# A Revision of Clausius Work on the Second Law. 1. On the Lack of Inner Consistency of Clausius Analysis Leading to the Law of Increasing Entropy

## Abstract

**:**

## Introduction

Hence, the second main principle of the mechanical Theory of Heat, which in this form may perhaps be called the principle of the Equivalence of Transformations, can be expressed in the following terms: if we call two transformations which may cancel each other without requiring any other permanent change to take place, Equivalent Transformations, then the generation out of work of the quantity of heat Q of temperature τ has the equivalence value Q/τ and the transference of the quantity of heat Q from temperature T1 to temperature T2 has the equivalence valuein which τ is a function of temperature independent of the kind of process by which the transformation is accomplished.

## The Nature of the Inconsistency

_{h}would have associated an entropy change equal to −Q/T

_{h}; while the entropy change associated to the transformation of an amount of heat Q' from temperature T

_{h}to temperature T

_{c}would be equal to Q'(T

_{h}− T

_{c})/ T

_{h}T

_{c}complying these values with the condition that..."in every reversible process of the kind given above the two transformations that take place must be equal in magnitude but of opposite sign, so that their algebraical sum is zero" [9]. We will demonstrate however, through a simple analysis on a reversible cyclical process, that none of the two transformations occurring in the said reversible cyclical process comply with the values required by Clausius principle of the Equivalence of Transformations. This analysis is shown next, and in going through it, it has to be understood that a transformation encompass all the bodies in any way involved in its occurrence. For example, the transformation of heat from high to low temperature, as it occurs in a reversible cyclical process, could not be possible if either the hot reservoir, the cold reservoir or the proper sequence of changes of the variable body were missing. It is the concerted concatenation of changes of each of these bodies what makes possible the occurrence of the said transformation. The value or entropy change of a particular transformation will thus be determined by the contributions associated to the changes each and every body in it participating has to sustain in order to accomplish it. Now, if by the universe of a process we understand the collection of bodies in any way participating in the said process it then follows that a given transformation is a universe in itself, and on this perspective the universe of a simple reversible cyclical process can be thought of as formed by the conjunction of the universes of the two transformations constituting it or vice versa.

## Analysis

_{h})→ w]

_{rev}and [Q'(T

_{h})→ Q'(T

_{h})]

_{rev}will represent the two transformations taking part in a simple reversible cyclical process, such as the one depicted in Figure 1 and commonly known as Carnot's ideal gas reversible engine. Thus, [Q(T

_{h})→ w]

_{rev}indicates the fact that an amount of heat Q of temperature T

_{h}has been reversibly converted into an equivalent amount of work (w), while [Q'(T

_{h})→ Q'(T

_{c})]

_{rev}indicates that an amount of heat Q' has been reversibly transformed from temperature T

_{h}to temperature T

_{c}. The entropy changes associated to these transformations will be represented as ΔS[Q(T

_{h})→ w]

_{rev}and ΔS[Q'(T

_{h})→ Q'(T

_{c})]

_{rev}, respectively [10].

_{h}from A to B, be equal to the amount by it released to the cold reservoir during its isothermal compression at T

_{c}from C to D. The first part of the said analysis will concern itself with the entropy change calculations for the processes taking place as the variable body evolves from state A to E.

#### Process 1. Reversible and isothermal (T_{h}) expansion of the working substance from state A to state B

_{h}.

_{h}.

#### Process 2. Reversible and adiabatic expansion of the working substance from state B to state C

#### Process 3. Reversible and isothermal (T_{c}) compression of the working substance from state C to D

_{c}

_{c}

#### Process 4. Adiabatic and reversible compression of the working substance from state D to state E

_{C}the heat Q’ previously absorbed from the hot reservoir at T

_{h,}it thus follows that the said sequence has brought about transformation [Q'(T

_{h})→ Q'(T

_{C})]

_{rev}with an entropy change equal to the sum of the entropy changes of processes 1 through 4 which are the ones defining the referred sequence, i.e.

_{h})→ Q'(T

_{c})]

_{rev}= 0 (1)

**Figure 1.**Splitting of the reversible operation of Carnot's engine into its two constitutive transformations: [Q'(T

_{h})→Q'(T

_{C})]

_{rev}and [Q(T

_{h})→w]rev.

_{h})→ w]

_{rev}. Through it, an amount of heat Q of temperature T

_{h}is quantitatively transformed into work. The entropy change associated to this transformation will be given in the following calculation:

#### Process 5. Isothermal and reversible expansion of the working substance from state E to state A

_{h}.

_{h}.

_{h})→ w]

_{rev}= 0 (2)

_{h})→ Q'(T

_{c})]

_{rev}= Q'(T

_{h}− T

_{c})/T

_{h}T

_{c}

_{h})→ w]

_{rev}= −Q/T

_{h}

## Discussion

_{h})→w]

_{rev}and [Q'(T

_{h})→Q'(T

_{c})]

_{rev}are the non zero quantities respectively represented as −Q/T

_{h}and Q'(T

_{h}− T

_{c})/T

_{h}T

_{c}, the latter affirms that the entropy change for the universe of any reversible process is equal to zero. As can be clearly seen from this contrast, the problem here arising is that the veracity of any of these statements imply the falsity of the other. These statements are thus not equivalent. They are contradictory.

_{h}to T

_{c}is equal to zero, that of the irreversible transfer is equal to Q'(T

_{h}−T

_{c})/T

_{h}T

_{c}. Looking back now at the principle of the equivalence of transformations one can only qualify as peculiar the fact that the entropy change in it associated to the reversible transfer of heat taking place in the reversible cyclical process to which such a principle refers, instead of being zero, as should correspond to a reversible process, be Q'(T

_{h}−T

_{c})/T

_{h}T

_{c}which is the one associated to an irreversible heat transfer.

## References and Notes

- Clausius, R. The Mechanical Theory of Heat; MacMillan: London, 1879; p. 78. [Google Scholar]
- Clausius, R. The Mechanical Theory of Heat; MacMillan: London, 1879; p. 92. [Google Scholar]
- Clausius, R. The Mechanical Theory of Heat; MacMillan: London, 1879; p. 91. [Google Scholar]
- Clausius, R. The Mechanical Theory of Heat; MacMillan: London, 1879; p. 100. [Google Scholar]
- Clausius, R. The Mechanical Theory of Heat; MacMillan: London, 1879; p. 102. [Google Scholar]
- Clausius, R. The Mechanical Theory of Heat; MacMillan: London, 1879; p. 108. [Google Scholar]
- Clausius, R. The Mechanical Theory of Heat; MacMillan: London, 1879; p. 213. [Google Scholar]
- Clausius, R. The Mechanical Theory of Heat; MacMillan: London, 1879; p. 109. [Google Scholar]
- Clausius, R. The Mechanical Theory of Heat; MacMillan: London, 1879; p. 98. [Google Scholar]
- The square brackets, it should be recognized, enclose the representation of a physical process. When preceded by the ∆S function (or any other thermodynamic function, for that matter) they become the argument of such a function and the whole expression “∆S[ ]” is ultimately given by a number called the entropy change of such a process. An arbitrary concatenation of those “∆S[ ]” numbers does not necessarily imply the existence i.e. the physical possibility of the process defined by the concatenation of their arguments.

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**MDPI and ACS Style**

Iñiguez, J.C.
A Revision of Clausius Work on the Second Law. 1. On the Lack of Inner Consistency of Clausius Analysis Leading to the Law of Increasing Entropy. *Entropy* **1999**, *1*, 111-117.
https://doi.org/10.3390/e1040111

**AMA Style**

Iñiguez JC.
A Revision of Clausius Work on the Second Law. 1. On the Lack of Inner Consistency of Clausius Analysis Leading to the Law of Increasing Entropy. *Entropy*. 1999; 1(4):111-117.
https://doi.org/10.3390/e1040111

**Chicago/Turabian Style**

Iñiguez, José C.
1999. "A Revision of Clausius Work on the Second Law. 1. On the Lack of Inner Consistency of Clausius Analysis Leading to the Law of Increasing Entropy" *Entropy* 1, no. 4: 111-117.
https://doi.org/10.3390/e1040111