# Second Law and Non-Equilibrium Entropy of Schottky Systems—Doubts and Verification–

## Abstract

**:**

## 1. Introduction

- Kelvin: There is no Thomson process (but friction processes exist),
- Clausius: There is no Clausius process (but heat conduction processes exist).
- Accepting additionally
- Carnot [5]: Reversible Carnot processes exist (not really, but as a mathematical closure of irreversible processes),

- What is the meaning of the parameter t in connection with the < and = signs in Clausius inequality?
- On what state space characterizing the system does the cyclic process operate?
- How can the inequality be extended to open systems?
- What are the relationships between Clausius inequality and entropies?

## 2. Schottky Systems

#### 2.1. Exchanges and Partitions

**a**of the system

#### 2.2. State Spaces and Processes

**Z**onto the equilibrium subspace

#### 2.3. The First Law

## 3. The Second Law

#### 3.1. Doubts: Non-Equilibrium Entropy

#### 3.2. Defining Inequalities

**Definition**

**1.**

**Ψ**through $\partial \mathcal{G}$ and a production $\mathbf{R}$ in $\mathcal{G}$.

**Axiom**

**1.**

**Axiom**

**2.**

**Axiom**

**3.**

**s**can be decomposed into molar enthalpies

**h**and chemical potentials $\mathit{\mu}$ [28]

**Axiom**

**4.**

**h**and $\mathit{\mu}$ are place holders in the dissipation inequality (25). Their definitions have to be compatible with adiabatic processes $(\stackrel{{}_{*}}{Q}=0)$ and closed systems $({\stackrel{*}{\mathit{n}}}^{e}=0)$. Taking into consideration that the entropy production $\mathsf{\Sigma}$ is independent of the exchange rates which are also independent of each other, the definitions of the place holders have to be introduced in such a way that the dissipation inequality (25) is satisfied for all these special cases. This is achieved by setting

**Axiom**

**5.**

**h**and $\mathit{\mu}$ of $\mathcal{G}$

## 4. Contact Quantities

#### 4.1. Contact Temperature

**Definition**

**2.**

#### 4.2. Non-Equilibrium Molar Enthalpies and Chemical Potentials

#### 4.3. Non-Equilibrium Molar Entropies

## 5. Verification: Non-Equilibrium Entropy

#### 5.1. A Non-Equilibrium State Space

**a**, the mole numbers

**n**and the internal energy U, these state variables appear also in the non-equilibrium state space. If the First Law (17) and (24)${}_{2}$ are inserted, (20) results in

#### 5.2. Embedding Theorem

#### 5.3. Adiabatical Uniqueness

**Definition**

**3.**

#### 5.4. The Integrability Conditions

## 6. Summary

- Introduce a large state space [18] which is composed of an equilibrium subspace and a non-equilibrium part
- The variables of the equilibrium subspace are determined by the Zeroth Law: work variables, mole numbers and internal energy
- Processes are trajectories on the state space
- Reversible processes are projections of non-equilibrium processes onto the equilibrium subspace. An accompanying process [19] with time as a path parameter is generated by projection of the corresponding non-equilibrium process onto the equilibrium subspace
- The time rate of the internal energy is introduced by the First Law
- The entropy is in equilibrium and in non-equilibrium a balanceable quantity
- The entropies of partial systems are additive
- Introduction of the Second Law for isolated systems and entropy productions [14]
- The defining inequalities for contact temperature, non-equilibrium molar enthalpies and chemical potentials resulting in the non-equilibrium molar entropy
- The embedding theorem enforcing compatibility of a non-equilibrium entropy with the equilibrium one
- Adiabatic uniqueness guaranteeing that the non-equilibrium entropy is a state function on the non-equilibrium state space.

## 7. Closure

## Funding

## Acknowledgments

## Conflicts of Interest

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Muschik, Wolfgang. 2018. "Second Law and Non-Equilibrium Entropy of Schottky Systems—Doubts and Verification–" *Entropy* 20, no. 10: 740.
https://doi.org/10.3390/e20100740