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第73行：
第73行： − + − − − 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. − − 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>. − − − 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. − − 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: 第110行：
第92行： − − 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: 第119行：
第97行： − 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: 第127行：
第102行： − − 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> 第136行：
第107行： − 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, 第145行：
第113行： − − In particular, − − In particular, 第155行：
第119行： − − which was to be proven. − − which was to be proven. − − − 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), 第178行：
第133行： − 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, 第188行：
第140行： − 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. − −
→Proof 证据
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。
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 ==
== 证据 ==
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。
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。
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> 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> dS_{Total}=dS_{Sys} +dS_{Res} \geq 0 </math>。
当系统吸收无穷小的热量<math>\delta Q_{1}</math>(<math>\geq 0</math>)时，为了使此过程中的熵<math>dS_{Total_{1}}</math>的净变量为正，“热”库<math>T_{Hot}</math>的温度必须稍大于该时刻的系统温度。
当系统吸收无穷小的热量<math>\delta Q_{1}</math>(<math>\geq 0</math>)时，为了使此过程中的熵<math>dS_{Total_{1}}</math>的净变量为正，“热”库<math>T_{Hot}</math>的温度必须稍大于该时刻的系统温度。
如果系统温度在该时刻由<math>T_{1}</math>给出，则<math> dS_{Sys_{1}}=\frac{\delta Q_{1}}{T_{1}}</math>和<math>T_{Hot}\geq T_{1}</math>迫使我们具有：
如果系统温度在该时刻由<math>T_{1}</math>给出，则<math> dS_{Sys_{1}}=\frac{\delta Q_{1}}{T_{1}}</math>和<math>T_{Hot}\geq T_{1}</math>迫使我们具有：
:<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>
这意味着来自热库的熵“损失”的大小，即<math> |dS_{Res_{1}}|=\frac{\delta Q_{1}}{T_{Hot}} </math>小于了系统熵增加的大小<math>dS_{Sys_{1}}</math>(<math>\geq 0</math>)：
这意味着来自热库的熵“损失”的大小，即<math> |dS_{Res_{1}}|=\frac{\delta Q_{1}}{T_{Hot}} </math>小于了系统熵增加的大小<math>dS_{Sys_{1}}</math>(<math>\geq 0</math>)：
类似地，当温度为<math>T_{2}</math>的系统在瞬间发生的过程内将热量<math>-\delta Q_{2}</math> (<math>\delta Q_{2}\leq 0</math>)排入较冷的热库（温度<math>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：
类似地，当温度为<math>T_{2}</math>的系统在瞬间发生的过程内将热量<math>-\delta Q_{2}</math> (<math>\delta Q_{2}\leq 0</math>)排入较冷的热库（温度<math>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 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>
这里系统“吸收”的热量由<math>\delta Q_{2}</math>(<math>\leq 0</math>)给出，表示热量从系统传递到热库，且<math>dS_{Sys_{2}}\leq 0</math>。由热库获得的熵大小<math> dS_{Res_{2}}=\frac{|\delta Q_{2}|}{T_{cold}}</math>，大于系统熵损失的大小<math> |dS_{Sys_{2}}|</math>。
这里系统“吸收”的热量由<math>\delta Q_{2}</math>(<math>\leq 0</math>)给出，表示热量从系统传递到热库，且<math>dS_{Sys_{2}}\leq 0</math>。由热库获得的熵大小<math> dS_{Res_{2}}=\frac{|\delta Q_{2}|}{T_{cold}}</math>，大于系统熵损失的大小<math> |dS_{Sys_{2}}|</math>。
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>T_{surr}</math>，可得出：
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>T_{surr}</math>，可得出：
:<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>
尤其是：
尤其是：
:<math>\oint \frac{\delta Q}{T_{surr}}\leq 0, </math>
:<math>\oint \frac{\delta Q}{T_{surr}}\leq 0, </math>
得到了证明。
得到了证明。
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），
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），
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：
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：
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。
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 ==
== See also 其他参考资料 ==