卡诺定理的普适性
The universality of the Carnot theorem
摘要：
在许多热力学教科书中，通过使用气体、顺磁体和其他简单热力学系统的不同状态方程来说明卡诺定理是很常见的。众所周知，卡诺效率的普遍性很容易在温熵图中得到证明，这意味着卡诺效率与工作物质无关。在目前的工作中，我们注意到卡诺定理的普遍性超出了传统的状态方程，它是由气体状态方程，不表现为理想气体在稀释极限时，体积是无限的。这些非传统的状态方程中的一些具有某些热力学“反常”，但并不妨碍它们服从卡诺定理。我们讨论如何产生这种非常普遍的行为从麦克斯韦关系，这是连接到一个几何性质表示通过保持面积变换。提出了一种利用保面积关系计算热力学系统麦克斯韦关系的规则。以这种方式，可以通过给出包含在相应的吉布斯方程中的所有热力学变量对之间的可能的雅可比恒等式的数目来计算可能的保持区域映射的数目。这项工作是为本科生和热力学及相关领域的专家。
英文摘要：
It is common in many thermodynamic textbooks to illustrate the Carnot theorem through the usage of diverse state equations for gases, paramagnets, and other simple thermodynamic systems. As it is well-known, the universality of the Carnot efficiency is easily demonstrated in a temperature-entropy diagram, which means that the Carnot efficiecy is independent of the working substance. In this present work we remark that the universality of the Carnot theorem goes beyond the conventional state equations, and it is fulfilled by gas state equations that do not behave as ideal gas in the dilution limit when the volume is infinite. Some of these unconventional state equations have certain thermodynamic "anomalies" that nonetheless do not forbid them from obeying the Carnot theorem. We discuss how this very general behaviour arises from the Maxwell relations, which are connected with a geometrical property expressed through preserving area transformations. A rule is proposed to calculate the Maxwell relations associated with a thermodynamic system by using the preserving area relationships. In this way it is possible to calculate the number of possible preserving area mappings by giving the number of possible Jacobian identities between all pairs of thermodynamic variables included in the corresponding Gibbs equation. This work is intended for undergraduate and specialists in thermodynamics and related areas.
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