# 热力学第二定律

## 简介

$\displaystyle{ \mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {（封闭系统中理想状态下的可逆过程）} }$

$\displaystyle{ \mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {（封闭系统中理想状态下的可逆过程）。} }$

$\displaystyle{ \mathrm dS = \frac{\delta Q}{T} \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {（实际上可能的,不改变成分的准静态不可逆性）。} }$

$\displaystyle{ \mathrm dS = \frac{\delta Q}{T} - \frac{1}{T} \sum_{j} \, \Xi_{j} \,\delta \xi_j \,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {（封闭系统，实际可能的准静态不可逆过程）} }$

## 热力学第二定律的不同表述 Various statements of the law

### 卡诺原理 Carnot's principle

"...wherever there exists a difference of temperature, motive power can be produced.[23]"
...只要有温差，就能产生动力。

"The production of motive power is then due in steam engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body ...[24]"

"The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of caloric.[25]"

"The efficiency of a quasi-static or reversible Carnot cycle depends only on the temperatures of the two heat reservoirs, and is the same, whatever the working substance. A Carnot engine operated in this way is the most efficient possible heat engine using those two temperatures."

### 克劳修斯表述 Clausius statement

1850年，德国科学家鲁道夫·克劳修斯 Rudolf Clausius 通过研究热传递和功之间的关系，为热力学第二定律奠定了基础。他在1854年用德语发表的论文中所提及的热力学第二定律定义被称为克劳修斯表述:

### 开尔文表述 Kelvin statements

Lord Kelvin expressed the second law in several wordings.

### 克劳修斯和开尔文表述的等价性 Equivalence of the Clausius and the Kelvin statements

Derive Kelvin Statement from Clausius Statement从克劳修斯表述推导出开尔文表述

### 普朗克命题 Planck's proposition

"It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and cooling of a heat reservoir.[33][34]"
"不可能建造一台发动机，使其在一个完整的循环中工作，并且除了提高重量和冷却热储以外不会产生任何效果。[35][36]

### 开尔文表述与普朗克命题的关系 Relation between Kelvin's statement and Planck's proposition

"It is impossible to devise a cyclically operating device, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.[38]"

### 普朗克表述 Planck's statement

"Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limit, i.e. for reversible processes, the sum of the entropies remains unchanged.[39][40][41]"

"... in an irreversible or spontaneous change from one equilibrium state to another (as for example the equalization of temperature of two bodies A and B, when brought in contact) the entropy always increases.[42]"

…在从一个平衡态到另一个平衡态的不可逆或自发的变化中（例如，当两个物体 A 和 B 接触时的温度平衡过程），熵总是增加。[43]

### 卡拉西奥多里原理 Principle of Carathéodory

In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.

### 普朗克原理 Planck's principle

1926年，马克斯·普朗克 Max Planck写了一篇关于热力学基础的重要论文。[47][49] 他指出了以下原理

"The internal energy of a closed system is increased by an adiabatic process, throughout the duration of which, the volume of the system remains constant.[13]模板:Sfnp"

Borgnakke 和 Sonntag 提出的以下表述在某种意义上是对普朗克原理的补充。他们没有将其作为第二定律的完整表述:

"... there is only one way in which the entropy of a [closed] system can be decreased, and that is to transfer heat from the system.[54]"

…只有一种方法可以减少（封闭）系统的熵，将热从系统中转移出去。[55]"

## 推论

### 第二类永动机 Perpetual motion of the second kind

"All irreversible heat engines between two heat reservoirs are less efficient than a Carnot engine"

"All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs."

### 克劳修斯不等式 Clausius inequality

$\displaystyle{ \oint \frac{\delta Q}{T} \leq 0. }$

### 热力学温度 Thermodynamic temperature

$\displaystyle{ \eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1) }$

$\displaystyle{ \frac {q_C}{q_H} = f(T_H,T_C)\qquad (2). }$

$\displaystyle{ f(T_1,T_3) = \frac{q_3}{q_1} = \frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3). }$

$\displaystyle{ f(T_2,T_3) = \frac{f(T_1,T_3)}{f(T_1,T_2)} = \frac{273.16 \cdot f(T_1,T_3)}{273.16 \cdot f(T_1,T_2)}. }$

$\displaystyle{ T = 273.16 \cdot f(T_1,T) \, }$

$\displaystyle{ f(T_2,T_3) = \frac{T_3}{T_2}, }$

### 熵 Entropy

$\displaystyle{ \oint \frac{\delta Q}{T}=0 }$

$\displaystyle{ dS = \frac{\delta Q}{T} \! }$

$\displaystyle{ -\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}\lt 0 }$

$\displaystyle{ \Delta S \ge \int \frac{\delta Q}{T} \,\! }$

### 能量，可用的有用工作 Energy, available useful work

$\displaystyle{ dS_{\mathrm{tot}}= dS + dS_R \ge 0 }$

$\displaystyle{ dU = \delta q - \delta w + d(\sum \mu_{iR}N_i) \, }$

$\displaystyle{ \delta q = T_R (-dS_R) \le T_R dS }$

$\displaystyle{ \delta w \le - dU + T_R dS + \sum \mu_{iR} dN_i \, }$

$\displaystyle{ \delta w_u \le -d (U - T_R S + p_R V - \sum \mu_{iR} N_i )\, }$

$\displaystyle{ E = U - T_R S + p_R V - \sum \mu_{iR} N_i }$

$\displaystyle{ dE + \delta w_u \le 0 \, }$

$\displaystyle{ dS_{tot} \ge 0 }$ 等价于 $\displaystyle{ dE + \delta w_u \le 0 }$

### 化学热力学的第二定律 second law in chemical thermodynamics

$\displaystyle{ \Delta G \lt 0 }$

## 历史

19世纪提出的开尔文-普朗克第二陈述 Kelvin-Planck 表示：“任何循环运行的设备都不可能从单个热源接收热并产生净功。”这被证明与克劳修斯的陈述等价。

Clausius还提出了一种传统的学说，他认为熵可以被理解为宏观系统中的分子“无序”molecular 'disorder' ，但这种学说已经过时了。[59][60][61]

### 克劳修斯的描述

1856年，德国物理学家鲁道夫·克劳修斯 Rudolf Clausius阐述了他所谓的“热力学理论中的第二个基本定理 second fundamental theorem in the mechanical theory of heat” ，其形式如下：

$\displaystyle{ \int \frac{\delta Q}{T} = -N }$

The entropy of the universe tends to a maximum.

$\displaystyle{ \frac{dS}{dt} \ge 0 }$

$\displaystyle{ \frac{dS}{dt} = \dot S_{i} }$ with $\displaystyle{ \dot S_{i} \ge 0 }$

$\displaystyle{ \dot S_{i} }$表示系统内所有进程熵产生 Entropy Production的速率之和。这个公式的优点是它显示了熵产生的效果。熵产生率是一个非常重要的概念，因为它决定（或限制）热机的效率。乘以环境温度$\displaystyle{ T_{a} }$，它给出所谓的耗散能$\displaystyle{ P_{diss}=T_{a}\dot S_{i} }$

$\displaystyle{ \frac{dS}{dt} = \frac{\dot Q}{T}+\dot S_{i} }$ with $\displaystyle{ \dot S_{i} \ge 0 }$

$\displaystyle{ \frac{dS}{dt} = \frac{\dot Q}{T}+\dot S+\dot S_{i} }$ with $\displaystyle{ \dot S_{i} \ge 0 }$

## 从统计力学导出 Derivation from statistical mechanics

$\displaystyle{ S = k_{\mathrm B} \ln\left[\Omega\left(E\right)\right]\, }$

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## 延申阅读

• Goldstein, Martin, and Inge F., 1993. The Refrigerator and the Universe. Harvard Univ. Press. Chpts. 4–9 contain an introduction to the Second Law, one a bit less technical than this entry.
• Leff, Harvey S., and Rex, Andrew F. (eds.) 2003. Maxwell's Demon 2 : Entropy, classical and quantum information, computing. Bristol UK; Philadelphia PA: Institute of Physics.
• Halliwell, J.J. (1994). Physical Origins of Time Asymmetry. Cambridge. ISBN 978-0-521-56837-1. (technical).
• Carnot, Sadi (1890). Robert Henry Thurston. ed. Reflections on the Motive Power of Heat and on Machines Fitted to Develop That Power. New York: J. Wiley & Sons.  (full text of 1897 ed.) (html)
• Stephen Jay Kline (1999). The Low-Down on Entropy and Interpretive Thermodynamics, La Cañada, CA: DCW Industries. .

## 外部链接

• E.T. Jaynes, 1988, "The evolution of Carnot's principle," in G. J. Erickson and C. R. Smith (eds.)Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol 1: p. 267.