride in acetic acid (w) + water (1 − w) mixed solutions with various mass fractions at different temperature: diamond, w = 61.15 %; inverted triangle, w = 51.20 %; triangle, w = 41.16 %; circle, w = 31.02 %; square, w = 20.78 %

Fig. 3

Mole fraction solubility x of 3,3′,4,4′-oxydiphthalic anhydride in propionic acid (w) + water (1 − w) mixed solutions with various mass fractions at different temperature: diamond, w = 59.84 %; inverted triangle, w = 49.83 %; triangle, w = 39.84 %; circle, w = 29.86 %; square, w = 19.89 %

From the solubility data presented in Tables 2 and 3, it can be seen that the solubility of 3,3′,4,4′-oxydiphthalic anhydride in the binary solvent mixtures of (acetic acid + water) and (propionic acid + water) are functions of solvent composition and temperature. The solubility of 3,3′,4,4′-oxydiphthalic anhydride, in mole fraction, in mixed solvents increases with an increase in temperature, while it decreases with a decrease in amount of acetic acid or propionic acid in the solvent mixtures. This may result from the polarity of the binary solvent mixtures, i.e. intermolecular hydrogen bonding between 3,3′,4,4′-oxydiphthalic anhydride and water or acetic acid or propionic acid.

3.3 Solubility Data Correlations

To illustrate quantitatively and extend the use range of the solubility data, the relationship between the solubility of 3,3′,4,4′-oxydiphthalic anhydride in solvent mixtures and the temperature were fitted by the three parameter equation, the $\lambda h$ model, the NRTL model and two binary solvent models, the Ma model and the Sun model.

3.3.1 Three Parameter Equation

At a fixed temperature and pressure, when the liquid–solid system reaches equilibrium, the solid phase fugacity of the solute is equal to that in the liquid phase. For component A, this case is described as

$$ x_{\text{A}}^{\text{S}} \gamma_{\text{A}}^{\text{S}} f_{\text{A}}^{\text{S}} = x_{\text{A}}^{\text{L}} \gamma_{\text{A}}^{\text{L}} f_{\text{A}}^{\text{L}} $$

(4)

where S and L stand for the solid and liquid state, respectively. f _A represents the fugacity of component A, and $\gamma_{\text{A}}$ is the activity coefficient.

The solid–liquid phase equilibrium can be expressed as [24]

$$ \ln \left( {x_{\text{A}} \cdot \gamma_{\text{A}} } \right) = \frac{{\Delta H_{\text{t}} }}{R}\left( {\frac{1}{{T_{\text{tp}} }} - \frac{1}{T}} \right) - \frac{{\Delta C_{p} }}{R}\left( {\frac{1}{{T_{\text{tp}} }} - \frac{1}{T}} \right) - \frac{{\Delta C_{p} }}{R}\ln \left( {\frac{{T_{\text{tp}} }}{T}} \right) $$

(5)

where ΔH _t, T _tp, and ΔC _p are the molar fusion enthalpy of the solute, the triple-point temperature, and the difference in heat capacity of solute between the solid phase and the liquid phase at the melting temperature, respectively. Generally, the difference between the triple-point temperature and the normal melting point temperature is very small for many substances; the difference between the fusion enthalpies at the two temperatures is negligible. So, the temperature and fusion enthalpy at the triple point can be replaced by their values at the normal melting point. In Eq. 5, the last two terms almost offset each other.

The value of $\gamma_{A}$ can be expressed by Eq. 6 for regular solutions [25]:

$$ \ln \left( {\gamma_{\text{A}} } \right) = a + \frac{b}{T} $$

(6)

In Eq. 6 a and b are empirical parameters. Combining Eq. 5 with Eqs. 6 and 7 gives [26]:

$$ \ln \left( {x_{\text{A}} } \right) = \left[ {\frac{{\Delta_{\text{fus}} H}}{{RT_{\text{m}} }} + \frac{{\Delta C_{p} }}{R}\left( {1 + \ln T_{\text{m}} } \right) - a} \right] - \left[ {b + \left( {\frac{{\Delta_{\text{fus}} H}}{{RT_{\text{m}} }} + \frac{{\Delta C_{p} }}{R}} \right)T_{\text{m}} } \right]\frac{1}{T} - \frac{{\Delta C_{p} }}{R}\ln T $$

(7)

Then, the expression of the three parameter model describing solubility and temperature can be obtained [27]:

$$ \ln (x) = A + \frac{B}{(T)} + C\ln (T) $$

(8)

with

$$ A = \frac{{\Delta_{\text{fus}} H}}{{RT_{\text{m}} }} + \frac{{\Delta C_{p} }}{R}\left( {1 + \ln T_{\text{m}} } \right) - a $$

(9)

$$ B = - \left[ {b + \left( {\frac{{\Delta_{\text{fus}} H}}{{RT_{\text{m}} }} + \frac{{\Delta C_{p} }}{R}} \right)T_{\text{m}} } \right] $$

(10)

$$ C = - \frac{{\Delta C_{p} }}{R}\ln T $$

(11)

where T stands for the absolute temperature in Kelvin. A, B and C are parameters for the three parameter equation. $\Delta_{\text{fus}} H$ is the molar enthalpy of fusion of 3,3′,4,4′-oxydiphthalic anhydride at normal melting temperature T _m. The value of parameter C represents the effect of temperature on the enthalpy of fusion of the solid. The A and B values reveal the change in the solution nonidealities and the solute activity coefficient on the solubility of the solute, respectively. The parameters A, B and C can be obtained by regression of the experimental solubility data using multidimensional unconstrained nonlinear minimization.

3.3.2 λh Equation

Buchowski et al. proposed an equation to correlate solubility data [28, 29], which was named the $\lambda h$ equation and applied to hydrogen-bond-forming solutes. For this equation, two parameters ($\lambda$ and h) are needed to correlate the experimental solubility data. In the present work, the $\lambda h$ equation is employed to correlate the experimental data of 3,3′,4,4′-oxydiphthalic anhydride in binary solvents of (acetic acid + water) and (propionic acid + water):

$$ \ln \left[ {\left( {1 + \frac{\lambda (1 - x)}{x}} \right)} \right] = \lambda h\left( {\frac{1}{T} - \frac{1}{{T_{\text{m}} }}} \right). $$

(12)

3.3.3 NRTL Model

According to Eq. 7, if a solid–solid phase transition does not happen then it can be simplified to Eq. 13 [24]:

$$ \ln \left( {x_{i} \cdot \gamma_{i} } \right) = \frac{{\Delta_{\text{fus}} H}}{R}\left( {\frac{1}{{T_{\text{m}} }} - \frac{1}{T}} \right) $$

(13)

The NRTL model, first proposed by Renon and Prausnitz [30] and based on the local composition concept, has been widely used in the calculation and correlation of solid–liquid phase equilibria. According to this model, the activity coefficient for a component i can be described as follows:

$$ \ln \gamma_{i} = \frac{{\sum\limits_{j = 1}^{N} {\tau_{ji} G_{ji} x_{j} } }}{{\sum\limits_{i = 1}^{N} {G_{ij} x_{i} } }} + \sum\limits_{j = 1}^{N} {\frac{{x_{j} G_{ij} }}{{\sum\limits_{i = 1}^{N} {G_{ij} x_{i} } }}\left[ {\tau_{ij} - \frac{{\sum\limits_{i = 1}^{N} {x_{i} \tau_{ij} G_{ij} } }}{{\sum\limits_{i = 1}^{N} {G_{ij} x_{i} } }}} \right]} $$

(14)

$G_{ij}$ and $\tau_{ij}$ are parameters for NRTL model, and can be calculated by:

$$ G_{ji} = \exp ( - \alpha_{ji} \tau_{ji} ) $$

(15)

$$ \tau_{ij} = a_{i} + \frac{{b_{i} }}{T} $$

(16)

$$ \alpha_{ij} = \alpha_{ji} $$

(17)

Here $a_{i}$ and $b_{i}$ are the interaction energy parameters (J·mol⁻¹) and are regarded as temperature independent. $\alpha$ is the parameter related to the nonrandomness in the solution and usually in the range from 0.2 to 0.47. In this work, the value of $\alpha$ is taken as 0.3. The two parameters can be obtained by regression from experimental solubility data by combining Eqs. 13 and 14.

Equation 18 was proposed by Sun et al. [31] and is used to correlate solubility data in mixed solvents at different temperatures:

$$ \ln x_{m,T} = D_{1} + \frac{{D_{2} }}{T} + D_{3} m_{1} + D_{4} \frac{{m_{1} }}{T} + D_{5} \frac{{m_{1}^{2} }}{T} + D_{6} \frac{{m_{1}^{3} }}{T} + D_{7} \frac{{m_{1}^{4} }}{T} $$

(18)

However Ma et al. [32] put forward another model, which is expressed as:

$$ \ln x_{m,T} = E_{1} + \frac{{E_{2} }}{T} + E_{3} \ln T + E_{4} m_{1} + E_{5} \frac{{m_{1} }}{T} + E_{6} \frac{{m_{1}^{2} }}{T} + E_{7} \frac{{m_{1}^{3} }}{T} + E_{8} \frac{{m_{1}^{4} }}{T} + E_{9} m_{1} \ln T $$

(19)

where D ₁–D ₇ and E ₁–E ₉ are the model parameters. Detailed discussions of Eqs. 18 and 19 are given in references [31] and [32].

To estimate the deviation, the relative average deviation (RAD) and the root-mean-square relative deviation (Rmsd) were employed. The relative average deviation (RAD) is expressed as:

$$ {\text{RAD}} = \frac{1}{N}\sum {\left( {\frac{{\left| {x_{m,T}^{\text{c}} - x_{m,T}^{\text{e}} } \right|}}{{x_{m,T}^{\text{e}} }}} \right)} $$

(20)

and the root-mean-square relative deviation (Rmsrd) is described as Eq. 21:

$$ {\text{Rmsrd}} = \left[ {\frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {x_{m,T}^{\text{c}} - x_{m,T}^{\text{e} } } \right)^{2} } } \right]^{{\tfrac{1}{2}}} $$

(21)

In Eqs. 20 and 21, N is the number of data points. $x_{m,T}^{\text{c}}$ is the calculated solubility value in mole fraction, based on the solubility models at absolute temperature T, while $x_{m,T}^{\text{e}}$ is the corresponding experimental value.

The regressed parameter values for the five solubility modes, along with the corresponding Rmsd values are shown in Tables 4 and 5. The evaluated solubility values using these models and the calculated RAD values are listed in Tables 2 and 3. The calculated solubility results with the three parameter equation are plotted in Figs. 2 and 3.

Table 4 Parameters of the thermodynamic models (three parameter equation, $\lambda h$ equation and NRTL model for solubility of 3,3′,4,4′-oxydiphthalic anhydride in mixed solvents

Table 5 Parameters of the Ma and Sun models for 3,3′,4,4′-oxydiphthalic anhydride in the mixtures

We can find from Tables 4 and 5 that the differences between the measured solubility data of the 3,3′,4,4′-oxydiphthalic anhydride in binary solvents and the calculated solubility by three parameters’equation, $\lambda h$ equation, NRTL model and Sun model are all small, the largest value of Rmsd is 2.97 × 10⁴, and of RAD is 3 %. However, the Ma model shows the largest Rmsd value of 2.97 × 10⁴. From the solubility data presented in Tables 4 and 5, all the Rmsd values with the three parameter equation are smaller those with NRTL model for the system of acetic acid + water, nevertheless, for the acetic acid + water system, the situation is reversed. In general, the three parameter equation and the NRTL model give excellent correlation results for solubility data of 3,3′,4,4′-oxydiphthalic anhydride in the studied temperature range, thus they are the more suitable models to represent the solubility data of the studied system. From the practical point of view, the five models (three parameter equation, $\lambda h$ model, NRTL model, the Ma model and the Sun model) can be used to calculate solubility of 3,3′,4,4′-oxydiphthalic anhydride in binary solvents (acetic acid + water) and (propionic acid + water) when the temperature varies from 278.15 to 313.15 K.

3.4 Thermodynamic Properties of Dissolution

It can be seen from the discussion above that the dissolution process of 3,3′,4,4′-oxydiphthalic anhydride in mixed solvents of (acetic acid + water) and (propionic acid + water) depends on both the temperature and solvent composition. These results can provide the basis for a thermodynamic analysis of the dissolution process. From an energetic point of view, the dissolution process of 3,3′,4,4′-oxydiphthalic anhydride into a solvent is related to changes of some thermodynamic properties, for instance, the standard molar dissolution Gibbs energy ($\Delta_{\text{sol}} G^{\text{o}}$), standard molar dissolution enthalpy ($\Delta_{\text{sol}} H^{\text{o}}$) and standard molar dissolution entropy ($\Delta_{\text{sol}} S^{\text{o}}$). These thermodynamic properties can be evaluated by the linear relationship between $\ln (x^{e} )$ and (1/T − 1/T _hm). Supposing that in the aqueous phase the water activity coefficient is equal to 1, and combining the van’t Hoff equation and Gibbs equation, Eq. 22 can be deduced [33]:

$$ \Delta _{{{\text{sol}}}} H^{{\text{o}}} = - R\left( {\frac{{\partial \ln x_{{m,T}}^{{\text{e}}} }}{{\partial (1/T)}}} \right)_{p} = - R\left( {\frac{{\partial \ln x_{{m,T}}^{{\text{e}}} }}{{\partial [(1/T) - (1/T_{{{\text{hm}}}} )]}}} \right)_{p} $$

(22)

where T _hm is the mean harmonic temperature used in the van’t Hoff analysis, which can be calculated using Eq. 23:

$$ T_{\text{hm}} = \frac{N}{{\sum\limits_{i = 1}^{N} {\frac{1}{{T_{i} }}} }} $$

(23)

The $\Delta _{\text{sol}} H_{{}}^{\text{o}}$ can be obtained from the slope of the plot of $\ln x_{m,T}^{\text{e}}$ versus (1/T − 1/T _hm). Using Eq. 24, the values of standard molar Gibbs energy of dissolution ($\Delta _{\text{sol}} G_{{}}^{{_{\text{o}} }}$) can be obtained:

$$ \Delta_{\text{sol}} G_{{}}^{\text{o}} = - RT_{\text{hm}} \times intercept $$

(24)

where the intercept can be obtained from a plot of $\ln x_{m,T}^{\text{e}}$ versus (1/T − 1/T _hm).

The values of standard molar entropy of dissolution ($\Delta _{\text{sol}} S_{{}}^{\text{o}}$) are calculated based on the values of $\Delta_{\text{sol}} H_{{}}^{\text{o}}$ and $\Delta_{\text{sol}} G^{\text{o}}$:

$$ \Delta_{\text{sol}} S_{{}}^{\text{o}} = \frac{{\Delta_{\text{sol}} H_{{}}^{\text{o}} - \Delta_{\text{sol}} G_{{}}^{\text{o}} }}{{T_{\text{hm}} }} $$

(25)

In order to compare the contribution of enthalpy and entropy to the Gibbs energy in the dissolution process, Eqs. 26 and 27 are used [22, 34]:

$$ \% \, \xi_{H} = \frac{{\left| {\Delta _{\text{sol}} H_{{}}^{{_{\text{o}} }} } \right|}}{{\left| {\Delta _{\text{sol}} H_{{}}^{{_{\text{o}} }} } \right| + \left| {T\Delta _{\text{sol}} S_{{}}^{{_{\text{o}} }} } \right|}} \times 100 $$

(26)

$$ \% \, \xi_{TS} = \frac{{\left| {T\Delta _{\text{sol}} S_{{}}^{{_{\text{o}} }} } \right|}}{{\left| {\Delta _{\text{sol}} H_{{}}^{{_{\text{o}} }} } \right| + \left| {T\Delta _{\text{sol}} S_{{}}^{{_{\text{o}} }} } \right|}} \times 100 $$

(27)

The relationship between $\ln x_{m,T}^{{}}$ and (1/T – 1/T _hm) for 3,3′,4,4′-oxydiphthalic anhydride in (acetic acid + water) and (propionic acid + water) mixed solutions are plotted in Figs. 4 and 5. The values of $\Delta_{\text{sol}} H_{{}}^{\text{o}}$, $\Delta_{\text{sol}} G_{{}}^{\text{o}}$, $\Delta_{\text{sol}} S_{{}}^{\text{o}}$,  %ζ _H and  %ζ _TS were calculated using Eqs. 22 and 24–27, respectively, and are listed in Table 6. The values of $\Delta_{\text{sol}} H_{{}}^{\text{o}}$ of dissolution are positive, which illustrates that the dissolution process is endothermic. The values of standard molar entropies are positive for all cases, which demonstrates that the driving force for the dissolution process is entropy. The dissolution enthalpy versus mole fraction of acetic acid and propionic acid are also plotted in Fig. 6. It can be found from Fig. 6 that the plot of molar enthalpy of dissolution of 3,3′,4,4′-oxydiphthalic anhydride with cosolvent ratio is approximately linear. The molar dissolution enthalpies of 3,3′,4,4′-oxydiphthalic anhydride in (acetic acid + water) mixtures are larger than those in (propionic acid + water) mixtures. The larger the dissolution enthalpy change, the more energy is required to overcome the cohesive force between the solute and the solvent in the dissolution process, which also illustrates the stronger dependence of solubility on temperature. The lower value of $\Delta_{\text{sol}} H_{{}}^{\text{o}}$ demonstrates that the dissolution process of 3,3′,4,4′-oxydiphthalic anhydride in (propionic acid + water) mixtures is easier than that in the mixed solvents of (acetic acid + water). Although the $\Delta_{\text{sol}} G_{{}}^{\text{o}}$ values are positive, the dissolution process of 3,3′,4,4′-oxydiphthalic anhydride in solutions is spontaneous. The real criterion is $\Delta_{\text{sol}} G$ of non-equilibrium, which is calculated as $\Delta_{\text{sol}} G = \Delta_{\text{sol}} G_{{}}^{\text{o}} + RT\ln Q$, where Q is the solute concentration below saturation. Because x is always lower than 1.0, $\Delta_{\text{sol}} G$ will be negative and the dissolution process will occur.

Fig. 4

The van’t Hoff plots of the mole fraction solubility (ln x) of 3,3′,4,4′-oxydiphthalic anhydride in acetic acid (w) + water (1 – w) mixed solutions versus (1/T − 1/T _hm) with a straight line to correlate the data: diamond, w = 61.15 %; inverted triangle, w = 51.20 %; triangle, w = 41.16 %; circle, w = 31.02 %; square, w = 20.78 %

Fig. 5

The van’t Hoff plots of the mole fraction solubility (ln x) of 3,3′,4,4′-oxydiphthalic anhydride in propionic acid (w) + water (1 − w) mixed solutions verse (1/T − 1/T _hm) with a straight line to correlate the data: diamond, w = 59.84 %; inverted triangle, w = 49.83 %; triangle, w = 39.84 %; circle, w = 29.86 %; square, w = 19.89 %

Table 6 Thermodynamic properties of the dissolution of 3,3′,4,4′-oxydiphthalic anhydride in acetic acid + water and propionic acid + water solvents at various temperatures

Fig. 6

Enthalpy of dissolution of solubility of 3,3′,4,4′-oxydiphthalic anhydride in binary solvents of acetic acid and propionic acid with different ratio. square, acetic acid + water; circle, propionic acid + water; w is the mass fraction of acetic acid or propionic acid in mixed solvents

To illustrate the relative contribution to the standard Gibbs energy in the dissolution process, the relative contribution values of enthalpy ($\% \xi_{H}$) and entropy ($\% \xi_{TS}$) were calculated and are presented in Table 6. It can be seen that the values of  %ζ _H are greater than 50, the standard molar enthalpy is thus the main contribution to the standard Gibbs energy during the dissolution process of 3,3′,4,4′-oxydiphthalic anhydride under the studied conditions.

4 Conclusions

The solubility data of 3,3′,4,4′-oxydiphthalic anhydride in (acetic acid + water) and (propionic acid + water) mixed solvents were determined in temperature range from 278.15 to 313.15 K. The solubility values of 3,3′,4,4′-oxydiphthalic anhydride in the aqueous solutions of acetic acid and propionic acid increased with increasing in temperature and mass fraction of acetic acid or propionic acid. Five models were employed to fit the solubility data for the 3,3′,4,4′-oxydiphthalic anhydride. The calculated solubilities agreed very well with the corresponding experimental values for 3,3′,4,4′-oxydiphthalic anhydride in binary solvent of (acetic acid + water). Thus the five models were suitable to represent the solubility of 3,3′,4,4′-oxydiphthalic anhydride in mixed solvents of (acetic acid + water) and (propionic acid + water). Thermodynamic properties of the dissolution process were evaluated by the van’t Hoff equation and the Gibbs equation. For the two mixed solvents, the values of both standard molar enthalpy change and standard molar Gibbs energy change of dissolution are all positive. This indicates that the dissolution process of 3,3′,4,4′-oxydiphthalic anhydride in (acetic acid + water) and (propionic acid + water) mixed solvents is endothermic. In addition, entropy was the driving force in the dissolution process.