1.7.9: Compressions- Isentropic and Isothermal- Solutions- Approximate Limiting Estimates
- Page ID
- 374151
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The Newton Laplace Equation relates the speed of sound \(\mathrm{u}\) in an aqueous solution, density \(\rho(\mathrm{aq})\) and isentropic compressibility \(\kappa_{\mathrm{S}}(\mathrm{aq})\); equation (a).
\[\mathrm{u}^{2}=\left[\kappa_{\mathrm{s}}(\mathrm{aq}) \, \rho(\mathrm{aq})\right]^{-1} \nonumber \]
The differential dependence of sound velocity \(\mathrm{u}\) on \(\kappa_{\mathrm{S}}(\mathrm{aq})\) and \(\rho(\mathrm{aq})\) is given by equation (b).
\[\begin{aligned}
&2 \, u(a q) \, d u(a q)= \\
&\quad-\frac{1}{\left[\kappa_{\mathrm{s}}(a q)\right]^{2} \, \rho(a q)} \, d \kappa_{s}(a q)-\frac{1}{\left.\kappa_{s}(a q)\right] \,[\rho(a q)]^{2}} \, d \rho(a q)
\end{aligned} \nonumber \]
We divide equation (b) by equation (a).
\[2 \, \frac{\mathrm{du}(\mathrm{aq})}{\mathrm{u}(\mathrm{aq})}=-\frac{\mathrm{d} \kappa_{\mathrm{s}}(\mathrm{aq})}{\kappa_{\mathrm{s}}(\mathrm{aq})}-\frac{\mathrm{d} \rho(\mathrm{aq})}{\rho(\mathrm{aq})} \nonumber \]
We explore three approaches based on equation (c)
Analysis I
Two extra-thermodynamic assumptions are made.
- Sound velocity \(\mathrm{u}(\mathrm{aq})\) is a linear function of solute concentration, \(\mathrm{c}_{j}\).
\[\text { Thus[1] } \quad \mathrm{u}(\mathrm{aq})=\mathrm{u}_{1}^{*}(\ell)+\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}} \nonumber \]
\[\text { By definition, } \quad \mathrm{du}(\mathrm{aq})=\mathrm{u}(\mathrm{aq})-\mathrm{u}_{1}^{*}(\ell)=\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}} \nonumber \]
- Density \(\rho(\mathrm{aq})\) is a linear function of concentration \(\mathrm{c}_{j}\).
\[\text { Thus[1] } \quad \rho(\mathrm{aq})=\rho_{1}^{*}(\ell)+\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}} \nonumber \]
\[2 \, \frac{\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}(\mathrm{aq})}=-\frac{\mathrm{d} \kappa_{\mathrm{S}}(\mathrm{aq})}{\mathrm{K}_{\mathrm{s}}(\mathrm{aq})}-\frac{\mathrm{A}_{\mathrm{\rho}} \, \mathrm{c}_{\mathrm{j}}}{\rho(\mathrm{aq})} \nonumber \]
\[\frac{\mathrm{d} \kappa_{\mathrm{S}}(\mathrm{aq})}{\kappa_{\mathrm{S}}(\mathrm{aq})}=-2 \, \frac{\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}(\mathrm{aq})}-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho(\mathrm{aq})} \nonumber \]
In principle the change in \(\kappa_{\mathrm{S}}(\mathrm{aq})\) resulting from addition of a solute \(j\) to form a solution concentration \(\mathrm{c}_{j}\) can be obtained from the experimentally determined parameters \(\mathrm{A}_{\rho}\) and \(\mathrm{A}_{\mathrm{u}}\).
Analysis II
Another approach expresses the two dependences using a general polynomial in \(\mathrm{c}_{j}\).
\[\text { By definition, } \quad \mathrm{A}_{\mathrm{u}}^{\infty}=\operatorname{limit}\left(\mathrm{c}_{\mathrm{j}} \rightarrow 0\right)\left(\frac{\partial \mathrm{u}(\mathrm{aq})}{\partial \mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}} \nonumber \]
\[\text { and } \mathrm{A}_{\rho}^{\infty}=\operatorname{limit}\left(\mathrm{c}_{\mathrm{j}} \rightarrow 0\right)\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}} \nonumber \]
The assumption is made that both \(\mathrm{A}_{\mathrm{u}}^{\infty}\) and \(\mathrm{A}_{\rho}^{\infty}\) are finite.
\[\text { Similarly } \operatorname{limit}\left(\mathrm{c}_{\mathrm{j}} \rightarrow 0\right)\left(\frac{\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{S}}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}=\left(\frac{\partial \kappa_{\mathrm{S}}(\mathrm{aq})}{\partial \mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}^{\infty} \nonumber \]
Analysis III
The procedures described above are incorporated into the following equation for \(\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right)\).
\[\text { Thus } \phi\left(\mathrm{K}_{\mathrm{s} j} ; \text { def }\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{s} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{s} 1}^{*}(\ell) \nonumber \]
Hence using equation (h) with \(\mathrm{d}_{\mathrm{s}}(\mathrm{aq})=\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{s} 1}^{*}(\ell)\)
\[\begin{aligned}
&\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)= \\
&\qquad\left[\kappa_{\mathrm{S}}(\mathrm{aq}) / \mathrm{c}_{\mathrm{j}}\right] \,\left[-\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}(\mathrm{aq})}-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho(\mathrm{aq})}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \mathrm{K}_{\mathrm{S} 1}^{\mathrm{*}}(\ell)
\end{aligned} \nonumber \]
If we assume that \(\kappa_{\mathrm{S}}(\mathrm{aq})\) is close to \(\kappa_{\mathrm{S} 1}^{*}(\ell)\), then [2]
\[\phi\left(\mathrm{K}_{\mathrm{Sj}_{j}} ; \operatorname{def}\right)=\kappa_{\mathrm{s}}(\mathrm{aq}) \,\left[-\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}(\mathrm{aq})}-\frac{\mathrm{A}_{\rho}}{\rho(\mathrm{aq})}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \nonumber \]
Equation (n) is complicated in the sense that the properties \(\kappa_{\mathrm{S}}(\mathrm{aq})\), \(\mathrm{u}(\mathrm{aq})\), \(\rho(\mathrm{aq})\) and \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) depend on concentration \(\mathrm{c}_{j}\). With respect to \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\), the following equation is exact.
\[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell)-\rho(\mathrm{aq})\right]+\mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell) \nonumber \]
\[\text { Using equation }(f), \phi\left(V_{j}\right)=-\frac{A_{\rho}}{\rho_{1}^{*}(\ell)}+\frac{M_{j}}{\rho_{1}^{*}(\ell)} \nonumber \]
\[\text { Or, }-\frac{A_{\rho}}{\rho_{1}^{*}(\ell)}=\phi\left(V_{j}\right)-\frac{M_{j}}{\rho_{1}^{*}(\ell)} \nonumber \]
Equation (q) is multiplied by the ratio, \(\rho_{1}^{*}(\ell) / \rho(\mathrm{aq})\).
\[\text { Thus }-\frac{\mathrm{A}_{\rho}}{\rho(\mathrm{aq})}=\frac{\rho_{1}^{*}(\ell)}{\rho(\mathrm{aq})} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)-\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})} \nonumber \]
Combination of equations (n) and (r) yields equation (s).
\[\begin{aligned}
&\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)= \\
&\kappa_{\mathrm{s}} \,\left[-\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}(\mathrm{aq})}+\frac{\rho_{1}^{*}(\ell)}{\rho(\mathrm{aq})} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)-\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]
\end{aligned} \nonumber \]
The argument is advanced that \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)\) can be meaningfully extrapolated to infinite dilution.
\[\operatorname{limit}\left(c_{j} \rightarrow 0\right) \phi\left(K_{\mathrm{Sj}_{j}} ; \operatorname{def}\right)=\phi\left(\mathrm{K}_{\mathrm{sj}} ; \operatorname{def}\right)^{\infty} \nonumber \]
In the same limit \(\rho_{1}^{*}(\ell) / \rho(\mathrm{aq})=1.0\) and \(\mathrm{K}_{\mathrm{S}}(\mathrm{aq})=\mathrm{K}_{\mathrm{S}}^{*}(\ell)\).
\[\phi\left(\mathrm{K}_{\mathrm{S}_{j}} ; \operatorname{def}\right)^{\infty}=\kappa_{\mathrm{s} 1}^{*}(\ell) \,\left[2 \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}-\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}-\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}\right] \nonumber \]
\[\text { But from equation }(\mathrm{d}), \mathrm{A}_{\mathrm{u}}=\left[\mathrm{u}(\mathrm{aq})-\mathrm{u}_{1}^{*}(\ell)\right] / \mathrm{c}_{\mathrm{j}} \nonumber \]
\[\phi\left(\mathrm{K}_{\mathrm{sj}} ; \operatorname{def}\right)^{\infty}=\kappa_{\mathrm{sl}}^{*}(\ell) \,\left[2 \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}-2 \, \mathrm{U}-\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right] \nonumber \]
where (cf. equation (v)),
\[\mathrm{U}=\left[\mathrm{u}(\mathrm{aq})-\mathrm{u}_{1}^{*}(\ell)\right] /\left[\mathrm{u}_{1}^{*}(\ell) \, \mathrm{c}_{\mathrm{j}}\right] \nonumber \]
The symbol \(\mathrm{U}\) identifies the relative molar increment of the speed of sound [3-9]. Equation (w) shows \(\phi\left(K_{S_{j}} ; \operatorname{def}\right)^{\infty}\) is obtained from \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}\) and the speed of sound in a solution concentration \(\mathrm{c}_{j}\).
\[\text { In this approach we assume that }\left(\frac{\partial \mathrm{u}}{\partial \mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}=\frac{\mathrm{u}(\mathrm{aq})-\mathrm{u}_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}} \nonumber \]
\[\text { Then, } U=\frac{1}{\mathrm{u}_{1}^{*}(\ell)} \,\left(\frac{\mathrm{du}(\mathrm{aq})}{\mathrm{dc}_{\mathrm{j}}}\right) \nonumber \]
However \(\left(\frac{\mathrm{du}(\mathrm{aq})}{\mathrm{dc}}\right)\) and similarly \(\left(\frac{\mathrm{du}(\mathrm{aq})}{\mathrm{dm}_{\mathrm{j}}}\right)\) are obtained using experimental results for real concentrations. Hence the estimated \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}\) is likely to be poor.
Analysis IV
The apparent molar isothermal compression of solute \(j\) is related to the concentration \(\mathrm{c}_{j}\) using the following exact equation.
\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \nonumber \]
\[\text { By definition. } \quad \delta(a q)=\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{S}}(\mathrm{aq}) \nonumber \]
\[\text { and } \delta_{1}^{*}(1)=\kappa_{\mathrm{T} 1}^{*}(\ell)-\kappa_{\mathrm{S} 1}^{*}(\ell) \nonumber \]
\[\text { For an aqueous solution, } \kappa_{\mathrm{T}}(\mathrm{aq})=\delta(\mathrm{aq})+\kappa_{\mathrm{s}}(\mathrm{aq}) \nonumber \]
According to the Newton-Laplace Equation.
\[[u(\mathrm{aq})]^{2}=\left[\kappa_{\mathrm{s}}(\mathrm{aq}) \, \rho(\mathrm{aq})\right]^{-1} \nonumber \]
\[\text { From equation }(\mathrm{zd}), \kappa_{\mathrm{T}}(\mathrm{aq})=\delta(\mathrm{aq})+\left\{[\mathrm{u}(\mathrm{aq})]^{2} \, \rho(\mathrm{aq})\right\}^{-1} \nonumber \]
At this stage, assumptions are made concerning the dependences of \(\kappa_{\mathrm{T}}(\mathrm{aq})\) and \(\delta(\mathrm{aq})\) on concentration \(\mathrm{c}_{j}\).
\[\text { Thus } \quad \kappa_{\mathrm{T}}(\mathrm{aq})=\kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{A}_{\mathrm{KT}} \, \mathrm{c}_{\mathrm{j}} \nonumber \]
\[\text { and } \quad \delta(\mathrm{aq})=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \nonumber \]
Using equations (d), (f) and (zf),
\[\begin{aligned}
\kappa_{\mathrm{Tl}}^{*}(\ell)+\mathrm{A}_{\kappa \mathrm{T}} \, \mathrm{c}_{\mathrm{j}}=& \delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \\
&+\frac{1}{\left\{\mathrm{u}_{1}^{*}(\ell)+\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}\right\}^{2} \,\left\{\rho_{1}^{*}(\ell)+\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}\right\}}
\end{aligned} \nonumber \]
Or,
\[\begin{aligned}
&\kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{A}_{\kappa \mathrm{T}} \, \mathrm{c}_{\mathrm{j}}=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \\
&+\frac{1}{\left[\mathrm{u}_{1}^{*}(\ell)\right]^{2} \,\left\{1+\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}} / \mathrm{u}_{1}^{*}(\ell)\right\}^{2} \, \rho_{1}^{*}(\ell) \,\left\{1+\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}} / \rho_{1}^{*}(\ell)\right\}}
\end{aligned} \nonumber \]
Assuming \(\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}} / \mathrm{u}_{1}^{*}(\ell)<<1\) and \(A_{\rho} \, c_{j} / \rho_{1}^{*}(\ell)<<1\),
\[\begin{aligned}
&\kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{A}_{\mathrm{kT}} \, \mathrm{c}_{\mathrm{j}}=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \\
&+\frac{1}{\left[\mathrm{u}_{1}^{*}(\ell)\right]^{2} \, \rho_{1}^{*}(\ell)} \,\left[1-\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}_{1}^{*}(\ell)}\right] \,\left[1-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right]
\end{aligned} \nonumber \]
\[\text { We assume that }\left[\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}_{1}^{*}(\ell)}\right] \,\left[\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right]<<1 \nonumber \]
\[\begin{aligned}
&\text { Therefore, } \\
&\kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{A}_{\mathrm{KT}} \, \mathrm{c}_{\mathrm{j}}=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \\
&+\frac{1}{\left[\mathrm{u}_{1}^{*}(\ell)\right]^{2} \, \rho_{1}^{*}(\ell)} \,\left[1-\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}_{1}^{*}(\ell)}-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right]
\end{aligned} \nonumber \]
\[\text { But } \kappa_{\mathrm{s} 1}^{*}(\ell)=\left\{\left[u_{1}^{*}(\ell)\right]^{2} \, \rho_{1}^{*}(\ell)\right\}^{-1} \nonumber \]
\[\text { and } \kappa_{\mathrm{T} 1}^{*}(\ell)=\delta_{1}^{*}(\ell)+\kappa_{\mathrm{S} 1}^{*}(\ell) \nonumber \]
Then,
\[\begin{aligned}
&\delta_{1}^{*}(\ell)+\kappa_{\mathrm{S} 1}^{*}(\ell)+\mathrm{A}_{\mathrm{KT}} \, \mathrm{c}_{\mathrm{j}}=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \\
&+\kappa_{\mathrm{Sl}}^{*}(\ell) \,\left[1-\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}_{1}^{*}(\ell)}-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right]
\end{aligned} \nonumber \]
\[\text { Or } \mathrm{A}_{\mathrm{K}}=\mathrm{A}_{\delta}-\kappa_{\mathrm{S} 1}^{*}(\ell) \,\left[\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}+\frac{\mathrm{A}_{\rho}}{\rho_{1}^{*}(\ell)}\right] \nonumber \]
From equations (za) and (zg),
\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{A}_{\mathrm{KT}}+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{Tl}}^{*}(\ell) \nonumber \]
Equations (zq) and (zr) yield equation (as),
\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{A}_{\delta}-\kappa_{\mathrm{S} 1}^{*}(\ell) \,\left[\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}+\frac{\mathrm{A}_{\rho}}{\rho_{1}^{*}(\ell)}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \nonumber \]
Or, using equation (q)
\[\begin{aligned}
\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{A}_{\delta}-\mathrm{K}_{\mathrm{S} 1}^{*}(\ell) \,[&\left.\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}-\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\
&+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T} 1}^{*}(\ell)
\end{aligned} \nonumber \]
Using equation (zc),
\[\begin{aligned}
\phi\left(\mathrm{K}_{\mathrm{Tj}_{\mathrm{j}}}\right)=& \mathrm{A}_{\delta}-\kappa_{\mathrm{S} 1}^{*}(\ell) \,\left[\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}-\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\
&+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \delta_{1}^{*}(\ell)+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{S} 1}^{*}(\ell)
\end{aligned} \nonumber \]
Or,
\[\begin{aligned}
&\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{A}_{\delta}+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \delta_{1}^{*}(\ell) \\
&+\kappa_{\mathrm{Sl}}^{*}(\ell) \,\left[2 \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)-\frac{\mathrm{M}_{\mathrm{j}}}{\left.\rho_{1}^{*} \ell\right)}-\frac{2 \, \mathrm{A}_{u}}{\left.\mathrm{u}_{1}^{*} \ell\right)}\right]
\end{aligned} \nonumber \]
The latter is the Owen-Simons Equation[4] which takes the following form in the limit of infinite dilution.
\[\begin{aligned}
\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}=\left[\mathrm{A}_{\delta}\right.&\left.+\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} \, \delta_{1}^{*}(\ell)\right] \\
&+\kappa_{\mathrm{Sl}}^{*}(\ell) \,\left[2 \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}-\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}-\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}\right]
\end{aligned} \nonumber \]
The term \(\left[\mathrm{A}_{\delta}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} \, \delta_{1}^{*}(\ell)\right]\) is not negligibly small. Using equation (u), equation (zw) takes the following form,
\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}=\left[\mathrm{A}_{\delta}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} \, \delta_{1}^{*}(\ell)\right]+\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty} \nonumber \]
Clearly the approximation which sets \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}\) equal to \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}\) is poor although often made. In fact Hedwig and Hoiland [10] show that for N-acetylamino acids in aqueous solution at \(298.15 \mathrm{~K} \mathrm{} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}\) and \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}\) can have different signs, offering convincing evidence that the assumption is untenable.
Footnotes
[1] \(\begin{aligned}
&A_{u}=\left[\frac{m}{s}\right] \,\left[\frac{m^{3}}{m o l}\right]=\left[m^{4} \mathrm{~s}^{-1} \mathrm{~mol}^{-1}\right] \\
&A_{\rho}=\left[\frac{k g}{m^{3}}\right] \,\left[\frac{m^{3}}{m o l}\right]=\left[k g \mathrm{~mol}^{-1}\right]
\end{aligned}\)
[2] \(\begin{aligned}
&2 \, \frac{\mathrm{A}_{\mathrm{u}}}{\mathrm{u}} \, \mathrm{K}_{\mathrm{S}}(\mathrm{aq})=[1] \, \frac{1}{\left[\mathrm{~m} \mathrm{~s}^{-1}\right]} \,\left[\mathrm{m}^{4} \mathrm{~s}^{-1} \mathrm{~mol}^{-1}\right] \, \frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]} \\
&\frac{\mathrm{A}_{\rho}}{\rho} \, \kappa_{\mathrm{S}}(\mathrm{aq})=\frac{\left[\mathrm{kg} \mathrm{m}^{-3}\right]}{\left[\mathrm{mol} \mathrm{m}^{-3}\right]} \, \frac{1}{\left.\mathrm{~kg} \mathrm{~m}^{-3}\right]} \, \frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{Nm}^{-2}\right]} \\
&\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{s}}(\mathrm{aq})=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \, \frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]}
\end{aligned}\)
[3] S. Barnatt, J. Chem. Phys.,1952,20,278.
[4] B. B. Owen and H. L. Simons, J. Phys.Chem.,1957,61,479.
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