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Mitigating Complexity: Cohesion Parameters and Related Topics. I: The Hildebrand Solubility Parameter

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Abstract

The objective of this review in two parts is to present a compact overview of the development of the solubility parameter \(\left( {\delta_{i} } \right)\) concept: from the seminal work of van Laar in 1910, to the contributions of Scatchard, Hildebrand, Scott and Prausnitz, leading finally to the generalized multi-component (multi-dimensional) cohesion parameters, with the Hansen solubility parameter being the most prominent representative. In this first part, physico-chemical aspects concerning \(\delta_{i}\)-related models in solution chemistry and chemical engineering will be presented, and recent theoretical efforts in this field, that is, equation-of-state approaches and computer simulation methods for the estimation of solubility parameters, will be indicated. Indeed, prediction of thermodynamic properties of liquid nonelectrolyte solutions from properties of the corresponding pure constituents has come a long way since the classic studies by Hildebrand and by Scatchard leading to regular solution theory (RST), in which the solubility parameter is the property of central importance. Selected aspects of RST will be discussed, including the influence of T and of P on \(\delta_{i}\) and their reliable estimation, thereby clearing up misconceptions and pointing out pitfalls not generally recognized. Extending the dicussion to supercritical conditions, the use of solubility parameters in supercritical fluid (SCF) technologies will be indicated, focusing on practical implications of some of the unique phenomena happening in the near supercritical region, which provide the basis of SCF extraction in industries devoted to food-processing, nutraceuticals, pharmaceuticals and biotechnology.

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Notes

  1. Max Margules was an Austrian physicist. Born in Brody, Galicia, a former crown-land of the Austro-Hungarian Empire, on April 23, 1856, he started his studies in mathematics, physics and chemistry at the University of Wien (Vienna), Austria, in 1872. Among his teachers were Ludwig Boltzmann and Josef Loschmidt. After a few years as Assistant at the Central Institute of Meteorology in Wien, he went to Berlin for additional studies in mathematical physics (1879/1880), and was habilitated at the University of Wien after his return. However, in 1882 Margules resigned from this academic position and rejoined the Central Institute of Meteorology. During the first years at this institution, he continued to pursue physical and physical–chemical research parallel to his work on theoretical meteorology. In 1906 he voluntarily retired at the age of fifty on a very modest pension. This was also the year of his last meteorological publication, dedicated to the theory of storms: Margules, M.: Zur Sturmtheorie. Meteorolog. Z. 23, 481–497 (1906). His small pension and the inflation after the end of World War I led to a life in poverty, which was compounded by the general state of malnutrition of the Austrian population and his refusal to accept help from colleagues and/or the Austrian Meteorological Society (he was awarded the Hann Medal in 1919 but declined the associated honorarium). Max Margules died of starvation in Perchtoldsdorf near Wien (Vienna), Austria, on October 4, 1920. He contributed significantly and lastingly to meteorology and thermodynamics [55]. Obituaries were prepared by F. M. Exner (Meteorolog. Z. 37, 322–324 (1920)) and E. Gold (Nature 106, 286–287 (1920)).

  2. At low temperatures, the second virial coefficient \(B_{i}\) is negative. As a rough estimate, \(B_{i}\) for vapors of common organic liquids at \(T_{{{\text{bp,}}i}}\) is in the range of about − 1000 cm3·mol−1 to − 2000 cm3·mol−1. With increasing temperature \(B_{i}\) becomes less negative, i.e., \({{{\text{d}}B_{i} } \mathord{\left/ {\vphantom {{{\text{d}}B_{i} } {{\text{d}}T}}} \right. \kern-0pt} {{\text{d}}T}}\) is positive, and at the Boyle temperature \(B_{i} = 0\). For simple fluids, \(T_{{{\text{Boyle,}}i}}\) corresponds to a reduced temperature of ca. \(T_{{{\text{r}},i}} = 2.7\). At higher temperatures, repulsive intermolecular interactions dominate and \(B_{i}\) becomes positive, increasing slowly with increasing T. Eventually, \(B_{i}\) will pass through a very flat maximum, though such a maximum has only been observed for fluids with very low critical temperatures, such as helium and hydrogen.

  3. This terminology is not related to the concept of residual properties presented below and in some detail in Appendix 2.

  4. \(\left( {\frac{\text{cal}}{{{\text{cm}}^{3} {\cdot} {\text{atm}}}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} = \left( {\frac{4.184}{{10^{ - 6} \times 1.01325 \times 10^{5} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \left( {\frac{\text{J}}{{{\text{m}}^{3} {\cdot} {\text{Pa}}}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} = 6.42595\left( {\frac{\text{J}}{{{\text{m}}^{3} {\cdot} {\text{Pa}}}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} .\)

  5. The frequently given reference quotation “Berthelot, D.: Sur une méthode purement physique pour la détermination des poids moléculaires des gaz et des poids atomiques des leurs éléments. J. Phys. 8, 263–274 (1899)”, is incorrect: in this article Berthelot uses the van der Waals EOS (cf. p. 265, Eq. 2).

  6. At the critical point \(\left( {T_{\text{c}} ,P_{\text{c}} } \right)\), \(\gamma_{V}\) and the slope of the vapor-pressure curve \({{{\text{d}}P_{\sigma } } \mathord{\left/ {\vphantom {{{\text{d}}P_{\sigma } } {{\text{d}}T}}} \right. \kern-0pt} {{\text{d}}T}} = \left( {{{\partial P} \mathord{\left/ {\vphantom {{\partial P} {\partial T}}} \right. \kern-0pt} {\partial T}}} \right)_{\sigma } \equiv \gamma_{\sigma }\) become equal. Even here is \(T\gamma_{V}\) significantly larger than \(P = P_{\text{c}}\): \({\text{Ar}}:T_{\text{c}} \gamma_{{V,{\text{c}}}} = 27.15{\text{ MPa}}\), \(P_{\text{c}} = 4.90{\text{ MPa}}\); \({\text{CCl}}_{ 4} :T_{\text{c}} \gamma_{{V,{\text{c}}}} = 32.27{\text{ MPa}}\), \(P_{\text{c}} = 4.56{\text{ MPa}}\); \(n{\text{ - C}}_{ 7} {\text{H}}_{ 1 6} :T_{\text{c}} \gamma_{{V,{\text{c}}}} = 18.91{\text{ MPa}}\), \(P_{\text{c}} = 2.74{\text{ MPa}}\); \({\text{C}}_{ 6} {\text{H}}_{ 6} :T_{\text{c}} \gamma_{{V,{\text{c}}}} = 34.85{\text{ MPa}}\), \(P_{\text{c}} = 4.90{\text{ MPa}}\); \({\text{H}}_{ 2} {\text{O}}:T_{\text{c}} \gamma_{{V,{\text{c}}}} = 170.83{\text{ MPa}}\), \(P_{\text{c}} = 22.06{\text{ MPa}}\).

  7. The SI unit for \(\alpha_{\text{pol}}\) is \({\text{C}}^{2} {\cdot} {\text{J}}^{ - 1} {\cdot} {\text{m}}^{2} = {\text{F}} {\cdot} {\text{m}}^{2}\) [406], but values of this scalar property known as average molecular polarizability are commonly quoted as the value \({{\alpha_{\text{pol}} } \mathord{\left/ {\vphantom {{\alpha_{\text{pol}} } {4\pi }}} \right. \kern-0pt} {4\pi }}\varepsilon_{0}\), which has the dimension of a volume \(\left( {{\text{m}}^{3} } \right)\) since the permittivity of vacuum (or the electric constant) is \(\varepsilon_{0} = 8.854{ 187 } \ldots \times 1 0^{ - 12} {\text{ C}}^{2}{ \cdot} {\text{J}}^{ - 1} {\cdot} {\text{m}}^{ - 1}\), with \({\text{C}}^{2} {\cdot} {\text{J}}^{ - 1} {\cdot} {\text{m}}^{ - 1} = {\text{F}}{\cdot} {\text{m}}^{ - 1}\). The quantity \({{\alpha _{{{\text{pol}}}} } \mathord{\left/ {\vphantom {{\alpha _{{{\text{pol}}}} } {4\pi \varepsilon _{0} }}} \right. \kern-\nulldelimiterspace} {4\pi \varepsilon _{0} }}\) is thus frequently called the polarizability volume; most conveniently, the tabulated numerical values [396] are directly comparable with values given in older tables that report molecular polarizabilities in units \(10^{ - 24} {\text{ cm}}^{3}\) or Å3 (Å = 10−8 cm).

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Wilhelm, E. Mitigating Complexity: Cohesion Parameters and Related Topics. I: The Hildebrand Solubility Parameter. J Solution Chem 47, 1626–1709 (2018). https://doi.org/10.1007/s10953-018-0821-1

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