1.15.4: Heat Capacities: Isobaric: Salt Solutions
The excess enthalpy \(\mathrm{H}^{\mathrm{E}}\) of an aqueous salt solution prepared using \(1 \mathrm{~kg}\) of water and \(\mathrm{m}_{j}\) moles of a 1:1 salt is related to \(\mathrm{m}_{j}\) using the DHLL. Because \({\mathrm{C}_{\mathrm{p}}}^{\mathrm{E}}\) is the isobaric temperature dependence of \(\mathrm{H}^{\mathrm{E}}\), then \({\mathrm{C}_{\mathrm{p}}}^{\mathrm{E}}\) for this aqueous solution is given by equation (a) [1].
\[\mathrm{C}_{\mathrm{p}}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=-(4 / 3) \, \mathrm{R} \, \mathrm{m}_{\mathrm{j}}^{(3 / 2)} \,\left(\mathrm{m}^{0}\right)^{-1 / 2}\left[2 \, \mathrm{T} \, \mathrm{S}_{\mathrm{H}}+\mathrm{T}^{2} \,\left(\partial \mathrm{S}_{\mathrm{H}} / \partial \mathrm{T}\right)_{\mathrm{p}}\right] \nonumber \]
\[\mathrm{S}_{\mathrm{Cp}_{\mathrm{p}}}=2 \, \mathrm{T} \, \mathrm{S}_{\mathrm{H}}+\mathrm{T}^{2} \,\left(\partial \mathrm{S}_{\mathrm{H}} / \partial \mathrm{T}\right) \nonumber \]
\(\mathrm{S}_{\mathrm{Cp}}\) is the DHLL factor in the equation for the isobaric heat capacity.
\[C_{p}^{E}\left(a q ; w_{1}=1 \mathrm{~kg}\right)=-(4 / 3) \, R \, S_{C p} \, m_{j}^{(3 / 2)} \,\left(m^{0}\right)^{-1 / 2} \nonumber \]
Using equation (c) [2],
\[\phi\left(\mathrm{J}_{\mathrm{j}}\right)=-(4 / 3) \, \mathrm{R} \, \mathrm{S}_{\mathrm{Cp}_{\mathrm{p}}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2} \nonumber \]
We could perhaps have anticipated that according to DHLL, \(\phi \left(\mathrm{J}_{j}}\right)\) is a linear function of \(\left(\mathrm{m}_{\mathrm{j}}\right)^{1 / 2}\). An extensive literature describes the limiting partial molar isobaric heat capacities of ions in aqueous solution. One of the earliest investigations of the isobaric heat capacities of salt solutions was made by Randall and Ramage[3] and later by Randall and Taylor [4]. The groups lead by Hepler [5,6] and by Desnoyers [7] have made significant contributions in this area. However no agreement has been reached on a scale of absolute values. Hepler reported relative estimates based on \(\mathrm{C}_{\mathrm{p}}^{\infty}\left(\mathrm{H}^{+} ; \mathrm{aq} ; 298 \mathrm{~K}\right)\) equal to zero. Perhaps most attention has been directed at salts formed by alkylammonium cation [8,9] and hydrophobic anions; e.g. amino acids [10], phenylcarboxylates, t-butylcarboxlates[11] and cryptates[12]. Data [7] for \(\mathrm{R}_{4} \mathrm{~N}^{+} \mathrm{Br}^{-}(\mathrm{aq})\) show that \(\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\) increases with increase in hydrophobic character of the R-group. French and Criss argue [13] in favour of a scale which sets \(\mathrm{C}_{\mathrm{p}}^{\infty}\left(\mathrm{Br}^{-} ; \mathrm{aq}\right)\) at \(– 68 \mathrm{~J K}^{-1} \mathrm{~mol}^{-1}\). An attempt[14] has identified the various contributions to \(\mathrm{C}_{\mathrm{p}}^{\infty}(\text { ion; aq })\). Certainly trends in \(\mathrm{C}_{\mathrm{p}}^{\infty}(\text { ion; aq })\) point to characteristic features associated with the properties of ions in aqueous solution. Nevertheless, interpretation is not straightforward [14].
Footnotes
[1]
\[\mathrm{S}_{\mathrm{Cp}}=2 \, \mathrm{T} \, \mathrm{S}_{\mathrm{H}}+\mathrm{T}^{2} \,\left(\partial \mathrm{S}_{\mathrm{H}} / \partial \mathrm{T}\right)_{\mathrm{p}}=[1] \,[\mathrm{K}] \,\left[\mathrm{K}^{-1}\right]+[\mathrm{K}]^{2} \,\left[\mathrm{K}^{-1}\right] \,[\mathrm{K}]^{-1}=[1] \nonumber \]
[2]
\[\phi\left(\mathrm{J}_{\mathrm{j}}\right)=[1] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[1] \,[1]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \nonumber \]
[3] M. Randall and W.D.Ramage, J.Am.Chem.Soc.,1927, 49 ,93
[4] M.Randall and M.D.Taylor, J. Am. Chem. Soc.,1941, 45 ,959.
[5] I. K. Hovey, L. G. Hepler and P. R. Tremaine, J. Phys. Chem.,1988, 92 , 1323; Thermochim. Acta, 1988, 126 , 245; J. Chem. Thermodyn.,1988, 20 , 595.
[6] J. J. Spitzer, I. V. Oloffson, P. P. Singh and L. G. Hepler, Thermochim. Acta,1979, 28 , 155.
[7] J.-L. Fortier, P.-A. Leduc and J. E. Desnoyers, J. Solution Chem, 1974, 3 , 323.
[8] B. Chawla and J. C. Ahluwalia, J. Phys. Chem.,1972 76 , 2582.
[9] E. M. Arnett and J. J. Campion, J. Am. Chem. Soc.,1970, 92 , 7097.
[10] J. C. Ahluwalia, C. Ostiguy, G. Perron and J. E. Desnoyers, Can. J. Chem., 1977, 55 , 3364, and 3368.
[11] M. Lucas and H. Le Bail, J. Phys. Chem., 1976, 80 , 2620.
[12] N. Morel-Desrosiers and J. P. Morel, J. Phys. Chem., 1985, 89 , 1541.
[13] R. N. French and C. M. Criss, J. Solution Chem.,1982, 11 , 625.
[14] C. Shin, I. Worsley and C.M.Criss, J. Solution Chem.,1976, 5 , 867.
[15] For further details of heat capacities of salt solutions see—
- NaOH(aq); G. Conti, P. Gianni, A. Papini and E. Matteoli, J. Solution Chem.,1988, 17 ,481.
- Alkali metal halides(aq);also solutions in D2O; J.-L. Fortier, P. R. Philip and J.E.Desnoyers, J. Solution Chem.,1974, 3 ,523.
- NaBPh4(aq); and in urea(aq); B. Chawla, S. Subramanian and J. C. Ahluwalia, J.Chem.Thermodyn.,1972, 4 ,575.
- Bu4NBr(aq) and NaBPh4(aq); and in aqueous mixtures; S. Subramanian and J. C.Ahluwalia, Trans. Faraday Soc.,1971, 67 ,305.
- R4NBr(aq; 382 to 363) K; M.J.Mastroianni and C. M. Criss, J. Chem. Thermodyn, 1972, 4 ,321.
- R4NBr(aq); E.M.Arnett and J. Campion, J. Am. Chem. Soc., 1970, 92 , 7097.
- R4NCl(aq); K.Tamaki, S. Yoshikawa and M.Kushida, Bull. Chem.Soc.,Jpn, 1975, 48 ,3018.
- R4N+ R’COO- (aq); P.-A. Leduc and J. E. Desnoyers, Can. J.Chem. 1973, 51 ,2993.
- Amino acids(aq); G. C. Kresheck J.Chem.Phys.,1970, 52 ,5966.
- MCl(aq + 2-methylpropan-2-ol); G. T. Hefter, J.-P. E. Grolier and A. H. Roux, J. Solution Chem.,1989, 18 ,229.
- Am4NBr(aq + 2-methylpropan-2-ol); R. K. Mohanty, S. Sunder and J. C. Ahluwalia, J. Phys.Chem.,1972, 76 ,2577.
- CsI(aq; 273 – 373 K); R. E. Mitchell, and J. W. Cobble, J. Am. Chem. Soc., 1964, 86 ,5401.
- Amino acids(aq) and (aq. + urea); C. Jolicoeur, B. Riedl, D. Desrochers, L. L. Lemelin, R. Zamojska and O. Enea, J. Solution Chem.,1986, 15 ,109.
- Bu4NBr(aq); NaBPh4(aq); ( also binary aq. mixtures); R. K. Mohanty, T. S. Sarma, S. Subrahamian and J. C. Ahluwalia, Trans. Faraday Soc., 1971, 67 , 305.
- NaCl(aq; 274 to 318 K); G. Perron, J.-L. Fortier and J. E. Desnoyers, J. Chem. Thermodynamics, 1975, 7 ,1177.
- Salts (aq .and mixed solvents); J. E. Desnoyers, O. Kiyohara , G. Perron and L. Avedikian, Adv. Chem. Series, No 155. 1976.
- Bu4N+ butyrate(aq; 283 to 323 K); A. S. Levine and S. Lindenbaum, J. Solution Chem., 1973, 2 ,445.
- NaCl(aq; 320 to 600 K); D. Smith-Magowan and R. H. Wood, J. Chem. Thermodyn., 1981, 13 ,1047.
- CaCl2(aq; 306 to 603 K; 17.4 Mpa); D. E. White, A. L. Doberstein, J. A. Gates, D. M. Tillett and R. H. Wood, J. Chem. Thermodyn., 1987, 19 ,251.
- NaCl(aq; 273 to 313 K; 0 to 1000 bar); C.-T. A. Chen, J. Chem. Eng. Data, 1982, 27 ,356.
- CH3COOH(aq); CH3COONa(aq), NH3(aq); NH4Cl(aq); 283, 298 and 313 K; G. Allred and E. M Woolley, J. Chem. Thermodyn., 1981, 13 , 155.
- Group III metal perchlorates;
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R. A. Marriott, A.W. Hakin and J. A. Rard, J.Chem.Thermodyn.,2001,33,643.
- A. W. Hakin, M. J. Lukacs, J. L. Liu, K. Erickson and A. Madhavji, J.Chem.Thermodyn.,2003, 35 ,775.