An ideal gas and the increasing (total) entropy principle: what is allowed in the PV plane and what is not

Abstract

In a recent article in this journal, Brown and Singh present the results of an extensive in-class survey of student difficulties with the second law of thermodynamics. Here, we discuss in detail some issues identified by them in an attempt to resolve some of the problems. We do this by making clear the distinction between the ‘system entropy’, ‘reservoir entropy’, ‘total entropy’, and the ‘entropy of the universe’. We identify, without ambiguity, which quasistatic processes are ‘internally reversible’, which are ‘totally reversible’, and which are ‘irreversible’. We discuss the meaning of quasistatic processes represented by curves in the PV plane. We show that the process P=P(V) that takes an ideal gas from initial state (P_A V_A) to final state (P_B V_B) always takes the gas away from the adiabat PV^γ=P_A V_A^γ. We establish the increasing (total) entropy principle as the quantitative expression of the second law of thermodynamics. This establishes the fundamental role of the total entropy in macroscopic thermodynamics. Expressing the increasing (total) entropy principle in the form 〖(T〗_res-T_gas)d(PV^γ)>0 reveals which processes are allowed, which are not, and which points in the PV plane are accessible from a given initial state. Students can now observe the limitations on individual processes mandated by the second law. Such second law analysis is suitable for the introductory physics course.

Revisiting and comparing the Carnot Cycle and the Otto Cycle

“…The processes b C → c C and d C → a C are also reversible since the entropy of the gas remains unchanged as no heat is transferred from it to the surroundings. Therefore, the whole cycle is considered reversible, meaning the universe's entropy remains unchanged after the gas completes the cycle [10,11]. As a consequence of the Second Law of Thermodynamics, it is possible to conclude that a reversible heat engine is the most efficient one operating between two given thermal reservoirs.…”

Comment on ‘An ideal gas and the increasing (total) entropy principle: what is allowed in the PV plane and what is not’

“…Considering surroundings consisting of reservoirs, the heat dQ and the dissipative work dW D (which accounts for mechanical dissipative effects) are given by [2][3][4] The term / dW T D e of (5) is missing in [1], which is relevant in the subsequent analysis leading to the conclusion that the term…”

“…In a recent paper [1], Marcella used an ideal gas to deal with some issues regarding the second law of thermodynamics. Despite the merit of promoting a useful discussion on a topic often considered a difficult one, that paper conveys the incorrect statement that 'a quasistatic process is necessarily free of dissipative effects.'…”

Free expansion of an ideal gas by kinetic gas theory

“…Classical thermodynamics prescribes no change of the average internal energy in the adiabatic expansion above. However, classical thermodynamics is a theory of equilibrium states [4]. If the gas expands into a finite volume, rather than into an infinite void as we have assumed, the reverse flow of gas that grows gradually more important as time progresses will ensure both that (1) the fast 'hot' molecules in the emission return to the original vessel at a higher rate than the slow 'cold' ones, so that the temperature eventually equilibrates at one and the same starting value in both volumes, and that (2) there is no net center-of-mass translation for the escaping gas after it is reflected in the opposing wall.…”