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 ${\lim }_{N\to \infty }\Delta S^{(0)}=0$, 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.

Keywords:
second law of thermodynamics; heat transfer; entropy variation