Introduction

As challenge emerged in the burgeoning chemical industries and environmental pollution due to use of volatile organic solvents, researchers were confined in a box what they could and could not do. They explored ways to use new compounds like ionic liquids in place of traditional volatile and corrosive organic liquids to minimize their use. Creating new thermodynamic data on liquid mixtures containing ionic liquid due to their unique properties [1, 2] will foster new opportunities for their use in chemical industries. Heat capacity of liquid or liquid mixtures is one of the properties for use in thermal applications. Accurate heat capacity and excess heat capacity data of liquid/liquid mixtures are crucial (i) for the design and operation of heat transfer processes [3, 4] and (ii) for units such as fractionation tower used to separate mixture [5]. The sensitivity of tower’s segment is determined by an energy balance which in turn involves liquid/liquid mixture enthalpy or heat capacity data. The use of ionic liquids due to their unique properties especially low volatility or their mixtures with organic liquids can be used as alternative sources in designing of equipments or environmental treatment technologies [6–9]. Ionic liquids having imidazolium cations are considered to be efficient ionic liquids for high absorption of carbon dioxide, diesel extractive desulfurization (EDS), oxidative desulfurization (ODS) and catalytic oxidative desulfurization (ECODS) and seem to be a good alternative for diesel desulfurization [10–14]. Further, several investigations on imidazolium-based ionic liquids possessing tetrafluoroborate anion have been carried out for their use in capacitors, solar cells, fuel cells and batteries [15–20]. Pyrrolidin-2-one and 1-methyl pyrrolidin-2-one have potential to be used in the solvent extraction process for separating polar substances from nonpolar substances, petrochemicals and biological applications [21, 22]. Cyclopentanone and cyclohexanone are used as safety solvents and important intermediates for the synthesis of many organic compounds which are used in chemical, pharmaceutical and cosmetic industries [22–25]. Liquid mixtures consisting of 1-ethyl-3-methylimidazolium tetrafluoroborate, pyrrolidin-2-one, 1-methyl pyrrolidin-2-one, cyclopentanone and cyclohexanone may, therefore, comprise a class of mixtures of importance in chemical, pharmaceutical and biological industries. In continuation of our earlier studies on thermodynamic properties of binary/ternary liquid mixtures containing 1-ethyl-3-methylimidazolium tetrafluoroborate as one of the component [26–29], we report here excess heat capacity, \( (C_{\text{P}}^{\text{E}} )_{123} \), data of 1-ethyl-3-methylimidazolium tetrafluoroborate (1) + pyrrolidin-2-one or 1-methyl pyrrolidin-2-one (2) + cyclopentanone or cyclohexanone (3) ternary mixtures.

Experimental

1-Ethyl-3-methylimidazolium tetrafluoroborate [emim][BF4] (mass fraction: 0.980) was used without further purification. The water content in ionic liquid was regularly checked using Karl Fischer titration [30] and found to be less than 320 ppm. Pyrrolidin-2-one (2-Py) (mass fraction: 0.994) was purified by vacuum distillation over calcium oxide [31], 1-methyl pyrrolidin-2-one (NMP) (mass fraction: 0.992) was purified by fractional distillation under reduced pressure [32], and cyclopentanone (Fluka, mass fraction: 0.991) and cyclohexanone (Fluka, mass fraction 0.988) were purified by fractional distillation [33]. The source of liquids, along with methods of purification and final purity, is reported in Table 1. Densities, ρ, speeds of sound, u, and heat capacities, C P, of the pure liquids at studied temperatures are listed in Table 2 and also compared with literature values [23, 25, 33–53]. The ρ and u values of the purified liquids were measured using a density and sound analyzer apparatus (Anton Paar DSA 5000) in the manner as described elsewhere [54, 55]. The uncertainties in the density and speed of sound measurements are ±0.5 kg m−3 and 0.1 m s−1, respectively. Further, uncertainty in the temperature measurement is ±0.01 K.

Table 1 Details of studied chemicals, CAS number, source, purification method, purity and analysis method
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Table 2 Comparison of densities, ρ, speeds of sound, u, and heat capacities,\( C_{\text{P}} \), of pure liquids with their literature values at T = (293.15 to 308.15) K
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The molar heat capacities, C P, of pure liquids and the present mixtures were measured by differential scanning calorimeter Micro DSC (Model—μDSC 7 Evo) manufactured by M/S SETARAM instrumentation, France, as described elsewhere [56]. The calibration of equipment was done by Joule effect method which in turn was controlled by SETARAM software. The joule effect calibration was checked by measuring heat of fusion of naphthalene (147.78 J g−1) [57]. The heat capacity of a liquid or their mixtures was measured in a standard batch cell (Hastalloy C276) composed of a cylinder of 6.4 mm of internal diameter and 19.5 mm height and had a capacity of containing 1 cm3 of a liquid. The reference experimental cell was filled with water (equivalent to the mass of liquid in a standard batch cell). The mole fraction of each mixture was made by measuring the masses of the components of mixtures in airtight glass bottles using an electric balance (Mettler AX-205 Delta) with an uncertainty of ±10−5 g. For a scanning sequence, the initial and final temperatures were supplied along with heating rate of 0.4 K min−1. The temperature cycle and scanning rate of isothermal level were maintained by software provided by SETARAM instrumentation. The uncertainty in measured heat capacity, C P, values is 0.3 %.The uncertainty in mole fraction is 1 × 10−4. The uncertainty in the temperature measurement is ±0.02 K.

Results

The molar heat capacities, C P, of [emim][BF4] (1) + 2-Py or NMP (2) + cyclopentanone or cyclohexanone (3) ternary mixtures are listed in Table 3. The excess heat capacities, \( (C_{\text{P}}^{\text{E}} )_{123} \), for (1 + 2 + 3) mixtures were calculated by Eq. 1

$$ (C_{\text{P}}^{\text{E}} )_{123} = (C_{\text{P}} ) -\sum\limits_{i = 1}^{3} {x_{\text{i}} (C_{\text{P}} )_{\text{i}} }$$
(1)

where (C P), (C P)i (i = 1 or 2 or 3) and x i (i = 1 or 2 or 3) denote molar heat capacity of the ternary mixtures, molar heat capacity and mole fraction of pure components, respectively. Such \( (C_{\text{P}}^{\text{E}} )_{123} \) values for the present (1 + 2 + 3) mixtures are listed in Table 3. The \( \left( {C_{\text{P}}^{\text{E}} } \right)_{123} \) were fitted to Redlich–Kister [58] equation

$$\begin{array}{lll} (C_{\text{P}}^{\text{E}} )_{123} = x_{1} x_{2} \left[ {\mathop \sum \limits_{n = 0}^{2} \left( {C_{\text{P}} } \right)_{12}^{(\rm{n})} (x_{1} - x_{2} )^{\rm{n}} } \right] \\ \qquad \qquad \, + x_{2} x_{3\,\,} \left[ {\mathop \sum \limits_{n = 0}^{2} \left( {C_{\text{P}} } \right)_{23}^{(\rm{n})} (x_{2} - x_{3} )^{\rm{n}} } \right]\\ \qquad \qquad \, + x_{1} x_{3} \left[ {\mathop \sum \limits_{n = 0}^{2} \left( {C_{\text{P}} } \right)_{13}^{(\rm{n})} (x_{3} - x_{1} )^{\rm{n}} } \right] \\ \qquad \qquad \, + x_{1} x_{2} x_{3} \left[ {\mathop \sum \limits_{n = 0}^{2} \left( {C_{\text{P}} } \right)_{123}^{(\rm{n})} (x_{2} - x_{3} )^{\rm{n}} x_{1}^{\rm{n}} } \right] \end{array} $$
(2)

where \( (C_{\text{P}} )_{123}^{(\text{n})} (n = 0 - 2) \), etc., are characteristic parameters of binaries (1 + 2), (2 + 3) and (1 + 3) of (1 + 2 + 3) mixtures and were taken from literature [59–61]. The \( (C_{\text{P}} )_{123}^{(\text{n})} (n = 0 - 2) \), etc., are ternary adjustable parameters of the (1 + 2 + 3) ternary mixtures and were calculated by least-square optimization of these parameters. Such parameters along with standard deviations, \( \sigma (C_{\text{P}}^{\text{E}} )_{123} \), defined by Eq. 3

$$ \sigma \left( {(C_{\rm{P}}^{\rm{E}} )_{123} } \right) = \left[ {{{\sum {\left( {\left( {C_{\rm{P}}^{\rm{E}} } \right)_{{123\{ \exp tl\} }} - \left( {C_{\rm{P}}^{\rm{E}} } \right)_{{123\{ calc.equation(2)\} }} } \right)}^{2} } \mathord{\left/ {\vphantom {{\sum {\left( {\left( {C_{\rm{P}}^{\rm{E}} } \right)_{{123\{ \exp tl\} }} - \left( {C_{\rm{P}}^{\rm{E}} } \right)_{{123\{ calc.equation(2)\} }} } \right)}^{2} } {(m - n)}}} \right. \kern-0pt} {(m - n)}}} \right]^{0.5} $$
(3)

(where m is the number of data points and n is the number of adjustable parameters of Eq. 2) are listed in supporting Table 1S. Various surfaces generated by \( (C_{\text{P}}^{\text{E}} )_{123} \) data are shown in Figs. 1–4. In Fig. 1, \( (C_{\text{P}}^{\text{E}} )_{123} \) values (corresponding to 1–2 axis) were obtained by keeping x 3 constant and varying the values of x 1 and x 2 (shown as green line); \( (C_{\text{P}}^{\text{E}} )_{123} \) values (corresponding to 1–3 axis) were obtained by keeping x 2 constant and varying the values x 1 and x 3 (shown as red line).

Table 3 Comparison of experimental \( \left( {C_{\text{P}} } \right)_{123} \) data for the various (1 + 2 + 3) ternary mixtures with values evaluated from the Graph and Flory theories at T = (293.15–308.15) K
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Fig. 1
figure 1

Excess heat capacities, \( \left( {C_{\text{P}}^{\text{E}} } \right)_{123} \), for 1-ethyl-3-methylimidazolium tetrafluoroborate (1) + pyrrolidin-2-one (2) + cyclopentanone (3) mixture at 298.15 K. (Color figure online)

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Fig. 2
figure 2

Excess heat capacities, \( \left( {C_{\text{P}}^{\text{E}} } \right)_{123} \), for 1-ethyl-3-methylimidazolium tetrafluoroborate (1) + pyrrolidin-2-one (2) + cyclohexanone (3) mixture at 298.15 K. (Color figure online)

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Fig. 3
figure 3

Excess heat capacities, \( \left( {C_{\text{P}}^{\text{E}} } \right)_{123} \), for 1-ethyl-3-methylimidazolium tetrafluoroborate (1) + 1-methylpyrrolidin-2-one (2) + cyclopentanone (3) mixture at 298.15 K. (Color figure online)

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Fig. 4
figure 4

Excess heat capacities, \( \left( {C_{\text{P}}^{\text{E}} } \right)_{123} \), for 1-ethyl-3-methylimidazolium tetrafluoroborate (1) + 1-methylpyrrolidin-2-one (2) + cyclohexanone (3) ternary mixture at 298.15 K. (Color figure online)

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Discussion

The excess heat capacities, \( (C_{\text{P}}^{\text{E}} )_{123} \), of the [emim][BF4] (1) + NMP or 2-Py (2) + cyclopentanone or cyclohexanone (3) mixtures are not available in literature for comparison with measured results. The \( (C_{\text{P}}^{\text{E}} )_{123} \) of the present (1 + 2 + 3) mixtures are positive over entire mole fraction of (1) and (2). The positive \( (C_{\text{P}}^{\text{E}} )_{123} \) values suggest that capability of cyclohexanone or cyclopentanone molecules to build a non-random structure in mixed state (due to interactions with [emim][BF4]:NMP or 2-Py molecular entities) is superior to the effect caused by disruption of associated NMP or 2-Py entities, and interactions between [emim][BF4]:NMP or 2-Py molecular entities [37]. The \( (C_{\text{P}}^{\text{E}} )_{123} \) values of [emim][BF4] (1) + NMP or 2-Py (2) + cyclohexanone (3) mixtures are higher than those of [emim][BF4] (1) + NMP or 2-Py (2) + cyclopentanone (3) mixtures. It may be due to reason that cyclohexanone is more basic in character than cyclopentanone [62] and also possesses chair form with almost no strain. Thus, cyclohexanone will give strong interactions and more compact structure with [emim][BF4]: NMP or 2-Py molecular entities as compared to cyclopentanone. Higher\( (C_{\text{P}}^{\text{E}} )_{123} \) for [emim][BF4] (1) + 2-Py (2) + cyclopentanone or cyclohexanone (3) mixtures than those for [emim][BF4] (1) + NMP (2) + cyclopentanone or cyclohexanone (3) mixtures may be due to lesser mixing of 2-Py with [emim][BF4] and cyclohexanone because of the preference of 2-Py molecules to hydrogen bond with themselves. The \( \left( {\frac{{\partial C_{\text{P}}^{\text{E}} }}{\partial T}} \right) \) for [emim][BF4] (1) + 2-Py (2) + cyclopentanone or cyclohexanone (3) mixtures are positive which in turn suggest strong interactions occurring in mixed state due to disruption of self-associated entities of 2-Py and ion–dipole interactions in [emim][BF4]. However, \( \left( {\frac{{\partial C_{\text{P}}^{\text{E}} }}{\partial T}} \right) \) for [emim][BF4] (1) + NMP (2) + cyclopentanone or cyclohexanone (3) mixtures are negative. Decreasing \( (C_{\text{P}}^{\text{E}} )_{123} \) values with increasing temperature can be associated with decrease of molecular interactions between like molecules compared with unlike molecules in mixed state.

The \( (C_{\text{P}}^{\text{E}} )_{123} \) data were next analyzed in terms of Graph and Flory’s theories.

Graph theory

Excess heat capacities of ternary mixtures

The analysis of excess molar volumes, V E, excess isentropic compressibilities, \( \kappa_{\text{S}}^{\text{E}} \), excess molar enthalpies, H E, and excess heat capacities, \( C_{\text{P}}^{\text{E}} \), and IR studies of [emim][BF4] (1) + 2-Py or NMP or cyclopentanone or cyclohexanone (2); 2-Py or NMP (1) + cyclopentanone or cyclohexanone (2) mixtures [37–39, 63] have shown that (1) [emim][BF4] exists as monomer; (2) 2-Py or NMP exists as associated molecular entities; and (3) cyclopentanone or cyclohexanone is characterized by dipole–dipole interactions. The [emim][BF4] (1) + 2-Py or NMP (2) + cyclopentanone or cyclohexanone (3) mixtures can, therefore, be assumed to involve processes; (a) establishment of unlike (i) 1 − 2n (n = 2), (ii) 2n − 3n (n = 2) and (iii) 1 − 3n (n = 2) contacts; (b) unlike contact formation between the constituent molecules ruptures self-association or dipole–dipole interactions with (i) 2n and (ii) 3n molecules to form 2 and 3 molecules and enhances the randomness in mixed state; (c) 1, 2 and 3 constituent molecules undergo interactions to form (i) 1:2, (ii) 2:3 and (iii) 1:3 molecular complexes, which in turn leads to non-randomness in mixed state as compared to pure state. If χ 12, χ 23, χ 13; χ 22, \( \chi_{{\text{33}}} \) and \( \chi_{12}^{{\prime }} ,\chi_{12}^{{\prime \prime }} ,\chi_{12}^{{{\prime \prime \prime }}} \) are molar interaction energy parameters for 1 − 2n, 2n − 3n, 1 − 3n unlike contacts, respectively, rupture of associated molecular entities 2n and 3n, and molecular interactions between 1, 2 and 3 constituent molecules to yield randomness and non-randomness, respectively, in mixed state, then change in thermodynamic property, \( \left( {\Delta C_{\bf{P}} } \right) \), due to processes (a) (i–iii), (b) (i–ii) and (c) (i–iii) were given [56, 64–69] by relation:

$$ \begin{aligned} \left( {C_{\text{P}}^{\text{E}} } \right)_{123} = & \left[ {\frac{{x_{1} x_{2} v_{2} }}{{\sum\nolimits_{i = 1}^{2} {x_{1} v_{1} } }}} \right]\left[ {\chi_{12} + x_{1} \chi_{22} + x_{2} \chi_{12}^{{\prime }} } \right] + \left[ {\frac{{x_{2} x_{3} v_{3} }}{{\sum\nolimits_{j = 2}^{3} {x_{2} v_{2} } }}} \right]\left[ {\chi_{23} + x_{3} \chi_{12}^{{\prime \prime }} } \right] \\ \quad + \left[ {\frac{{x_{3} x_{1} v_{3} }}{{\sum\nolimits_{k = 3}^{1} {x_{2} v_{2} } }}} \right]\left[ {\chi_{13} + x_{3} \chi_{33} + x_{1} \chi_{12}^{{{\prime \prime \prime }}} } \right] \\ \end{aligned} $$
(4)

As \( v_{ 2} /v_{ 1} = ^{ 3} \xi_{ 1} /^{ 3} \xi_{ 2} \), where (\( ^{3} \xi_{i} \)), (\( ^{3} \xi_{i} \))m (i = 1 or 2 or 3), etc., are connectivity parameters of third degree of molecules in pure as well as mixed state and are defined [70] by

$$ ^{3} \xi = \sum\limits_{{{\text{m}} < {\text{n}} < {\text{o}} < {\text{p}}}} {(\delta_{\text{m}}^{\nu } \delta_{\text{n}}^{\nu } \delta_{\text{o}}^{\nu } \delta_{\text{p}}^{\nu } )^{ - 0.5} } $$
(5)

The \( \delta_{\text{m}}^{\nu } \), etc., values reflect the valency of the atoms forming the bond and are expressed as [71] \( \delta^{v} \) = Z m − h, where Z m is the maximum valency of the atom and h is the number of hydrogen atom attached to it. The \( ^{3} \xi \) values for the constituent molecules were taken from literature [37, 61, 63] consequently; Eq. 4 was reduced to

$$ \begin{aligned} \left( {C_{\text{P}}^{\text{E}} } \right)_{123} = & \left[ {\frac{{x_{1} x_{2} \left( {{}^{3}\xi_{1} /\xi_{2} } \right)}}{{x_{1} + x_{2} \left( {{}^{3}\xi_{1} /{}^{3}\xi_{2} } \right)}}} \right]\left[ {\chi_{12} + x_{1} \chi_{22} + x_{1} \chi_{12}^{{\prime }} } \right] + \left[ {\frac{{x_{2} x_{3} \left( {{}^{3}\xi_{2} /{}^{3}\xi_{3} } \right)}}{{x_{2} + x_{3} \left( {{}^{3}\xi_{2} /{}^{3}\xi_{3} } \right)}}} \right]\left[ {\chi_{23} + x_{3} \chi_{12}^{{\prime \prime }} } \right] \\ \quad + \left[ {\frac{{x_{3} x_{1} \left( {{}^{3}\xi_{3} /{}^{3}\xi_{1} } \right)}}{{x_{3} + x_{1} \left( {{}^{3}\xi_{3} /{}^{3}\xi_{1} } \right)}}} \right]\left[ {\chi_{13} + x_{3} \chi_{33} + \chi_{12}^{{{\prime \prime \prime }}} } \right] \\ \end{aligned} $$
(6)

For the present mixtures, we assumed that \( \chi_{ 1 2} \cong \chi^{\prime}_{ 1 2} = \chi_{12}^{*} ;\chi_{ 2 3} \cong \chi^{\prime\prime}_{12} = \chi_{ 2 3}^{*} ;\chi_{ 1 3} \cong \chi^{\prime\prime\prime}_{12} = \chi_{13}^{*} \) and χ 22 ≅ χ 33 = χ * and then Eq. 6 was reduced to

$$ \begin{aligned} \left( {C_{\text{P}}^{\text{E}} } \right)_{123} = & \left[ {\frac{{x_{1} x_{2} \left( {{}^{3}\xi_{1} /{}^{3}\xi_{2} } \right)}}{{x_{1} + x_{2} \left( {{}^{3}\xi_{1} /{}^{3}\xi_{2} } \right)}}} \right]\left[ {(1 + x_{2} )\chi_{12}^{*} + x_{1} \chi^{*} } \right] + \left[ {\frac{{x_{2} x_{3} \left( {{}^{3}\xi_{2} /{}^{3}\xi_{3} } \right)}}{{x_{2} + x_{3} \left( {{}^{3}\xi_{2} /{}^{3}\xi_{3} } \right)}}} \right]\left[ {\left( {1 + x_{3} } \right)\chi_{23}^{ * } } \right] \\ \quad + \left[ {\frac{{x_{3} x_{1} \left( {{}^{3}\xi_{3} /{}^{3}\xi_{1} } \right)}}{{x_{3} + x_{1} \left( {{}^{3}\xi_{3} /{}^{3}\xi_{1} } \right)}}} \right]\left[ {\left( {1 + x_{1} } \right)\chi_{13}^{ * } + x_{3} \chi^{ * } } \right] \\ \end{aligned} $$
(7)

Equation 7 contains four unknown χ *12 , χ *23 , χ *13 and χ *parameters. These parameters were commuted utilizing \( (C_{\text{P}}^{\text{E}} )_{123} \) data at four arbitrary compositions and then subsequently used to predict \( (C_{\text{P}}^{\text{E}} )_{123} \) data at other values of x 1 and x 2. Such \( (C_{\text{P}}^{\text{E}} )_{123} \) values are listed in Table 3 and also compared with their corresponding experimental values. The χ *12 , χ *23 , χ *13 and χ * parameters and mean deviations between \( (C_{\text{P}}^{\text{E}} )_{123} \) values and \( (C_{\text{P}}^{\text{E}} )_{123} \) values calculated by Graph theory, \( \sigma (C_{\text{P}}^{\text{E}} )_{{123\;{\text{Graph}}}} \), are also reported in supporting Table 2S. Perusal of data in Table 3 indicates that \( (C_{\text{P}}^{\text{E}} )_{123} \) values determined by Graph theory are in agreement with experimental data which in turn support various assumptions in deriving Eq. 7.

Flory’s theory

Differentiating Flory’s expression for excess molar enthalpies [72, 73] for binary and ternary mixtures with respect to the temperature, T, and excess heat capacities, \( (C_{\text{P}}^{\text{E}} )_{123} \), for ternary mixtures was expressed by

$$ \left( {C_{\text{P}}^{\text{E}} } \right)_{123} = - \sum\limits_{i = 1}^{3} {\frac{{x_{\rm{i}} P_{\rm{i}}^{*} \tilde{v}_{\rm{i}}^{*} \alpha_{\rm{i}} }}{{\tilde{v}_{\rm{i}} }} + \left( {\frac{\alpha }{{\tilde{v}}}} \right)\left[ {\sum\limits_{i = 1}^{3} {x_{\rm{i}} P_{\rm{i}}^{*} v_{\rm{i}}^{*} - \sum\limits_{i = 1}^{3} {x_{\rm{i}} v_{\rm{i}}^{*} \theta_{\rm{j}} \chi_{12}^{**} } } } \right]} $$
(8)

where \( \tilde{v}_{\text{i}}^{*} \),\( P_{\text{i}}^{*} \) and \( \tilde{v}_{\text{i}}^{{}} \)(i = 1 or 2 or 3) are the characteristic volume, characteristic pressure and reduced volume of pure component (i) and \( \tilde{v} \) is reduced volume of mixture and all the terms have the same significance as described elsewhere [72, 73]. The Flory parameters for the liquids under investigations are taken from literature [61, 63]. Flory assumed that interaction energy parameters, \( \chi_{12}^{**} \), etc., for sub-binaries of (1 + 2 + 3) ternary mixtures, which in turn are evaluated by using their H E data at equimolar composition, were assumed to be independent of temperature by Flory. However, Benson and D’ Arcy [74] assumed that \( \chi_{12}^{**} \), etc., parameters for binary mixtures should be a function of temperature. Consequently, \( (C_{\text{P}}^{\text{E}} )_{123} \) values for ternary mixtures were then expressed by relation

$$ \left( {C_{\text{P}}^{\text{E}} } \right)_{123} = - \sum\limits_{i = 1}^{3} {\frac{{x_{\rm{i}} P_{\rm{i}}^{*} \tilde{v}_{\rm{i}}^{*} \alpha_{\rm{i}} }}{{\tilde{v}_{\rm{i}} }} + \left( {\frac{\alpha }{{\tilde{v}}}} \right)\left[ {\sum\limits_{i = 1}^{3} {x_{\rm{i}} P_{\rm{i}}^{*} v_{\rm{i}}^{*} - \sum\limits_{i = 1}^{3} {x_{\rm{i}} v_{\rm{i}}^{*} \theta_{\rm{j}} \chi_{12}^{**} } } } \right]} + \sum\limits_{i = 1}^{3} {\frac{{x_{\rm{i}} v_{\rm{i}}^{*} \theta_{\rm{j}} }}{{\tilde{v}}}\left( {\frac{{\partial \chi_{12}^{**} }}{\partial T}} \right)} $$
(9)

The reduced volumes, \( \tilde{v} \), and thermal coefficient, α, of ternary mixtures were calculated using

$$ \tilde{v} = {{\left( {V_{123}^{\text{E}} + \sum\limits_{i = 1}^{3} {x_{\rm{i}} v_{\rm{i}} } } \right)} \mathord{\left/ {\vphantom {{\left( {V_{123}^{\text{E}} + \sum\limits_{i = 1}^{3} {x_{i} v_{i} } } \right)} {\sum\limits_{i = 1}^{3} {x_{\rm{i}} v_{\rm{i}}^{*} } }}} \right. \kern-0pt} {\sum\limits_{i = 1}^{3} {x_{\rm{i}} v_{\rm{i}}^{*} } }} $$
(10)
$$ \alpha = \sum\limits_{i = 1}^{3} {x_{\rm{i}} \alpha_{\rm{i}} } $$
(11)

where \( V_{{\hbox{123}}}^{\hbox{E}} \) represent excess molar volumes of ternary (1 + 2 + 3) mixtures. Such \( \chi_{12}^{**} \), etc., values for the various binaries were calculated using H E value at equimolar composition that was taken from literature [37, 61, 63]. The calculated \( \left( {C_{\text{P}}^{\text{E}} } \right)_{123} \) values via Eqs. 8–11 for the present mixtures are compared with experimental values and presented in Table 3. The values of \( \chi_{12}^{**} \), etc., are recorded in supporting Table 2S. Examination of data in Table 3 indicates that Flory’s theory correctly predicts the sign of \( \left( {C_{\text{P}}^{\text{E}} } \right)_{123} \) values. However, quantitative agreement is poor. The failure of theory to correctly predict the sign of \( \left( {C_{\text{P}}^{\text{E}} } \right)_{123} \) may be due to strong interactions between unlike molecules.

Conclusions

The excess heat capacity, \( \left( {C_{\text{P}}^{\text{E}} } \right)_{123} \), values of the 1-ethyl-3-methylimidazolium tetrafluoroborate (1) + pyrrolidin-2-one or 1-methylpyrrolidin-2-one (2) + cyclopentanone or cyclohexanone (3) mixtures have been evaluated by using their molar heat capacities, \( C_{\varvec{P}} \). The \( \left( {C_{\text{P}}^{\text{E}} } \right)_{123} \) of the studied ternary mixtures are positive over entire mole fraction of (1) and (2) which in turn suggest that capability of cyclohexanone or cyclopentanone molecules to build a non-random structure in mixed state is superior to the effect caused by disruption of associated NMP or 2-Py entities, and interactions between [emim][BF4]:NMP or 2-Py molecular entities. While \( \left( {\frac{{\partial C_{\text{P}}^{\text{E}} }}{\partial T}} \right) \) for [emim][BF4] (1) + 2-Py (2) + cyclopentanone or cyclohexanone (3) mixtures are positive, those for [emim][BF4] (1) + NMP (2) + cyclopentanone or cyclohexanone (3) mixtures are negative. The \( \left( {C_{\text{P}}^{\text{E}} } \right)_{123} \) data have been analyzed in terms of Graph and Flory theories. It has been observed that \( \left( {C_{\text{P}}^{\text{E}} } \right)_{123} \) values obtained from Graph theory compare reasonably well with the experimental values.