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Estimation of Vapor Pressures of Solvent + Salt Systems with Quadratic Solvation Relationships

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Abstract

The paper studies quadratic solvation models (QSMs) applied to the numerical simulation of the vapor pressures of solvent systems with a salt effect. Two useful general quadratic solvation relationships are presented within an integration framework, incorporating cumulative physical indices for components and boundary constraints, in conjunction with the vapor pressures of monomolecular fluids calculated from Antoine, Senol, Frost-Kalkwarf and Xiang-Tan equations. Literature data for the vapor pressures of 18 diverse binary (solvent + salt) and ternary (solvent 1 + solvent 2 + salt 3) vapor–liquid equilibrium systems are subjected to the statistical analysis of QSMs via a logarithmic-ratio objective function and cumulative frequency distribution. Essentially, the examined QSMs with twelve (QSM12) and six (QSM6) adjustable coefficients are quite accurate in yielding overall design factors (Fod) lower than 1.015 and 1.08, respectively. The concentration-dependent model (CM) also simulates precisely the observed data with Fod = 1.013 as far as salt effects are concerned. QSM12 models have proven reasonably successful in predicting the vapor pressures of ternary systems with a mean deviation of 2.2%.

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Abbreviations

\(C_{{\text{s}}}\) :

Salt concentration (wt %)

\(cu\) :

Cumulative descriptor

\(\overline{e}\) :

Mean relative error, \( \bar{e}{\text{ = }}\left( {{{{\text{100}}} \mathord{\left/ {\vphantom {{{\text{100}}} {\text{N}}}} \right. \kern-\nulldelimiterspace} {\text{N}}}} \right)\sum\nolimits_{{i = 1}}^{N} {\left| {{{\left( {P_{{i{\text{,obs}}}} - P_{{i{\text{,mod}}}} } \right)} \mathord{\left/ {\vphantom {{\left( {P_{{i{\text{,obs}}}} - P_{{i{\text{,mod}}}} } \right)} {P_{{i{\text{,obs}}}} }}} \right. \kern-\nulldelimiterspace} {P_{{i{\text{,obs}}}} }}} \right|} \) (\(\%\))

\(F\) :

Concentration dependent factor

\(F_{{\text{m}}}\) :

Model normalization factor

\(F_{{{\text{od}}}}\) :

Overall design factor

\(F_{{\text{s}}}\) :

Safety factor

\(N\) :

Number of observations

\(n_{i}\) :

Stoichiometric number of the ion in the salt structure

\(P_{i}\) :

Partial pressure of a component in the solvent mixture (kPa)

\(P\) :

Vapor pressure of (solvent + salt) system (kPa)

\(P_{0}\) :

Vapor pressure of a pure solvent (kPa)

\(P_{{\text{c}}}\) :

Critical vapor pressure of a pure solvent (kPa)

\(P_{{\text{t}}}^{{\prime}}\) :

Total vapor pressure of a solvent mixture involving a salt (kPa)

\(r\) :

Crystal ionic radius of the ion (nm)

\(r^{{\prime}}\) :

The normalized reciprocal of the crystal ionic radius (nm−1)

\(R_{{\text{D}}}\); \(R_{{\text{D}}}^{{\prime}}\) :

Molar refractivity of the ion and its normalized value (cm3·mol−1)

\(S\) :

Standard deviation

\(T\) :

Temperature (\({\text{K}}\))

\(T_{{\text{c}}}\) :

Critical temperature of a pure solvent (K)

\(t\) :

Student’s t parameter

\(V\) :

Molar volume of the component (dm3·mol−1)

\(v\); \(v^{{\prime}}\) :

Volume occupied by the ion and its normalized value (nm3)

w solv :

Mass fraction of solvent

\(X\) :

Objective function of model reliability

\(\overline{X}\) :

Mean of objective function

\(x_{i}\) :

Mole fraction of the component in the liquid phase

\(x_{W}^{{\prime}}\) :

The argument of the Lambert W function

\(Y\) :

Independent variable of objective function

\(y_{i}\) :

Mole fraction of the component in the vapor phase

\(z\) :

The charge of the ion

α; α*:

Solvatochromic parameters

β; β*:

Solvatochromic parameters

δ; δ*:

Solvatochromic parameters

δH; δH*:

Hildebrand solubility parameter (MPa0.5)

ϕ; ϕ0 :

Fugacity coefficients of the component

γ :

Activity coefficient of the component

π; π*:

Solvatochromic parameters

σ :

Root mean square deviation, \( \sigma {\text{ = }}\left[ {{{\sum\nolimits_{{i = 1}}^{N} {\left( {Y_{{i,{\text{obs}}}} - Y_{{i,{\text{mod}}}} } \right)^{2} } } \mathord{\left/ {\vphantom {{\sum\nolimits_{{i = 1}}^{N} {\left( {Y_{{i,{\text{obs}}}} - Y_{{i,{\text{mod}}}} } \right)^{2} } } N}} \right. \kern-\nulldelimiterspace} N}} \right]^{{0.5}} \)

σ p :

Softness parameter of the ion

desn:

Design property

mod:

Modeled property

obs:

Observed property

s:

Salt

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  1. Department of Chemical Engineering, Faculty of Engineering, Istanbul University-Cerrahpasa, 34320, Avcilar, Istanbul, Turkey

    Aynur Senol

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The coefficients of Eqs. 6–9, complete set of the regression coefficients of Eqs. 10–16and the deviation statistics of model reliability analysis relating to systems I–XVI areprovided in the supplementary Tables S1–S11. This material is available free of charge viathe Internet at the Web site of the journal. (DOCX 375 kb)

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Senol, A. Estimation of Vapor Pressures of Solvent + Salt Systems with Quadratic Solvation Relationships. J Solution Chem 49, 559–582 (2020). https://doi.org/10.1007/s10953-020-00983-3

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