Formulation of Entropy through Work by Carnot Machine and Direct Derivation of Law of Entropy Non-Decrease from Kelvin-Planck Principle

TL;DR

The paper addresses how to derive the entropy non-decrease law directly from the Kelvin-Planck principle without invoking the Clausius inequality. It introduces an entropy formulation based on the maximum reversible work a Carnot machine can extract between a system and a single reservoir at fixed temperature, yielding a path-independent entropy change $S_1=S_0+rac{1}{ }\/\int_0^1 \frac{\delta Q}{T}$ and a corresponding maximum-work expression $W_{0\to1}^{\rm rev}=U_0-U_1-T_o(S_0-S_1)$. The main contributions are the derivation of the non-decrease law for both simple and compound systems, the demonstration that this entropy is equivalent to Clausius and Gyftopoulos–Beretta entropies, and the interpretation of entropy as an extra thermodynamic cost to create nonuniformity. The work provides a direct Kelvin-Planck–based foundation for entropy and yields insights potentially applicable to nonequilibrium entropy notions, while maintaining thermodynamic consistency with established entropy concepts.

Abstract

We derive the law of entropy non-decrease directly from the Kelvin-Planck principle for simple and compound systems without using the Clausius inequality. A key of the derivation is a new formulation of entropy in terms of work by a Carnot machine operating between a system and a single heat reservoir at fixed temperature, which is equivalent to Clausius entropy based on heat and Gyftopoulos-Beretta entropy based on work. We also show that we may characterize entropy as an extra thermodynamic cost that needs to be paid to create nonuniformity in the system.

Figures (3)

• Figure 1: A reversible path $\gamma$ from $A_0$ to $A_1$ and a reversible path $\gamma'$ from $A_1$ to $A_0$ on the temperature-volume ($T$--$V$) plane. They consist of the reversible adiabatic processes (arrows along the adiabatic curves) and the reversible temperature-changing processes with a Carnot machine operating between the system with its volume being kept fixed and a heat reservoir at temperature $T_o$ (vertical arrows).
• Figure 2: An arbitrary adiabatic process from one equilibrium state $A_1$ of a simple system to another equilibrium state $A_2$ (dashed curve), where $A_1$ and $A_2$ are created reversibly from a reference equilibrium state $A_0$ (solid curves). The Kelvin-Planck principle Eq. (\ref{['eq.Kelvin']}) is applied to the cyclic process $A_0 \stackrel{\rm rev}{\longrightarrow} A_1 \stackrel{\rm ad}{\longrightarrow} A_2 \stackrel{\rm rev}{\longrightarrow} A_0$.
• Figure 3: A compound system at state $A_1$ and a simple system at state $A_{1'}$ of the same internal energy and volume are created from a simple system at state $A_0$. An adiabatic process $A_1 \stackrel{\rm ad}{\longrightarrow} A_{1'}$, which occurs spontaneously after the removal of the insulated walls separating the subsystems of the compound system, is realized only when Eq. (\ref{['eq.W_he']}) is satisfied.