**Clausius II principle (***CII***)**

*There is a function of the state of the thermodynamic system called entropy, which the dependence of the value on time allows to determine the system's following of the thermodynamic equilibrium. Namely, in thermally insulated systems, the only possible processes are those in which the total entropy of the system (two heat reservoirs) does not decrease:*

$$
\Delta S = \Delta S\_1 + \Delta S\_2 \ge 0. \tag{7}
$$

*For reversible processes, the increase in total entropy is zero, and for irreversible processes it is greater than zero. The change in entropy of a reservoir is here understood as the ratio of the heat introduced into it to its temperature.*

Thus, when using standard notations, Clausius II principle takes the following form for the processes considered here:

$$
\Delta S = \frac{-Q\_1}{T\_1} + \frac{Q\_2}{T\_2} \ge 0. \tag{8}
$$

It is assumed that a specific device operating cyclically between heat reservoirs does not change its entropy by definition of cyclicity. Additionally, similarly for the *C*0 principle, the absolute temperature scale is assumed here.

The Clausius II principle is the most elegant formulation of the II law of thermodynamics. Some textbooks only introduce an entropy formulation of this principle (e.g., [34]). Only the definition of entropy presents some difficulty and challenge here. Clausius's definition of entropy (integral of heat divided by temperature) is not always general enough in thermodynamics. However, we assume that it is sufficient for the process diagrams considered here. Another, usually overlooked, subtlety of the II law of thermodynamics (Clausius II principle and others) is that it actually speaks of states beyond the thermodynamic equilibrium (*T*<sup>1</sup> = *T*2) to which the system is approaching but which it does not necessarily achieve (e.g., on diagrams with unlimited heat reservoirs—thermostats).

The year 1865 of the publication of work [35] is regarded to be the year when principle *CII* was created, even though in the years 1854 and 1856 in Clausisus works [19,20] there were expressions of the type *N* = ∑ *Q*/*T* or *N* = *dQ*/*T* with the correct analysis of signs (*N* > 0 for irreversible processes, *N* < 0 for impossible and *N* = 0 for reversible). The use of inequalities in physics would be something completely new, so in the 1850s in Clausius works they do not appear explicitly. Inequalities in the mathematical formula can be found in Clausius, for example, in the textbook [22] from 1867. In this textbook, Clausius also changes the notation of *N* to *S* by calling this quantity a transformation value (process measure). Then, translating *transformation* into Greek (*τρoπη*), Clausius introduces the term *entropy*. Of course, Clausius distinguished the entropy change in a process from its absolute value. The concept of entropy entered the canon of physics, and the principle of its growth (more precisely, its not decreasing) constitutes, for example, the term *entropy production* [36]. This form of the principle is valid in elementary systems of statistical mechanics [37].

There are also other attempts to formulate the II law of thermodynamics. The formulation by Caratheodory is sometimes given: *"In the surroundings of each thermodynamic state, there are states that cannot be achieved by adiabatic processes"*. However, such a formulation is expressed in a completely different language than that considered in this work. Moreover, this statement sounds as tautological as the statement: *there are places in the mountains that are inaccessible to hiking trails*. At the same time, Caratheodory's statement is given very often without strict proof of equivalence with other formulations (see [29,31]). From a contemporary critique of Caratheodory's formulation in preprint [38] we also learn that this formulation was greatly criticized by Planck.

The second law of thermodynamics is also formulated in the language of statistical physics. The outstanding physicist Ludwig Boltzmann was the first to undertake this task. Unfortunately, this article is limited to phenomenological thermodynamics only.

In the context of thermodynamic considerations, the abstract concept of *perpetuum mobile* often appears. It is, therefore, worthwhile to clarify and differentiate this concept as follows.
