1.13.6: Equilibrium - Depression of Freezing Point of a Solvent by a Solute
- Page ID
- 375626
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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 homogeneous liquid system (at pressure \(\mathrm{p}\)) comprises solvent \(\mathrm{i}\) and solute \(\mathrm{j}\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). In the absence of solute \(\mathrm{j}\), the freezing point of the solvent is \(\mathrm{T}_{1}^{0}\). But in the presence of solute \(\mathrm{j}\) the freezing point is temperature \(\mathrm{T}\) where \(\mathrm{T} < \mathrm{T}_{1}^{0}\). The depression of freezing point \(\theta\left[=\mathrm{T}_{1}^{0}-\mathrm{T}\right]\) is recorded for a solution where the mole fraction of solvent is \(\mathrm{x}_{1}(\mathrm{sln})\). If the solution is dilute, we can assume that the thermodynamic properties of the solution are ideal. From the Schroeder-van Laar equation,
\[-\ln \left[x_{1}(s \ln )\right]=\frac{\left[\Delta_{f} H_{1}^{0}(T)\right]}{R} \,\left[\frac{1}{T}-\frac{1}{T_{1}^{0}}\right] \nonumber \]
\[-\ln \left[\mathrm{x}_{1}(\mathrm{~s} \ln )\right]=\frac{\Delta_{\mathrm{f}} \mathrm{H}_{1}^{0}}{\mathrm{R}} \, \frac{\theta}{\left(\mathrm{T}_{1}^{0}-\theta\right) \, \mathrm{T}_{1}^{0}} \nonumber \]
If
\[\mathrm{T}_{1}^{0}-\theta \cong \mathrm{T}_{1}^{0},-\ln \left[\mathrm{x}_{1}(\mathrm{~s} \ln )\right]=\frac{\Delta_{\mathrm{f}} \mathrm{H}_{1}^{0}}{\mathrm{R}} \, \frac{\theta}{\left(\mathrm{T}_{1}^{0}\right)^{2}} \nonumber \]
Or,
\[\ln \left[\frac{1}{x_{1}(s \ln )}\right]=\frac{\Delta_{\mathrm{f}} H_{1}^{0}}{R} \, \frac{\theta}{\left(T_{1}^{0}\right)^{2}} \nonumber \]
Hence [2]
\[\theta=\left[\frac{\mathrm{R} \,\left(\mathrm{T}_{1}^{0}\right)^{2} \, \mathrm{M}_{1}}{\Delta_{\mathrm{f}} \mathrm{H}_{1}^{0}}\right] \, \mathrm{m}_{\mathrm{j}} \nonumber \]
The quantity enclosed in the […] brackets is characteristic of the solvent.
Footnotes
[1] \(\theta=\mathrm{T}_{1}^{0}-\mathrm{T}\); \(\frac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{1}^{0}}=\frac{\mathrm{T}_{1}^{0}-\mathrm{T}}{\mathrm{T} \, \mathrm{T}_{1}^{0}}=\frac{\mathrm{T}_{1}^{0}-\mathrm{T}}{\left(\mathrm{T}_{1}^{0}-\theta\right) \, \mathrm{T}_{1}^{0}}=\frac{\theta}{\left(\mathrm{T}_{1}^{0}-\theta\right) \, \mathrm{T}_{1}^{0}}\)
[2] \(\frac{1}{x_{1}}=\frac{1}{1-x_{j}}=\frac{1}{1-\left[n_{j} /\left(n_{1}+n_{j}\right)\right]}=\frac{n_{1}+n_{j}}{n_{1}+n_{j}-n_{j}}\) For a solution where the molality of solute \(j=m_{j}\) \(\mathrm{m}_{\mathrm{j}}=\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1} \, \mathrm{M}_{1}}\)
Then, \(\frac{1}{\mathrm{x}_{1}}=\frac{\mathrm{n}_{1}+\mathrm{n}_{1} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}}{\mathrm{n}_{1}}\)
\(-\ln \left[\mathrm{x}_{1}(\mathrm{~s} \ln )\right]=-\ln \left[1+\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]\);
\(-\ln \left[x_{1}(\operatorname{sln})\right]=-\ln \left[1-x_{j}(s \ln )\right] \approx x_{j}\)
\(x_{j}=\frac{m_{j}}{\left(1 / M_{1}\right)+m_{j}} \approx m_{j} \, M_{1}\)
[3] see I Prigogine and R Defay, Chemical Thermodynamics, trans. D. H. Everett, Longmans Green, London, 1953.