# 熵

## 熵的熱力學定義

${\displaystyle \Delta S={\frac {\Delta Q}{T}}}$

1923年，德國科學家普朗克到中國講學用到entropy這個詞，胡剛復教授翻譯時靈機一動，把「商」字加火旁來意譯「entropy」這個字，創造了「熵」字（音讀：商）[3]，因為熵是Q（熱量）除以T（溫度）的商數[4]

### 熵的增减与热机

1. 系統对外所做的功（等于外界对系统做功的相反数）
2. 高溫熱庫之間的熱能傳遞

${\displaystyle {\frac {Q}{T}}={\frac {Q'}{T'}}}$

${\displaystyle \sum _{i=1}^{N}{\frac {Q_{i}}{T_{i}}}\leq 0}$

${\displaystyle Q_{0,j}=T_{0}{\frac {Q_{j}}{T_{J}}}}$

${\displaystyle Q_{0}=\sum _{j=1}^{N}Q_{0,j}=T_{0}\sum _{j=1}^{N}{\frac {Q_{j}}{T_{j}}}}$

${\displaystyle \sum _{i=1}^{N}{\frac {Q_{i}}{T_{i}}}\leq 0}$

${\displaystyle {\frac {Q_{j}}{T_{j}}}\leq {\frac {Q_{j}}{T}}}$

${\displaystyle \oint {\frac {\delta Q}{T}}\equiv \oint dS=0}$（可逆循環）

### 熵作為狀態函數

${\displaystyle S_{X}=S_{R}+\int _{R}^{X}{\frac {\delta Q}{T}}}$

${\displaystyle \Delta S\geq \int {\frac {\delta Q}{T}}}$

## 熵的统计学定义，玻尔兹曼原理

1877年，玻尔兹曼發現單一系統中的熵跟構成熱力學性質的微觀狀態數量相關。可以考慮情況如：一個容器內的理想气体。微觀狀態可以以每個組成的原子的位置及動量予以表達。為了一致性起見，只需考慮包含以下條件的微觀狀態：（i）所有粒子的位置皆在容器的體積範圍內；（ii）所有原子的動能總和等於該氣體的總能量值。玻尔兹曼並假設：

${\displaystyle S=k(\ln \Omega )}$

## 熵的圖繪

${\displaystyle S=nR\ \ln(1+P^{C_{V} \over R}V^{C_{P} \over R})}$

## 熵的測量

${\displaystyle C_{X}=T\left({\frac {\partial S}{\partial T}}\right)_{X}}$

${\displaystyle \Delta S=\int {\frac {C_{X}}{T}}dT}$

${\displaystyle S(P,V)=S(P,V_{0})+\int _{T(P,V_{0})}^{T(P,V)}{\frac {C_{P}(P,V(T,P))}{T}}dT}$

## 注釋

1. In certain types of advanced system configurations, such as at the critical point of water or when salt is added to an ice-water mixture, entropy can either increase or decrease depending on system parameters, such as temperature and pressure. For example, if the spontaneous crystallization of a supercooled liquid takes place under adiabatic conditions the entropy of the resulting crystal will be greater than that of the supercooled liquid (Denbigh, K. (1982). The Principles of Chemical Equilibrium, 4th Ed.). In general, however, when ice melts, the entropy of the two adjoined systems, i.e. the adjacent hot and cold bodies, when thought of as one "universe", increases. Here are some further tutorials: Entropy and Ice-melting - Michigan State University (course page); Ice-meltingJCE example; Ice-melting and Entropy Change – example; Ice-melting and Entropy Change – discussions
2. 音同“[2]
3. 系統「內向」與心理「內向」的概念無綠，後者在相當於拉丁文introversio一詞、iintroversio與entropia語義內涵相同而外延相異。

## 参考文献

### 引用

1. Clausius, Rudolf (1862). "On the Application of the Theorem of the Equivalence of Transformations to Interior Work.]"向蘇黎士自然研究會（Naturforschende Gesellschaft）1862年01月27日發佈;刊登在該会的季刊（Vierteljahrschrift of this Society）vol. vii.第48頁;又在Poggendorff’s Annalen, 1862年5月，第cxvi冊第73頁;在哲學雜誌（Philosophical Magazine）, S. 4. vol. xxiv. pp. 81, 201;在巴黎數學刊物 （Journal des Mathematiques）S. 2. vol. vii. P. 209.
2. （中文）.
3. 熵字的解释---在线新华字典
4. 秦允豪. . 高等教育出版社. : 169. ISBN 978-7-04-013790-3.
5. . 清华大学出版社有限公司. 2000: 144. ISBN 978-7-302-04197-9.
6. . Wolfram Research. 2007 [2010-02-24].
7. Clausius, Rudolf. . Poggendorff's Annalen der Physick, LXXIX (Dover Reprint). 1850. ISBN 0-486-59065-8.

### 来源

• 恩里科·費米. . Prentice Hall. 1937.
• Reif, F., . . McGraw-Hill. 1965.

## 外部链接

 查询維基詞典中的。