Total entropy variation for an object in contact with heat reservoirs: the path to reversibility

Abstract The second law of thermodynamics is one of the least understood fundamental physical laws, even among science and engineering students and professionals, possibly due to its subtleness and several seemingly different statements. Here we investigate the entropy variation of an isolated system composed of an object of heat capacity C(T) and one or more heat reservoirs, as the absolute temperature of the object varies due to heat exchange with subsequent reservoirs. We obtain a general expression for the total entropy variation, ΔS(0), in terms of C(T) and the number of reservoirs N. We numerically show that ΔS(0) decreases as N increases, and considering that as N → ∞ the temperature difference between subsequent reservoirs becomes infinitesimal, we analytically show that lim N → ∞ Δ S ( 0 ) = 0, in accordance with the second law of thermodynamics for a reversible quasi-static process. We conclude by proposing an undergraduate exam problem based on the demonstration that the total entropy variation vanishes in the quasi-static limit.

摘要 热力学第二定律（second law of thermodynamics）是最不易被理解的基础物理定律之一，即便在理工科学生与专业从业者中亦是如此，这或许源于其概念的微妙性以及若干看似迥异的表述形式。本文针对由热容（heat capacity）为C(T)的物体与一个或多个热源（heat reservoir）组成的孤立系统（isolated system），探究该物体因与后续热源进行热交换而导致绝对温度（absolute temperature）变化时的总熵变（entropy variation）情况。我们推导得到了总熵变ΔS(0)关于C(T)与热源数量N的通用表达式。数值计算结果表明，ΔS(0)随N的增大而减小；进一步考虑当N→∞时，相邻热源间的温度差将趋近于无穷小，我们通过解析方法证明了当N→∞时ΔS(0)=0，这符合可逆准静态过程（quasi-static process）下的热力学第二定律。最后，我们基于“准静态极限下总熵变趋于零”这一论证结论，提出了一道本科物理考题。