## Studies

The Table of Contents of planned Studies

### A history of heat and thermodynamics

Development of the concept of entropy

by Eri YAGI

Professor Emerita of Physics

(Faculty of Engineering, Toyo University)

Director of History of Science Institute of Physics

email: eri.yagi.blog@gmail.com

(written April 1996)

(revised July 1997)(modified email address, Mar. 2012)

## Introduction

I have been working on Rudolf Clausius’s mechanical theory of heat by focusing on his thermodynamics rather than his gas theory. The latter approach has been a popular means of understanding statistical concepts of entropy and a mean free path in the general history of physics. Although no book on R. Clausius has been published yet, a few articles have been written about him. Generally, Clausius was evaluated by his presentation of the second law of thermodynamics within the concept of entropy. However, I would like to evaluate him as the presentator of both the first and second laws. In addition, I would like to make a comparative study among W. Thomson, Rankine and Clausius from a non- westerner’s view-point. The following is the line of my research (This plan was first presented at the summer colloquium of Dibner Institute, MIT, July 14,1994)

**Figures:**

0-0) People around R. Clausius

0-1) Chronological table around the years(1822-1888)

[Historical background of the mechanical theory of heat is shown]

## I. Clausius’s Theory of Light and Heat: his skillful treatment with differentials.

**I.1 Clausius’s early papers on light propagation (1847-1849)** : Here Clausius’s cosmological interest in physics and mathematics is mentioned. He showed(1849) his skillful treatment of differentials of three variables along the x and y axes, where the first differential was canceled out by taking a difference between flow quantities, in and out, and only the second differential remained.

**I.2 Clausius’s first paper on the mechanical theory of heat(1850)** : Here Clausius calculated the heat-flow difference between two quantities, in and out, in an infinitesimal Carnot cycle. By the use of the above mathematical method, only the second differential remained, from which Clausius’s first law of thermodynamics emerged.

**I.3 J. Fourier’s influence** : The above mathematical method came from Fourier’s analytical theory of heat(1822). This influence was also confirmed through my manuscript search at the library of the Deutshes Museum, Munich. There can be found Clausius’s own additional drawing in his manuscript (1848), “Aus Wärmetheorie von Fourier”

**Figures:**

I-1) Similar calculation method by Clausius’s light-flow and Fourier’s heat-flow along the x-axis

I-2) Clausius’s heat-flow calculation (1850, Note in 1864)

I-3) Fourier on Clausius’s manuscript.

**References:**

E. Yagi, “Fourier on Clausius I (in Japanese),” Abstract, Annual meeting of the Physical Society of Japan (March 28, 1987), “Fourier on Clausius II (in Japanese),” Abstract, Annual meeting of the Physical Society of Japan (March 31,1996) & “Fourier on Clausius III (in Japanese)”, Abstract, ibid (March 29, 1997)

## II. Clausius’s first law of Thermodynamics: his adoption of Clapeyron’s presentations.

**II.1 Clausius adopted Clapeyron’s graphical and analytical expressions of 1834** : Clausius’s diagrams of the Carnot cycle together with the infinitesimal cycle were very similar to Clapeyron’s. By the use of these diagrams, Clausius obtained the work produced during the whole cycle process as Clapeyron did on the basis of the caloric theory of heat.

**II.2 The first analytical (differential) expression based on the mechanical theory of heat was presented by Clausius** : He accepted Clapeyron’s expression as the first order approximation in a mathematical sense for the infinitesimal Carnot cycle. Ratio equations between work and heat played the essential role in these calculations. See the following three figures, comparatively presented: (Figs. from E. Yagi, “Clausius’s mathematical method,” HSPS, 1984, 15:1, 177-195)

**Figures:**

II-1) Clapeyron’s diagrams vs. those of Clausius

II-2) Illustration of ratio equations with the infinitesimal Carnot cycle

## III. Clausius’s approach to the 1st and 2nd laws of Thermodynamics as a related set

**III.1 Clausius’s series of papers on thermodynamics (between 1850 and 1865)** : The first and second laws of thermodynamics were handled by Clausius as a related set of analytical equations. This was found out by creating a database, including almost all equations from Clausius’s papers on the mechanical theory of heat. See Eri Yagi, On the Formation of Entropy, (a database), 1989, the Japanese Ministry of Education, Project No. 61580086, Japan.

**III.2 Clausius’s approach to reversible and irreversible processes** : For the reversible process, dU (the expression for the energy change of a gas) was correspondingly presented to dS (that for entropy change) in 1865. Before that time, dU was not put on the left hand side of the equation for the 1st law, while dQ was always put on the left hand side.

For the irreversible process, expressions for these two laws were also treated as a related set. The well-known nature of heat, namely, heat-flows from higher to lower temperatures in the case of heat conduction, played the most important role in Clausius’s approach to the irreversible process.

**Figures:**

III-1) Clausius’s first and second laws in his papers in 1854

III-2) Clausius’s first and second laws in his papers in 1862

III-3) Clausius’s first and second laws in his papers in 1865

III-4) Clausius vs. Sommerfelt (traditional German expression)

**References:**

E. Yagi, “R. Clausius and entropy (in Japanese),” in Yagi eds., Netsugaku Daini-Hosoku no Tenkai (development of the second law of thermodynamics) (Tokyo, Asakura, 1990) 69-95.

## IV. Clausius’s analytical approach with his confusing usage of the term “complete differential”

The heat change was regarded by Clausius as a non-complete differential as we would call it today. Therefore it was changed to a complete differential dS by the division of absolute temperature T, called an integrating denominator. However, Clausius regarded a heat change as the complete differential of the first order (die vollständige Differentialgleichung erste Ordung). This had been a source of confusion. I have been clarifying the confusion. Despite his confusing usage for the complete differential, mentioned in the chapter of the mathematical introduction in his collected papers on the mathematical theory of heat (1864), Clausius correctly treated dQ/T in his paper of 1854 and dH/T in 1862 (where H meant the quantity of heat presented in a gas) as the complete differential.

**Figures:**

IV-1) Clausius’s confusing equation (1850, Note 1864)

## V. Clausius’s gas theory vs. thermodynamics

In his gas theory, Clausius brought over some important expressions and concepts from his thermodynamics. These were the first law of thermodynamics (in the work-unit by the use of 1/A), and two kinds of specific heat (per unit volume and per unit weight) under a constant volume or under a constant pressure.

By the use of the first law and specific heats, Clausius indicated that the ratio between the total kinetic energy due to the translation motion of a gas particle and the total heat of the gas is constant, and that the former energy is less than the total heat in the work-unit. Clausius suggested the existence of some other movement in addition to the translation movement of the gas particle.

**Figures:**

V-1) Clausius’s first law in his paper on gas theory in 1857.

V-2) Clausius’s ratio between the total kinetic energy due to the translation motion of a gas particle K, and the total heat of the gas H.

**References:**

E. Yagi, “R. Clausius’s gas theory (in Japanese),” Abstract, Annual Meeting of the Physical Society of Japan (March 29, 1994)

## VI. The first and second laws by William Thomson (Lord Kelvin) and by W. J. M. Rankine

As a non-westerner, I am planning to find out some useful method to compare contributions made by Clausius, W. Thomson, and Rankine.

### Remarks

Having studied Clausius’s series of papers on the mechanical theory of heat from the views of his interest in thermodynamics, I would like to emphasize his role as the presenter of the first and second laws of thermodynamics. He was the first person to present the first law in an analytical differential form on the basis of the mechanical theory of heat. The entropy dS for a reversible process was correspondingly proposed to be a complete differential just like the energy of a gas dU. The first law played an essential role in Clausius’s gas theory.

## A list of E.Yagi’s publications on Thermodynamics

- E. Yagi, “Analytical approach to Clausius’s first memoir on the mechanical theory of heat, “Historia Scientiarum”, Tokyo, 20,77-94, 1981
- E. Yagi, “Clausius’s mathematical method,” HSPS, Berkeley, 15:1, 177-195, 1984
- E. Yagi, “Clausius’s mechanical theory of heat, “17th International Congress of the History of Science, Berkeley, Act 1, Pj7A, 1985
- E. Yagi, “Fourier on Clausius I (in Japanese),” Abstract, Annual meeting of the Physical Society of Japan (March 28, 1987)
- E. Yagi, On the Formation of Entropy, the Ministry of Education, Project No. 61580086, Japan, 1989
- E. Yagi, “Clausius’s entropy and irreversibility,” 18th International Congress of the History of Science, Hamburg-München, Abstract, R3-8, 1989
- E. Yagi, “Clausius and entropy (in Japanese),” in Yagi eds., Netsugaku Daini-Hosoku no Tenkai (development of the second law of thermodynamics) (Tokyo, Asakura, 1990) 69-95, where a list of papers and books by Clausius (compiled by Yagi) is included.
- E. Yagi, “Clausius’s gas theory (in Japanese),” Abstract, Annual Meeting of the Physical Society of Japan (March 29, 1994)
- E. Yagi, part 9.5, “Thermodynamics”, in I. Grattan-Guinness ed., Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, 2,1171-1182, Routledge, U.K, 1994
- E. Yagi,”Clausius’s mathematical background (in Japanese),” Abstract, Annual Meeting of the Physical Society of Japan (March 29, 1995)
- E. Yagi, “Fourier on Clausius II (in Japanese),” Abstract, Annual meeting of the Physical Society of Japan (March 31,1996)
- E. Yagi, “Fourier on Clausius III (in Japanese),” Abstract, Annual meeting of the Physical Society of Japan (March 29,1997)

## Recent Comments