1.7.20: Compressions- Isothermal- Salt Solutions
- Page ID
- 374452
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In 1933 Gucker [1-3] reviewed attempts to measure the apparent isothermal molar compressions of salts in aqueous solution, these attempts dating back to the earliest reliable measurements by Rontgen and Schneider [4] in 1886 and 1887. Gucker showed [2] that for several aqueous salt solutions the apparent isothermal molar compression, \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is a linear function of the square root of the salt concentration.
\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}+\mathrm{a} \, \mathrm{c}_{\mathrm{j}}^{1 / 2} \nonumber \]
This general equation holds for \(\mathrm{CaCl}_{2}(\mathrm{aq})\) at 60 Celsius. In general terms, \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) for salts is negative becoming less negative as the salt concentration increases. Gibson described an interesting approach which characterises salt solutions in terms of effective pressures, \(\mathrm{p}_{\mathrm{c}}\) exerted by the salt on the solvent [5]. This effective pressure is expressed as a linear function of the product of salt and solvent concentrations. The constant of proportionality is characteristic of the salt. Leyendekkers based an analysis using the Tammann-Tait-Gibson (TTG) model, on the assertion that solutes, salts and organic solutes, exert an excess pressure on water in aqueous solution [6,7]. The TTG approach described by Leyendekkers is intuitively attractive but the analysis is based on an extra - thermodynamic assumption [8]. Calculation of an excess pressure requires an estimate of the volume of solute molecules, \(\phi_{j}\) in solution. If this property is independent of solute molality \(\mathrm{m}_{j}\), the dependence of the volume of a solution (in \(1 \mathrm{~kg}\) of water) on solute molality is described by the dependence of the ‘partial molar volume' of water. The difference between \(\left[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{m}_{\mathrm{j}} \, \phi(\mathrm{Vj})\right]\) and \(\mathrm{M}_{1}^{-1} \, \mathrm{V}_{1}^{*}(\ell)\) is understood in terms of an effective pressure on the solvent. The assumptions underlying this calculation are not trivial. Furthermore from a thermodynamic viewpoint, the pressure is the same in every volume element of a solution [8].
Footnotes
[1] F. T. Gucker, J. Am. Chem.Soc.,1933,55,2709.
[2] F. T. Gucker, Chem.Rev.,1933,13,111.
[3] F. T. Gucker, F. W. Lamb, G. A. Marsh and R. M. Haag, J.Am. Chem. Soc.,1950,72,310.
[4] W. C. Rontgen and J. Schneider, Wied. Ann., 1886,29,165;1887,31,36.
[5] R. E. Gibson, J.Am.Chem.Soc.,1934,56,4.;1935,57,284.
[6] J. V. Leyendekkers, J. Chem. Soc. Faraday Trans.1,1981,77,1529; 1982,78,357; 1988,84, 397,1653.
[7] J. V. Leyendekkers, Aust. J. Chem.,1981,34,1785.
[8] M. J. Blandamer, J. Burgess and A. Hakin, J. Chem. Soc. Faraday Trans.1,1986, 82,3681.