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## A Summary of Calculation Formula of the State Function in Physical Chemistry

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 基金资助: 国家精品资源共享课；吉林省高等教育教学改革立项课题.  吉教高字[2017]

 Fund supported: 国家精品资源共享课；吉林省高等教育教学改革立项课题.  吉教高字[2017]

Abstract

In this paper, the calculation formulas of the state function are summarized in three forms, including the basic equations, differential form and integral form.The students can learn and understand from a similar formula for classification easily.

Keywords： Physical chemistry ; Calculation formula ; Differential expression ; Integral formula

YANG Hua. A Summary of Calculation Formula of the State Function in Physical Chemistry. University Chemistry[J], 2017, 32(9): 59-65 doi:10.3866/PKU.DXHX201704014

### 1.1 热力学基本关系式

1)热力学第一定律涉及能量守恒与转化，涉及到的主要计算公式有：

$\Delta U = Q + W$

$\Delta H = \Delta U + \Delta \left( {pV} \right)$

2)热力学第二定律涉及变化的方向与限度，数学表示式为：

$\Delta S = \int_{{T_1}}^{{T_2}} {\frac{{\delta {Q_{\text{R}}}}}{T}}$

$\Delta A = \Delta U-\left( {{T_2}{S_2}-{T_1}{S_1}} \right)$

$\Delta G = \Delta H-\left( {{T_2}{S_2}-{T_1}{S_1}} \right)$

### 1.2 动力学基本关系式

$-\frac{{{\text{d}}c}}{{{\text{d}}t}} = k{c^n}$

$n = 0, {c_{{\text{A}}0}}-{c_{\text{A}}} = kt, {t_{1/2}} = \frac{{{c_{{\text{A}}0}}}}{{2k}}$

$n = 1, \ln \frac{{{c_{{\text{A}}0}}}}{{{c_{\text{A}}}}} = kt, {t_{1/2}} = \frac{{\ln 2}}{k}$

$n = 2, \frac{1}{{{c_{\text{A}}}}}-\frac{1}{{{c_{{\text{A}}0}}}} = kt, {t_{1/2}} = \frac{1}{{k{c_{{\text{A}}0}}}}$

$n = 3, \frac{1}{2}\left( {\frac{1}{{c_{\text{A}}^2}}-\frac{1}{{c_{{\text{A0}}}^2}}} \right) = kt, {t_{1/2}} = \frac{3}{{2kc_{{\text{A0}}}^2}}$

### 1.3 电化学基本关系式

$Q = \frac{m}{M}ZF = nZF$

$E = {E^\Theta }-\frac{{RT}}{{nF}}\ln \frac{{a_{\text{G}}^{\text{g}}a_{\text{H}}^{\text{h}}}}{{a_{\text{A}}^{\text{a}}a_{\text{B}}^{\text{b}}}}$

$-{\Delta _{\text{r}}}G =-{W_{\text{R}}}^\prime$

${W_{\text{R}}}^\prime =-nFE$

$-{\Delta _{\text{r}}}{G_{\text{m}}} = nFE;-{\Delta _{\text{r}}}G_{\text{m}}^\Theta = nF{E^\Theta }$

${\Delta _{\text{r}}}{S_{\text{m}}} = nF{\left( {\frac{{\partial E}}{{\partial T}}} \right)_p}$

${\Delta _{\text{r}}}{H_{\text{m}}} =-nFE + nET{\left( {\frac{{\partial E}}{{\partial T}}} \right)_p}$

### 2 微分计算式

#### 2.1.1 状态变化

${\left( {\frac{{\partial U}}{{\partial T}}} \right)_V} = {C_V}$

${\left( {\frac{{\partial H}}{{\partial T}}} \right)_p} = {C_p}$

${\left( {\frac{{\partial S}}{{\partial T}}} \right)_V} = \frac{{{C_V}}}{T}$

${\left( {\frac{{\partial S}}{{\partial T}}} \right)_p} = \frac{{{C_p}}}{T}$

##### 2.1.2.2 压力、浓度和平衡常数随温度变化

1)单组分体系气液两相平衡时，平衡压力p与温度T的关系式：

$\frac{{{\text{d}}\ln p}}{{{\text{d}}T}} = \frac{{\Delta {H_{\text{m}}}}}{{R{T^2}}}$

2)二组分体系液固两相平衡时，液相浓度x与温度T的关系式：

$\frac{{{\text{d}}\ln x}}{{{\text{d}}T}} = \frac{{\Delta {H_{\text{m}}}}}{{R{T^2}}}$

$\frac{{{\text{d}}\ln a}}{{{\text{d}}T}} = \frac{{\Delta {H_{\text{m}}}}}{{R{T^2}}}$

3)标准平衡常数随温度变化。

${\left( {\frac{{\partial \ln K_a^\Theta }}{{\partial T}}} \right)_p} = \frac{{{\Delta _{\text{r}}}H_{\text{m}}^\Theta }}{{R{T^2}}}$

${\left( {\frac{{\partial \ln {K_x}}}{{\partial T}}} \right)_p} = \frac{{{\Delta _{\text{r}}}H_{{\text{m, }}x}^\Theta }}{{R{T^2}}}$

${\left( {\frac{{\partial \ln K_m^\Theta }}{{\partial T}}} \right)_p} = \frac{{{\Delta _{\text{r}}}H_{{\text{m, }}m}^\Theta }}{{R{T^2}}}$

${\left( {\frac{{\partial \ln K_c^\Theta }}{{\partial T}}} \right)_p} = \frac{{{\Delta _{\text{r}}}H_{{\text{m, }}c}^\Theta }}{{R{T^2}}}$

##### 2.2 动力学微分计算式

$\frac{{{\text{d}}\ln k}}{{{\text{d}}T}} = \frac{{{E_{\text{a}}}}}{{R{T^2}}}$

##### 2.3 电化学微分计算式

${\left( {\frac{{\partial E}}{{\partial T}}} \right)_p} = \frac{{{\Delta _{\text{r}}}S}}{{nF}}$

### 3 积分计算式

#### 3.1.1 状态变化

$\Delta U = {C_V}\left( {{T_2}-{T_1}} \right)$

$\Delta H = {C_p}\left( {{T_2}-{T_1}} \right)$

$\Delta S = {C_p}\ln \frac{{{T_2}}}{{{T_1}}}-nR\ln \frac{{{p_2}}}{{{p_1}}}$

$\Delta S = {C_V}\ln \frac{{{T_2}}}{{{T_1}}} + nR\ln \frac{{{V_2}}}{{{V_1}}}$

#### 3.1.2 相变化和化学变化

${\Delta _{\text{r}}}{U_2} = {\Delta _{\text{r}}}{H_1} + \int_{{T_1}}^{{T_2}} {\Delta {C_V}{\text{d}}T}$

${\Delta _{\text{r}}}{H_2} = {\Delta _{\text{r}}}{H_1} + \int_{{T_1}}^{{T_2}} {\Delta {C_p}{\text{d}}T}$

${\Delta _{\text{r}}}{S_2} = {\Delta _{\text{r}}}{S_1} + \int_{{T_1}}^{{T_2}} {\frac{{\Delta {C_p}}}{T}{\text{d}}T}$

${\Delta _{\text{r}}}{S_2} = {\Delta _{\text{r}}}{S_1} + \int_{{T_1}}^{{T_2}} {\frac{{\Delta {C_V}}}{T}{\text{d}}T}$

${\Delta _r}{U_2} = {\Delta _r}{U_1} + \Delta {C_V}\left( {{T_2}-{T_1}} \right)$

${\Delta _r}{H_2} = {\Delta _r}{H_1} + \Delta {C_p}\left( {{T_2}-{T_1}} \right)$

${\Delta _r}{S_2} = {\Delta _r}{S_1} + \Delta {C_p}\ln \frac{{{T_2}}}{{{T_1}}}$

${\Delta _r}{S_2} = {\Delta _r}{S_1} + \Delta {C_V}\ln \frac{{{T_2}}}{{{T_1}}}$

$\frac{{{\Delta _{\text{r}}}{G_2}}}{{{T_2}}}-\frac{{{\Delta _r}{G_1}}}{{{T_1}}} = \int_{{T_1}}^{{T_2}} {-\frac{{{\Delta _{\text{r}}}H}}{{{T_2}}}{\text{d}}T}$

$\frac{{{\Delta _{\text{r}}}{A_2}}}{{{T_2}}}-\frac{{{\Delta _r}{A_1}}}{{{T_1}}} = \int_{{T_1}}^{{T_2}} {-\frac{{{\Delta _{\text{r}}}U}}{{{T_2}}}{\text{d}}T}$

$\ln \frac{{{p_2}}}{{{p_1}}} = \frac{{{\Delta _{\text{r}}}H_{\text{m}}^\Theta }}{R}\left( {\frac{1}{{{T_1}}}-\frac{1}{{{T_2}}}} \right)$

$\ln \frac{{{x_2}}}{{{x_1}}} = \frac{{{\Delta _{\text{r}}}H_{\text{m}}^\Theta }}{R}\left( {\frac{1}{{{T_1}}}-\frac{1}{{{T_2}}}} \right)$

$\ln \frac{{{a_2}}}{{{a_1}}} = \frac{{{\Delta _{\text{r}}}H_{\text{m}}^\Theta }}{R}\left( {\frac{1}{{{T_1}}}-\frac{1}{{{T_2}}}} \right)$

$\ln \frac{{K_2^\Theta }}{{K_1^\Theta }} = \frac{{{\Delta _{\text{r}}}H_{\text{m}}^\Theta }}{R}\left( {\frac{1}{{{T_1}}}-\frac{1}{{{T_2}}}} \right)$

#### 3.2 动力学积分计算式

$\ln \frac{{{k_2}}}{{{k_1}}} = \frac{{{E_{\text{a}}}}}{R}\left( {\frac{1}{{{T_1}}}-\frac{1}{{{T_2}}}} \right)$

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