1.6.3: Composition- Scale Conversion- Solvent Mixtures
- Page ID
- 373590
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A given mixed solvent is prepared (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) by mixing \(\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \mathrm{m}^{3}\) of liquid 1 and \(\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell) \mathrm{m}^{3}\) of liquid 2. We will assume that the thermodynamic properties of the mixture are ideal.
\[\text { Then volume } \mathrm{V}=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell) \nonumber \]
Then volume% of liquid 2 in the mixture is given by equation (b).
\[\mathrm{V}_{2} \%=\left[10^{2} \, \mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right] /\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right] \nonumber \]
The mass of a given mixed solvent system equals \(\mathrm{w}_{\mathrm{s}}\). Further mass% of liquid 2 is \(\mathrm{w}_{2}\)%.
\[\text { Thus } \mathrm{w}_{2} \%=\mathrm{w}_{2} \, 10^{2} /\left(\mathrm{w}_{1}+\mathrm{w}_{2}\right) \nonumber \]
\[\text { Mole fraction } \mathrm{x}_{2}=\left(\mathrm{w}_{2} \% / \mathrm{M}_{2}\right) /\left[\frac{\left(10^{2}-\mathrm{w}_{2} \%\right)}{\mathrm{M}_{1}}+\frac{\mathrm{w}_{2} \%}{\mathrm{M}_{2}}\right] \nonumber \]
\[\text { Also } \left.\mathrm{V}_{2} \%(\mathrm{mix} ; \text { id })=\left[10^{2} \, \mathrm{w}_{2} / \rho_{2}^{*}(\ell)\right] / \frac{\mathrm{w}_{1}}{\rho_{1}^{*}(\ell)}+\frac{\mathrm{w}_{2}}{\rho_{2}^{*}(\ell)}\right] \nonumber \]
If \(\left(\mathrm{w}_{1}+\mathrm{w}_{2}\right)=100 \mathrm{~kg}\),
\[\mathrm{V}_{2} \%(\operatorname{mix} ; \mathrm{id})=\left[10^{2} \, \mathrm{w}_{2} \% / \rho_{2}^{*}(\ell)\right] /\left[\frac{\left(10^{2}-\mathrm{w}_{2} \%\right)}{\rho_{1}^{*}(\ell)}+\frac{\mathrm{w}_{2} \%}{\rho_{2}^{*}(\ell)}\right] \nonumber \]
Molality and Mole fraction
A given solvent mixture has mass \(10^{2} \mathrm{~kg}\) is prepared using \(\mathrm{w}_{2} \mathrm{~kg}\left[=\mathrm{w}_{2} \% \right]\) of liquid 2; nj moles of solute are dissolved in this mixture.
\[\text { Molality } \mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{} \mathrm{kg}^{-1}=\mathrm{n}_{\mathrm{j}} / 10^{2} \nonumber \]
\[\text { Mole fraction, } \mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left\{\left[\left(10^{2}-\mathrm{w}_{2} \%\right) / \mathrm{M}_{1}\right]+\left[\mathrm{w}_{2} \% / \mathrm{M}_{2}\right]+\mathrm{n}_{\mathrm{j}}\right\} \nonumber \]
For dilute solutions, \(\mathrm{n}_{\mathrm{j}}<<\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)\)
\[\text { Then, } \quad \mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left\{\left[\left(10^{2}-\mathrm{w}_{2} \%\right) / \mathrm{M}_{1}\right]+\left[\mathrm{w}_{2} \% / \mathrm{M}_{2}\right]\right\} \nonumber \]
\[\text { Or, } \left.\mathrm{x}_{\mathrm{j}}=10^{2} \, \mathrm{m}_{\mathrm{j}} /\left\{\left[10^{2}-\mathrm{w}_{2} \%\right] / \mathrm{M}_{1}\right]+\left[\mathrm{w}_{2} \% / \mathrm{M}_{2}\right]\right\} \nonumber \]
Concentration and Molality
A given solution is prepared (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) using \(\mathrm{n}_{1}\) moles of liquid 1, \(\mathrm{n}_{2}\) moles of liquid 2 and \(\mathrm{n}_{j}\) moles of a simple solute (e.g. urea) where \(n_{j}<<\left(n_{1}+n_{2}\right)\).
\[\text { Mass of mixed solvent } \mathrm{w}_{\mathrm{s}}=\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2} \nonumber \]
\[\text { Mass of system, } w=n_{j} \, M_{j}+n_{1} \, M_{1}+n_{2} \, M_{2} \nonumber \]
\[\text { Molality of solute } \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{w}_{1}+\mathrm{w}_{2}\right] \nonumber \]
\[\text { Or, } \quad \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{\mathrm{s}} \nonumber \]
Density of solution \(= \rho\) Mass of solution \(= \mathrm{w}\)
\[\text { Volume of solution } \mathrm{V}=\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}+\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right] / \rho \nonumber \]
\[\text { Concentration of solute } j, \mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} \, \rho /\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}+\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right] \nonumber \]
\[\text { For dilute solutions, } \mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}} \ll\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right] \nonumber \]
\[\text { Then, } \mathrm{c}_{\mathrm{j}} \cong \mathrm{n}_{\mathrm{j}} \, \rho /\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right] \nonumber \]
If the solution is dilute, the density of the solution is approx. equal to density of the solvent \(\rho_{s}\) at the same \(\mathrm{T}\) and \(\mathrm{p}\).
\[\text { Hence } \mathrm{c}_{\mathrm{j}} \cong \mathrm{n}_{\mathrm{j}} \, \rho_{\mathrm{s}} /\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right] \nonumber \]
\[\text { Molality of solute } \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right] \nonumber \]
Then \(c_{j} \cong m_{j} \, p_{s}\)