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===Relation between Kelvin's statement and Planck's proposition 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 almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.
It is almost customary in textbooks to speak of the "Kelvin-Planck statement" of the law, as for example in the text by ter Haar and Wergeland.
教科书中几乎惯常提及该定律的“'''开尔文-普朗克表述 Kelvin-Planck Statement'''” ，例如'''德克·特哈尔 Diek ter Haar''' 和'''哈拉尔德·沃格兰 Harald Wergeland''' 在文中就是这样表述的。
--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）Diek ter Haar 中间是小写的
The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that
The Kelvin–Planck statement (or the heat engine statement) of the second law of thermodynamics states that
热力学第二定律的开尔文-普朗克表述（或称“热机表述 Heat Engine Statement”）指出
::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref>
::It is impossible to devise a [[thermodynamic cycle|cyclically]] operating device, the sole effect of which is to absorb energy in the form of heat from a single [[heat reservoir|thermal reservoir]] and to deliver an equivalent amount of [[Work (physics)|work]].<ref name="Rao">{{cite book|last=Rao|first=Y. V. C.|title=Chemical Engineering Thermodynamics|publisher=Universities Press|isbn=978-81-7371-048-3|page=158|year=1997}}</ref>
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.
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.
设计一种唯一效果是从单一热源吸收热量并提供等量的功的循环运行装置是不可能的。
===Planck's statement 普朗克表述===
Planck stated the second law as follows.
Planck stated the second law as follows.
普朗克表述第二定律如下。
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.
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.
自然界中发生的任一过程都是沿参与其中的所有物体的熵之和增加的方向进行的。在某些限制下——即对于可逆过程，熵的总和保持不变。
Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.
Rather like Planck's statement is that of Uhlenbeck and Ford for irreversible phenomena.
与普朗克表述非常相似的是'''乌伦贝克 Uhlenbeck'''和'''福特 Ford'''关于不可逆现象的表述。
... 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.
... 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.
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。
===Principle of Carathéodory 卡拉西奥多里原理===
<!-- [[Caratheodory's principle]] redirects here -->
<!-- [[Caratheodory's principle]] redirects here -->
Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:
Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:
'''康斯坦丁·卡拉西奥多里 Constantin Carathéodory'''在纯数学公理的基础上进行了热力学公理化。他对第二定律的陈述被称为'''卡拉西奥多里原理 Principle of Carathéodory'''，可以这样表述:
<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote>
<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.<ref>Buchdahl, H.A. (1966), p. 68.</ref></blockquote>
<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote>
<blockquote>In every neighborhood of any state S of an adiabatically enclosed system there are states inaccessible from S.</blockquote>
在绝热封闭系统的任意状态 S 附近，总有从 S 出发不可达的状态。
With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}
With this formulation, he described the concept of [[adiabatic accessibility]] for the first time and provided the foundation for a new subfield of classical thermodynamics, often called [[Ruppeiner geometry|geometrical thermodynamics]]. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic [[process function]], in other words, <math>\delta Q=TdS</math>.<ref name="Sychev1991">{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1-56032-121-7}}</ref> {{clarify|date=February 2014}}
With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>.
With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics. It follows from Carathéodory's principle that quantity of energy quasi-statically transferred as heat is a holonomic process function, in other words, <math>\delta Q=TdS</math>.
公理化处理后，他第一次描述了'''绝热可达性 Adiabatic Accessibility'''的概念，并为经典热力学的一个新的子领域，即通常所说的'''几何热力学 Geometrical Thermodynamics'''奠定了基础。由卡拉西奥多里原理可以推出，准静态转移的热量值是一个可积过程函数，即<math>\delta Q=TdS</math>。
--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）准静态转移的热量值是一个可积过程函数 存疑
Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.
Though it is almost customary in textbooks to say that Carathéodory's principle expresses the second law and to treat it as equivalent to the Clausius or to the Kelvin-Planck statements, such is not the case. To get all the content of the second law, Carathéodory's principle needs to be supplemented by Planck's principle, that isochoric work always increases the internal energy of a closed system that was initially in its own internal thermodynamic equilibrium.
尽管在教科书几乎惯称卡拉西奥多里原理表述了第二定律，并认为其与克劳修斯表述或开尔文-普朗克表述等价，但事实并非如此。为了得到第二定律的所有内容，需要对卡拉西奥多里原理补充普朗克表述，即等量功总是增加一个最初处于自身内部热力学平衡的封闭系统的内部能量。
===Planck's principle===
===Planck's principle 普朗克原理===
In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle
In 1926, [[Max Planck]] wrote an important paper on the basics of thermodynamics.<ref name="Planck 1926"/><ref>Uffink, J. (2003), pp. 129–132.</ref> He indicated the principle
In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the principle
In 1926, Max Planck wrote an important paper on the basics of thermodynamics. He indicated the 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。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.
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.
一个封闭系统的内部能量因绝热过程增加，在整个过程中，系统的体积保持不变。
This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."
This formulation does not mention heat and does not mention temperature, nor even entropy, and does not necessarily implicitly rely on those concepts, but it implies the content of the second law. A closely related statement is that "Frictional pressure never does positive work." Planck wrote: "The production of heat by friction is irreversible."
这个公式没有提到热，没有提到温度，甚至没有提到熵，也不一定隐含地依赖于这些概念，但它暗示了第二定律的内容。一个密切相关的表述为，“摩擦力从来不做正功。”普朗克写道: “摩擦生热是不可逆的。”
Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.
Not mentioning entropy, this principle of Planck is stated in physical terms. It is very closely related to the Kelvin statement given just above. It is relevant that for a system at constant volume and mole numbers, the entropy is a monotonic function of the internal energy. Nevertheless, this principle of Planck is not actually Planck's preferred statement of the second law, which is quoted above, in a previous sub-section of the present section of this present article, and relies on the concept of entropy.
更不用说熵了，这个普朗克定理是用物理术语来表述的。它与上面给出的开尔文表述密切相关。相关的是，对于恒定体积和摩尔数的系统，熵是内能的单调函数。然而，普朗克的这个原理实际上并不是普朗克对第二定律的首选表述（见前面小节），而是依赖于熵的概念。
A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:
A statement that in a sense is complementary to Planck's principle is made by Borgnakke and Sonntag. They do not offer it as a full statement of the second law:
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.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref>
::... 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.<ref>Borgnakke, C., Sonntag., R.E. (2009), p. 304.</ref>
... 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.
... 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.
只有一种方法可以减少[闭合]系统的熵，那就是从系统中传递热量。
只有一种方法可以减少[闭合]系统的熵——从系统中转移热量。
Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.
Differing from Planck's just foregoing principle, this one is explicitly in terms of entropy change. Removal of matter from a system can also decrease its entropy.
与普朗克之前提出的原理不同，这一原理明确地用熵的变化来表示。从系统中去除物质也可以减少系统的熵。
===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables===
===Statement for a system that has a known expression of its internal energy as a function of its extensive state variables 一个其内能有已知表达式（其扩展状态变量的函数）的系统的表述===
The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...).
The second law has been shown to be equivalent to the internal energy U being a weakly convex function, when written as a function of extensive properties (mass, volume, entropy, ...).
第二定律被证明等价于弱凸函数内能 U （写成广泛性质（质量，体积，熵，...）的函数时）。
--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）存疑
==Corollaries 推论==
===Perpetual motion of the second kind 第二类永动机===
{{main article|Perpetual motion}}
{{main article|Perpetual motion}}
Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.
Before the establishment of the second law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of first law of thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.
在第二定律建立之前，许多志在发明永动机的人试图通过提取环境的巨大内能作为机器动力来突破'''热力学第一定律 First Law of Thermodynamics''' 的限制。这种机器被称为“第二类永动机 Perpetual Motion Machine of the Second Kind”。第二定律表明这种机器是不可能的。
===Carnot theorem===
===Carnot theorem 卡诺定理===
[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:
[[Carnot theorem (thermodynamics)|Carnot's theorem]] (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:
Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:
Carnot's theorem (1824) is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:
卡诺定理（1824）是一条限制任何可能的发动机的最大效率的原理。效率完全取决于热库和冷库之间的温差。卡诺定理指出:
*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.
*All irreversible heat engines between two heat reservoirs are less efficient than a [[Carnot engine]] operating between the same reservoirs.
* 所有不可逆热机的效率低于在两相同热源之间工作的'''卡诺机 Carnot engine'''。
* 所有可逆热机的效率与在两相同热源之间工作的卡诺机相等。
In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.
In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot, however, further postulated that some caloric is lost, not being converted to mechanical work. Hence, no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.
在他的理想模型中，热转换成功可以通过逆转循环的运动而恢复，这个概念后来被称为'''热力学可逆性 Thermodynamic Reversibility'''。然而，卡诺进一步假定，一些热量损失了，并没有转化为机械功。因此，没有一个真实的热机能够实现'''卡诺循环 Carnot cycle'''的可逆性，并且被认为效率较低。
Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.
Though formulated in terms of caloric (see the obsolete caloric theory), rather than entropy, this was an early insight into the second law.
该理论尽管是用热量表述的（见被取代的'''热质说 Caloric Theory'''），而不是熵，但是它是对第二定律的早期认识。
===Clausius inequality===
===Clausius inequality 克劳修斯不等式===
The [[Clausius theorem]] (1854) states that in a cyclic process
The [[Clausius theorem]] (1854) states that in a cyclic process
The Clausius theorem (1854) states that in a cyclic process
The Clausius theorem (1854) states that in a cyclic process
克劳修斯定理（1854）指出，在一个循环的过程中
: <math>\oint \frac{\delta Q}{T} \leq 0.</math>
: <math>\oint \frac{\delta Q}{T} \leq 0.</math>
The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.
The equality holds in the reversible case and the strict inequality holds in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero from state functionality.
等号在可逆情况下成立，严格不等号在不可逆情况下成立。可逆情况下引入状态函数熵。这是因为在循环过程中，状态函数的变化为零。
--[[用户:Dorr|Dorr]]（[[用户讨论:Dorr|讨论]]）末句存疑
===Thermodynamic temperature 热力学温度===
{{main article|Thermodynamic temperature}}
{{main article|Thermodynamic temperature}}
For an arbitrary heat engine, the efficiency is:
For an arbitrary heat engine, the efficiency is:
对于任意热机，效率为:
: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math>
: <math>\eta = \frac {W_n}{q_H} = \frac{q_H-q_C}{q_H} = 1 - \frac{q_C}{q_H} \qquad (1)</math>
where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.
where ''W''<sub>n</sub> is for the net work done per cycle. Thus the efficiency depends only on q<sub>C</sub>/q<sub>H</sub>.
其中 ''W''<sub>n</sub> 表示每个循环所做的净功。因此效率只取决于 q<sub>C</sub>/q<sub>H</sub>。
Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only:
Thus, any reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>2</sub> must have the same efficiency, that is to say, the efficiency is the function of temperatures only:
因此，任何在温度 t 小于1 / sub 和 t 小于2 / sub 之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:
因此，任何在温度''T''<sub>1</sub>和''T''<sub>2</sub>之间运行的可逆热机必须具有相同的效率，也就是说，效率只是温度的函数:
<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math>
<math>\frac{q_C}{q_H} = f(T_H,T_C)\qquad (2).</math>
In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if
In addition, a reversible heat engine operating between temperatures T<sub>1</sub> and T<sub>3</sub> must have the same efficiency as one consisting of two cycles, one between T<sub>1</sub> and another (intermediate) temperature T<sub>2</sub>, and the second between T<sub>2</sub> andT<sub>3</sub>. This can only be the case if
另外，在温度 ''T''<sub>1</sub> 和 ''T''<sub>3</sub> 之间工作的可逆热机必须具有与由分别在温度''T''<sub>1</sub> 和（中间）温度 ''T''<sub>2</sub>之间、在 ''T''<sub>2</sub> 和 ''T''<sub>3</sub> 之间的两个循环组成的系统效率相同。只有下式成立才会出现这种情况:
: <math>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).</math>
Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,
Now consider the case where <math>T_1</math> is a fixed reference temperature: the temperature of the triple point of water. Then for any T<sub>2</sub> and T<sub>3</sub>,
现在考虑如下情形，<math>T_1</math> 是一个固定的参考温度: 水的三相点的温度。则对于任意 T<sub>2</sub> 和 T<sub>3</sub>,
: <math>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)}.</math>
: <math>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)}.</math>
: <math>T = 273.16 \cdot f(T_1,T) \,</math>
: <math>T = 273.16 \cdot f(T_1,T) \,</math>
then the function f, viewed as a function of thermodynamic temperature, is simply
then the function f, viewed as a function of thermodynamic temperature, is simply
那么函数 f 作为热力学温度的函数
那么函数 ''f'' 作为热力学温度的函数，为
: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math>
: <math>f(T_2,T_3) = \frac{T_3}{T_2},</math>
and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be usedthe choice here corresponds to the Kelvin scale.)
and the reference temperature T<sub>1</sub> will have the value 273.16. (Any reference temperature and any positive numerical value could be usedthe choice here corresponds to the Kelvin scale.)
参考温度 t 小于1 / sub 的值为273.16。(任何参考温度和任何正数值都可以用这里的选择对应开尔文标度。)
参考温度 ''T''<sub>1</sub> 的值为273.16。（任何参考温度和任何正值均可用——此处的选择对应开尔文标度。）
===Entropy===
===Entropy 熵===
{{main article|Entropy (classical thermodynamics)}}
{{main article|Entropy (classical thermodynamics)}}
According to the Clausius equality, for a reversible process
According to the Clausius equality, for a reversible process
根据克劳修斯定理，对可逆过程有
: <math>\oint \frac{\delta Q}{T}=0</math>
: <math>\oint \frac{\delta Q}{T}=0</math>
That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.
That means the line integral <math>\int_L \frac{\delta Q}{T}</math> is path independent for reversible processes.
这意味着线积分 math int l frac { delta q }{ t } / math 对于可逆过程是路径无关的。
这意味着线积分 <math>\int_L \frac{\delta Q}{T}</math> 对于可逆过程是路径无关的。
So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies
So we can define a state function S called entropy, which for a reversible process or for pure heat transfer satisfies
所以我们可以定义一个叫做熵的状态函数 S，对于可逆过程或者纯热传递满足
: <math>dS = \frac{\delta Q}{T} \!</math>
: <math>dS = \frac{\delta Q}{T} \!</math>
With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.
With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the third law of thermodynamics, which states that S = 0 at absolute zero for perfect crystals.
据此，只有对上述公式进行积分，才能得到熵的差值。为了获得绝对值，我们需要'''热力学第三定律 Third Law of Thermodynamics'''，它指出绝对零度下完美晶体的 ''S'' = 0。
For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.
For any irreversible process, since entropy is a state function, we can always connect the initial and terminal states with an imaginary reversible process and integrating on that path to calculate the difference in entropy.
对于任意不可逆过程，由于熵是一个状态函数，我们总是可以将初始状态和最终状态与一个虚拟的可逆过程联系起来，并在这条路径上积分以计算熵的差值。
Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,
Now reverse the reversible process and combine it with the said irreversible process. Applying the Clausius inequality on this loop,
现在把可逆过程逆过来，将其与上述不可逆过程结合。把克劳修斯不等式应用到这个循环,
: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math>
: <math>-\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}< 0</math>
Thus,
Thus,
故,
: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math>
: <math>\Delta S \ge \int \frac{\delta Q}{T} \,\!</math>
where the equality holds if the transformation is reversible.
where the equality holds if the transformation is reversible.
如果变换可逆，等号成立。
Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.
Notice that if the process is an adiabatic process, then <math>\delta Q=0</math>, so <math>\Delta S\ge 0</math>.
注意，若该过程是一个'''绝热过程 Adiabatic Process'''，则<math>\delta Q=0</math>，故<math>\Delta S\ge 0</math>。