1.12.16: Expansions- Solutions- Apparent Molar Isentropic and Isobaric
- Page ID
- 377809
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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 the context of the properties of aqueous solutions the concept of apparent molar properties is important with respect to the analysis of experimental results; e.g. apparent molar volume for solute \(j\) \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) calculated from the densities of a given solution and solvent at fixed \(\mathrm{T}\) and \(\mathrm{p}\). Similarly apparent molar isobaric expansions \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) characterise the dependence of \phi\left(\mathrm{V}_{j}\right) on temperature at fixed pressure. Nevertheless problems emerge when we turn attention to comparable isentropic properties. The way ahead involves definition of apparent molar isentropic expansions \(\phi\left(\mathrm{E}_{\mathrm{sj}} ; \text { def }\right)\) and apparent molar isentropic compressions \(\phi\left(\mathrm{K}_{\mathrm{sj}} ; \text { def }\right)\). These two properties are related [1]; equation(a).
\[\begin{aligned}
&\phi\left(\mathrm{E}_{\mathrm{sj}} ; \mathrm{def}\right)= \\
&-\frac{\alpha_{\mathrm{S} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)+\frac{\alpha_{\mathrm{s}}(\mathrm{aq})}{\kappa_{\mathrm{s}}(\mathrm{aq})} \, \phi\left(\mathrm{K}_{\mathrm{s}}\right)+\frac{\alpha_{\mathrm{s}}(\mathrm{aq}) \, \kappa_{\mathrm{S} 1}^{*}(\ell)}{\kappa_{\mathrm{s}}(\mathrm{aq}) \, \sigma(\mathrm{aq})} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \\
&\quad+\left[\alpha_{\mathrm{p} 1}^{*}(\ell) \,\left(1+\frac{\alpha_{\mathrm{pl} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}\right)-\frac{\alpha_{\mathrm{s}}(\mathrm{aq}) \, \kappa_{\mathrm{S} 1}^{*}(\ell)}{\kappa_{\mathrm{s}}(\mathrm{aq})} \,\left(1+\frac{\sigma_{1}^{*}(\ell)}{\sigma(\mathrm{aq})}\right)\right] \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
Equation (b) relates the corresponding properties at infinite dilution [1].
\[\frac{\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}}{\alpha_{\mathrm{S} 1}^{*}(\ell)}=-\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p} 1}^{*}(\ell)}+\frac{\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}}{\kappa_{\mathrm{S} 1}^{*}(\ell)}+\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma_{1}^{*}(\ell)} \nonumber \]
‘Semi’ apparent molar isentropic expansions and compressions are related using equation (c)
\[\frac{1}{\alpha_{\mathrm{S}}(\mathrm{aq})} \,\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right]_{\mathrm{S}(\mathrm{aq})}=-\frac{1}{\kappa_{\mathrm{S}}(\mathrm{aq})} \,\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})} \nonumber \]
Equations (a) , (b) and (c) illustrate the power of thermodynamics in drawing together and relating the several properties of a solution.
Footnotes
[1] In the following we simplify the algebra by omitting the descriptors (aq) and (\(\ell\)). The starting point is the following equation.
\[\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}} \, \sigma} \,\left(\kappa_{\mathrm{s}} \, \sigma-\kappa_{\mathrm{s}}^{*} \, \sigma^{*}\right)-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \,\left(\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right) \nonumber \]
The latter equation is effectively an identity. From equation (a),
\[\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\alpha_{\mathrm{s}}-\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}} \, \sigma} \, \kappa_{\mathrm{s}}^{*} \, \sigma^{*}-\alpha_{\mathrm{s}}^{*}+\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \, \alpha_{\mathrm{p}}^{*} \nonumber \]
From equation (b), \(\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}+\frac{\kappa_{\mathrm{s}} \, \sigma}{\mathrm{T} \, \alpha_{\mathrm{p}} \, \kappa_{\mathrm{s}} \, \sigma} \, \kappa_{\mathrm{s}}^{*} \, \sigma^{*}-\frac{\kappa_{\mathrm{S}}^{*} \, \sigma^{*}}{\alpha_{\mathrm{p}} \, \mathrm{T}}\) or, \(\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}\)
But as an identity,
\[\kappa_{\mathrm{S}} \, \sigma-\kappa_{\mathrm{S}}^{*} \, \sigma^{*}=\sigma \,\left(\kappa_{\mathrm{S}}-\kappa_{\mathrm{S}}^{*}\right)+\kappa_{\mathrm{S}}^{*} \,\left(\sigma-\sigma^{*}\right) \nonumber \]
From equations (a) and (c). \(\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \,\left(\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \,\left(\sigma-\sigma^{*}\right)-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}}\left(\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right)\)
But,
\[\begin{aligned}
&\phi\left(\mathrm{E}_{\mathrm{s} j} ; \text { def }\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s}}^{*} \\
&\phi\left(\mathrm{E}_{\mathrm{S}_{\mathrm{j}}} ; \operatorname{def}\right)=\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \, \frac{1}{\mathrm{c}_{\mathrm{j}}} \,\left(\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \, \frac{1}{\mathrm{c}_{\mathrm{j}}} \,\left(\sigma-\sigma^{*}\right) \\
&-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \, \frac{1}{\mathrm{c}_{\mathrm{j}}} \,\left(\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right)+\alpha_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
But for the isobaric heat capacities \(\phi\left(\mathrm{C}_{\mathrm{pj}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\sigma-\sigma^{*}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \sigma^{*}\)
Also,
\[\begin{array}{r}
\phi\left(\mathrm{K}_{\mathrm{s} j} ; \text { def }\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{s}}^{*} \\
\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \text { def }\right)=\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{S}}} \,\left[\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)-\kappa_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]
\end{array} \nonumber \]
Hence,
\[\begin{aligned}
&+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\sigma^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\
&-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \,\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\alpha_{\mathrm{p}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\alpha_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
With a little reorganisation,
\[\begin{aligned}
\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)=-& \frac{\alpha_{\mathrm{S}}^{*}}{\alpha_{\mathrm{p}}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)+\frac{\alpha_{\mathrm{S}}}{\kappa_{\mathrm{S}}} \, \phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{S}} \, \sigma} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \\
&+\left[\alpha_{\mathrm{s}}^{*} \,\left(1+\frac{\alpha_{\mathrm{p}}^{*}}{\alpha_{\mathrm{p}}}\right)-\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{S}}}\left(1+\frac{\sigma^{*}}{\sigma}\right)\right] \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)
\end{aligned} \nonumber \]
Hence in the limit of infinite dilution,
\[\frac{\phi\left(E_{\mathrm{S} j} ; \mathrm{def}\right)^{\infty}}{\alpha_{\mathrm{s}}^{*}}=-\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p}}^{*}}+\frac{\phi\left(\mathrm{K}_{\mathrm{s} j} ; \mathrm{def}\right)^{\infty}}{\kappa_{\mathrm{s}}^{*}}+\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma^{*}} \nonumber \]
[2]
\[\begin{aligned}
&{\frac{\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}}{\alpha_{\mathrm{s}}^{*}}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]}{\left[\mathrm{K}^{-1}\right]}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]} \\
&\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p}}^{*}}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]}{\left[\mathrm{K}^{-1}\right]}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \\
&\frac{\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}}{\kappa_{\mathrm{S}}^{*}}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} / \mathrm{N} \mathrm{m}^{-2}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \\
&\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma^{*}}=\frac{\left[\mathrm{J} \mathrm{mol}^{-1}\right]}{\left[\mathrm{J} \mathrm{m}^{-3}\right]}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]
\end{aligned} \nonumber \]