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## The Physical Significance and the Role of the Molecular Partition Function

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Abstract

We discussed the physical meaning of the molecular partition function, and the role of both the molecular partition function and the partition function of the system in the calculation of the thermodynamic properties.It was shown that the molecular partition function is neither an extensive property nor an intensive one, but only a tie to connect the thermodynamic properties with the microscopic information of the system.

Keywords： Molecular partition function ; Canonical partition function ; Thermodynamic function ; Intensive property ; Extensive property

LIU Guo-Jie, SHI Ji-Bin. The Physical Significance and the Role of the Molecular Partition Function. University Chemistry[J], 2016, 31(1): 75-78 doi:10.3866/pku.DXHX20160175

## 1 子的配分函数的定义

$q\underline{\underline{\text{def}}}\sum\limits_{i}{{{\text{e}}^{-{{\varepsilon }_{i}}/kT}}}$

$q\underline{\underline{\text{def}}}\sum\limits_{j}{{{g}_{j}}{{\text{e}}^{-{{\varepsilon }_{j}}/kT}}}$

${{q}_{0}}\underline{\underline{\text{def}}}\sum\limits_{j}{{{g}_{j}}{{\text{e}}^{-({{\varepsilon }_{j}}-{{\varepsilon }_{0}})/kT}}}$

$q={{q}_{0}}{{\text{e}}^{-{{\varepsilon }_{\text{0}}}/kT}}$

### 2.2 从Boltzmann能量分布定律理解

${{P}_{j}}=\frac{{{N}_{j}}}{N}=\frac{{{g}_{j}}{{\text{e}}^{\text{-}({{\varepsilon }_{j}}-{{\varepsilon }_{0}})/kT}}}{{{q}_{0}}}\quad \quad(j=0,1,2,\cdots)$

${{q}_{0}}=\frac{{{g}_{j}}{{\text{e}}^{\text{-}({{\varepsilon }_{j}}-{{\varepsilon }_{0}})/kT}}}{{{P}_{j}}}\quad \quad(j=0,1,2,\cdots)$

${{q}_{0}}=\frac{N}{{{N}_{0}}}$

①由于N0是处于基态能级的子数，它的值不可能大于子的总数N，故由式(7)可见，子的配分函数q0的值不可能小于1。

②温度越高，子从基态能级逃逸到较高能级的程度越大，故子的配分函数q0的值是随温度升高而增大的。

③在相同的温度下，子的相邻能级的间隔越小，子越容易逃逸到较高能级，故q0值也越大，这就是说，平动子的q0最大，振子的q0最小，转子的q0介于其间。

## 3 子的配分函数属于一个子所有

$q={{q}_{\text{t}}}\cdot {{q}_{\text{r}}}\cdot {{q}_{\text{v}}}\cdot {{q}_{\text{e}}}\cdot {{q}_{\text{n}}}$

\begin{align} &q={{q}_{\text{t}}}\cdot {{q}_{\text{r}}}\cdot {{q}_{\text{v}}}\cdot {{q}_{\text{e}}}\cdot {{q}_{\text{n}}} \\ &\ \ \=V{{\left(\frac{2\pi mkT}{{{h}^{2}}} \right)}^{3/2}}\left(\frac{\sqrt{\pi }{{\left(8{{\pi }^{2}}kT \right)}^{3/2}}}{\sigma {{h}^{3}}}{{\left({{I}_{\text{A}}}{{I}_{\text{B}}}{{I}_{\text{C}}} \right)}^{1/2}} \right)\left(\prod\limits_{i=1}^{3n-6}{\frac{{{\text{e}}^{-h{{\nu }_{i}}/2kT}}}{1-{{\text{e}}^{\text{-}h{{\nu }_{i}}/kT}}}} \right)\cdot {{g}_{0,\text{e}}}\cdot {{g}_{0,\text{n}}} \\ \end{align}

## 4 子的配分函数与系统的热力学函数

$E=k{{T}^{2}}{{\left(\partial \ln Z/\partial T \right)}_{V,N}}$

$S=kT{{\left(\partial \ln Z/\partial T \right)}_{V,N}}+k\ln Z$

$A=- k{\rm{Tln}}Z$

$p=kT{\left({\partial \ln Z/\partial V} \right)_{T,N}}$

$\mu=- LkT{\left({\partial \ln Z/\partial N} \right)_{T,V}}$

$Z={q^N}$

$Z=\frac{{{q^N}}}{{N!}} \approx {\left({\frac{{q{\rm{e}}}}{N}} \right)^N}$

$定域子系统:\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A=-kT\ln {{q}^{N}}=-NkT\ln q$

$离域子系统:\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A=-kT\ln {{\left(\frac{q\text{e}}{N} \right)}^{N}}=-NkT\ln \frac{q}{N}-NkT$

$q=V{{\left(\frac{2\pi mkT}{{{h}^{2}}} \right)}^{3/2}}$

$\frac{q}{N}=\left({{\left(\frac{2\pi mkT}{{{h}^{2}}} \right)}^{3\text{/}2}}\text{/}{{\left(\frac{2\pi mkT}{{{h}^{2}}} \right)}^{3\text{/}2}}\left(\frac{N}{V} \right)\left(\frac{N}{V} \right)\right){{q}_{\text{I}}}$

## 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

Tolman R.C. The Principles of Statistical Mechanics Oxford: Oxford University Presss, 1938,

Gasser, R. P. H. ; Richards, W. G.熵与能级.曾实,译.北京:人民教育出版社, 1981.

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