熵增原理

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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,

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,

克劳修斯定理Clausius theorem（1855）指出，对于 热力学系统Thermodynamic system （例如，热机或热泵），当其与 外部热库External reservoirs进行热交换并经历 热力学循环Thermodynamic cycle时，

[math]\displaystyle{ \oint \frac{\delta Q}{T_{\text{surr}}} \leq 0, }[/math]

where [math]\displaystyle{ \delta Q }[/math] is the infinitesimal amount of heat absorbed by the system from the reservoir and [math]\displaystyle{ 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]\displaystyle{ \delta Q }[/math] is the infinitesimal amount of heat absorbed by the system from the reservoir and [math]\displaystyle{ 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]\displaystyle{ \delta Q }[/math]是系统从热库吸收的热量极小值，[math]\displaystyle{ T_{\text{surr}} }[/math]是特定时间点外部热库（周围环境）的温度。该表达式是指沿着从初始/最终状态到相同的初始/最终状态下 热力学过程路径Thermodynamic process path执行闭合积分。原则上，该闭合积分可以沿路径的任意点开始和结束。

If there are multiple reservoirs with different temperatures [math]\displaystyle{ \left(T_1,T_2, \cdots T_n\right) }[/math], then Clausius inequality reads:

If there are multiple reservoirs with different temperatures [math]\displaystyle{ \left(T_1,T_2, \cdots T_n\right) }[/math], then Clausius inequality reads:

如果存在有多个具有不同温度[math]\displaystyle{ \left(T_1,T_2, \cdots T_n\right) }[/math]的热库，则克劳修斯不等式为：

[math]\displaystyle{ \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]

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.

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.

当在过程可逆的特殊情况下，该等式成立。其可逆过程可用于引入 熵Entropy 状态函数State function 。这是因为在循环过程中，状态函数的变化为零。换句话说， 克劳修斯表述Clausius statement指出，不可能构造出一种装置，使其仅仅将热从低温热库传递至高温热库而不引起其他变化。相当于说，热量只能自发地从高温物体流向相对低温的物体，反向则不行。

The generalized "inequality of Clausius"

The generalized "inequality of Clausius"

克劳修斯的广义不等式为：

[math]\displaystyle{ 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.

对于熵趋近于无穷小变化时，S不仅适用于循环过程，而且适用于封闭系统中发生的任何过程。

History 历史

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 热力学第二定律Second law of thermodynamics的数学解释。它是由鲁道夫·克劳修斯Rudolf Clausius提出的。他的目的是解释系统中的热量传递与系统及其周围环境熵之间的关系。当初他为了解释并定量熵而逐步发展出了该公式。更直接地讲，该定理为我们提供了一种确定热循环过程是否可逆的方法，为理解第二定律提供了一个定量公式。

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 (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

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，甚至负责为其命名。关于现阶段“克劳修斯定理”的称呼最初出现在1862年克劳修斯的第六本回忆录《关于转换等价定理在定量物质内做功的应用》中。克劳修斯试图表达熵与通过加热（其热量表示为δQ）进入系统的能量流之间的比例关系。在系统中，这种热能可以转化为功，并且功也可以通过循环过程转化为热。克劳修斯写道：“在一个循环过程中发生的所有转换的代数和只能小于零，或者说在极端情况下等于零。”也就是如下等式：

[math]\displaystyle{ \oint \frac{\delta Q}{T} = 0 }[/math]

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".

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".

其中𝛿Q是由于加热而从外界流入系统的能量，T是吸收能量时该主体的绝对温度，该等式对于任何周期性且可逆的过程均成立。之后，克劳修斯进一步扩展并确定，对于任何可能可逆的或不可逆的周期性过程，必须满足以下关系，即“克劳修斯不等式”。。

[math]\displaystyle{ \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

于是现在明确了克劳修斯不等式和熵之间的必然联系。而其周期性过程中所增加的熵量S为：

[math]\displaystyle{ \Delta S {{=}} \oint \frac{\delta Q}{T} }[/math]

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]\displaystyle{ \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]\displaystyle{ (\Delta S_{surr}\gt 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]\displaystyle{ \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]\displaystyle{ (\Delta S_{surr}\gt 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.

如热力学第二定律所述，熵已经确定是一个状态函数：它仅取决于系统所处的状态，而不取决于系统传递热量的过程路径。这与通过加热（𝛿Q）和作功（𝛿W）增加的能量是不同的，后者随路径的变化而变化。因此，在循环过程中，无论其是可逆还是不可逆的，系统在循环开始时的熵必须等于循环结束时的熵，即[math]\displaystyle{ \Delta S=0 }[/math]。在不可逆的情况下，系统会产生熵，而且其提取的熵量会大于已添加的熵量[math]\displaystyle{ (\Delta S_{surr}\gt 0) }[/math]，这样才能使系统回到其原始状态。而在循环过程可逆的情况下，系统则不会产生熵，其所添加熵的量等于其提取的量。

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.

循环过程中，如果能测量出因加热而增加的能量和其温度，那么通过对克劳修斯不等式进行积分，就能确定其过程是否可逆。

Proof 证据

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.

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.

将克劳修斯不等式积分，其被积函数分母中的温度实际上是系统与之交换热量的外部热库的温度。注意热量传递过程的每个瞬间，系统都与外部热库接触。

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]\displaystyle{ 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]\displaystyle{ dS_{Total}=dS_{Sys} +dS_{Res} \geq 0 }[/math].

根据热力学第二定律，在系统和热库之间，每个无穷小的热交换过程中，其总体系熵的净变化为[math]\displaystyle{ dS_{Total}=dS_{Sys} +dS_{Res} \geq 0 }[/math]。

When the system takes in heat by an infinitesimal amount [math]\displaystyle{ \delta Q_{1} }[/math]([math]\displaystyle{ \geq 0 }[/math]), for the net change in entropy [math]\displaystyle{ dS_{Total_{1}} }[/math] in this step to be positive, the temperature of the "hot" reservoir [math]\displaystyle{ 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]\displaystyle{ \delta Q_{1} }[/math]([math]\displaystyle{ \geq 0 }[/math]), for the net change in entropy [math]\displaystyle{ dS_{Total_{1}} }[/math] in this step to be positive, the temperature of the "hot" reservoir [math]\displaystyle{ T_{Hot} }[/math] needs to be slightly greater than the temperature of the system at that instant.

当系统吸收无穷小的热量[math]\displaystyle{ \delta Q_{1} }[/math]([math]\displaystyle{ \geq 0 }[/math])时，为了使此过程中的熵[math]\displaystyle{ dS_{Total_{1}} }[/math]的净变量为正，“热”库[math]\displaystyle{ T_{Hot} }[/math]的温度必须稍大于该时刻的系统温度。

If the temperature of the system is given by [math]\displaystyle{ T_{1} }[/math] at that instant, then [math]\displaystyle{ dS_{Sys_{1}}=\frac{\delta Q_{1}}{T_{1}} }[/math], and [math]\displaystyle{ T_{Hot}\geq T_{1} }[/math] forces us to have:

If the temperature of the system is given by [math]\displaystyle{ T_{1} }[/math] at that instant, then [math]\displaystyle{ dS_{Sys_{1}}=\frac{\delta Q_{1}}{T_{1}} }[/math], and [math]\displaystyle{ T_{Hot}\geq T_{1} }[/math] forces us to have:

如果系统温度在该时刻由[math]\displaystyle{ T_{1} }[/math]给出，则[math]\displaystyle{ dS_{Sys_{1}}=\frac{\delta Q_{1}}{T_{1}} }[/math]和[math]\displaystyle{ T_{Hot}\geq T_{1} }[/math]迫使我们具有：

[math]\displaystyle{ -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]\displaystyle{ |dS_{Res_{1}}|=\frac{\delta Q_{1}}{T_{Hot}} }[/math] is less than the magnitude of the entropy gain [math]\displaystyle{ dS_{Sys_{1}} }[/math]([math]\displaystyle{ \geq 0 }[/math]) by the system:

This means the magnitude of the entropy "loss" from the reservoir, [math]\displaystyle{ |dS_{Res_{1}}|=\frac{\delta Q_{1}}{T_{Hot}} }[/math] is less than the magnitude of the entropy gain [math]\displaystyle{ dS_{Sys_{1}} }[/math]([math]\displaystyle{ \geq 0 }[/math]) by the system:

这意味着来自热库的熵“损失”的大小，即[math]\displaystyle{ |dS_{Res_{1}}|=\frac{\delta Q_{1}}{T_{Hot}} }[/math]小于了系统熵增加的大小[math]\displaystyle{ dS_{Sys_{1}} }[/math]([math]\displaystyle{ \geq 0 }[/math])：

Similarly, when the system at temperature [math]\displaystyle{ T_{2} }[/math] expels heat in magnitude [math]\displaystyle{ -\delta Q_{2} }[/math] ([math]\displaystyle{ \delta Q_{2}\leq 0 }[/math]) into a colder reservoir (at temperature [math]\displaystyle{ 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]\displaystyle{ T_{2} }[/math] expels heat in magnitude [math]\displaystyle{ -\delta Q_{2} }[/math] ([math]\displaystyle{ \delta Q_{2}\leq 0 }[/math]) into a colder reservoir (at temperature [math]\displaystyle{ 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，当温度为[math]\displaystyle{ T_{2} }[/math]的系统在瞬间发生的过程内将热量[math]\displaystyle{ -\delta Q_{2} }[/math] ([math]\displaystyle{ \delta Q_{2}\leq 0 }[/math])排入较冷的热库（温度[math]\displaystyle{ T_{Cold}\leq T_{2} }[/math]）时，必须以同上完全相似的方式来满足热力学第二定律：

[math]\displaystyle{ -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]\displaystyle{ \delta Q_{2} }[/math]([math]\displaystyle{ \leq 0 }[/math]), signifying that heat is transferring from the system to the reservoir, with [math]\displaystyle{ dS_{Sys_{2}}\leq 0 }[/math]. The magnitude of the entropy gained by the reservoir, [math]\displaystyle{ dS_{Res_{2}}=\frac{|\delta Q_{2}|}{T_{cold}} }[/math] is greater than the magnitude of the entropy loss of the system [math]\displaystyle{ |dS_{Sys_{2}}| }[/math]

Here, the amount of heat 'absorbed' by the system is given by [math]\displaystyle{ \delta Q_{2} }[/math]([math]\displaystyle{ \leq 0 }[/math]), signifying that heat is transferring from the system to the reservoir, with [math]\displaystyle{ dS_{Sys_{2}}\leq 0 }[/math]. The magnitude of the entropy gained by the reservoir, [math]\displaystyle{ dS_{Res_{2}}=\frac{|\delta Q_{2}|}{T_{cold}} }[/math] is greater than the magnitude of the entropy loss of the system [math]\displaystyle{ |dS_{Sys_{2}}| }[/math]

这里系统“吸收”的热量由[math]\displaystyle{ \delta Q_{2} }[/math]([math]\displaystyle{ \leq 0 }[/math])给出，表示热量从系统传递到热库，且[math]\displaystyle{ dS_{Sys_{2}}\leq 0 }[/math]。由热库获得的熵大小[math]\displaystyle{ dS_{Res_{2}}=\frac{|\delta Q_{2}|}{T_{cold}} }[/math]，大于系统熵损失的大小[math]\displaystyle{ |dS_{Sys_{2}}| }[/math]。

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]\displaystyle{ 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]\displaystyle{ T_{surr} }[/math], one gets,

由于系统在循环过程中熵总量不变，因此，如果将前面两个方程式表示的所有从热库吸收和排放的热，分解成无穷小的阶段再相加，然后在定义出每个时刻给定热库温度[math]\displaystyle{ T_{surr} }[/math]，可得出：

[math]\displaystyle{ -\oint dS_{Res}= \oint \frac{\delta Q}{T_{surr}}\leq \oint dS_{Sys}=0 }[/math]

In particular,

In particular,

尤其是：

[math]\displaystyle{ \oint \frac{\delta Q}{T_{surr}}\leq 0, }[/math]

which was to be proven.

which was to be proven.

得到了证明。

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),

综上所述，我们得出（下面第三条陈述中的不等式显然来自于热力学第二定律，这是我们计算的基础），

[math]\displaystyle{ \oint dS_{Res}\geq 0 }[/math]

[math]\displaystyle{ \oint dS_{Sys}=0 }[/math] (as hypothesized)

[math]\displaystyle{ \oint dS_{Total}=\oint dS_{Res}+\oint dS_{Sys}\geq 0 }[/math]

For a reversible 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，在每个无穷小的传热阶段中都不会产生熵，因此以下等式成立：

[math]\displaystyle{ \oint \frac{\delta Q_{rev}}{T}=0. }[/math]

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，克劳修斯不等式是基于热力学第二定律并应用在热传递过程中每个无穷小阶段的结果，从某种意义上说，它是热力学第二定律的弱条件。

See also 其他参考资料

• Kelvin-Planck statement

• Carnot's theorem (thermodynamics)

• Carnot heat engine

• Introduction to entropy

• [[ 开尔文－普朗克表述Kelvin-Planck statement ]]
• [[ 卡诺定理（热力学）Carnot's theorem (thermodynamics )]]
• [[ 卡诺热机Carnot heat engine ]]
• [[ 熵的介绍Introduction to entropy ]]

External links 相关链接

• Judith McGovern (2004-03-17). "Proof of Clausius's theorem". Archived from the original on July 19, 2011. Retrieved October 4, 2010.

• "The Clausius Inequality And The Mathematical Statement Of The Second Law" (PDF). Retrieved October 5, 2010.

• The Mechanical Theory of Heat (eBook). https://books.google.com/books?id=8LIEAAAAYAAJ. Retrieved December 1, 2011. 

Judith McGovern (2004-03-17). " 克劳修斯定理的证明Proof of Clausius's theorem ". Archived from the original on July 19, 2011. Retrieved October 4, 2010.

" 克劳修斯不等式和热力学第二定律的数学表述The Clausius Inequality And The Mathematical Statement Of The Second Law " (PDF). Retrieved October 5, 2010.

热力学原理The Mechanical Theory of Heat (eBook). Retrieved December 1, 2011.

Category:Laws of thermodynamics

类别: 热力学定律

Category:Physics theorems

范畴: 物理学定理

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