1.11.2: Ecuación de Gibbs-Duhem- Soluciones salinas- Coeficientes osmóticos y de actividad
- Page ID
- 80565
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Consideramos una solución acuosa preparada usando\(1 \mathrm{~kg}\) solvente, agua, a temperatura\(\mathrm{T}\) y presión\(\mathrm{p}\) (\(\cong p^{0}\)). Para una solución salina acuosa, el potencial químico\(\mu_{j}(\mathrm{aq})\) para la sal\(j\) a la molalidad\(\mathrm{m}_{j}\) viene dado por la ecuación (a) donde\(\gamma_{\pm}\) está el coeficiente medio de actividad iónica de la sal.
\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\]
Por definición, en absoluto\(\mathrm{T}\) y\(\mathrm{p}\),
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1\]
Para el solvente, agua,
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{\star}(\ell)-\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]
Por definición, en absoluto\(\mathrm{T}\) y\(\mathrm{p}\),
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0\]
Potenciales químicos\(\mu_{j}(\mathrm{aq})\) y\(\mu_{1}(\mathrm{aq})\) están vinculados por la ecuación de Gibbs-Duhem. Esto,
\[\left(1 / M_{1}\right) \, d \mu_{1}(a q)+v \, m_{j} \, d \mu_{j}(a q)=0\]
\ [\ begin {alineado}
&\ left (1/\ mathrm {M} _ {1}\ derecha)\,\ mathrm {d}\ izquierda [\ mu_ {1} ^ {*} (\ ell) -\ mathrm {v}\,\ mathrm {R}\,\ mathrm {T}\,\ phi\,\ mathrm {M} _ {1}\,\ mathrm rm {m} _ {\ mathrm {j}}\ derecha]\\
&\ quad+\ mathrm {v}\,\ mathrm {m} _ {\ mathrm {j}}\,\ mathrm {d}\ izquierda [\ mu_ {\ mathrm {j}} ^ {0} (\ mathrm {aq}) +\ mathrm {v}\,\ mathrm {R}\,\ mathrm {T}\,\ ln\ left (\ mathrm {Q}\,\ mathrm {m} _ {\ mathrm {j}}\,\ gamma_ {\ pm}/\ mathrm {m} ^ {0}\ derecha] =0\ derecha.
\ end {alineado}\]
\ [\ comenzar {alineado}
&-v\, R\, T\, d\ izquierda [\ phi\, m_ {j}\ derecha]\\
&+v\, m_ {j}\, v\, R\, T\, d\ izquierda [\ ln (Q) +\ ln\ izquierda (m_ {j}\ derecha) +\ ln\ izquierda (\ gamma_ {\ pm}\ derecha) -\ ln\ izquierda (m^ {0}\ derecha)\ derecha] =0
\ final {alineado}\]
\[-\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{v} \, \mathrm{m}_{\mathrm{j}} \,\left\{\mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right)+\mathrm{d} \ln \left(\gamma_{\pm}\right)\right\}=0\]
\[-\phi \, d m_{j}-m_{j} \, d \phi+v \, m_{j} \, d \ln \left(m_{j}\right)+v \, m_{j} \, d \ln \left(\gamma_{\pm}\right)=0\]
Entonces,
\[-\phi \, d m_{j}-m_{j} \, d \phi+v \, d m_{j}+v \, m_{j} \, d \ln \left(\gamma_{\pm}\right)=0\]
\[v \, m_{j} \, d \ln \left(\gamma_{\pm}\right)=\phi \, d m_{j}-v \, d m_{j}+m_{j} \, d \phi\]
O,
\[\mathrm{d} \ln \left(\gamma_{\pm}\right)=(\phi-v) \, \frac{\mathrm{dm}_{\mathrm{j}}}{\mathrm{v} \, \mathrm{m}_{\mathrm{j}}} + \frac{\mathrm{d} \phi}{\mathrm{v}}\]
Para un soluto donde un mol de soluto puro forma un mol de soluto en solución,
\[\mathrm{d} \ln \left(\gamma\right)=(\phi-1) \, \frac{\mathrm{dm}_{\mathrm{j}}}{\mathrm{m}_{\mathrm{j}}} + \mathrm{d} \phi\]
Entonces,
\[\ln (\gamma)=(\phi-1) \,+\int_{0}^{\mathrm{m}(\mathrm{j})}(\phi-1) \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right)\]