## Calculation of Adsorption Thermodynamics Parameters for Adsorption on the Solid-Liquid Interface

Wang Weitao,, Chen Xiangli, Yang Baiqin

 基金资助: 陕西科技大学教学改革研究项目.  19Y069

Abstract

The adsorption on the solid-liquid interface is important for industry and daily life. To investigate the adsorption process, the adsorption thermodynamics usually are essential. The adsorption thermodynamics are analyzed with the parameters of the change of Gibbs free energy (ΔG), enthalpy (ΔH), and entropy (ΔS). In this presentation, the different methods for calculation of the adsorption thermodynamics parameters are introduced. Moreover, examples for calculations and applications of the adsorption thermodynamics are presented.

Keywords： Adsorption ; Thermodynamics ; Thermodynamics parameter

Wang Weitao. Calculation of Adsorption Thermodynamics Parameters for Adsorption on the Solid-Liquid Interface. University Chemistry[J], 2021, 36(2): 2003032-0 doi:10.3866/PKU.DXHX202003032

## 1 固-液界面吸附过程化学势

${\mu }_{i}={\mu }_{i}^{\ominus}+RT\mathrm{l}\mathrm{n}\left({a}_{i}\right)={\mu }_{i}^{\ominus}+RT\mathrm{l}\mathrm{n}\left({f}_{i}{c}_{i}\right)$

${\mu }_{\mathrm{a}\mathrm{d}\mathrm{s}, i}={\mu }_{\mathrm{a}\mathrm{d}\mathrm{s}, i}^{\ominus}+RT\mathrm{l}\mathrm{n}\left({a}_{\mathrm{a}\mathrm{d}\mathrm{s}, i}\right)={\mu }_{\mathrm{a}\mathrm{d}\mathrm{s}, i}^{\ominus}+RT\mathrm{l}\mathrm{n}\left({f}_{\mathrm{a}\mathrm{d}\mathrm{s}, i}{c}_{\mathrm{a}\mathrm{d}\mathrm{s}, i}\right)$

${\mu }_{i}^{\ominus}+RT\mathrm{l}\mathrm{n}\left({a}_{i}\right)={\mu }_{\mathrm{a}\mathrm{d}\mathrm{s}, i}^{\ominus}+RT\mathrm{l}\mathrm{n}\left({a}_{\mathrm{a}\mathrm{d}\mathrm{s}, i}\right)$

${\mu }_{\mathrm{a}\mathrm{d}\mathrm{s}, i}^{\ominus}-{\mu }_{i}^{\ominus}=-[RT\mathrm{l}\mathrm{n}\left({a}_{\mathrm{a}\mathrm{d}\mathrm{s}, i}\right)-RT\mathrm{l}\mathrm{n}\left({a}_{i}\right)]$

$∆{G}^{\ominus}=-[RT\mathrm{l}\mathrm{n}\left({a}_{\mathrm{a}\mathrm{d}\mathrm{s}, i}\right)-RT\mathrm{l}\mathrm{n}\left({a}_{i}\right)]$

$∆{G}^{\ominus}=-RT\mathrm{l}\mathrm{n}\left(\frac{{a}_{\mathrm{a}\mathrm{d}\mathrm{s}, i}}{{a}_{i}}\right)=-RT\mathrm{l}\mathrm{n}\left(\frac{{f}_{\mathrm{a}\mathrm{d}\mathrm{s}, i}{c}_{\mathrm{a}\mathrm{d}\mathrm{s}, i}}{{f}_{i}{c}_{i}}\right)$

$∆{G}^{\ominus}=-RT\mathrm{l}\mathrm{n}K$

### 2.1 极限法获得K计算ΔG

$\underset{{c}_{\mathrm{ads}, i}\to 0}{\mathrm{lim}}\frac{{c}_{\mathrm{a}\mathrm{d}\mathrm{s}, i}}{{c}_{i}}=\frac{{a}_{\mathrm{a}\mathrm{d}\mathrm{s}, i}}{{a}_{i}}=K$

${c}_{\mathrm{a}\mathrm{d}\mathrm{s}, i}=\frac{({\rho }_{1}/{M}_{1}){A}_{1}}{\frac{s}{N(x/m)}}\frac{1}{{M}_{2}}·{10}^{-3}$

### 2.2 利用平衡浓度获得K计算ΔG

$K=\frac{{c}_{0}-{c}_{\mathrm{e}}}{{c}_{\mathrm{e}}}$

$K=\frac{{q}_{\mathrm{e}}}{{c}_{\mathrm{e}}}$

${q}_{\mathrm{e}}=\frac{{(c}_{0}-{c}_{\mathrm{e}})V}{m}$

### 2.3 利用等温吸附模型中的常数计算

 模型名称 模型方程表达式 Langmuir方程 ${q}_{\mathrm{e}}=\frac{{q}_{m}{K}_{\mathrm{L}}{c}_{\mathrm{e}}}{1+{K}_{\mathrm{L}}{c}_{\mathrm{e}}}$ Freundlich方程 ${q}_{\mathrm{e}}={K}_{\mathrm{F}}{c}_{\mathrm{e}}^{1/n}$ Herry方程 ${q}_{\mathrm{e}}={K}_{\mathrm{H}}{c}_{\mathrm{e}}$ Redlich-Peterson方程 ${q}_{\mathrm{e}}=\frac{A{c}_{\mathrm{e}}}{1+B{c}_{\mathrm{e}}^{g}}$ Temkin方程 ${q}_{\mathrm{e}}=A+B\mathrm{l}\mathrm{n}{c}_{\mathrm{e}}$ Kobe-Corrigan方程 ${q}_{\mathrm{e}}=\frac{A{c}_{\mathrm{e}}^{n}}{1+B{c}_{\mathrm{e}}^{n}}$

### 2.4 利用Freundlich公式参数法计算ΔG

Gibbs吸附等温式可以写成[15]:

$\Delta G' = - RT\int_0^a {N\frac{{{\rm{d}}a}}{a}}$

$\Delta G'' = - RT\int_0^a {q\frac{{{\rm{d}}a}}{a}}$

$\Delta G'' = - RT\int_0^x {q\frac{{{\rm{d}}x}}{x}}$

$\Delta G'' = - RT\int_0^x {{K_{\rm{F}}}{x^{\left( {\frac{1}{n} - 1} \right)}}\frac{{{\rm{d}}x}}{x}}$

$\Delta G'' = - \frac{{RT{K_{\rm{F}}}{x^{1/n}}}}{{1/n}} = - nRTq$

$∆{G}^{\text{'}\text{'}}$除以q，可以得到每摩尔溶质吸附的吉布斯自由能变(J·mol–1)，即

$\Delta G' = - \frac{{\Delta G''}}{q} = - nRT$

### 3.1 利用平衡常数K求ΔH

${\rm{\Delta }}H = - R\frac{{{\rm{d}}\ln K}}{{{\rm{d}}\left( {\frac{1}{T}} \right)}}$

$\ln \frac{{{K_2}}}{{{K_1}}} = - \frac{{\Delta H}}{R}\left( {\frac{1}{{{T_2}}} - \frac{1}{{{T_1}}}} \right)$

$\ln K = - \frac{{\Delta H}}{{RT}} + C$

$- RT\ln K = \Delta H - T\Delta S$

$\ln K = - \frac{{\Delta H}}{{RT}} + \frac{{\Delta S}}{R}$

$\frac{{\partial \lg K}}{{\partial \left( {1/T} \right)}} = - \frac{{\Delta H}}{{2.303R}}$

### 3.2 利用平衡浓度ce求ΔH

$K\propto \frac{1}{{{x_{\rm{e}}}}} \propto \frac{1}{{{c_{\rm{e}}}}}$

$\ln \frac{{{c_{{\rm{e, }}1}}}}{{{c_{{\rm{e, }}2}}}} = - \frac{{\Delta H}}{R}\left( {\frac{1}{{{T_2}}} - \frac{1}{{{T_1}}}} \right)$

$\ln {c_{\rm{e}}} = \frac{{\Delta H}}{{RT}} + C'$

$\Delta H = \Delta G - T{\left( {\frac{{\partial \Delta G}}{{\partial T}}} \right)_p}$

## 4 固-液界面吸附过程ΔS计算

$\Delta S = - \left( {\frac{{\partial \Delta G}}{{\partial T}}} \right)$

$\Delta S = - \frac{{\Delta H - \Delta G}}{T}$

### 5.1 活性炭自水溶液中吸附对二甲苯

KC-8活性炭自水溶液中吸附对二甲苯，通过对其吸附等温线拟合，发现其符合Freundlich吸附模型，因此可以根据式(18)计算ΔG，根据式(27)计算ΔH，根据式(30)计算ΔS，得到的吸附热力学参数如表 2所示[22]。该吸附过程的ΔG < 0且其值在-20 – 0 kJ∙mol-1范围内，表明该吸附过程是一个自发的物理吸附过程。该吸附过程的ΔH > 0，表明该吸附过程是吸热过程；根据ΔH值的大小并结合ΔG值可以判断，该吸附作用力不可能为化学键作用力。吸附熵变ΔS > 0，因此可以推断该吸附过程伴随着溶剂的脱附过程。当活性炭吸附对二甲苯时，其自由度减少，对应的熵减少；当溶剂分子(水)脱附时，熵值增大。当吸附一个对二甲苯分子时，就会有多个溶剂水分子脱附，因而该吸附过程总熵变大于0。

 ΔG/(kJ∙mol-1) ΔH/(kJ∙mol-1) ΔS/(kJ∙mol-1∙K-1) -15.66 ± 4.84 31.53 ± 0.01 150.67 ± 15.50

### 5.2 硅胶对同系物的吸附

 吸附质 C2H5OH C3H7OH C4H9OH C5H11OH C6H13OH C8H17OH ΔG/(kJ∙mol-1) -20.8 -19.3 -18.2 -17.4 -16.7 -15.5

 吸附质 ΔG/(kJ∙mol-1) ΔH/(kJ∙mol-1) ΔS/(kJ∙mol-1∙K-1) 苯 -9.28 -9.03 -6.27 萘 -8.61 -15.7 -25.0 蒽 -8.44 17.5 -33.4 菲 -8.27 -16.0 -27.2

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