Densities of Pure Ionic Liquids and Mixtures: Modeling and Data Analysis

Abstract

Our two-parameter corresponding states model for liquid densities and compressibilities has been extended to more pure ionic liquids and to their mixtures with one or two solvents. A total of 19 new group contributions (5 new cations and 14 new anions) have been obtained for predicting pressure effects over wide ranges of temperature and pressure. Comparisons of the technique with contemporary treatments based on equations of state show that it compares favorably with all other approaches.

Appendix: Model Equations

The form for the DCFI model is based on hard-sphere properties with a linear density perturbation assuming that the DCFI can be expressed as functions of (reduced) temperature, density and composition as follows:

$$ C_{ij} (\tilde{T},\tilde{\rho }) = C_{ij}^{hs} (\tilde{\varvec{\rho }}) - 2\rho \left[ {b_{ij} (\tilde{T}) - b_{ij}^{hs} (\tilde{T})} \right]. $$

(15)

Superscript ‘hs’ means from a hard-sphere equation such as the Percus/Yevick, ‘Hypernetted Chain’, or Carnahan/Starling forms [39]. The ~ denotes a reduced temperature,${\tilde{T}}$= T/ T ^*, and a set of reduced densities, ${\tilde{\rho}}_{i}$, (defined below). The quantities b are a matrix of values that for simple substances resemble second virial coefficients. We use the hard spheres expressions given by Reed and Gubbins [39], and generalized equations for b empirically determined from pure component data [16]. The pressure can be found analytically by integrating Eq. 4 with Eq. 15. This gives the density variation with pressure and activity coefficient,

$$ \frac{{p - p^{\text{ref}} }}{RT} = \left( {\frac{{p^{\text{hs}} }}{RT}} \right) - \left( {\frac{{P^{\text{hs}} }}{RT}} \right) + \sum\limits_{i = 1}^{C} {\sum\limits_{j = 1}^{C} {\left( {x_{j} x_{i} \rho^{2} - x_{i}^{\text{ref}} x_{j}^{\text{ref}} \rho^{{{\text{ref}}2}} } \right)\left( {b_{ij} - b_{ij}^{\text{hs}} } \right)}}. $$

(16)

The full standard state specification includes pressure, p ^ref, and solvent density, ρ ^ref, at T. Typically an experimental measurement at low pressure is used, but the density could also be estimated by a predictive model. Details are given in [17]. Applications to supercritical solutes in aqueous/organic mixtures are summarized in [18]. Applications to supercritical solutes in ionic liquids are described in [9] and ionic liquid phase behavior in [10, 11].

Various quantities given in the above equations must be modeled. The matrices b, b ^hs, and other ‘hard sphere’ quantities use characteristic quantities T ^* and V ^* for each mixture component, as well as binary parameters k _ij , characterizing component pairs. Characteristics T ^* and V ^* values for many gaseous and nonionic liquid substances are available from Campanella [18]. Values for the ionic liquids have previously been found by regression of density data [9–11]. Here we have predicted these using a group contribution method based on the van der Waals volumes for V ^* as shown in Eq. 5. For ionic liquids the characteristic temperature was kept fixed at T ^* = 755 K. The elements of b ^hs are expressed as,

$$ b_{ij}^{hs} = \frac{{b_{ii}^{hs} + b_{jj}^{hs} }}{2},\,\quad b_{ii}^{hs} = V_{i}^{ * } \widetilde{b}_{ii}^{hs},\quad \widetilde{b}_{ii}^{hs} = \left\{ {\begin{array}{ll} {\frac{{\alpha_{1} }}{{\tilde{T}_{ii}^{{\alpha_{\text{2}} }} }}} & {\tilde{T}_{ii} = \frac{T}{{T_{i}^{*} }} > 0.73} \\ {\beta_{\text{1}} e^{{\beta_{\text{2}} \tilde{T}_{ii} }} } & {\tilde{T}_{ii} = \frac{T}{{T_{i}^{*} }} < 0.73} \\ \end{array} } \right., $$

(17)

where α and β are

$$ \varvec{\alpha}\text{ = }\left( {\begin{array}{*{20}c} {\text{0}\text{.65386227}} \\ {\text{0}\text{.16067976}} \\ \end{array} } \right), \quad\varvec{\beta}\text{ = }\left( {\begin{array}{*{20}c} {\text{0}\text{.807662393}} \\ { - \text{0}\text{.22010926}} \\ \end{array} } \right). $$

(18)

The quantities in b are expressed as

$$ b_{ij} = \tilde{b}_{ij} V_{ij}^{*} $$

(19)

$$\begin{aligned} \tilde{b}_{ij} = c_{\text{1}} \text{ + }\frac{{c_{\text{2}} }}{{\tilde{T}_{ij}^{{}} }}\text{ + }\frac{{c_{\text{3}} }}{{\tilde{T}_{ij}^{2} }}\text{ + }\frac{{c_{\text{4}} }}{{\tilde{T}_{ij}^{3} }}\text{ + }\frac{{c_{\text{5}} }}{{\tilde{T}_{ij}^{8} }} \\ c^{\text{T}} \text{ = }\left[ {\begin{array}{*{20}c} {\text{0}\text{.3625065}} & { - \text{ 0}\text{.7140666}} & { - \text{1}\text{.7543882}} & {\text{0}\text{.47075}} & { - \text{0}\text{.0041793}} \\ \end{array} } \right], \\ \end{aligned} $$

(20)

with $V_{ij}^{*}$ and ${\tilde{T}_{ij}}$ from

$$ V_{ij}^{*} \text{ = }\frac{{\left( {\sqrt[\text{3}]{{V_{i}^{*} }}\text{ + }\sqrt[\text{3}]{{V_{j}^{*} }}} \right)^{\text{3}} }}{\text{8}} $$

(21)

$$ \tilde{T}_{ij} \text{ = }\frac{T}{{\left( {\text{1} - k_{ij} } \right)\sqrt {T_{i}^{*} T_{j}^{*}}}}. $$

(22)

The hard-sphere terms are

$$ \begin{gathered} \left( {\frac{{p^{hs} }}{RT}} \right) = \frac{6}{\pi }\left( {\frac{{\xi_{0}^{{}} }}{{1 - \xi_{3}^{{}} }} + \frac{{3\xi_{1} \xi_{2} }}{{\left( {1 - \xi_{3} } \right)^{2} }} + \frac{{\xi_{2}^{3} \left( {3 - \xi_{3} } \right)}}{{\left( {1 - \xi_{3} } \right)^{3} }}} \right) \hfill \\ \left( {\frac{{p^{hs} }}{RT}} \right)^{ref} = \frac{6}{\pi }\left( {\frac{{\xi_{0}^{ref} }}{{1 - \xi_{3}^{ref} }} + \frac{{3\xi_{1}^{ref} \xi_{2}^{ref} }}{{\left( {1 - \xi_{3}^{ref} } \right)^{2} }} + \frac{{\xi_{2}^{ref3} \left( {3 - \xi_{3}^{ref} } \right)}}{{\left( {1 - \xi_{3}^{ref} } \right)^{3} }}} \right). \hfill \\ \end{gathered} $$

(23)

These involve the hard-sphere quantities

$$ \begin{gathered} \xi_{q} = \frac{\pi }{6}\sum\limits_{j = 1}^{C} {\rho_{j} \sigma_{j}^{q} } = \frac{\pi }{6}\rho \sum\limits_{j = 1}^{C} {x_{j} \sigma_{j}^{q} };\quad \quad q = 0,1,2,3 \hfill \\ \xi_{q}^{ref} = \frac{\pi }{6}\sum\limits_{j = 1}^{C} {\rho_{j}^{ref} \sigma_{j}^{q} } = \frac{\pi }{6}\rho^{ref} \sum\limits_{j = 1}^{C} {x_{j}^{ref} \sigma_{j}^{q} };\quad \quad q = 0,1,2,3, \hfill \\ \end{gathered} $$

(24)

where

$$ \sigma_{i} = \sqrt[3]{{\frac{3}{2\pi }b_{ii}^{hs}}}. $$

(25)

In previous applications to binary systems (gases in ionic liquids), the standard state pressure was p ^ref = 0 bar (due to the non-volatile nature of the solvent) and the reference density was the density of the solvent at the given temperature extrapolated to 0 bar, i.e. ρ ^ref = ρ (T, p = 0 bar). In practice, density data given at 1 bar can also be used, with negligible difference from zero pressure. The binary parameter, k _ij , appearing in Eq. 22 is determined from binary data. For pairs involving a supercritical gas and a condensable component, the value is best determined from phase equilibrium data. For any supercritical species, there is a constraint relating its Henry’s law constants in liquids with different compositions that can be used to find the k _ij value in one solvent from that in another [18]. We have developed this form for ionic liquids [9] (both mixed and pure).