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Moved page from wikipedia:en:Clausius theorem (history)
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{{short description|A version of the second law of thermodynamics}}
{{Thermodynamics|cTopic=[[Thermodynamic equations|Equations]]}}
The '''Clausius theorem''' (1855) states that for a [[thermodynamic system]] (e.g. [[heat engine]] or [[Heat pump and refrigeration cycle|heat pump]]) exchanging heat with [[Thermal reservoir|external reservoirs]] and undergoing a [[thermodynamic cycle]],
The Clausius theorem (1855) states that for a thermodynamic system (e.g. heat engine or heat pump) exchanging heat with external reservoirs and undergoing a thermodynamic cycle,
1855年的《美国克劳修斯定理法典》中规定，对于热力学系统来说。热机或热泵)与外部储热器交换热量，并经历一个热力学循环,
:<math>\oint \frac{\delta Q}{T_{\text{surr}}} \leq 0,</math>
<math>\oint \frac{\delta Q}{T_{\text{surr}}} \leq 0,</math>
0，</math > oint frac { delta q }{ t _ { text { surr }} leq 0，</math >
where <math>\delta Q</math> is the infinitesimal amount of heat absorbed by the system from the reservoir and <math>T_{\text{surr}}</math> is the [[temperature]] of the external reservoir (surroundings) at a particular instant in time. The closed integral is carried out along a [[thermodynamic process path]] from the initial/final state to the same initial/final state. In principle, the closed integral can start and end at an arbitrary point along the path.
where <math>\delta Q</math> is the infinitesimal amount of heat absorbed by the system from the reservoir and <math>T_{\text{surr}}</math> is the temperature of the external reservoir (surroundings) at a particular instant in time. The closed integral is carried out along a thermodynamic process path from the initial/final state to the same initial/final state. In principle, the closed integral can start and end at an arbitrary point along the path.
其中“ math” > delta q </math > 是系统从蓄水池吸收的无穷小的热量，而“ math” > t _ { text { surr } </math > 是外部蓄水池(周围环境)在某一特定时刻的温度。封闭积分是沿着一条热力学过程路径进行的，从初始/终止状态到相同的初始/终止状态。原则上，闭积分可以在路径上的任意点开始和结束。
If there are multiple reservoirs with different temperatures <math>\left(T_1,T_2, \cdots T_n\right)</math>, then Clausius inequality reads:
If there are multiple reservoirs with different temperatures <math>\left(T_1,T_2, \cdots T_n\right)</math>, then Clausius inequality reads:
如果存在多个温度不同的储层，左(t1，t2，cdots t _ n 右) </math > ，那么克劳修斯不等式如下:
:<math>\oint \left(\frac{\delta Q_1}{T_1}+\frac{\delta Q_2}{T_2}+\cdots+\frac{\delta Q_n}{T_n}\right) \leq 0.</math>
<math>\oint \left(\frac{\delta Q_1}{T_1}+\frac{\delta Q_2}{T_2}+\cdots+\frac{\delta Q_n}{T_n}\right) \leq 0.</math>
左(frac { delta q _ 1}{ t _ 1} + frac { delta q _ 2}{ t _ 2} + cdots + frac { delta q _ n }{ t _ n }右) leq 0. </math >
In the special case of a reversible process, the equality holds.<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html '''Clausius theorem'''] at [[Wolfram Research]]</ref> The reversible case is used to introduce the [[state function]] known as [[entropy]]. This is because in a cyclic process the variation of a state function is zero. In other words, the Clausius statement states that it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir.<ref>Finn, Colin B. P. ''Thermal Physics''. 2nd ed., CRC Press, 1993. </ref> Equivalently, heat spontaneously flows from a hot body to a cooler one, not the other way around.<ref>Giancoli, Douglas C. ''Physics: Principles with Applications''. 6th ed., Pearson/Prentice Hall, 2005. </ref>
In the special case of a reversible process, the equality holds. The reversible case is used to introduce the state function known as entropy. This is because in a cyclic process the variation of a state function is zero. In other words, the Clausius statement states that it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir. Equivalently, heat spontaneously flows from a hot body to a cooler one, not the other way around.
在可逆过程的特殊情况下，平等是成立的。可逆情形用来引入状态函数，即熵。这是因为在循环过程中，状态函数的变化为零。换句话说，克劳修斯的声明指出，不可能建造一个装置，其唯一的作用是将热量从冷库转移到热库。相当于，热量自发地从一个热的物体流向一个较冷的物体，而不是相反。
The generalized "inequality of Clausius"<ref>Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref>
The generalized "inequality of Clausius"
推广的“克劳修斯不等式”
:<math>dS_{\text{sys}} \geq \frac{\delta Q}{T_{\text{surr}}} </math>
<math>dS_{\text{sys}} \geq \frac{\delta Q}{T_{\text{surr}}} </math>
[数学 > dS { text { sys } geq frac { delta q }{ t _ { text { surr }}} </math >
for an infinitesimal change in entropy ''S'' applies not only to cyclic processes, but to any process that occurs in a closed system.
for an infinitesimal change in entropy S applies not only to cyclic processes, but to any process that occurs in a closed system.
熵 s 中的无穷小的变化不仅适用于循环过程，而且适用于发生在封闭系统中的任何过程。
==History==
The Clausius theorem is a mathematical explanation of the [[second law of thermodynamics]]. It was developed by [[Rudolf Clausius]] who intended to explain the relationship between the heat flow in a system and the entropy of the system and its surroundings. Clausius developed this in his efforts to explain entropy and define it quantitatively. In more direct terms, the theorem gives us a way to determine if a cyclical process is reversible or irreversible. The Clausius theorem provides a quantitative formula for understanding the second law.
The Clausius theorem is a mathematical explanation of the second law of thermodynamics. It was developed by Rudolf Clausius who intended to explain the relationship between the heat flow in a system and the entropy of the system and its surroundings. Clausius developed this in his efforts to explain entropy and define it quantitatively. In more direct terms, the theorem gives us a way to determine if a cyclical process is reversible or irreversible. The Clausius theorem provides a quantitative formula for understanding the second law.
克劳修斯定理是对热力学第二定律的一种数学解释。它是由鲁道夫 · 克劳修斯提出的，他试图解释系统中的热流与系统及其周围环境的熵之间的关系。克劳修斯在解释熵和定义熵的过程中发展了这个理论。用更直接的术语来说，这个定理给我们提供了一种方法，来确定一个循环过程是可逆的还是不可逆的。克劳修斯定理提供了一个理解第二定律的定量公式。
Clausius was one of the first to work on the idea of entropy and is even responsible for giving it that name. What is now known as the Clausius theorem was first published in 1862 in Clausius' sixth memoir, "On the Application of the Theorem of the Equivalence of Transformations to Interior Work". Clausius sought to show a proportional relationship between entropy and the energy flow by heating (δ''Q'') into a system. In a system, this heat energy can be transformed into work, and work can be transformed into heat through a cyclical process. Clausius writes that "The algebraic sum of all the transformations occurring in a cyclical process can only be less than zero, or, as an extreme case, equal to nothing." In other words, the equation
Clausius was one of the first to work on the idea of entropy and is even responsible for giving it that name. What is now known as the Clausius theorem was first published in 1862 in Clausius' sixth memoir, "On the Application of the Theorem of the Equivalence of Transformations to Interior Work". Clausius sought to show a proportional relationship between entropy and the energy flow by heating (δQ) into a system. In a system, this heat energy can be transformed into work, and work can be transformed into heat through a cyclical process. Clausius writes that "The algebraic sum of all the transformations occurring in a cyclical process can only be less than zero, or, as an extreme case, equal to nothing." In other words, the equation
克劳修斯是最早研究熵概念的人之一，他甚至给熵起了这个名字。1862年，克劳修斯在他的第六本回忆录《论等效变换定理在室内工作中的应用》中首次出版了现在被称为《克劳修斯定理。克劳修斯试图通过加热(δq)到一个系统来表明熵和能量流之间的比例关系。在一个系统中，热能可以转化为功，功可以通过循环过程转化为热。克劳修斯写道: “在循环过程中发生的所有变换的代数和只能小于零，或者，在极端情况下，等于零。”换句话说，方程式
:<math>\oint \frac{\delta Q}{T} = 0</math>
<math>\oint \frac{\delta Q}{T} = 0</math>
0 </math > oint frac { delta q }{ t } = 0
with 𝛿''Q'' being energy flow into the system due to heating and ''T'' being absolute temperature of the body when that energy is absorbed, is found to be true for any process that is cyclical and reversible. Clausius then took this a step further and determined that the following relation must be found true for any cyclical process that is possible, reversible or not. This relation is the "Clausius inequality".
with 𝛿Q being energy flow into the system due to heating and T being absolute temperature of the body when that energy is absorbed, is found to be true for any process that is cyclical and reversible. Clausius then took this a step further and determined that the following relation must be found true for any cyclical process that is possible, reversible or not. This relation is the "Clausius inequality".
因为 δq 是由于加热而流入系统的能量，t 是体内吸收能量时的绝对温度，对于任何周期性和可逆的过程都是正确的。克劳修斯又进一步确定，对于任何可能的循环过程，无论可逆与否，都必须找到以下关系式。这种关系就是“克劳修斯不等式”。
:<math>\oint \frac{\delta Q}{T_{surr}} \leq 0</math>
<math>\oint \frac{\delta Q}{T_{surr}} \leq 0</math>
[ math > oint frac { delta q }{ t _ { surr } leq 0 </math >
Now that this is known, there must be a relation developed between the Clausius inequality and entropy. The amount of entropy S added to the system during the cycle is defined as
Now that this is known, there must be a relation developed between the Clausius inequality and entropy. The amount of entropy S added to the system during the cycle is defined as
既然已经知道了这一点，那么克劳修斯不等式和熵之间一定存在着某种关系。在循环过程中添加到系统中的熵 s 的量被定义为
:<math>\Delta S {{=}} \oint \frac{\delta Q}{T}</math>
<math>\Delta S \oint \frac{\delta Q}{T}</math>
数学，数学，数学
It has been determined, as stated in the [[second law of thermodynamics]], that the entropy is a state function: It depends only upon the state that the system is in, and not what path the system took to get there. This is in contrast to the amount of energy added as heat (𝛿''Q'') and as work (𝛿''W''), which may vary depending on the path. In a cyclic process, therefore, the entropy of the system at the beginning of the cycle must equal the entropy at the end of the cycle, <math>\Delta S=0</math>, regardless of whether the process is reversible or irreversible. In the irreversible case, entropy will be created in the system, and more entropy must be extracted than was added <math>(\Delta S_{surr}>0)</math> in order to return the system to its original state. In the reversible case, no entropy is created and the amount of entropy added is equal to the amount extracted.
It has been determined, as stated in the second law of thermodynamics, that the entropy is a state function: It depends only upon the state that the system is in, and not what path the system took to get there. This is in contrast to the amount of energy added as heat (𝛿Q) and as work (𝛿W), which may vary depending on the path. In a cyclic process, therefore, the entropy of the system at the beginning of the cycle must equal the entropy at the end of the cycle, <math>\Delta S=0</math>, regardless of whether the process is reversible or irreversible. In the irreversible case, entropy will be created in the system, and more entropy must be extracted than was added <math>(\Delta S_{surr}>0)</math> in order to return the system to its original state. In the reversible case, no entropy is created and the amount of entropy added is equal to the amount extracted.
正如《热力学第二定律所说，熵是一个状态函数: 它只取决于系统所处的状态，而不取决于系统到达那个状态所走的路径。这与作为热(δq)和作为功(δw)加入的能量相反，后者可能随路径而变化。因此，在循环过程中，系统在循环开始时的熵必须等于循环结束时的熵，< math > Delta s = 0 </math > ，不管这个过程是可逆的还是不可逆的。在不可逆的情况下，系统会产生熵，为了使系统恢复到原来的状态，必须提取比加入 math (Delta s { surr } > 0)更多的熵。在可逆的情况下，不会产生熵，熵的增加量等于提取的量。
If the amount of energy added by heating can be measured during the process, and the temperature can be measured during the process, the Clausius inequality can be used to determine whether the process is reversible or irreversible by carrying out the integration in the Clausius inequality.
If the amount of energy added by heating can be measured during the process, and the temperature can be measured during the process, the Clausius inequality can be used to determine whether the process is reversible or irreversible by carrying out the integration in the Clausius inequality.
如果可以测量加热过程中所增加的能量，并且可以测量过程中的温度，则可以用克劳修斯不等式进行克劳修斯不等式的积分，来确定过程是可逆的还是不可逆的。
==Proof==
The temperature that enters in the denominator of the integrand in the Clausius inequality is actually the temperature of the [[Thermal reservoir|external reservoir]] with which the system exchanges heat. At each instant of the process, the system is in contact with an external reservoir.
The temperature that enters in the denominator of the integrand in the Clausius inequality is actually the temperature of the external reservoir with which the system exchanges heat. At each instant of the process, the system is in contact with an external reservoir.
在克劳修斯不等式中，进入被积函数分母的温度，实际上是系统与之交换热量的外部热库的温度。在这个过程的每一个瞬间，系统都与外部储存器接触。
Because of the Second Law of Thermodynamics, in each infinitesimal heat exchange process between the system and the reservoir, the net change in entropy of the "universe", so to say, is <math> dS_{Total}=dS_{Sys} +dS_{Res} \geq 0 </math>.
Because of the Second Law of Thermodynamics, in each infinitesimal heat exchange process between the system and the reservoir, the net change in entropy of the "universe", so to say, is <math> dS_{Total}=dS_{Sys} +dS_{Res} \geq 0 </math>.
由于热力学第二定律的存在，在系统和蓄热体之间每一个微小的热交换过程中，“宇宙”熵的净变化，也就是说，是。
When the system takes in heat by an infinitesimal amount <math>\delta Q_{1}</math>(<math>\geq 0</math>), for the net change in entropy <math>dS_{Total_{1}}</math> in this step to be positive, the temperature of the "hot" [[Thermal reservoir|reservoir]] <math>T_{Hot}</math> needs to be slightly greater than the temperature of the system at that instant.
When the system takes in heat by an infinitesimal amount <math>\delta Q_{1}</math>(<math>\geq 0</math>), for the net change in entropy <math>dS_{Total_{1}}</math> in this step to be positive, the temperature of the "hot" reservoir <math>T_{Hot}</math> needs to be slightly greater than the temperature of the system at that instant.
当系统以无限小的量吸收热量时(< math > delta q {1} </math > (< math > geq 0 </math >) ，因为在这一步中熵的净变化为正，“热”库 < t { Hot } </math > 需要比系统当时的温度稍高。
If the temperature of the system is given by <math>T_{1}</math> at that instant, then <math> dS_{Sys_{1}}=\frac{\delta Q_{1}}{T_{1}}</math>, and <math>T_{Hot}\geq T_{1}</math> forces us to have:
If the temperature of the system is given by <math>T_{1}</math> at that instant, then <math> dS_{Sys_{1}}=\frac{\delta Q_{1}}{T_{1}}</math>, and <math>T_{Hot}\geq T_{1}</math> forces us to have:
如果系统的温度是由 < math > t _ {1} </math > 给出的，那么 < math > dS _ { Sys _ 1} = frac { delta q _ {1}{ t _ {1} </math > ，< math > t _ { Hot } geq _ {1} </math > 强迫我们:
:<math> -dS_{Res_{1}} =\frac{\delta Q_{1}}{T_{Hot}}\leq \frac{\delta Q_{1}}{T_{1}} = dS_{Sys_{1}} </math>
<math> -dS_{Res_{1}} =\frac{\delta Q_{1}}{T_{Hot}}\leq \frac{\delta Q_{1}}{T_{1}} = dS_{Sys_{1}} </math>
< math >-dS _ Res _ {1} = frac { delta q _ {1}{ t _ { Hot } leq frac { delta q _ {1}{ t _ {1}} = dS _ { Sys _ {1} </math >
This means the magnitude of the entropy "loss" from the reservoir, <math> |dS_{Res_{1}}|=\frac{\delta Q_{1}}{T_{Hot}} </math> is less than the magnitude of the entropy gain <math>dS_{Sys_{1}}</math>(<math>\geq 0</math>) by the system:
This means the magnitude of the entropy "loss" from the reservoir, <math> |dS_{Res_{1}}|=\frac{\delta Q_{1}}{T_{Hot}} </math> is less than the magnitude of the entropy gain <math>dS_{Sys_{1}}</math>(<math>\geq 0</math>) by the system:
这意味着从水库的熵“损失”的大小，| dS { Res {1} | = frac { delta q {1}{ t { Hot } </math > 小于系统的熵增 < dS { Sys {1} </math > (< math > geq 0 </math >) :
Similarly, when the system at temperature <math>T_{2}</math> expels heat in magnitude <math>-\delta Q_{2}</math> (<math>\delta Q_{2}\leq 0</math>) into a colder [[Thermal reservoir|reservoir]] (at temperature <math>T_{Cold}\leq T_{2}</math>) in an infinitesimal step, then again, for the Second Law of Thermodynamics to hold, one would have, in an exactly similar manner:
Similarly, when the system at temperature <math>T_{2}</math> expels heat in magnitude <math>-\delta Q_{2}</math> (<math>\delta Q_{2}\leq 0</math>) into a colder reservoir (at temperature <math>T_{Cold}\leq T_{2}</math>) in an infinitesimal step, then again, for the Second Law of Thermodynamics to hold, one would have, in an exactly similar manner:
类似地，当系统在温度 < math > t {2} </math > 将热量以数量级 < math >-delta q {2} </math > (< math > delta q {2} leq 0 </math >)排放到一个较冷的水库中(在温度 < math > t { Cold } leq {2} </math >) ，然后再一次，对于热力学第二定律来说，人们会以完全相同的方式:
:<math display="block"> -dS_{Res_{2}}=\frac{\delta Q_{2}}{T_{Cold}}\leq \frac{\delta Q_{2}}{T_{2}}= dS_{Sys_{2}} </math>
<math display="block"> -dS_{Res_{2}}=\frac{\delta Q_{2}}{T_{Cold}}\leq \frac{\delta Q_{2}}{T_{2}}= dS_{Sys_{2}} </math>
< math display = " block" >-dS _ Res _ {2} = frac { delta q _ {2}{ t _ { Cold } leq frac { delta q _ {2}{ t _ {2}} = dS _ { Sys _ {2} </math >
Here, the amount of heat 'absorbed' by the system is given by <math>\delta Q_{2}</math>(<math>\leq 0</math>), signifying that heat is transferring from the system to the reservoir, with <math>dS_{Sys_{2}}\leq 0</math>. The magnitude of the entropy gained by the reservoir, <math> dS_{Res_{2}}=\frac{|\delta Q_{2}|}{T_{cold}}</math> is greater than the magnitude of the entropy loss of the system <math> |dS_{Sys_{2}}|</math>
Here, the amount of heat 'absorbed' by the system is given by <math>\delta Q_{2}</math>(<math>\leq 0</math>), signifying that heat is transferring from the system to the reservoir, with <math>dS_{Sys_{2}}\leq 0</math>. The magnitude of the entropy gained by the reservoir, <math> dS_{Res_{2}}=\frac{|\delta Q_{2}|}{T_{cold}}</math> is greater than the magnitude of the entropy loss of the system <math> |dS_{Sys_{2}}|</math>
在这里，系统所“吸收”的热量是通过 < math > delta q _ {2} </math > (< math > leq 0 </math >)给出的，表示热量正从系统传递到储存器，而 < math > dS _ Sys _ {2} leq 0 </math > 。水库所获得的熵的大小，大于系统的熵损失的大小
Since the total change in entropy for the system is 0 in a cyclic process, if one adds all the infinitesimal steps of heat intake and heat expulsion from the reservoir, signified by the previous two equations, with the temperature of the reservoir at each instant given by <math>T_{surr}</math>, one gets,
Since the total change in entropy for the system is 0 in a cyclic process, if one adds all the infinitesimal steps of heat intake and heat expulsion from the reservoir, signified by the previous two equations, with the temperature of the reservoir at each instant given by <math>T_{surr}</math>, one gets,
由于系统在循环过程中熵的总变化为0，如果加上前面两个方程所表示的储热器吸热和排热的所有无穷小的步骤，再加上每一瞬间储热器的温度，就得到,
:<math> -\oint dS_{Res}= \oint \frac{\delta Q}{T_{surr}}\leq \oint dS_{Sys}=0 </math>
<math> -\oint dS_{Res}= \oint \frac{\delta Q}{T_{surr}}\leq \oint dS_{Sys}=0 </math>
[数学]-oint dS { Res } = oint frac { delta q }{ t _ { surr } leq oint dS { Sys } = 0 </math >
In particular,
In particular,
特别是,
:<math>\oint \frac{\delta Q}{T_{surr}}\leq 0, </math>
<math>\oint \frac{\delta Q}{T_{surr}}\leq 0, </math>
0，</math > oint frac { delta q }{ t _ { surr } leq 0，</math >
which was to be proven.
which was to be proven.
这是有待证实的。
In summary, (the inequality in the third statement below, being obviously guaranteed by the [[second law of thermodynamics]], which is the basis of our calculation),
In summary, (the inequality in the third statement below, being obviously guaranteed by the second law of thermodynamics, which is the basis of our calculation),
总之，(下面第三个陈述中的不平等，显然得到了我们计算的基础---- 美国热力学第二定律的保证) ,
:<math>\oint dS_{Res}\geq 0 </math>
<math>\oint dS_{Res}\geq 0 </math>
0 </math > oint dS { Res } geq 0
:<math>\oint dS_{Sys}=0 </math> (as hypothesized)
<math>\oint dS_{Sys}=0 </math> (as hypothesized)
[ math > oint dS _ { Sys } = 0
:<math>\oint dS_{Total}=\oint dS_{Res}+\oint dS_{Sys}\geq 0</math>
<math>\oint dS_{Total}=\oint dS_{Res}+\oint dS_{Sys}\geq 0</math>
[数学][数学][数学][数学]
For a [[Reversible process (thermodynamics)|reversible]] [[Thermodynamic cycle|cyclic]] process, there is no generation of entropy in each of the infinitesimal heat transfer processes, so the following equality holds,
For a reversible cyclic process, there is no generation of entropy in each of the infinitesimal heat transfer processes, so the following equality holds,
对于可逆循环过程，在每一个无限小的传热过程中都不会产生熵，因此下面的等式成立,
:<math>\oint \frac{\delta Q_{rev}}{T}=0. </math>
<math>\oint \frac{\delta Q_{rev}}{T}=0. </math>
0.0.0.数学
Thus, the Clausius inequality is a consequence of applying the [[second law of thermodynamics]] at each infinitesimal stage of heat transfer, and is thus in a sense a weaker condition than the Second Law itself.
Thus, the Clausius inequality is a consequence of applying the second law of thermodynamics at each infinitesimal stage of heat transfer, and is thus in a sense a weaker condition than the Second Law itself.
因此，克劳修斯不等式是在传热的每一个无穷小阶段应用热力学第二定律的结果，因此在某种意义上比第二定律本身更弱。
==See also==
* [[Kelvin-Planck statement]]
* [[Carnot's theorem (thermodynamics)]]
* [[Carnot heat engine]]
* [[Introduction to entropy]]
==References==
{{reflist}}
{{refbegin}}
==Further reading==
*Morton, A. S., and P.J. Beckett. ''Basic Thermodynamics''. New York: Philosophical Library Inc., 1969. Print.
*Saad, Michel A. ''Thermodynamics for Engineers''. Englewood Cliffs: Prentice-Hall, 1966. Print.
*Hsieh, Jui Sheng. ''Principles of Thermodynamics''. Washington, D.C.: Scripta Book Company, 1975. Print.
*Zemansky, Mark W. ''Heat and Thermodynamics''. 4th ed. New York: McGwaw-Hill Book Company, 1957. Print.
*Clausius, Rudolf. ''The Mechanical Theory of Heat''. London: Taylor and Francis, 1867. eBook
{{refend}}
==External links==
*{{cite web|title=Proof of Clausius's theorem|url=http://theory.ph.man.ac.uk/~judith/stat_therm/node30.html|accessdate=October 4, 2010|author=Judith McGovern|date=2004-03-17|archive-url=https://web.archive.org/web/20110719052220/http://theory.ph.man.ac.uk/~judith/stat_therm/node30.html|archive-date=July 19, 2011|url-status=dead}}
*{{cite web|title=The Clausius Inequality And The Mathematical Statement Of The Second Law|url=http://ronispc.chem.mcgill.ca/ronis/chem213/hnd10.pdf|accessdate=October 5, 2010}}
*{{cite book|title=The Mechanical Theory of Heat (eBook)|url=https://books.google.com/books?id=8LIEAAAAYAAJ |accessdate=December 1, 2011}}
{{DEFAULTSORT:Clausius Theorem}}
[[Category:Laws of thermodynamics]]
Category:Laws of thermodynamics
类别: 热力学定律
[[Category:Physics theorems]]
Category:Physics theorems
范畴: 物理学定理
<noinclude>
<small>This page was moved from [[wikipedia:en:Clausius theorem]]. Its edit history can be viewed at [[熵增原理/edithistory]]</small></noinclude>
[[Category:待整理页面]]
{{short description|A version of the second law of thermodynamics}}
{{Thermodynamics|cTopic=[[Thermodynamic equations|Equations]]}}
The '''Clausius theorem''' (1855) states that for a [[thermodynamic system]] (e.g. [[heat engine]] or [[Heat pump and refrigeration cycle|heat pump]]) exchanging heat with [[Thermal reservoir|external reservoirs]] and undergoing a [[thermodynamic cycle]],
The Clausius theorem (1855) states that for a thermodynamic system (e.g. heat engine or heat pump) exchanging heat with external reservoirs and undergoing a thermodynamic cycle,
1855年的《美国克劳修斯定理法典》中规定，对于热力学系统来说。热机或热泵)与外部储热器交换热量，并经历一个热力学循环,
:<math>\oint \frac{\delta Q}{T_{\text{surr}}} \leq 0,</math>
<math>\oint \frac{\delta Q}{T_{\text{surr}}} \leq 0,</math>
0，</math > oint frac { delta q }{ t _ { text { surr }} leq 0，</math >
where <math>\delta Q</math> is the infinitesimal amount of heat absorbed by the system from the reservoir and <math>T_{\text{surr}}</math> is the [[temperature]] of the external reservoir (surroundings) at a particular instant in time. The closed integral is carried out along a [[thermodynamic process path]] from the initial/final state to the same initial/final state. In principle, the closed integral can start and end at an arbitrary point along the path.
where <math>\delta Q</math> is the infinitesimal amount of heat absorbed by the system from the reservoir and <math>T_{\text{surr}}</math> is the temperature of the external reservoir (surroundings) at a particular instant in time. The closed integral is carried out along a thermodynamic process path from the initial/final state to the same initial/final state. In principle, the closed integral can start and end at an arbitrary point along the path.
其中“ math” > delta q </math > 是系统从蓄水池吸收的无穷小的热量，而“ math” > t _ { text { surr } </math > 是外部蓄水池(周围环境)在某一特定时刻的温度。封闭积分是沿着一条热力学过程路径进行的，从初始/终止状态到相同的初始/终止状态。原则上，闭积分可以在路径上的任意点开始和结束。
If there are multiple reservoirs with different temperatures <math>\left(T_1,T_2, \cdots T_n\right)</math>, then Clausius inequality reads:
If there are multiple reservoirs with different temperatures <math>\left(T_1,T_2, \cdots T_n\right)</math>, then Clausius inequality reads:
如果存在多个温度不同的储层，左(t1，t2，cdots t _ n 右) </math > ，那么克劳修斯不等式如下:
:<math>\oint \left(\frac{\delta Q_1}{T_1}+\frac{\delta Q_2}{T_2}+\cdots+\frac{\delta Q_n}{T_n}\right) \leq 0.</math>
<math>\oint \left(\frac{\delta Q_1}{T_1}+\frac{\delta Q_2}{T_2}+\cdots+\frac{\delta Q_n}{T_n}\right) \leq 0.</math>
左(frac { delta q _ 1}{ t _ 1} + frac { delta q _ 2}{ t _ 2} + cdots + frac { delta q _ n }{ t _ n }右) leq 0. </math >
In the special case of a reversible process, the equality holds.<ref>[http://scienceworld.wolfram.com/physics/ClausiusTheorem.html '''Clausius theorem'''] at [[Wolfram Research]]</ref> The reversible case is used to introduce the [[state function]] known as [[entropy]]. This is because in a cyclic process the variation of a state function is zero. In other words, the Clausius statement states that it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir.<ref>Finn, Colin B. P. ''Thermal Physics''. 2nd ed., CRC Press, 1993. </ref> Equivalently, heat spontaneously flows from a hot body to a cooler one, not the other way around.<ref>Giancoli, Douglas C. ''Physics: Principles with Applications''. 6th ed., Pearson/Prentice Hall, 2005. </ref>
In the special case of a reversible process, the equality holds. The reversible case is used to introduce the state function known as entropy. This is because in a cyclic process the variation of a state function is zero. In other words, the Clausius statement states that it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir. Equivalently, heat spontaneously flows from a hot body to a cooler one, not the other way around.
在可逆过程的特殊情况下，平等是成立的。可逆情形用来引入状态函数，即熵。这是因为在循环过程中，状态函数的变化为零。换句话说，克劳修斯的声明指出，不可能建造一个装置，其唯一的作用是将热量从冷库转移到热库。相当于，热量自发地从一个热的物体流向一个较冷的物体，而不是相反。
The generalized "inequality of Clausius"<ref>Mortimer, R. G. ''Physical Chemistry''. 3rd ed., p. 120, Academic Press, 2008. </ref>
The generalized "inequality of Clausius"
推广的“克劳修斯不等式”
:<math>dS_{\text{sys}} \geq \frac{\delta Q}{T_{\text{surr}}} </math>
<math>dS_{\text{sys}} \geq \frac{\delta Q}{T_{\text{surr}}} </math>
[数学 > dS { text { sys } geq frac { delta q }{ t _ { text { surr }}} </math >
for an infinitesimal change in entropy ''S'' applies not only to cyclic processes, but to any process that occurs in a closed system.
for an infinitesimal change in entropy S applies not only to cyclic processes, but to any process that occurs in a closed system.
熵 s 中的无穷小的变化不仅适用于循环过程，而且适用于发生在封闭系统中的任何过程。
==History==
The Clausius theorem is a mathematical explanation of the [[second law of thermodynamics]]. It was developed by [[Rudolf Clausius]] who intended to explain the relationship between the heat flow in a system and the entropy of the system and its surroundings. Clausius developed this in his efforts to explain entropy and define it quantitatively. In more direct terms, the theorem gives us a way to determine if a cyclical process is reversible or irreversible. The Clausius theorem provides a quantitative formula for understanding the second law.
The Clausius theorem is a mathematical explanation of the second law of thermodynamics. It was developed by Rudolf Clausius who intended to explain the relationship between the heat flow in a system and the entropy of the system and its surroundings. Clausius developed this in his efforts to explain entropy and define it quantitatively. In more direct terms, the theorem gives us a way to determine if a cyclical process is reversible or irreversible. The Clausius theorem provides a quantitative formula for understanding the second law.
克劳修斯定理是对热力学第二定律的一种数学解释。它是由鲁道夫 · 克劳修斯提出的，他试图解释系统中的热流与系统及其周围环境的熵之间的关系。克劳修斯在解释熵和定义熵的过程中发展了这个理论。用更直接的术语来说，这个定理给我们提供了一种方法，来确定一个循环过程是可逆的还是不可逆的。克劳修斯定理提供了一个理解第二定律的定量公式。
Clausius was one of the first to work on the idea of entropy and is even responsible for giving it that name. What is now known as the Clausius theorem was first published in 1862 in Clausius' sixth memoir, "On the Application of the Theorem of the Equivalence of Transformations to Interior Work". Clausius sought to show a proportional relationship between entropy and the energy flow by heating (δ''Q'') into a system. In a system, this heat energy can be transformed into work, and work can be transformed into heat through a cyclical process. Clausius writes that "The algebraic sum of all the transformations occurring in a cyclical process can only be less than zero, or, as an extreme case, equal to nothing." In other words, the equation
Clausius was one of the first to work on the idea of entropy and is even responsible for giving it that name. What is now known as the Clausius theorem was first published in 1862 in Clausius' sixth memoir, "On the Application of the Theorem of the Equivalence of Transformations to Interior Work". Clausius sought to show a proportional relationship between entropy and the energy flow by heating (δQ) into a system. In a system, this heat energy can be transformed into work, and work can be transformed into heat through a cyclical process. Clausius writes that "The algebraic sum of all the transformations occurring in a cyclical process can only be less than zero, or, as an extreme case, equal to nothing." In other words, the equation
克劳修斯是最早研究熵概念的人之一，他甚至给熵起了这个名字。1862年，克劳修斯在他的第六本回忆录《论等效变换定理在室内工作中的应用》中首次出版了现在被称为《克劳修斯定理。克劳修斯试图通过加热(δq)到一个系统来表明熵和能量流之间的比例关系。在一个系统中，热能可以转化为功，功可以通过循环过程转化为热。克劳修斯写道: “在循环过程中发生的所有变换的代数和只能小于零，或者，在极端情况下，等于零。”换句话说，方程式
:<math>\oint \frac{\delta Q}{T} = 0</math>
<math>\oint \frac{\delta Q}{T} = 0</math>
0 </math > oint frac { delta q }{ t } = 0
with 𝛿''Q'' being energy flow into the system due to heating and ''T'' being absolute temperature of the body when that energy is absorbed, is found to be true for any process that is cyclical and reversible. Clausius then took this a step further and determined that the following relation must be found true for any cyclical process that is possible, reversible or not. This relation is the "Clausius inequality".
with 𝛿Q being energy flow into the system due to heating and T being absolute temperature of the body when that energy is absorbed, is found to be true for any process that is cyclical and reversible. Clausius then took this a step further and determined that the following relation must be found true for any cyclical process that is possible, reversible or not. This relation is the "Clausius inequality".
因为 δq 是由于加热而流入系统的能量，t 是体内吸收能量时的绝对温度，对于任何周期性和可逆的过程都是正确的。克劳修斯又进一步确定，对于任何可能的循环过程，无论可逆与否，都必须找到以下关系式。这种关系就是“克劳修斯不等式”。
:<math>\oint \frac{\delta Q}{T_{surr}} \leq 0</math>
<math>\oint \frac{\delta Q}{T_{surr}} \leq 0</math>
[ math > oint frac { delta q }{ t _ { surr } leq 0 </math >
Now that this is known, there must be a relation developed between the Clausius inequality and entropy. The amount of entropy S added to the system during the cycle is defined as
Now that this is known, there must be a relation developed between the Clausius inequality and entropy. The amount of entropy S added to the system during the cycle is defined as
既然已经知道了这一点，那么克劳修斯不等式和熵之间一定存在着某种关系。在循环过程中添加到系统中的熵 s 的量被定义为
:<math>\Delta S {{=}} \oint \frac{\delta Q}{T}</math>
<math>\Delta S \oint \frac{\delta Q}{T}</math>
数学，数学，数学
It has been determined, as stated in the [[second law of thermodynamics]], that the entropy is a state function: It depends only upon the state that the system is in, and not what path the system took to get there. This is in contrast to the amount of energy added as heat (𝛿''Q'') and as work (𝛿''W''), which may vary depending on the path. In a cyclic process, therefore, the entropy of the system at the beginning of the cycle must equal the entropy at the end of the cycle, <math>\Delta S=0</math>, regardless of whether the process is reversible or irreversible. In the irreversible case, entropy will be created in the system, and more entropy must be extracted than was added <math>(\Delta S_{surr}>0)</math> in order to return the system to its original state. In the reversible case, no entropy is created and the amount of entropy added is equal to the amount extracted.
It has been determined, as stated in the second law of thermodynamics, that the entropy is a state function: It depends only upon the state that the system is in, and not what path the system took to get there. This is in contrast to the amount of energy added as heat (𝛿Q) and as work (𝛿W), which may vary depending on the path. In a cyclic process, therefore, the entropy of the system at the beginning of the cycle must equal the entropy at the end of the cycle, <math>\Delta S=0</math>, regardless of whether the process is reversible or irreversible. In the irreversible case, entropy will be created in the system, and more entropy must be extracted than was added <math>(\Delta S_{surr}>0)</math> in order to return the system to its original state. In the reversible case, no entropy is created and the amount of entropy added is equal to the amount extracted.
正如《热力学第二定律所说，熵是一个状态函数: 它只取决于系统所处的状态，而不取决于系统到达那个状态所走的路径。这与作为热(δq)和作为功(δw)加入的能量相反，后者可能随路径而变化。因此，在循环过程中，系统在循环开始时的熵必须等于循环结束时的熵，< math > Delta s = 0 </math > ，不管这个过程是可逆的还是不可逆的。在不可逆的情况下，系统会产生熵，为了使系统恢复到原来的状态，必须提取比加入 math (Delta s { surr } > 0)更多的熵。在可逆的情况下，不会产生熵，熵的增加量等于提取的量。
If the amount of energy added by heating can be measured during the process, and the temperature can be measured during the process, the Clausius inequality can be used to determine whether the process is reversible or irreversible by carrying out the integration in the Clausius inequality.
If the amount of energy added by heating can be measured during the process, and the temperature can be measured during the process, the Clausius inequality can be used to determine whether the process is reversible or irreversible by carrying out the integration in the Clausius inequality.
如果可以测量加热过程中所增加的能量，并且可以测量过程中的温度，则可以用克劳修斯不等式进行克劳修斯不等式的积分，来确定过程是可逆的还是不可逆的。
==Proof==
The temperature that enters in the denominator of the integrand in the Clausius inequality is actually the temperature of the [[Thermal reservoir|external reservoir]] with which the system exchanges heat. At each instant of the process, the system is in contact with an external reservoir.
The temperature that enters in the denominator of the integrand in the Clausius inequality is actually the temperature of the external reservoir with which the system exchanges heat. At each instant of the process, the system is in contact with an external reservoir.
在克劳修斯不等式中，进入被积函数分母的温度，实际上是系统与之交换热量的外部热库的温度。在这个过程的每一个瞬间，系统都与外部储存器接触。
Because of the Second Law of Thermodynamics, in each infinitesimal heat exchange process between the system and the reservoir, the net change in entropy of the "universe", so to say, is <math> dS_{Total}=dS_{Sys} +dS_{Res} \geq 0 </math>.
Because of the Second Law of Thermodynamics, in each infinitesimal heat exchange process between the system and the reservoir, the net change in entropy of the "universe", so to say, is <math> dS_{Total}=dS_{Sys} +dS_{Res} \geq 0 </math>.
由于热力学第二定律的存在，在系统和蓄热体之间每一个微小的热交换过程中，“宇宙”熵的净变化，也就是说，是。
When the system takes in heat by an infinitesimal amount <math>\delta Q_{1}</math>(<math>\geq 0</math>), for the net change in entropy <math>dS_{Total_{1}}</math> in this step to be positive, the temperature of the "hot" [[Thermal reservoir|reservoir]] <math>T_{Hot}</math> needs to be slightly greater than the temperature of the system at that instant.
When the system takes in heat by an infinitesimal amount <math>\delta Q_{1}</math>(<math>\geq 0</math>), for the net change in entropy <math>dS_{Total_{1}}</math> in this step to be positive, the temperature of the "hot" reservoir <math>T_{Hot}</math> needs to be slightly greater than the temperature of the system at that instant.
当系统以无限小的量吸收热量时(< math > delta q {1} </math > (< math > geq 0 </math >) ，因为在这一步中熵的净变化为正，“热”库 < t { Hot } </math > 需要比系统当时的温度稍高。
If the temperature of the system is given by <math>T_{1}</math> at that instant, then <math> dS_{Sys_{1}}=\frac{\delta Q_{1}}{T_{1}}</math>, and <math>T_{Hot}\geq T_{1}</math> forces us to have:
If the temperature of the system is given by <math>T_{1}</math> at that instant, then <math> dS_{Sys_{1}}=\frac{\delta Q_{1}}{T_{1}}</math>, and <math>T_{Hot}\geq T_{1}</math> forces us to have:
如果系统的温度是由 < math > t _ {1} </math > 给出的，那么 < math > dS _ { Sys _ 1} = frac { delta q _ {1}{ t _ {1} </math > ，< math > t _ { Hot } geq _ {1} </math > 强迫我们:
:<math> -dS_{Res_{1}} =\frac{\delta Q_{1}}{T_{Hot}}\leq \frac{\delta Q_{1}}{T_{1}} = dS_{Sys_{1}} </math>
<math> -dS_{Res_{1}} =\frac{\delta Q_{1}}{T_{Hot}}\leq \frac{\delta Q_{1}}{T_{1}} = dS_{Sys_{1}} </math>
< math >-dS _ Res _ {1} = frac { delta q _ {1}{ t _ { Hot } leq frac { delta q _ {1}{ t _ {1}} = dS _ { Sys _ {1} </math >
This means the magnitude of the entropy "loss" from the reservoir, <math> |dS_{Res_{1}}|=\frac{\delta Q_{1}}{T_{Hot}} </math> is less than the magnitude of the entropy gain <math>dS_{Sys_{1}}</math>(<math>\geq 0</math>) by the system:
This means the magnitude of the entropy "loss" from the reservoir, <math> |dS_{Res_{1}}|=\frac{\delta Q_{1}}{T_{Hot}} </math> is less than the magnitude of the entropy gain <math>dS_{Sys_{1}}</math>(<math>\geq 0</math>) by the system:
这意味着从水库的熵“损失”的大小，| dS { Res {1} | = frac { delta q {1}{ t { Hot } </math > 小于系统的熵增 < dS { Sys {1} </math > (< math > geq 0 </math >) :
Similarly, when the system at temperature <math>T_{2}</math> expels heat in magnitude <math>-\delta Q_{2}</math> (<math>\delta Q_{2}\leq 0</math>) into a colder [[Thermal reservoir|reservoir]] (at temperature <math>T_{Cold}\leq T_{2}</math>) in an infinitesimal step, then again, for the Second Law of Thermodynamics to hold, one would have, in an exactly similar manner:
Similarly, when the system at temperature <math>T_{2}</math> expels heat in magnitude <math>-\delta Q_{2}</math> (<math>\delta Q_{2}\leq 0</math>) into a colder reservoir (at temperature <math>T_{Cold}\leq T_{2}</math>) in an infinitesimal step, then again, for the Second Law of Thermodynamics to hold, one would have, in an exactly similar manner:
类似地，当系统在温度 < math > t {2} </math > 将热量以数量级 < math >-delta q {2} </math > (< math > delta q {2} leq 0 </math >)排放到一个较冷的水库中(在温度 < math > t { Cold } leq {2} </math >) ，然后再一次，对于热力学第二定律来说，人们会以完全相同的方式:
:<math display="block"> -dS_{Res_{2}}=\frac{\delta Q_{2}}{T_{Cold}}\leq \frac{\delta Q_{2}}{T_{2}}= dS_{Sys_{2}} </math>
<math display="block"> -dS_{Res_{2}}=\frac{\delta Q_{2}}{T_{Cold}}\leq \frac{\delta Q_{2}}{T_{2}}= dS_{Sys_{2}} </math>
< math display = " block" >-dS _ Res _ {2} = frac { delta q _ {2}{ t _ { Cold } leq frac { delta q _ {2}{ t _ {2}} = dS _ { Sys _ {2} </math >
Here, the amount of heat 'absorbed' by the system is given by <math>\delta Q_{2}</math>(<math>\leq 0</math>), signifying that heat is transferring from the system to the reservoir, with <math>dS_{Sys_{2}}\leq 0</math>. The magnitude of the entropy gained by the reservoir, <math> dS_{Res_{2}}=\frac{|\delta Q_{2}|}{T_{cold}}</math> is greater than the magnitude of the entropy loss of the system <math> |dS_{Sys_{2}}|</math>
Here, the amount of heat 'absorbed' by the system is given by <math>\delta Q_{2}</math>(<math>\leq 0</math>), signifying that heat is transferring from the system to the reservoir, with <math>dS_{Sys_{2}}\leq 0</math>. The magnitude of the entropy gained by the reservoir, <math> dS_{Res_{2}}=\frac{|\delta Q_{2}|}{T_{cold}}</math> is greater than the magnitude of the entropy loss of the system <math> |dS_{Sys_{2}}|</math>
在这里，系统所“吸收”的热量是通过 < math > delta q _ {2} </math > (< math > leq 0 </math >)给出的，表示热量正从系统传递到储存器，而 < math > dS _ Sys _ {2} leq 0 </math > 。水库所获得的熵的大小，大于系统的熵损失的大小
Since the total change in entropy for the system is 0 in a cyclic process, if one adds all the infinitesimal steps of heat intake and heat expulsion from the reservoir, signified by the previous two equations, with the temperature of the reservoir at each instant given by <math>T_{surr}</math>, one gets,
Since the total change in entropy for the system is 0 in a cyclic process, if one adds all the infinitesimal steps of heat intake and heat expulsion from the reservoir, signified by the previous two equations, with the temperature of the reservoir at each instant given by <math>T_{surr}</math>, one gets,
由于系统在循环过程中熵的总变化为0，如果加上前面两个方程所表示的储热器吸热和排热的所有无穷小的步骤，再加上每一瞬间储热器的温度，就得到,
:<math> -\oint dS_{Res}= \oint \frac{\delta Q}{T_{surr}}\leq \oint dS_{Sys}=0 </math>
<math> -\oint dS_{Res}= \oint \frac{\delta Q}{T_{surr}}\leq \oint dS_{Sys}=0 </math>
[数学]-oint dS { Res } = oint frac { delta q }{ t _ { surr } leq oint dS { Sys } = 0 </math >
In particular,
In particular,
特别是,
:<math>\oint \frac{\delta Q}{T_{surr}}\leq 0, </math>
<math>\oint \frac{\delta Q}{T_{surr}}\leq 0, </math>
0，</math > oint frac { delta q }{ t _ { surr } leq 0，</math >
which was to be proven.
which was to be proven.
这是有待证实的。
In summary, (the inequality in the third statement below, being obviously guaranteed by the [[second law of thermodynamics]], which is the basis of our calculation),
In summary, (the inequality in the third statement below, being obviously guaranteed by the second law of thermodynamics, which is the basis of our calculation),
总之，(下面第三个陈述中的不平等，显然得到了我们计算的基础---- 美国热力学第二定律的保证) ,
:<math>\oint dS_{Res}\geq 0 </math>
<math>\oint dS_{Res}\geq 0 </math>
0 </math > oint dS { Res } geq 0
:<math>\oint dS_{Sys}=0 </math> (as hypothesized)
<math>\oint dS_{Sys}=0 </math> (as hypothesized)
[ math > oint dS _ { Sys } = 0
:<math>\oint dS_{Total}=\oint dS_{Res}+\oint dS_{Sys}\geq 0</math>
<math>\oint dS_{Total}=\oint dS_{Res}+\oint dS_{Sys}\geq 0</math>
[数学][数学][数学][数学]
For a [[Reversible process (thermodynamics)|reversible]] [[Thermodynamic cycle|cyclic]] process, there is no generation of entropy in each of the infinitesimal heat transfer processes, so the following equality holds,
For a reversible cyclic process, there is no generation of entropy in each of the infinitesimal heat transfer processes, so the following equality holds,
对于可逆循环过程，在每一个无限小的传热过程中都不会产生熵，因此下面的等式成立,
:<math>\oint \frac{\delta Q_{rev}}{T}=0. </math>
<math>\oint \frac{\delta Q_{rev}}{T}=0. </math>
0.0.0.数学
Thus, the Clausius inequality is a consequence of applying the [[second law of thermodynamics]] at each infinitesimal stage of heat transfer, and is thus in a sense a weaker condition than the Second Law itself.
Thus, the Clausius inequality is a consequence of applying the second law of thermodynamics at each infinitesimal stage of heat transfer, and is thus in a sense a weaker condition than the Second Law itself.
因此，克劳修斯不等式是在传热的每一个无穷小阶段应用热力学第二定律的结果，因此在某种意义上比第二定律本身更弱。
==See also==
* [[Kelvin-Planck statement]]
* [[Carnot's theorem (thermodynamics)]]
* [[Carnot heat engine]]
* [[Introduction to entropy]]
==References==
{{reflist}}
{{refbegin}}
==Further reading==
*Morton, A. S., and P.J. Beckett. ''Basic Thermodynamics''. New York: Philosophical Library Inc., 1969. Print.
*Saad, Michel A. ''Thermodynamics for Engineers''. Englewood Cliffs: Prentice-Hall, 1966. Print.
*Hsieh, Jui Sheng. ''Principles of Thermodynamics''. Washington, D.C.: Scripta Book Company, 1975. Print.
*Zemansky, Mark W. ''Heat and Thermodynamics''. 4th ed. New York: McGwaw-Hill Book Company, 1957. Print.
*Clausius, Rudolf. ''The Mechanical Theory of Heat''. London: Taylor and Francis, 1867. eBook
{{refend}}
==External links==
*{{cite web|title=Proof of Clausius's theorem|url=http://theory.ph.man.ac.uk/~judith/stat_therm/node30.html|accessdate=October 4, 2010|author=Judith McGovern|date=2004-03-17|archive-url=https://web.archive.org/web/20110719052220/http://theory.ph.man.ac.uk/~judith/stat_therm/node30.html|archive-date=July 19, 2011|url-status=dead}}
*{{cite web|title=The Clausius Inequality And The Mathematical Statement Of The Second Law|url=http://ronispc.chem.mcgill.ca/ronis/chem213/hnd10.pdf|accessdate=October 5, 2010}}
*{{cite book|title=The Mechanical Theory of Heat (eBook)|url=https://books.google.com/books?id=8LIEAAAAYAAJ |accessdate=December 1, 2011}}
{{DEFAULTSORT:Clausius Theorem}}
[[Category:Laws of thermodynamics]]
Category:Laws of thermodynamics
类别: 热力学定律
[[Category:Physics theorems]]
Category:Physics theorems
范畴: 物理学定理
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