## 理想完全互溶双液系相图的数学解析

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## Mathematic Analysis of Phase Diagram of Ideal Completely Miscible Two Components Liquid System

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 基金资助: 衡阳师范学院教学改革研究项目.  JYKT201512，JYKT201423，JYKT201513

 Fund supported: 衡阳师范学院教学改革研究项目.  JYKT201512，JYKT201423，JYKT201513

Abstract

In an ideal completely miscible two components liquid system, the two components strictly abide by the Raoult's law, and their plane phase diagrams have a definite function relationship. Based on the Clausius-Clapeyron equation, the functional analytic formula ofp-xT-x and p-T phase diagrams in this ideal system were derived in the present paper.

Keywords： Ideal completely miscible two components liquid system ; Phase diagram ; Functional analytic formula ; Clausius-Clapeyron equation

LIU Xing, QU Jing-Nian, ZHOU Li-Jun, LAI Hua, ZENG Rong-Ying, LI Jun-Hua. Mathematic Analysis of Phase Diagram of Ideal Completely Miscible Two Components Liquid System. University Chemistry[J], 2016, 31(7): 96-100 doi:10.3866/PKU.DXHX201508010

## 1 p-x相图液相线、气相线的函数关系

p-x相图是指定T下得到的，设A、B为双液系的两个组分，以xAyA分别表示气液平衡时液相中与气相中A的含量(物质的量分数)，pApB分别表示气相中A、B的分压(p表示总压)，pA*pB*分别表示纯A、B的饱和蒸气压，且设pA* > pB*。由于理想完全互溶双液系的两个组分在全部浓度范围内均服从拉乌尔定律，所以

${{p}_{A}}={{x}_{A}}p_{_{A}}^{*}$

${{p}_{B}}={{x}_{B}}p_{_{B}}^{*}$

${{x}_{A}}+{{x}_{B}}=1$

$p={{p}_{A}}+{{p}_{B}}$

$p=\left( p_{_{A}}^{*}-p_{_{B}}^{*} \right){{x}_{A}}+p_{_{B}}^{*}$

${{p}_{A}}={{y}_{A}}p$

${{x}_{A}}p_{_{A}}^{*}={{y}_{A}}p$

${{x}_{A}}=\frac{p-p_{_{B}}^{*}}{p_{_{A}}^{*}-p_{_{B}}^{*}}$

$p=\frac{p_{_{A}}^{*}p_{_{B}}^{*}}{p_{_{A}}^{*}-{{y}_{A}}\left( p_{_{A}}^{*}-p_{_{B}}^{*} \right)}$

${p}'\left( {{y}_{A}} \right)=\frac{p_{_{A}}^{*}p_{_{B}}^{*}\left( p_{_{A}}^{*}-p_{_{B}}^{*} \right)}{{{\left[ p_{_{A}}^{*}-{{y}_{A}}\left( p_{_{A}}^{*}-p_{_{B}}^{*} \right) \right]}^{2}}}>0$

${{y}_{A}}=\frac{{{x}_{A}}p_{_{A}}^{*}}{p}$

${{y}_{B}}=\frac{{{x}_{B}}p_{_{B}}^{*}}{p}$

$\frac{{{y}_{A}}}{{{y}_{B}}}=\frac{p_{_{A}}^{*}}{p_{_{B}}^{*}}\cdot \frac{{{x}_{A}}}{{{x}_{B}}}$

## 2 T-x相图液相线、气相线的函数关系

T-x相图是指定p(设p = p)下得到的，由于A与B组成理想的双组分体系，对于液相是理想溶液，即A、B组分在全部浓度范围内均严格遵守Raoult定律，两个组分的分子大小及作用力情况与液态纯A、B时相同；而对于气相则是理想气体混合物，气相A与B均满足Dalton分压定律，混合气体中的每种组分与理想气体A、B单独存在时状态相同。由此可见，对于A与B组成理想完全互溶双液系，其中A、B的饱和蒸汽压pA*pB*与温度的关系可以用Clausius-Clapeyron方程[3-6]描述，如对A：

$\frac{\text{d}\ln p_{_{A}}^{*}}{\text{dT}}=\frac{{{\Delta }_{\text{vap}}}{{H}_{\text{m}}}}{R{{T}^{2}}}$

ΔvapHm是纯A的摩尔蒸发热，假设其受温度变化的影响很小，则其可以近似看作常数(可以由Troutonʹs规则估算)，以TA表示纯A在p下的正常沸点，对式(14)进行定积分，得：

$\ln \frac{p_{_{A}}^{*}}{{{p}^{\Theta }}}=\frac{{{\Delta }_{\text{vap}}}{{H}_{\text{m}}}}{R{{T}_{A}}T}$

$p_{_{A}}^{*}={{p}^{\Theta }}{{e}^{\frac{{{\Delta }_{\text{vap}}}{{H}_{\text{m}}}\left( T-{{T}_{A}} \right)}{R{{T}_{A}}T}}}$

${p_{\rm{B}}}* = {p^\Theta }{e^{\frac{{{\Delta _{{\rm{vap}}}}{H_{\rm{m}}}\left( {T - {T_B}} \right)}}{{R{T_B}T}}}}$

$1={{x}_{A}}{{e}^{\frac{{{\Delta }_{\text{vap}}}{{H}_{\text{m}}}\left( T-{{T}_{A}} \right)}{R{{T}_{A}}T}}}+\left( 1-{{x}_{A}} \right){{e}^{\frac{{{\Delta }_{\text{vap}}}{{{{H}'}}_{\text{m}}}\left( T-{{T}_{A}} \right)}{R{{T}_{A}}T}}}$

pA* > pB*，即：

$\left( 1-{{y}_{A}} \right){{e}^{\frac{{{\Delta }_{\text{vap}}}{{H}_{\text{m}}}\left( T-{{T}_{A}} \right)}{R{{T}_{A}}T}}}+{{y}_{A}}{{e}^{\frac{{{\Delta }_{\text{vap}}}{{{{H}'}}_{\text{m}}}\left( T-{{T}_{B}} \right)}{R{{T}_{B}}T}}}={{e}^{\frac{{{\Delta }_{\text{vap}}}{{H}_{\text{m}}}\left( T-{{T}_{A}} \right)}{R{{T}_{A}}T}}}+{{e}^{\frac{{{\Delta }_{\text{vap}}}{{{{H}'}}_{\text{m}}}\left( T-{{T}_{B}} \right)}{R{{T}_{B}}T}}}$

## 3 p-T相图液相线、气相线的函数关系

p-T相图是在指定组成的条件下得到的，在推导式(18)时，如把xA看作常数(xA ∈ [0, 1])，而把p当做变量，则有：

$p={{x}_{A}}{{p}^{\Theta }}{{e}^{\frac{{{\Delta }_{\text{vap}}}{{H}_{\text{m}}}\left( T-{{T}_{A}} \right)}{R{{T}_{A}}T}}}+\left( 1-{{x}_{A}} \right){{p}^{\Theta }}{{e}^{\frac{{{\Delta }_{\text{vap}}}{{{{H}'}}_{\text{m}}}\left( T-{{T}_{B}} \right)}{R{{T}_{B}}T}}}$

\begin{align} & \left[ (1-{{y}_{A}} \right){{e}^{\frac{{{\Delta }_{\text{vap}}}{{H}_{\text{m}}}\left( T-{{T}_{A}} \right)}{R{{T}_{A}}T}}}+{{y}_{A}}{{e}^{\frac{{{\Delta }_{\text{vap}}}{{{{H}'}}_{\text{m}}}\left( T-{{T}_{B}} \right)}{R{{T}_{B}}T}}}]p \\ & ={{p}^{\Theta }}{{e}^{\frac{{{\Delta }_{\text{vap}}}{{H}_{\text{m}}}\left( T-{{T}_{A}} \right)}{R{{T}_{A}}T}+\frac{{{\Delta }_{\text{vap}}}{{{{H}'}}_{\text{m}}}\left( T-{{T}_{B}} \right)}{R{{T}_{B}}T}}} \\ \end{align}

### 图3

(a) xA = 0.5；(b) yA = 0.5

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