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## Interpreting Freezing Point Depression from the Viewpoint of Entropy

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 基金资助: 吉林省教育厅优质教学团队建设项目

 Fund supported: 吉林省教育厅优质教学团队建设项目

Abstract

The equation for depression of the freezing point for dilute solution is derived on the basis of △G=△H-TS. The change of entropy in the solution before and after phase transition reveals that the depression of the freezing point is originated from the larger entropy change in the process of dissolution than that in fusion. The further application of this equation is discussed in a more general manner.

Keywords： Freezing point depression ; Colligative properties of dilute solution ; Phase transition entropy

XING Shuang-Xi, SUN Wen-Dong. Interpreting Freezing Point Depression from the Viewpoint of Entropy. University Chemistry[J], 2016, 31(12): 83-85 doi:10.3866/PKU.DXHX201606011

$\ln {x_{\rm{A}}} = \frac{{{\Delta _{{\rm{fus}}}}H_{{\rm{m, A}}}^*}}{R}\left( {\frac{1}{{T_{\rm{f}}^*}}-\frac{1}{{{T_{\rm{f}}}}}} \right)$

## 2 定量推导

$S_{{\text{m, A}}\left( {\text{1}} \right)}^*-S_{{\text{m, A}}\left( {\text{s}} \right)}^* = \frac{{{\Delta _{{\text{fus}}}}H_{{\text{m, A}}}^*}}{{T_{\text{f}}^*}}$

p不变，向上述平衡系统中加入少量溶质B，则液相A (l)变为稀溶液(其中A的摩尔分数为xA)，设固相仍为纯A (s)，在温度Tf时达平衡：${\text{A}}\left( {\text{s}} \right)\mathop \rightleftharpoons \limits^{{T_{\text{f}}}, p} {\text{A}}\left( {1, {x_{\text{A}}}} \right)$。则此时溶解熵为：

${S_{{\text{A}}\left( {\text{1}} \right)}}-S_{{\text{m, A}}\left( {\text{S}} \right)}^* = \frac{{{H_{{\text{A}}\left( {\text{1}} \right)}}-H_{{\text{m, A}}\left( {\text{s}} \right)}^*}}{{{T_{\text{f}}}}}$

${S_{{\text{A}}\left( {\text{1}} \right)}} = S_{{\text{m, A}}\left( {\text{1}} \right)}^*-R\ln {x_{\text{A}}}$

${H_{{\text{A}}\left( {\text{1}} \right)}} = H_{{\text{m, A}}\left( {\text{1}} \right)}^*$

$\left( {S_{{\text{m, A}}\left( {\text{1}} \right)}^*-S_{{\text{m, A}}\left( {\text{s}} \right)}^*} \right)-R\ln {x_{\text{A}}} = \frac{{{\Delta _{{\text{fus}}}}H_{{\text{m, A}}}^*}}{{{T_{\text{f}}}}}$

$\frac{{{\Delta _{{\text{fus}}}}H_{{\text{m, A}}}^*}}{{T_{\text{f}}^*}}-R\ln {x_{\text{A}}} = \frac{{{\Delta _{{\text{fus}}}}H_{{\text{m, A}}}^*}}{{{T_{\text{f}}}}}$

### 3.1 固相形成理想固溶体的情况：${\text{A}}\left( {{\text{s, }}x_{\text{A}}^{\text{S}}} \right)\mathop \rightleftharpoons \limits^{{T_{\text{f}}}, p} {\text{A}}\left( {{\text{1, }}x_{\text{A}}^{\text{1}}} \right)$

$\frac{{{\Delta _{{\text{fus}}}}H_{{\text{m, A}}}^*}}{{T_{\text{f}}^*}}-R\ln \frac{{x_{\text{A}}^{\text{l}}}}{{x_{\text{A}}^{\text{s}}}} = \frac{{{\Delta _{{\text{fus}}}}H_{{\text{m, A}}}^*}}{{{T_{\text{f}}}}}$

### 3.2 稀溶液的液-气平衡：${\text{A}}\left( {{\text{l, }}{x_{\text{A}}}} \right)\mathop \rightleftharpoons \limits^{{T_{\text{b}}}, p} {\text{A}}\left( {\text{g}} \right)$

$\frac{{{\Delta _{{\text{vap}}}}H_{{\text{m, A}}}^*}}{{T_{\text{b}}^*}}-R\ln {x_{\text{A}}} = \frac{{{\Delta _{{\text{vap}}}}H_{{\text{m, A}}}^*}}{{{T_{\text{b}}}}}$

## 4 结论

$\frac{{\Delta _\alpha ^\beta H_{{\text{m, A}}}^*}}{{{T^*}}}-R\ln \frac{{x_{\text{A}}^\beta }}{{x_{\text{A}}^\alpha }} = \frac{{\Delta _\alpha ^\beta H_{{\text{m, A}}}^*}}{T}$

TT*的相对大小取决于$-R\ln \frac{{x_{\text{A}}^\beta }}{{x_{\text{A}}^\alpha }}$的正负，即取决于相变熵ΔαβSm, A与纯A相变熵ΔαβSm, A*的相对大小。

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