## Analysis of the Temperature Deviation for Boiling of Liquid and Condensation of Vapor Using Kelvin Equation

Tao Zhibin, Zhang Shuyong,

Abstract

The discrepancies for calculating condensation temperature and boiling temperature for droplet and small bubble using Kelvin equation were discussed. The discrepancies can be ascribed to the neglect of variation of the vapor pressure of curved surface to plain surface with temperature, and the misusing of Kelvin equation for small bubble during boiling. The way to reduce the discrepancy is introduced while the applicability of Kelvin equation is discussed.

Keywords： Kelvin equation ; Curved surface ; Vapor pressure ; Boiling ; Condensing

Tao Zhibin. Analysis of the Temperature Deviation for Boiling of Liquid and Condensation of Vapor Using Kelvin Equation. University Chemistry[J], 2021, 36(2): 2002037-0 doi:10.3866/PKU.DXHX202002037

## 1 引言

$\ln \frac{{{p_{\rm{r}}}}}{{{p^*}}} = \frac{{2\sigma M}}{{RT\rho r}}$

### 2.1 凝聚形成小液滴时的蒸气冷凝温度

 步数 T/℃ pr/p* p**/kPa T’/℃ 1 30.00 2.779 1.0698 7.956 2 7.956 3.155 0.9423 6.107 3 6.107 3.192 0.9314 5.939 4 5.939 3.195 0.9305 5.925 5 5.925 3.196 0.9302 5.920 6 5.920 3.196 0.9302 5.920

T：环境温度；p**：对应的饱和蒸汽压；T’：对应温度

 T/℃ p*/kPa σ/(mN∙m−1) ρ/(kg∙m−3) 30.00 4.2470 71.19 995.61 7.956 1.0698 74.52 999.81 6.107 0.9423 74.78 999.90 5.939 0.9314 74.81 999.91 5.925 0.9305 74.81 999.91 5.920 0.9302 74.81 999.91

p*：水的饱和蒸汽压；σ：水的表面张力；ρ：水的密度

### 2.2 有小气泡存在时的液体沸腾温度

 步数 T/℃ pr/p* p**/kPa T’/℃ 1 100.00 0.999 101.431 100.0034 2 100.0034 0.999 101.431 100.0034

$\begin{array}{*{20}{c}} {\ln \frac{{{p_2}}}{{{p_1}}} = - \frac{{{\Delta _{{\rm{vap}}}}{H_{\rm{m}}}}}{R}\left( {\frac{1}{{{T_2}}} - \frac{1}{{{T_1}}}} \right)} \end{array}$

## 3 开尔文公式误差产生的原因分析

${p_{\rm{L}}} = {p_{\rm{G}}} + \sigma \frac{{{\rm{d}}{A_{\rm{ \mathsf{ σ} }}}}}{{{\rm{d}}{V_{\rm{L}}}}}$

${\mu _{\rm{L}}} = {\mu _{\rm{G}}}$

$\frac{{{\rm{d }}{A_{\rm{ \mathsf{ σ} }}}}}{{{\rm{d }}{V_{\rm{L}}}}} = 0$

${p_{\rm{L}}} = {p_{\rm{G}}} = {p^*}$

${V_{\rm{L}}} = \frac{{4{\rm{ \mathsf{ π} }}{r^3}}}{3}$

${A_{\rm{ \mathsf{ σ} }}} = 4{\rm{ \mathsf{ π} }}{r^2}$

${p_{\rm{L}}} - {p_{\rm{G}}} = \Delta p = \frac{{2\sigma }}{r}$

${\rm{d}}{\mu _{\rm{g}}} = RT{\rm{d}}\ln {p_{\rm{g}}}$

${\rm{d}}{\mu _{\rm{l}}} = RT{\rm{d}}\ln {p_{\rm{r}}} = {V_{\rm{m}}}{\rm{d}}{p_{\rm{L}}}$

$\int_{{p^*}}^{{p_{\rm{r}}}} {RT{\rm{d}}\ln p} = {\int_{{p_{\rm{G}}}}^{{p_{\rm{G}}} + \frac{{2\sigma }}{r}} {{V_{\rm{m}}}{\rm{ d}}p} _{\rm{L}}}$

$RT\ln \frac{{{p_{\rm{r}}}}}{{{p^*}}} = \frac{M}{\rho }\left( {\frac{{2\sigma }}{r}} \right)$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Haynes W. M. CRC Handbook of Chemistry and Physics 97th ed Boca Raton: CRC Press, 2017.

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