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## The Application of Taylor Series Expansion in Physical Chemistry

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 基金资助: 国家自然科学青年基金.  21501139西安工业大学2016年MOOC建设《无机化学》资助项目

 Fund supported: 国家自然科学青年基金.  21501139西安工业大学2016年MOOC建设《无机化学》资助项目

Abstract

Physical chemistry course is one of the basic theories of chemical science and chemical technology.The main contents of physical chemistry include two parts, the basic theories and scientific methods.The Taylor series expansion is one of the commonly used mathematical formulas.It is useful when dealing with complex problems.In this paper, the Taylor series expansion is introduced in the derivation of the Clapeyron equation, the freezing-point depression formula and the Kelvin formula.The basic idea of using Taylor series expansion to obtain the relationship between thermodynamic functions is summarized.This work provides a reference for the use of Taylor series expansion in physical chemistry.

Keywords： Taylor series expansion ; Thermodynamic function ; Physical chemistry

YANG Yan, ZHANG Gai, CHEN Shan-Chuan. The Application of Taylor Series Expansion in Physical Chemistry. University Chemistry[J], 2017, 32(7): 83-87 doi:10.3866/PKU.DXHX201612036

$f\left( x \right) = \frac{{f\left( {{x_0}} \right)}}{{0!}} + \frac{{f'\left( {{x_0}} \right)}}{{1!}}\left( {x - {x_0}} \right) + \frac{{f''\left( {{x_0}} \right)}}{{2!}}{\left( {x - {x_0}} \right)^2} + \cdots + \frac{{{f^{\left( n \right)}}\left( {{x_0}} \right)}}{{n!}}{\left( {x - {x_0}} \right)^n} + {R_n}\left( x \right)$

$\ln \left( {1 + x} \right) = x - \frac{1}{2}{x^2} + \frac{1}{3}{x^3} + \cdots + \frac{{{{\left( { - 1} \right)}^n}}}{{n + 1}}{x^{\left( {n + 1} \right)}}\;\;\;\;\left( {\left| x \right| < 1} \right)$

$\begin{array}{l}f\left( {x, y} \right) = f\left( {{x_0}, {y_0}} \right) + \left[{\left( {x-{x_0}} \right)\frac{\partial }{{\partial x}} + \left( {y-{y_0}} \right)\frac{\partial }{{\partial y}}} \right]f\left( {{x_0}, {y_0}} \right) + \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{{2!}}{\left[{\left( {x-{x_0}} \right)\frac{\partial }{{\partial x}} + \left( {y-{y_0}} \right)\frac{\partial }{{\partial y}}} \right]^2}f\left( {{x_0}, {y_0}} \right) + \cdots \end{array}$

0 < x < 1，应用泰勒展开式(2)，并舍去二次以上的高次项，即得：

## 1 泰勒展开式在推导克拉佩龙方程中的应用

$\begin{gathered} {\text{B}}\left( {\alpha, T + {\text{d}}T, p + {\text{d}}p} \right) \rightleftharpoons {\text{B}}\left( {\beta, T + {\text{d}}T, p + {\text{d}}p} \right) \hfill \\ G_{\text{m}}^\alpha \left( {T + {\text{d}}T, p + {\text{d}}p} \right) = G_{\text{m}}^\beta \left( {T + {\text{d}}T, p + {\text{d}}p} \right) \hfill \\ \end{gathered}$

$G_{\rm{m}}^\alpha \left( {T, p} \right) + {\left[{\frac{{\partial G_{\rm{m}}^\alpha \left( {T, p} \right)}}{{\partial p}}} \right]_p}{\rm{d}}T + {\left[{\frac{{\partial G_{\rm{m}}^\alpha \left( {T, p} \right)}}{{\partial p}}} \right]_T}{\rm{d}}p = G_{\rm{m}}^\beta \left( {T, p} \right) + {\left[{\frac{{\partial G_{\rm{m}}^\beta \left( {T, p} \right)}}{{\partial T}}} \right]_p}{\rm{d}}T + {\left[{\frac{{\partial G_{\rm{m}}^\beta \left( {T, p} \right)}}{{\partial p}}} \right]_T}{\rm{d}}p$

$- S_{\rm{m}}^\alpha {\rm{d}}T + V_{\rm{m}}^\alpha {\rm{d}}p = - S_{\rm{m}}^\beta {\rm{d}}T + V_{\rm{m}}^\beta {\rm{d}}p$

(此后的化简过程和教材类似[2]。)

$\Delta _\alpha ^\beta {S_{\rm{m}}} = S_{\rm{m}}^\beta-S_{\rm{m}}^\alpha, \Delta _\alpha ^\beta {V_{\rm{m}}} = V_{\rm{m}}^\beta-V_{\rm{m}}^\alpha$

$\frac{{{\rm{d}}p}}{{{\rm{d}}T}} = \frac{{\Delta _\alpha ^\beta {S_{\rm{m}}}}}{{\Delta _\alpha ^\beta {V_{\rm{m}}}}}$

$\frac{{{\rm{d}}p}}{{{\rm{d}}T}} = \frac{{\Delta _\alpha ^\beta {H_{\rm{m}}}}}{{T\Delta _\alpha ^\beta {V_{\rm{m}}}}}$

## 2 泰勒展开式在推导凝固点降低公式中的应用

${\mu _{{\rm{A}}\left( {\rm{l}} \right)}}\left( {T_{\rm{f}}^ * + {\rm{d}}T, {x_{{\rm{A}}\left( {\rm{l}} \right)}}{\rm{ + d}}{x_{{\rm{A}}\left( {\rm{l}} \right)}}} \right) = \mu _{{\rm{A}}\left( {\rm{s}} \right)}^ * \left( {T_{\rm{f}}^ * + {\rm{d}}T, {x_{{\rm{A}}\left( {\rm{s}} \right)}} + {\rm{d}}{x_{{\rm{A}}\left( {\rm{s}} \right)}}} \right)$

$\begin{array}{l}{\mu _{{\rm{A}}\left( {\rm{l}} \right)}}\left( {T_{\rm{f}}^ *, {x_{{\rm{A}}\left( {\rm{l}} \right)}}} \right) + {\left[{\frac{{\partial {\mu _{{\rm{A}}\left( {\rm{l}} \right)}}\left( {T_{\rm{f}}^ *, {x_{{\rm{A}}\left( {\rm{l}} \right)}}} \right)}}{{\partial T}}} \right]_{{x_{{\rm{A}}\left( {\rm{l}} \right)}}}}{\rm{d}}T + {\left[{\frac{{\partial {\mu _{{\rm{A}}\left( {\rm{l}} \right)}}\left( {T_{\rm{f}}^ *, {x_{{\rm{A}}\left( {\rm{l}} \right)}}} \right)}}{{\partial {\mu _{{\rm{A}}\left( {\rm{l}} \right)}}}}} \right]_T}{\rm{d}}{x_{{\rm{A}}\left( {\rm{l}} \right)}} = \\\mu _{{\rm{A}}\left( {\rm{s}} \right)}^ * \left( {T_{\rm{f}}^ *, {x_{{\rm{A}}\left( {\rm{s}} \right)}}} \right) + {\left[{\frac{{\partial \mu _{{\rm{A}}\left( {\rm{s}} \right)}^ * \left( {T_{\rm{f}}^ *, {x_{{\rm{A}}\left( {\rm{s}} \right)}}} \right)}}{{\partial T}}} \right]_{{x_{{\rm{A}}\left( {\rm{s}} \right)}}}}{\rm{d}}T + {\left[{\frac{{\partial \mu _{{\rm{A}}\left( {\rm{s}} \right)}^ * \left( {T_{\rm{f}}^ *, {x_{{\rm{A}}\left( {\rm{s}} \right)}}} \right)}}{{\partial {x_{{\rm{A}}\left( {\rm{s}} \right)}}}}} \right]_T}{\rm{d}}{x_{{\rm{A}}\left( {\rm{s}} \right)}}\end{array}$

$- S_{{\rm{m}}, {\rm{A}}\left( \rm{l} \right)}^ * {\rm{d}}T + {\left[{\frac{{\partial {\mu _{{\rm{A}}\left( {\rm{l}} \right)}}\left( {T_{\rm{f}}^ *, {x_{{\rm{A}}\left( {\rm{l}} \right)}}} \right)}}{{\partial {x _{{\rm{A}}\left( {\rm{l}} \right)}}}}} \right]_T}{\rm{d}}{x_{{\rm{A}}\left( {\rm{l}} \right)}} = - S_{{\rm{m}}, {\rm{A}}\left( {\rm{s}} \right)}^ * {\rm{d}}T$

(此后的化简过程和教材类似[3]。)

$\begin{array}{l} - S_{{\rm{m}}, {\rm{A}}\left( \rm{l} \right)}^ * {\rm{d}}T + RT{\rm{d}}\ln {x_{{\rm{A}}\left( {\rm{l}} \right)}} = - S_{{\rm{m}}, {\rm{A}}\left( {\rm{s}} \right)}^ * {\rm{d}}T\\RT{\rm{d}}\ln {x_{{\rm{A}}\left( {\rm{l}} \right)}} = \left( {S_{{\rm{m}}, {\rm{A}}\left( {\rm{l}} \right)}^ * - S_{{\rm{m}}, {\rm{A}}\left( {\rm{s}} \right)}^ * } \right){\rm{d}}T = \frac{{{\Delta _{{\rm{fus}}}}H_{{\rm{m}}, {\rm{A}}}^ * }}{T}{\rm{d}}T\\{\rm{d}}\ln {x_{{\rm{A}}\left( {\rm{l}} \right)}} = \frac{{{\Delta _{{\rm{fus}}}}H_{{\rm{m}}, {\rm{A}}}^ * }}{{R{T^2}}}{\rm{d}}T\end{array}$

$\begin{array}{l}\int_1^{{x_{{\rm{A}}\left( {\rm{l}} \right)}}} {{\rm{d}}\ln {x_{{\rm{A}}\left( {\rm{l}} \right)}}} = \int_{T_{\rm{f}}^ * }^{{T_{\rm{f}}}} {\frac{{{\Delta _{{\rm{fus}}}}H_{{\rm{m}}, {\rm{A}}}^ * }}{{R{T^2}}}{\rm{d}}T} \\\ln {x_{{\rm{A}}\left( {\rm{l}} \right)}} = - \frac{{{\Delta _{{\rm{fus}}}}H_{{\rm{m}}, {\rm{A}}}^ * }}{R}\left( {\frac{1}{{{T_{\rm{f}}}}} - \frac{1}{{T_{\rm{f}}^ * }}} \right) = - \frac{{{\Delta _{{\rm{fus}}}}H_{{\rm{m}}, {\rm{A}}}^ * }}{R}\left( {\frac{{T_{\rm{f}}^ * - {T_{\rm{f}}}}}{{{T_{\rm{f}}}T_{\rm{f}}^ * }}} \right)\end{array}$

$\ln \left( {1 - {x_{\rm{B}}}} \right) \approx - {x_{\rm{B}}} = - \frac{{{n_{\rm{B}}}}}{{{n_{\rm{A}}} + {n_{\rm{B}}}}} \approx - \frac{{{n_{\rm{B}}}}}{{{m_{\rm{A}}}/{M_{\rm{A}}}}} = - {b_{\rm{B}}}{M_{\rm{A}}}$

${K_{\rm{f}}} = \frac{{R{{\left( {T_{\rm{f}}^*} \right)}^2}{M_{\rm{A}}}}}{{{\Delta _{{\rm{fus}}}}H_{{\rm{m, A}}}^ \ominus }}$，称为凝固点降低系数，则：

$\Delta {T_{\rm{f}}} = {K_{\rm{f}}}{b_{\rm{B}}}$

## 3 泰勒展开式在推导开尔文公式中的应用

$\begin{gathered} {{\text{B}}_{\left( \rm{l} \right)}}\left( {p_{\text{r}}^ * + \frac{{2\gamma }}{r}} \right) \rightleftharpoons {{\text{B}}_{\left( {\text{g}} \right)}}\left( {p_{\text{r}}^ * } \right) \hfill \\ {\mu _{{\text{B}}\left( \rm{l} \right)}}\left( {p_{\text{r}}^ * + \frac{{2\gamma }}{r}} \right) = {\mu _{{\text{B}}\left( {\text{g}} \right)}}\left( {p_{\text{r}}^ * } \right) \hfill \\ \end{gathered}$

${\mu _{{\text{B}}\left( \rm{l} \right)}}\left( {{p^ * }} \right) + {\left[{\frac{{\partial {\mu _{{\text{B}}\left( \rm{l} \right)}}\left( {{p^ * }} \right)}}{{\partial {p_1}}}} \right]_T}{\text{d}}{p_1} = {\mu _{{\text{B}}\left( {\text{g}} \right)}}\left( {{p^ * }} \right) + {\left[{\frac{{\partial {\mu _{{\text{B}}\left( {\text{g}} \right)}}\left( {{p^ * }} \right)}}{{\partial {p_{\text{g}}}}}} \right]_T}{\text{d}}{p_{\text{g}}}$

μB(l)(p*) = μB(g)(p*)，所以式(17)化简为：

$\begin{array}{l}\frac{{\partial {\mu _{{\rm{B}}\left( \rm{l} \right)}}\left( {{p^ * }} \right)}}{{\partial {p_1}}}{\rm{d}}{p_1} = \frac{{\partial {\mu _{{\rm{B}}\left( {\rm{g}} \right)}}\left( {{p^ * }} \right)}}{{\partial {p_{\rm{g}}}}}{\rm{d}}{p_{\rm{g}}}\\{V_{{\rm{m}}\left( {\rm{l}} \right)}}{\rm{d}}{p_{\rm{l}}} = {V_{{\rm{m}}\left( {\rm{g}} \right)}}{\rm{d}}{p_{\rm{g}}}\end{array}$

(此后的化简过程同参考文献中的方法[4, 5]。)

$\begin{array}{l}\int_{{p^ * }}^{p_{\rm{r}}^ * + \frac{{2\gamma }}{r}} {{V_{{\rm{m}}\left( {\rm{l}} \right)}}{\rm{d}}{p_{\rm{l}}}} = \int_{{p^ * }}^{p_{\rm{r}}^ * } {\frac{{RT}}{{{p_{\rm{g}}}}}{\rm{d}}{p_{\rm{g}}}} \\{V_{{\rm{m}}\left( {\rm{l}} \right)}}\left( {p_{\rm{r}}^ * + \frac{{2\gamma }}{r} - {p^ * }} \right)RT\ln \frac{{p_{\rm{r}}^ * }}{{{p^ * }}}\end{array}$

$\ln \frac{{p_{\rm{r}}^ * }}{{{p^ * }}} = \frac{{2\gamma {V_{{\rm{m}}\left( {\rm{l}} \right)}}}}{{RTr}} = \frac{{2\gamma M}}{{RT\rho r}}$

## 4 结论

(1)将Gmμ表示为自变量的函数；

(2)利用相平衡关系得到联系两相Gmμ的等式；

(3)将Gmμ在已知的自变量值处进行泰勒展开；

(4)将Gmμ的一阶偏导数带入展开后的等式，进行化简，得到表达自变量关系的式子。

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