Introduction

Ionic liquids (ILs) are called “green solvent in the future”, because their special physical and chemical properties have great application potential in many fields [1,2,3,4,5,6,7,8,9,10]. In 2005 and 2006, the amino acid ionic liquids (AAILs) were synthesized by Kou et al. [11] and Fukumoto et al. [12], respectively. Since AAILs are derived from natural ions and heralded a new “natural ILs” or “bio-ILs”, both academia and industry have a widespread interest in AAILs as a new, easily biodegradable and environmentally friendly ILs [13,14,15].

One of the most attractive features of ILs is that the vapor pressure is negligible at room temperature. However, in many practical applications, ILs must be used at elevated temperatures, when their vapor pressure is close to a few Pa, and cannot be ignored in the long-term use [16, 17], so that stimulates interest in study on evaporation process of ILs. In addition, evaporation enthalpy and vapor pressure are not only the basic data for the practical application of ILs, but also the important parameters for the development of liquid theory and quantum chemistry calculation. However, experimental determination of the vapor pressure and vaporization enthalpy for ILs is a very difficult and challenging work, because the vapor pressure of ILs is very low at room temperature resulting in most of the traditional measurement techniques are not applicable. In order to meet the challenge, a new method for the determination of the enthalpy of vaporization of ionic liquids has been developed in recent years [18,19,20,21,22]. Among these new methods, isothermal thermogravimetry is an effective method for the rapid determination of vaporization enthalpy. This method is not only suitable for ionic liquids, but also for other substances with low vapor pressure which is great significance for chemistry and chemical engineering and for the research and design of radio, electronics, metallurgy, medicine, environmental protection and other fields of science and technology.

Using isothermal thermogravimetry, Tong et al. [23, 24] measured the vaporization enthalpy for ILs 1-alkyl-3-methylimidazolium propionate [Cnmim][Pro](n = 4, 5, 6) and put forward the evaporation enthalpy model of ILs and a new thermodynamic function—the molar surface free energy. Using the model, the empirical equation proposed by Kabo et al. [19] was proved. As a continuation of our research work, AAILs 1-alkyl-3-methylimidazolium threonine [Cnmim][Thr](n = 3, 5) were prepared and characterized by NMR spectroscopy and element analysis. The vaporization enthalpy for AAILs [Cnmim][Thr](n = 3, 5) and benzoic acid were determined by isothermal thermogravimetry, and the thermodynamic quantities of the evaporation process for the AAILs are discussed.

Experimental

Chemicals

Anion-exchange resin (type 717) was activated by the regular method [25]. The source and purity of the materials are listed in Table S1 of the Supporting Information.

Preparation of the ILs

The ILs, [Cnmim][Thr](n = 3, 5), were prepared by a neutralization method according to Fukumoto et al. [12]. The structures of the resulting materials were confirmed by 1H NMR and 13C NMR spectroscopy (see Section A and B in the Supporting Information). Element analysis (Flash EA1112 type produced by THERMO) showed that the purity of the synthesized ionic liquids is more than 0.99 (mass fraction) (see Section C in the Supporting Information). Thermogravimetry analysis (Switzerland METTLER TOLEDO Instruments TGA/SDTA851e) showed that the initial decomposition temperature of the synthesized ILs is about 220 °C that is 493.15 K (see Section D in the Supporting Information). The mass fractions of water in the ILs determined by a Karl Fischer moisture titrator (ZSD-2 type, Shanghai anting electronic instrument factory) were (0.00650, 0.00660) ± 0.0001, respectively.

Isothermal gravimetric analysis for the ILs and benzoic acid

In this work, a METTLER TOLEDO Instruments TGA/SDTA851e was used and calibrated for temperature according to Stewart’s method [26] using indium, tin, bismuth and lead. The accuracy of the temperature measurements was adjusted to be better than ± 0.2 K, the magnitude and linearity of the balance response was checked with standard milligram masses. Firstly, the conventional TGA curves of [Cnmim][Thr](n = 3, 5) were obtained by using METTLER TOLEDO Instruments TGA/SDTA851e (see Figs. D.1 and D.2 in the Supporting Information). According to these conventional TGA curves, the initial decomposition temperature of [Cnmim][Thr](n = 3, 5) is 490.15 K and 495.15 K, respectively. And then, the range of the isothermal gravimetric analysis was determined. Then, the isothermal gravimetric analysis curve for the ILs was measured according to optimal conditions recommended by Verevkin et al. [22] and our preliminary work [23, 24], some suitable experimental parameters are selected: a sample of about 50 mg was placed in a cylindrical platinum crucible, and argon was used as purge gas with a flow rate of 100 mL min−1. For each section of the constant temperature, the loss of the sample is controlled between 1.2 and 0.3 mg. The temperature jump rate between the constant temperature sections is 10 K min−1. According to the literature value of the vapor pressure of benzoic acid, the constant temperature range of benzoic acid was determined in the isothermal gravimetric analysis. Other contents of thermogravimetry program for the benzoic acid are similar to the AAILs.

According to the isothermal gravimetric experiments, plotting (m0 − m)/kg versus (t − t0)/s (m is the sample mass, t is the time, subscript 0 means the initial state) for the ILs and the benzoic acid at each constant temperature, a series of good straight lines were obtained (see Figs. 1, 2). These straight lines are typical time-course isothermal TGA mass loss curves, and the values of their slopes of the IL and the benzoic acid, − dm/dt, are listed in Table 1, respectively. As can be seen from Figs. 1 and 2, the mass loss curves are all rigorously linear with correlation coefficients exceeding 0.999. The high linearity associated with the isothermal TGA curves reveals zero-order mass loss kinetics, providing strong evidence that the observed decrease in mass over time at constant temperature results from vaporization of the IL and does not originate from evolution of thermal degradation product or from impurity [27].

Fig. 1
figure 1

Plot of 106 × (m0 − m)/kg versus (t − t0)/s for [Cnmim][Thr](n = 3, 5) from the bottom to top: 413.15 K, 423.15 K, 433.15 K, 443.15 K, 453.15 K, 463.15 K, respectively. a [C3mim][Thr] (filled square 413.15 K: y = 2.18 × 10−3 + 1.38 × 10−4x, r2 = 0.9999, sd = 3.31 × 10−4; filled circle 423.15 K: y = − 9.06 × 10−4 + 2.75 × 10−4x, r2 = 0.9999, sd = 1.26 × 10−4; filled triangle 433.15 K: y = − 1.83 × 10−3 + 6.03 × 10−4x, r2 = 0.9999, sd = 2.92 × 10−4; filled inverted triangle 443.15 K: y = 3.44 × 10−15 + 1.33 × 10−3x, r2 = 0.9999, sd = 3.82 × 10−15; filled diamond 453.15 K: y = − 2.50 × 10−5 + 2.60 × 10−3x, r2 = 0.9999, sd = 2.94 × 10−5; filled leftpoint triangle 463.15 K: y = − 6.67 × 10−3 + 4.69 × 10−3x, r2 = 0.9998, sd = 9.36 × 10−3) and b [C5mim][Thr] (filled square 413.15 K: y = − 1.04 × 10−3 + 1.21 × 10−4x, r2 = 0.9995, sd = 9.62 × 10−4; filled circle 423.15 K: y = − 2.44 × 10−3 + 2.30 × 10−4x, r2 = 0.9994, sd = 2.02 × 10−3; filled triangle 433.15 K: y = 3.36 × 10−3 + 4.87 × 10−4x, r2 = 0.9999, sd = 1.03 × 10−3; filled inverted triangle 443.15 K: y = 7.77 × 10−3 + 1.15 × 10−3x, r2 = 0.9998, sd = 4.30 × 10−3; filled diamond 453.15 K: y = − 3.54 × 10−2 + 2.64 × 10−3x, r2 = 0.9998, sd = 7.96 × 10−3; filled leftpoint triangle 463.15 K: y = 6.30 × 10−3 + 5.13 × 10−3x, r2 = 0.9997, sd = 8.49 × 10−3, x: (t – t0)/s; y: 106 × (m0 − m)/kg

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Fig. 2
figure 2

Plot of 106 × (m0 − m)/kg versus (t − t0)/s for benzoic acid from the bottom to top: 353.15 K, 358.15 K, 363.15 K, 368.15 K, 373.15 K, 378.15 K, 383.15 K, respectively (filled square 353.15 K: y = − 1.86 × 10−3 + 8.47 × 10−5x, r2 = 0.9998, sd = 6.08 × 10−4; filled circle 358.15 K: y = − 2.59 × 10−3 + 1.14 × 10−4x, r2 = 0.9998, sd = 8.23 × 10−4; filled triangle 363.15 K: y = − 3.92 × 10−3 + 1.81 × 10−4x, r2 = 0.9998, sd = 1.09 × 10−3; filled inverted triangle 368.15 K: y = − 5.56 × 10−3 + 2.75 × 10−4x, r2 = 0.9998, sd = 1.56 × 10−3; filled diamond 373.15 K: y = − 1.09 × 10−2 + 4.04 × 10−4x, r2 = 0.9995, sd = 3.38 × 10−3; filled leftpoint triangle 378.15 K: y = − 1.28 × 10−2 + 5.74 × 10−4x, r2 = 0.9992, sd = 5.75 × 10−3; filled rightpoint triangle 383.15 K: y = − 2.07 × 10−2 + 8.44 × 10−4x, r2 = 0.9996, sd = 5.90 × 10−3, x: (t – t0)/s; y: 106 × (m0 – m)/kg)

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Table 1 Values of −dm/dt and ln[T1/2·(−dm/dt)] for [Cnmim][Thr](n = 3, 5) from 413.15 to 463.15 K and for benzoic acid from 353.15 to 383.15 K
Full size table

Results and discussion

The vapor pressure of the ILs

Price [28] has shown that it is possible to use thermogravimetry to determine the vapor pressure using the Langmuir equation for free evaporation in vacuo:

$${-}{\text{d}}m/{\text{d}}t = p\alpha \left({{\text{M}}/2\uppi{\text{R}}T} \right)^{1/2}$$
(1)

where − dm/dt is the rate of mass loss per unit area, p is the vapor pressure, M is the molecular weight of the effusing vapor, R is the gas constant, T is the absolute temperature, and α is the vaporization coefficient (usually assumed to be 1).

In the case of the isothermal thermogravimetric experiment, the ILs volatilized into a flowing gas stream at one atmosphere rather than in vacuo, it can no longer be assumed to be unity. Rearranging Langmuir equation yields:

$$p = k\nu$$
(2)

where k = (2πR)1/2/α and ν = – dm/dt(T/M)1/2.

The literature values of the vapor pressure for benzoic acid are known and listed in Table 1. According to Eq. (2), the average value of k = (8.67 ± 0.05) × 109 was obtained using the values of p and ν of benzoic acid. So the values of the vapor pressure of [Cnmim][Thr](n = 3, 5) using the values of ν obtained by the isothermal thermogravimetric experiment were calculated in the temperature range from 413.15 to 463.15 K and are listed in Table 1. However, as can be seen from Tables 1 and S3, the vapor pressure values for the ILs are unrealistically high. This may indicate that benzoic acid may not be suitable for use as a calibration material for the TGA experiments of ILs for two reasons: (1) benzoic acid is crystalline and the ILs are liquid so that the mechanism of their vaporization may be different; and (2) the temperature range of the calibration and measurements is not overlap.

The vaporization enthalpies of the ILs

Dai et al. [20] first applied isothermal thermogravimetry to experimentally measure the evaporation enthalpy of ILs and derived a working equation for the calculation of vaporization enthalpies for the ILs combining Langmuir equation with Clausius–Clapeyron equation:

$$\ln \,[({-}{\text{d}}m/{\text{d}}t)T^{1/2}] = {\text{c}} - \Delta^{\rm{g}} {}_{\rm{l}} H^{\text{\rm{o}}}_{\text{m}}/{\text{R}}T$$
(3)

where c is an empirical parameter, \(\Delta^{\text{g}} _{\text{l}} H^{\text{o}} _{\text{m}}\) is the vaporization enthalpy of the ILs at the average temperature Tav (Tav = (Σ ni Ti)/n).

According to Eq. (1), the values of ln[(− dm/dt)T1/2] of [Cnmim][Thr](n = 3, 5) at different constant temperatures were calculated and are listed in Table 1. Plotting ln[(− dm/dt)T1/2] against 1/T for [Cnmim][Thr](n = 3, 5) and benzoic acid, three good straight lines with correlation coefficient square above 0.99 and standard deviation within the experimental error were obtained (see Fig. 3). The slopes of the straight lines for ILs were used to calculate the evaporation enthalpy of the ILs at the average temperature Tav:

$$\Delta^{\text{g}} _{\rm{l}} H^{\text{o}} _{\text{m}} \left({T_{\text{av}}} \right) = {-}{\text{R}}\cdot{\text{S}}_{\text{L}}$$
(4)

where SL is the slope. The values of SL for [Cnmim][Thr](n = 3, 5) are –1.398 × 104 and –1.493 × 104, and the values of SL for benzoic acid are − 1.075 × 104. According to Eq. (3), the calculated values of \(\Delta^{\text{g}} _{\text{l}} H^{\text{o}} _{\text{m}} \left({T_{\text{av}}} \right)\) for the ILs at Tav = 438.15 K and for benzoic acid at Tav = 368.15 K are listed in Table S2.

Fig. 3
figure 3

Plot of ln[(− dm/dt) T1/2] versus 1/T for [Cnmim][Thr](n = 3, 5) and benzoic acid (filled square [C3mim][Thr]: y = 14.11 − 1.398 × 104x, r2 = 0.9986, sd =5.099 × 10−2; filled circle [C5mim][Thr]: y = 16.18 − 1.493 × 104x, r2 = 0.9949, sd =1.042 × 10−1; filled triangle benzoic acid: y = 10.12 − 1.075 × 104x, r2 = 0.9979, sd =3.936 × 10−2, x: K/T; y: ln[(− dm/dt)T1/2/(kg s−1 K1/2)])

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In order to contrast the values of vaporization enthalpy obtained from different experimental methods, \(\Delta^{\text{g}} _{\text{l}} H^{\text{o}} _{\text{m}} \left({T_{\text{av}}} \right)\) should be converted into \(\Delta^{\text{g}} _{\text{l}} H^{\text{o}} _{\text{m}}\)(298.15) at the reference temperature, 298.15 K, using following equation:

$$\Delta^{\text{g}} _{\text{l}} H^{\text{o}} _{\text{m}} \left({298.15} \right) = \Delta^{\text{g}} _{\text{l}} H^{\text{o}} _{\text{m}} \left({T_{\text{av}}} \right) + \Delta^{\text{g}} _{\text{l}} C_{\text{p}} ^{\text{o}}{}_{\text{m}} \left({298.15 - T_{\text{av}}} \right)$$
(5)

where \(\Delta^{\text{g}}_{\text{l}} C_{\text{p}} ^{\text{o}}{}_{\text{m}}\) is the difference in heat capacity of the gaseous and liquid state of the IL at constant pressure (\(\Delta^{\text{g}}_{\text{l}} C_{\text{p}} ^{\text{o}}{}_{\text{m}} = C_{\text{P}} ^{\text{o}}{}_{\text{g}} {-}C_{\text{P}} ^{\text{o}}{}_{\text{l}}\)).

Recently, Verevkin et al. [29, 30] have suggested a new approach for estimating \(\Delta^{\text{g}}_{\text{l}} C_{\text{p}} ^{\text{o}}{}_{\text{m}}\) based on statistical thermodynamics and some auxiliary experimental data:

$$\Delta^{\text{g}}_{\text{l}} C_{\text{p}} ^{\text{o}}{}_{\text{m}} = {-}2{\text{R}}{-}\left({\alpha{}_{\text{p}}^{2}/\kappa{}_{\text{T}}} \right)V_{\text{m}} T$$
(6)

where αp is the thermal expansion coefficient, K−1, κT is the isothermal compressibility in Pa−1, and Vm is the molar volume in m3·mol−1. The molar volume, as well as thermal expansion coefficient, is usually derived from the density of ILs and the temperature dependence. The values of κT can be calculated from the speed of the sound W [31].

$$\kappa{}_{\text{T}} = \rho^{- 1} (W^{- 2} + {\text{M}}T\alpha{}_{\text{p}}^{2}/C_{\text{p}} ^{\text{o}}{}_{\text{m}})$$
(7)

where ρ/kg m−3 is the density [32], \(C_{\text{p}}^{\text{o}}{}_{\text{m}}\) is the heat capacity of the liquid state of the IL at constant pressure, and the value of W was calculated by following equation:

$$W = \left[{\gamma/\left({6.3 \times 10^{- 10} \rho} \right)} \right]^{2/3}$$
(8)

where γ/J cm−2 is surface tension of the IL. The data needed in the calculation of the values of \(\Delta^{\text{g}}_{\text{l}} C_{\text{p}} ^{\text{o}}{}_{\text{m}}\) are listed in Table S2 of the Supporting Information.

Using the values of \(\Delta^{\text{g}}_{\text{l}} C_{\text{p}} ^{\text{o}}{}_{\text{m}}\), the experimental vaporization enthalpy at Tav can be adjusted to the one at 298.15 K: \(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}}\)(298.15)/kJ mol−1 = 126.0 for [C3mim][Thr] and \(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}}\)(298.15)/kJ mol−1 = 134.0 for [C5mim][Thr]. Thus, it can be seen that the contribution of each methylene (–CH2–) group in the alkyl chains of the imidazolium-based ILs, \(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}}\)(–CH2–) = 4.0 kJ mol−1, which is between \(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}}\)(–CH2–) = 4.85 ± 0.3 kJ mol−1 recommended by Archer et al. [33] and \(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}}\)(–CH2–) = 3.89 kJ mol−1 obtained by Zaitsau et al. [30].

Using Chickos’s empirical equation [34] which is widely used in sublimation enthalpy, \(\Delta^{\text{g}}_{\text{s}} H^{\text{o}}_{\text{m}}\):

$$\Delta^{\text{g}}_{\text{s}} H^{\text{o}}_{\text{m}} \left({298.15} \right) = \Delta^{\text{g}}_{\text{s}} H^{\text{o}}_{\text{m}} \left(T \right) + 0.0320_{{}} (T{-}298.15)$$
(9)

the sublimation enthalpy of benzoic acid, \(\Delta^{\text{g}}_{\text{s}} H^{\text{o}}_{\text{m}}\)(298.15) = 91.45 kJ mol−1, at 298.15 K was calculated and the calculated value is in good agreement with (90.8 ± 0.6) kJ mol−1 measured by Price [28]. This fact further indicates the reliability of thermogravimetric method for determining the vaporization enthalpy of ILs.

The evaporation enthalpy model and molar surface Gibbs free energy for ILs

Tong et al. [23] proposed the evaporation enthalpy model for aprotic ILs, and the working equation is:

$$\gamma V_{\text{m}}^{2/3} = \left({1 - x} \right){\text{N}}^{2/3} \Delta \varepsilon \left({\text{kin}} \right) + \left({1 - x} \right){\text{N}}^{- 1/3} (\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}} - {\text{R}}T)$$
(10)

where γ is the surface tension, Vm is the molar volume, \(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}}\) is the vaporization enthalpy, N is Avogadro constant, Δε(kin) is the difference in kinetic energy of the gaseous and liquid state of a pair of positive and negative ions, x is the ratio of the coordination number between on the surface and in bulk phase. This model successfully proved the rationality of Kabo empirical equation [19].

Multiplied Eq. (10) by N1/3 yields:

$$g = \left({1 - x} \right)N\Delta \varepsilon \left({\text{kin}} \right) + \left({1 - x} \right)(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}} - {\text{R}}T)$$
(11)

where g = (γV 2/3m N1/3) is molar surface Gibbs free energy, NΔε(kin) is the difference in molar kinetic energy of the gaseous and liquid state of ILs so that Eq. (11) changed to:

$$g = A + B(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}} - {\text{R}}T)$$
(12)

where A and B are the empirical constants. Using Eq. (12), the predicted values: \(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}}\) = 125.7 kJ mol−1 for [C3mim][Thr] and \(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}}\) = 133.1 kJ mol−1 for [C5mim][Thr], were obtained in our previous article [35] and are in good agreement with the corresponding experimental values in this work. This fact shows that Tong’s model is reasonable.

The vaporization entropy of the ILs

In the previous article [32], the critical temperature, Tc = 1102 K for IL [C3mim][Thr] and Tc = 1044 K for IL [C5mim][Thr], was estimated. Rebelo et al. [36] pointed out that the hypothetical normal boiling point, Tb, of an IL is approximately equal to 0.60 of the critical temperature, that is, Tb ≈ 0.6Tc, so that Tb = 661 K for IL [C3mim][Thr] and Tb = 626 K for IL [C5mim][Thr]. These two ionic liquids exist in the form of hydrogen bonds acting on each other. Therefore, both of them have high boiling points. And the steric hindrance of [C5mim][Thr] is rather high, so the hydrogen bonds weaken and the boiling point drops. Therefore, the boiling point of [C3mim][Thr] is slightly higher than the boiling point of [C5mim][Thr].

At the hypothetical normal boiling point of an IL, the vapor pressure of the IL pb = 101.325 kPa, so that the vapor pressure of the IL, p1, at any temperature, T1, can be estimated according to Clausius–Clapeyron equation:

$$\ln \left({p_{\text{b}}/p_{1}} \right) = [\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}} \left(T \right)/{\text{R}}]\left({1/T_{1} - 1/T_{\text{b}}} \right)$$
(13)

where \(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}} \left(T \right)\) is the average vaporization enthalpy between T1 and Tb. The values of p1 are listed in Table S3 of the Supporting Information. Compared with the data in Table 1, the vapor pressure in Table S3 is much smaller at the similar temperature. This fact also shows that benzoic acid is not suitable as a reference substance for the vapor pressure of ILs.

At the hypothetical normal boiling point, the Gibbs free energy of ionic liquid evaporation is zero, \(\Delta^{\text{g}}_{\text{l}} G^{\text{o}}_{\text{m}} \left({T_{\text{b}}} \right) = 0\), so that the vaporization entropy \(\Delta^{\text{g}}_{\text{l}} S\left({T_{\text{b}}} \right)\) is:

$$\Delta^{\text{g}}_{\text{l}} S^{\text{o}}_{\text{m}} \left({T_{\text{b}}} \right) = \Delta^{\text{g}}{}_{\text{l}} H^{\text{o}}_{\text{m}} \left({T_{\text{b}}} \right)/T_{\text{b}}$$
(14)

The calculated values of \(\Delta^{\text{g}}_{\text{l}} S^{\text{o}}_{\text{m}}\)(Tb) and \(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}} \left({T_{\text{b}}} \right)\) are listed in Table S3 of the Supporting Information. In terms of the difference in heat capacity of the gaseous and liquid state of the IL at constant pressure, \(\Delta^{\text{g}}_{\text{l}} C_{\text{p}}^{\text{o}}{}_{\text{m}}\), \(\Delta^{\text{g}}_{\text{l}} S^{\text{o}}_{\text{m}} \left({T_{\text{b}}} \right)\) can be converted to \(\Delta^{\text{g}}_{\text{l}} S\)(298.15) which is the vaporization entropy at 298.15 K:

$$\Delta^{\text{g}}_{\text{l}} S^{\text{o}}_{\text{m}} \left({298.15} \right) = \Delta^{\text{g}}_{\text{l}} S^{\text{o}}_{\text{m}} \left({T_{\text{b}}} \right) + \Delta^{\text{g}}{}_{\text{l}} C_{\text{P}} \ln \left({298.15/T_{\text{b}}} \right)$$
(15)

At 298.15 K, the Gibbs free energy of ionic liquid evaporation, ΔvapG(298.15) is:

$$\Delta^{\text{g}}_{\text{l}} G^{\text{o}}_{\text{m}} \left({298.15} \right) = D^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}} \left({298.15} \right) - 298.15\Delta^{\text{g}}_{\text{l}} S^{\text{o}}_{\text{m}} \left({298.15} \right)$$
(16)

By using the similar method, the evaporation Gibbs free energy, \(\Delta^{\text{g}}_{\text{l}} G^{\text{o}}_{\text{m}} \left(T \right)\), and the evaporation entropy, \(\Delta^{\text{g}}_{\text{l}} S^{\text{o}}_{\text{m}} \left(T \right)\), for [Cnmim][Thr](n = 3, 5) can be calculated at any temperature. The values of \(\Delta^{\text{g}}_{\text{l}} G^{\text{o}}_{\text{m}} \left(T \right)\), \(\Delta^{\text{g}}_{\text{l}} S^{\text{o}}_{\text{m}} \left(T \right)\), \(T\Delta^{\text{g}}_{\text{l}} S^{\text{o}}_{\text{m}} \left(T \right)\) and \(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}} \left(T \right)\) are listed in Table S3 in the Supporting Information.

Using the data in Table S3, the curves of \(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}} \left(T \right)\) versus T, \(T\Delta^{\text{g}}_{\text{l}} S^{\text{o}}_{\text{m}} \left(T \right)\) versus T, \(\Delta^{\text{g}}_{\text{l}} G^{\text{o}}_{\text{m}} \left(T \right)\) versus T and p versus T are plotted in Fig. 4. Table S3 and Fig. 4 show that the evaporation Gibbs free energy of ILs [Cnmim][Thr](n = 3, 5) decreases with increasing temperature, reaching the hypothetical boiling point is zero, while the evaporation entropy increased with the temperature increasing. This fact shows that the evaporation entropy is the driving force of the evaporation process of the ILs.

Fig. 4
figure 4

Plot of \(\Delta^{\text{g}}{}_{\text{l}} H^{\text{o}}{}_{\text{m}}\)(T), T \(\Delta^{\text{g}}{}_{\text{l}} S^{\text{o}}{}_{\text{m}}\)(T), \(\Delta^{\text{g}}{}_{\text{l}} G^{\text{o}}{}_{\text{m}}\)(T) and p versus T for [Cnmim][Thr](n = 3, 5). a [C3mim][Thr] (filled square \(\Delta^{\text{g}}{}_{\text{l}} H^{\text{o}}{}_{\text{m}}\)(T) versus T: y = 146.74 − 0.0697x, r2 = 1.000, sd = 5.799 × 10−14; filled circle T \(\Delta^{\text{g}}{}_{\text{l}} S^{\text{o}}{}_{\text{m}}\)(T) versus T: y = 27.283 + 0.118x, r2 = 0.9980, sd = 3.513 × 10−1; filled triangle \(\Delta^{\text{g}}{}_{\text{l}} G^{\text{o}}{}_{\text{m}}\)(T) versus T: y = 119.46 − 0.188x, r2 = 0.9992, sd = 3.513 × 10−1, ○ p versus T: x: T; y: p) and b [C5mim][Thr] (filled square \(\Delta^{\text{g}}{}_{\text{l}} H^{\text{o}}{}_{\text{m}}\)(T) versus T: y = 154.99 − 0.0705x, r2 = 1.000, sd = 3.259 × 10−13; filled circle T \(\Delta^{\text{g}}{}_{\text{l}} S^{\text{o}}{}_{\text{m}}\)(T) versus T: y = 27.593 + 0.139x, r2 = 0.9985, sd = 3.552 × 10−1; filled triangle \(\Delta^{\text{g}}{}_{\text{l}} G^{\text{o}}{}_{\text{m}}\)(T) versus T: y = 127.40 − 0.209x, r2 = 0.9994, sd = 3.552 × 10−1; open circle p versus T: x: T; y: p)

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Conclusions

With the use of the isothermogravimetry, the vaporization enthalpies, \(\Delta^{\text{g}}_{\text{l}} H^{\text{o}}_{\text{m}}\)(298.15), of AAILs [Cnmim][Thr](n = 3, 5) and the sublimation enthalpy were determined. However, using benzoic acid as reference material, the determined vapor pressure and vaporization enthalpy for the AAILs were unrealistically high. This means that benzoic acid may not be suitable for use as a reference material for the TGA experiments of ILs. But the determined sublimation enthalpy is in good agreement with that measured by Price [28]. According to Rebelo et al. [36], the hypothetical normal boiling point, Tb, and the vaporization entropy, \(\Delta^{\text{g}}{}_{\text{l}} S\left({T_{\text{b}}} \right)\), of the AAILs were determined. Then, their evaporation enthalpy, evaporation entropy, evaporation free energy and vapor pressure were calculated at different temperatures. The results of this work show that thermostatic thermogravimetry is based on modern high-precision, highly automated thermogravimetric analyzer, which is an effective method for the rapid determination of enthalpy of ILs.