1.14.69: Temperature of Maximum Density: Aqueous Solutions
At ambient pressure, the molar volume of water (\(\ell\)) is a minimum near \(277 \mathrm{~K}\), the temperature of maximum density, \(\mathrm{TMD}\). The \(\mathrm{TMD}\) is sensitive to the concentration and nature of added solute. Generally added salts lower the \(\mathrm{TMD}\), the extent of lowering being often written \(\Delta \theta\). For dilute salt solutions, \(\mathrm{TMD}\) is a linear function of the molality of salt, \(\mathrm{m}_{j}\),\(\left(\partial \Delta \theta / \partial m_{\mathrm{j}}\right)\) being negative; Despretz Law. However considerable interest is generated by the observation that some organic solutes at low mole fractions raise the \(\mathrm{TMD}\); i.e. \(\Delta \theta > 0\); e.g. 2-methylpropan-2-ol.
Although the phenomenon of a shift in \(\mathrm{TMD}\) is straightforward from an experimental standpoint, explanations distinguish between possible contributions to the shift in \(\mathrm{TMD}\). Most treatments identify two contributions to the shift in \(\mathrm{TMD}\), an ‘ideal’ shift and a contribution which reflects the fact that the thermodynamic properties of the aqueous system are not ideal [1,2].
At the outset we assume that the molar volume of water at ambient pressure in the region of the \(\mathrm{TMD}\) is a quadratic function of the difference (\(\mathrm{T}-\mathrm{TMD}^{*}\)) where \(\mathrm{TMD}^{*}\) is the temperature of maximum density of water (\(\ell\)) [2]. At temperature \(\mathrm{T}\) the molar volume \(\mathrm{V}_{1}^{*}(\ell, \mathrm{T})\) is given by equation (a) where \(\chi_{1}\) is a dimensionless property of water (\(\ell\)) [3].
\[\mathrm{V}_{1}^{*}(\ell, \mathrm{T})=\mathrm{V}_{1}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \,\left\{1+\chi_{1} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right)^{2} /[\mathrm{K}]^{2}\right\} \nonumber \]
We consider briefly three types of systems;
- binary aqueous mixtures,
- aqueous solutions, and
- aqueous salt solutions.
(i) Binary Aqueous Mixtures
For the non-aqueous component, the dependence of molar volume \(\mathrm{V}_{2}^{*}(\ell, \mathrm{T})\) on temperature is given by equation (b) where \(\chi_{2}\) is a dimensionless property of the non-aqueous component.
\[\mathrm{V}_{2}^{*}(\ell, \mathrm{T})=\mathrm{V}_{2}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \,\left\{1+\chi_{2} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right) /[\mathrm{K}]\right\} \nonumber \]
\(\mathrm{V}_{2}^{*}\left(\ell, \mathrm{TMD}^{*}\right)\) is the molar volume of non-aqueous component at the temperature \(\mathrm{TMD}^{*}\). The volume of a mixture prepared using \(\mathrm{n}_{1}\) and \(\mathrm{n}_{2}\) moles of the two liquids at temperature \(\theta\) under the no-mix condition is given by equation (c).
\[\mathrm{V}(\mathrm{no}-\operatorname{mix} ; \theta)=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell ; \theta)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell ; \theta) \nonumber \]
Hence,
\[\begin{aligned}
\mathrm{V}(\mathrm{no}-\operatorname{mix} ; \theta) &=\mathrm{n}_{1} \,\left[\mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left\{1+\chi_{1} \,\left(\theta-\mathrm{TMD}^{*}\right)^{2} /[\mathrm{K}]^{2}\right\}\right] \\
&+\mathrm{n}_{2} \,\left[\mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left\{1+\chi_{2} \,\left(\theta-\mathrm{TMD}^{*}\right) /[\mathrm{K}]\right\}\right]
\end{aligned} \nonumber \]
At temperature \(\theta\), the volume of the real mixture \(\mathrm{V}(\operatorname{mix} ; \theta)\) is given by equation (e) where \(\mathrm{V}_{1}(\operatorname{mix} ; \theta)\) and \(\mathrm{V}_{2}(\operatorname{mix} ; \theta)\) are the partial molar volumes of the two components in the mixture at temperature \(\theta\).
\[\mathrm{V}(\operatorname{mix} ; \theta)=\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix} ; \theta)+\mathrm{n}_{2} \, \mathrm{V}_{2}(\operatorname{mix} ; \theta) \nonumber \]
The volume of mixing at temperature \(\theta\) is given by equation (f). By definition,
\[\Delta_{\text {mix }} V(\theta)=V(\operatorname{mix} ; \theta)-V(\text { no }-\operatorname{mix} ; \theta) \nonumber \]
Hence,
\[\begin{aligned}
\Delta_{\text {mix }} \mathrm{V}(\theta)=& \mathrm{n}_{1} \,\left\{\mathrm{V}_{1}(\operatorname{mix} ; \theta)-\mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left[1+\chi_{1} \,\left(\theta-\mathrm{TMD}^{*}\right)^{2} /[\mathrm{K}]^{2}\right]\right\} \\
&+\mathrm{n}_{2} \,\left\{\mathrm{V}_{2}(\operatorname{mix} ; \theta)-\mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left[1+\chi_{2} \,\left(\theta-\mathrm{TMD}^{*}\right) /[\mathrm{K}]\right]\right\}
\end{aligned} \nonumber \]
But the molar volume of mixing at temperature \(\theta\),
\[\Delta_{\text {mix }} \mathrm{V}_{\mathrm{m}}(\theta)=\Delta_{\text {mix }} \mathrm{V}(\theta) /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right) \nonumber \]
Hence,
\[\begin{aligned}
\Delta_{\operatorname{mix}} \mathrm{V}_{\mathrm{m}}(\theta) &=\mathrm{x}_{1} \, \mathrm{V}_{1}(\operatorname{mix} ; \theta)+\mathrm{x}_{2} \, \mathrm{V}_{2}(\operatorname{mix} ; \theta) \\
-\left\{\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{TMD}) \,\left[1+\chi_{1} \,\left(\theta-\mathrm{TMD}^{*}\right)^{2} /[\mathrm{K}]^{2}\right]\right\} \\
&-\left\{\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left[1+\chi_{2} \,\left(\theta-\mathrm{TMD}^{*}\right) /[\mathrm{K}]\right]\right\}
\end{aligned} \nonumber \]
Then the differential of \(\Delta_{\mathrm{mix}} \mathrm{~V}_{\mathrm{m}}\) is given by equation (j).
\[\begin{aligned}
\mathrm{d} \Delta_{\operatorname{mix}} \mathrm{V}_{\mathrm{m}}(\mathrm{T})=& \mathrm{x}_{1} \, \mathrm{dV}(\operatorname{mix} ; \mathrm{T})+\mathrm{V}_{1}(\operatorname{mix} ; \mathrm{T}) \, \mathrm{dx}{ }_{1} \\
&+\mathrm{x}_{2} \, \mathrm{dV}_{2}(\operatorname{mix} ; \mathrm{T})+\mathrm{V}_{2}(\mathrm{mix} ; \mathrm{T}) \, \mathrm{dx} \mathrm{x}_{2} \\
&-\mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left[1+\chi_{1} \,(\mathrm{T}-\mathrm{TMD})^{2} \,[\mathrm{K}]^{-2}\right] \, \mathrm{dx} \\
&\left.-\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{TMD}) \, 2 \, \chi_{1} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right) \,[\mathrm{K}]^{-2}\right] \, \mathrm{dT} \\
&-\mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left[1+\chi_{1} \,(\mathrm{T}-\mathrm{TMD})^{2} \,[\mathrm{K}]^{-2}\right] \, \mathrm{dx}_{2} \\
&-\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell ; \mathrm{TMD}) \, \chi_{2} \,[\mathrm{K}]^{-1} \, \mathrm{dT}
\end{aligned} \nonumber \]
But according to the Gibbs-Duhem Equation, at fixed pressure
\[x_{1} \, d V_{1}(\operatorname{mix} ; T)+x_{2} \, d V_{2}(\operatorname{mix} ; T)=E_{p m}(\operatorname{mix} ; T) \, d T \nonumber \]
In equation (k), \(\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{T})\) is the molar isobaric expansion of the mixture. Equation (j) can therefore be reorganized into equation (l).
\[\begin{aligned}
&\mathrm{d} \Delta_{\operatorname{mix}} \mathrm{V}_{\mathrm{m}}(\mathrm{T})=\\
&\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{T}) \, \mathrm{dT}\\
&+\left\{\mathrm{V}_{1}(\operatorname{mix} ; \mathrm{T})-\mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left[1+\chi_{1} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right)^{2} \,[\mathrm{K}]^{-2}\right\} \, \mathrm{dx} \mathrm{x}_{1}\right.\\
&+\left\{\mathrm{V}_{2}(\mathrm{mix} ; \mathrm{T})-\mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \,\left[1+\chi_{2} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right) \,[\mathrm{K}]^{-1}\right] \, \mathrm{dx}\right.\\
&\left.-\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \, 2 \, \chi_{1} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right) \,[\mathrm{K}]^{-2}\right] \, \mathrm{dT}\\
&-\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \, \chi_{2} \,[\mathrm{K}]^{-1} \, \mathrm{dT}
\end{aligned} \nonumber \]
\
We note that \(\mathrm{dx}_{1} = −\mathrm{dx}_{2}\) and that the coefficients of \(\mathrm{dx}_{1}\) and \(\mathrm{dx}_{2}\) are in fact excess partial molar volumes at temperature \(\mathrm{T}\). Hence,
\[\begin{aligned}
\mathrm{d} \Delta_{\operatorname{mix}} \mathrm{V}_{\mathrm{m}}(\mathrm{T})=& \\
\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{T}) \, & \mathrm{dT}+\left[\mathrm{V}_{2}^{\mathrm{E}}(\mathrm{T})-\mathrm{V}_{1}^{\mathrm{E}}(\mathrm{T})\right] \, \mathrm{dx} \mathrm{x}_{2} \\
&\left.-\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{TMD}) \, 2 \, \chi_{1} \,(\mathrm{T}-\mathrm{TMD}) \,[\mathrm{K}]^{-2}\right] \, \mathrm{dT} \\
&-\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell ; \mathrm{TMD}) \, \chi_{2} \,[\mathrm{K}]^{-1} \, \mathrm{dT}
\end{aligned} \nonumber \]
Or,
\[\begin{aligned}
\frac{\mathrm{d} \Delta_{\text {mix }} \mathrm{V}_{\mathrm{m}}(\mathrm{T})}{\mathrm{dT}} &=\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{T})+\left[\mathrm{V}_{2}^{\mathrm{E}}(\mathrm{T})-\mathrm{V}_{1}^{\mathrm{E}}(\mathrm{T})\right] \, \frac{\mathrm{dx}}{\mathrm{dT}} \\
&\left.-\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \, 2 \, \chi_{1} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right) \,[\mathrm{K}]^{-2}\right] \\
&-\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \, \chi_{2} \,[\mathrm{K}]^{-1}
\end{aligned} \nonumber \]
We use equation (n) at temperature ‘\(\mathrm{T} = \theta\)’ and at fixed composition; i.e. \(\mathrm{dx}_{2} = 0\). Moreover, by definition at the \(\mathrm{TMD}\), \(\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \theta)\) is zero. Hence the dependence of \(\Delta_{\text {mix }} \mathrm{V}_{\mathrm{m}}(\theta)\) on temperature at fixed pressure and composition is given by equation (o).
\[\begin{aligned}
\left(\frac{\partial \Delta_{\text {mix }} V_{\mathrm{m}}(\mathrm{T})}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{x}(2)}=&\left.-\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \, 2 \, \chi_{1} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right) \,[\mathrm{K}]^{-2}\right] \\
&-\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \, \chi_{2} \,[\mathrm{K}]^{-1}
\end{aligned} \nonumber \]
We identify temperature \(\theta\) with the recorded \(\mathrm{TMD}\). Hence from equation (o) with \(\Delta \theta=\theta-\mathrm{TMD}^{*}\) [4],
\[\begin{array}{r}
\Delta \theta=-\frac{\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \, \chi_{2} \,[\mathrm{K}]^{-1}}{2 \,\left(1-\mathrm{x}_{2}\right) \, \mathrm{V}_{1}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \, \chi_{1} \,[\mathrm{K}]^{-2}} \\
-\frac{\left[\partial \Delta_{\text {mix }} \mathrm{V}_{\mathrm{m}}(\theta) / \partial \mathrm{T}\right]_{\mathrm{p}, \mathrm{x}(2)}}{2 \,\left(1-\mathrm{x}_{2}\right) \, \mathrm{V}_{1}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \, \chi_{1} \,[\mathrm{K}]^{-2}}
\end{array} \nonumber \]
In other words, the shift in the \(\mathrm{TMD}\), \(\Delta \theta\), is made up of two contributions. For binary system having thermodynamic properties which are ideal, the second term on the r.h.s. of equation (p) is zero. The first term on the r.h.s. side of equation (p) predicts that \(\Delta \theta\) is negative in agreement with the Despretz rule. In summary therefore equation (p) can be written in the following simple form.
\[\Delta \theta=\Delta \theta(\text { ideal })+\Delta \theta(\text { struct }) \nonumber \]
The sign of \(\Delta \theta (\text{struct})\) is determined by the sign of \(\left[\partial \Delta_{\operatorname{mix}} V_{\mathrm{m}}(\theta) / \partial \mathrm{T}\right]\). If the latter term is negative, \(\Delta \theta (\text{struct})\) is positive and for some systems can be the dominant term. As noted above, this is the case at low mole fractions \(\mathrm{x}_{2}\) for 2-methylpropan-2-ol, a trend attributed to enhancement of water-water hydrogen bonding by the non-aqueous component.
(ii) Aqueous Solutions
The volume of a solution at temperature \(\mathrm{TMD}\), prepared using \(1 \mathrm{~kg}\) of solvent water and mj moles of a simple neutral solute is given by equation (r).
\[\mathrm{V}(\mathrm{aq} ; \mathrm{TMD})=\left(\mathrm{l} / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{TMD})+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}} ; \mathrm{TMD}\right) \nonumber \]
But at the \(\mathrm{TMD}\),
\[\left(1 / \mathrm{M}_{1}\right) \,\left[\partial \mathrm{V}_{1}^{*}(\ell ; \mathrm{TMD}) / \partial \mathrm{T}\right]=-\mathrm{m}_{\mathrm{j}} \,\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}} ; \mathrm{TMD}\right) / \partial \mathrm{T}\right] \nonumber \]
We use equation (a) to relate \(\mathrm{V}_{1}^{*}(\ell ; \mathrm{TMD})\) to \(\mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right)\) with \(\Delta \mathrm{T}\) representing (\(\mathrm{TMD} - \mathrm{~TMD}^{*}\)).
\[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{TMD}) \, \chi_{1} \, 2 \, \Delta \mathrm{T} /[\mathrm{K}]^{2}=-\mathrm{m}_{\mathrm{j}} \,\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}} ; \mathrm{TMD}\right) / \partial \mathrm{T}\right] \nonumber \]
We assume that for dilute real solutions \(\phi\left(\mathrm{V}_{j}\right)\) is a linear function of the molality of solute \(j\) and that the proportionality term is the pairwise volumetric interaction parameter \(\mathrm{v}_{jj}\). Thus,
\[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}+\mathrm{v}_{\mathrm{jj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
Then
\[\mathrm{d} \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \mathrm{dT}=\mathrm{d} \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \mathrm{dT}+\left[\partial \mathrm{v}_{\mathrm{ij}} / \partial \mathrm{T}\right]_{\mathrm{p}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \nonumber \]
Hence,
\[\begin{aligned}
&\left\{2 \, \mathrm{V}_{1}^{*}\left(\ell ; \mathrm{TMD}^{*}\right) \, \chi_{1} / \mathrm{M}_{1} \,[\mathrm{K}]^{2}\right\} \, \Delta \mathrm{T}= \\
&\quad-\mathrm{m}_{\mathrm{j}} \,\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\partial \mathrm{T}\right]-\left[\partial \mathrm{v}_{\mathrm{ij}} / \partial \mathrm{T}\right] \,\left(\mathrm{m}^{0}\right)^{-1} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2}
\end{aligned} \nonumber \]
We rewrite equation (w) as an equation for the change in \(\mathrm{TMD}\), \(\Delta \mathrm{T}\) using a quadratic in \(\mathrm{m}_{j}\) [5]. Thus,
\[\Delta \mathrm{T}=\mathrm{q}_{1} \, \mathrm{m}_{\mathrm{j}}+\mathrm{q}_{2} \, \mathrm{m}_{\mathrm{j}}^{2} \nonumber \]
Consequently a plot of \(\Delta \mathrm{T}\) \(\mathrm{m}_{j}\) is linear having intercept \(\mathrm{q}_{1}\) and slope \(\mathrm{q}_{2}\) [1]. If the solution is ideal [i.e. \(\mathrm{v}_{jj}\) is zero] then \(\mathrm{q}_{2}\) in zero and \(\left[\Delta \mathrm{T} / \mathrm{m}_{\mathrm{j}}\right]\) is constant independent of \(\mathrm{m}_{j}\) [6].
(iii) Aqueous Salt Solutions
The above analysis forms the basis for an analysis of the effects of salts on \(\mathrm{TMD}\) except that the dependence of \(\phi\left(\mathrm{V}_{j}\right)\) is expressed using the following equation where \(\mathrm{S}_{\mathrm{V}}\) is the Debye-Huckel Limiting Law volumetric parameter [7-12].
\[\phi\left(V_{j}\right)=\phi\left(V_{j}\right)^{\infty}+S_{v} \,\left(m_{j} / m^{0}\right)^{1 / 2}+b \,\left(m_{j} / m^{0}\right) \nonumber \]
The foregoing analysis has been extended to include consideration of isobaric expansions [13] and limiting partial molar expansions [14-17].
Footnotes
[1] C. Wada and S.Umeda, Bull. Chem. Soc. Jpn,1962, 35 ,646,1797.
[2] F. Franks and B. Watson, Trans. Faraday Soc.,1969, 65 ,2339.
[3] The symbol [K] indicates the unit of temperature, kelvin. The term \(\left\{1+\chi_{1} \,\left(\mathrm{T}-\mathrm{TMD}^{*}\right)^{2} /[\mathrm{K}]^{2}\right\}\) is dimensionless as required by equation (a).
[4] As noted \(\chi_{1}\) and \(\chi_{2}\) are dimensionless and characteristic properties of the two components.
\[\begin{aligned}
&\frac{\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \, \chi_{2} \,[\mathrm{K}]^{-1}}{2 \,\left(1-\mathrm{x}_{2}\right) \, \mathrm{V}_{1}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \, \chi_{1} \,[\mathrm{K}]^{-2}} \\
&=\frac{[1] \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,[1] \,[\mathrm{K}]^{-1}}{[1] \,[1] \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,[1] \,[\mathrm{K}]^{-2}}=[\mathrm{K}] \\
&\frac{\left[\partial \Delta_{\text {mix }} \mathrm{V}_{\mathrm{m}}(\theta) / \partial \theta\right]}{2 \,\left(1-\mathrm{x}_{2}\right) \, \mathrm{V}_{1}^{*}\left(\ell, \mathrm{TMD}^{*}\right) \, \chi_{1} \,[\mathrm{K}]^{-2}}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]}{[1] \,[1] \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,[1] \,[\mathrm{K}]^{-2}}=[\mathrm{K}]
\end{aligned} \nonumber \]
[5] M. V. Kaulgud, J. Chem. Soc. Faraday Trans.1,1979, 75 ,2246; 1990, 86 ,911.
[6] J. R. Kuppers, J.Phys.Chem.,1974, 78 ,1041.
[7] T. H. Lilley and S. Murphy, J.Chem. Thermodyn., 1973, 5 ,467.
[8] T. Wakabayashi and K. Takazuimi, Bull. Chem. Soc. Jpn., 1982, 55 ,2239.
[9] T. Wakabayashi and K. Takazuimi, Bull. Chem. Soc. Jpn., 1982, 55 ,3073.
[10] For comments on salts in D2O, see A. J. Darnell and J. Greyson, J. Phys. Chem.,1968, 73 ,3032.
[11] G. Wada and M. Miura, Bull. Chem. Soc. Jpn., 1969, 42 ,2498.
[12] J. R. Kuppers, J. Phys. Chem.,1975, 79 ,2105.
[13] D. A. Armitage, M. J. Blandamer, K. W. Morcom and N. C. Treloar, Nature, 1968, 219 ,718.
[14] J. E. Garrod and T. M. Herrington, J. Phys.Chem.,1970, 74 ,363.
[15] T. M. Herrington and E. L. Mole, J. Chem. Soc. Faraday Trans.1,1982, 78 ,213.
[16] D. D. Macdonald and J. B. Hyne, Can. J. Chem.,1976, 54 ,3073.
[17] D. D. Macdonald, B. Dolan and J. B. Hyne, J. Solution Chem.,1976, 5 ,405.