# A History of Thermodynamics: The Missing Manual

## Abstract

**:**

## 1. Introduction

**Sources.**The bulk of this article discusses specific fundamental primary sources, but most of the background “pre-history” was obtained from a few important secondary sources. Mendoza’s 1960 A Sketch for a History of Early Thermodynamics [5], written by a co-inventor of the helium dilution refrigerator [6], explains for contemporary scientists and engineers how the seemingly naive concept of a conserved heat fluid, caloric, could have been accepted for nearly seventy-five years. It also explains how a (now known to be non-physical) heat function, written as $Q(p,V)$, could in Laplace’s hands yield a recognizably modern, if incomplete, theoretical structure. $Q(p,V)$ is not a state function, as assumed by Laplace, since its value depends on its path in $(p,v)$ space. Mendoza’s sketch deliberately restricts its scope, with little on thermometry and heat, and little following Carnot’s classic analysis of 1824.

**Analysis.**Besides simply giving the basics of the early history, from Carnot onward I discuss derivations that originally appeared in a number of fundamental papers. In discussing all articles, when available I employ in the equation itself the original numbering; however, in addition I give automatic Latex numbering at the margins. Unfortunately, many of the original articles have no numbered equations at all. In these cases I try to cite equations in such papers by approximate location within a given section (when section numbers exist), or a reprint, as for Carnot [8,12,13] and for Clapeyron [12,14]. Müller [15], Weinberger [16], and Truesdell [17] had helpful bibliographies of original papers. In such cases I use brackets following the equation itself and the Latex numbering; thus the un-numbered 3rd equation in the 12th numbered paragraph (or numbered section) would have an appended $\left[12.3\right]$. I also use brackets to convert partial equations into full equations, so that something like $m{c}^{2}$ would become $[E=]m{c}^{2}$.

**On notation:**different works use volume V or v, and temperature T or t (or sometimes even $\theta $ or $\tau $). The present work tries to employ consistent usage. However, to avoid being excessively pedantic, I expect the reader to understand the appropriate usage in a number of equations. Typically, specific heats are per unit mass.

**Motivation:**The author’s motivation for the present study began on witnessing, in 1972, following a conference presentation, a disagreement between two distinguished low temperature physicists (John Wheatley—later to receive the 1975 London Prize in Low Temperature Physics—and Robert Richardson—later to receive the 1996 Nobel Prize in Physics). Each employed both noise and magnetic susceptibility thermometry in the millikelvin range, but they disagreed on which was more accurate. In 2018 I realized that, by studying the deviations from the entropy uniqueness condition $\Delta S=\oint dQ/T=0$ for all paths in p-v space, one can determine which of two or more temperature scales is “better” [18]. This resolution led me to study how Clausius’s entropy became established in the lore of physics, and the meaningful study of Clausius led to the study of his predecessors, and the present work.

## 2. An Overview

- (A)
- the meaning of temperature;
- (B)
- (C)
- the history of thermodynamics proper, treated with two distinct assumptions:
- (1)
- (2)

#### 2.1. Temperature—Material-Independent, Unique, Absolute, Agreement with Ideal Gas

“Is there any principle on which an absolute thermometric scale can be founded? It appears to me that Carnot’s theory of the motive power of heat enables us to give an affirmative answer.”

#### 2.2. Outline of Thermodynamics Prehistory

#### 2.3. Outline of Thermodynamics Based on Conservation of Heat

#### 2.4. Outline of Thermodynamics Based on Conservation of Energy

_{g}, for an arbitrary number of coupled Carnot cycles; Thomson did not take the limit to obtain the integral form.

#### 2.5. Additional Topics

## 3. On Definitions

#### 3.1. Force

#### 3.2. Energy and Work

#### 3.3. Temperature

#### 3.4. Heat

## 4. Friction, Fire, Phlogiston, Caloric

- (1)
- The non-quantitative idea that work produces heat; non-quantitative because neither work nor heat had any precise meaning. Fire was not in the picture. This is exemplified by early observations of Francis Bacon (1617), and later by Newton, Leibnitz and others [7].
- (2)
- The conjoining of fire and heat, in the late 1600s and early 1700s, with the idea that a substance called phlogiston, having mass (weight), is released on burning. There were also multiple, less chemically-based fire theories, with massive or massless particles of fire entering an object as it heats.
- (3)
- The separation of fire and heat, in the caloric idea of a massless conserved heat fluid, and the placement of fire within the domain of chemistry and chemical reactions.

#### 4.1. Gases

#### 4.2. Heat as Fluid, Phlogiston and ‘Fire’

#### 4.3. Heat as Fluid—Caloric

#### 4.4. Heat as Motion—Vis Viva

## 5. On the Steam Engine

## 6. Carnot

#### 6.1. Analyzing the Efficiency of Heat Engines

#### 6.2. Difficulties with Carnot’s Analysis

## 7. Clapeyron

#### 7.1. Ideal Gas and Heat Function Q

#### 7.2. Ideal Gas and Carnot Efficiency

#### 7.3. Ideal Gas and dependence of Q on v

#### 7.4. Ideal Gas and Specific Heats

#### 7.5. Vapor-Liquid (Steam)

#### 7.6. General Case

## 8. Thomson 1848 and 1849

#### 8.1. Thomson 1848

#### 8.2. Thomson 1849

#### 8.2.1. The Thomson Brothers and the Ice-Water Transition

#### 8.2.2. Thomson’s 1849 Paper Itself

#### 8.2.3. Thomson’s Four-Part Appendix

## 9. On Energy Conservation

**Mayer.**Mayer, a physician, in 1842 developed a theoretical argument for the mechanical equivalent of heat, based on the difference in specific heats at constant volume and at constant pressure. He attributed this difference to the work done at constant pressure as the volume expanded [58]. Perhaps without realizing it, he assumed, implicitly but correctly, that the internal energy of a real gas is basically independent of the gas interactions, and therefore is independent of the gas density. Without showing details of his calculation, he obtained a reasonable value for the mechanical equivalent of heat. Mayer performed no experiments himself, although he did employ specific heat data from Gay-Lussac [59]. Belatedly, but during his lifetime, Mayer’s work received the recognition it deserved [7]. Hutchinson discusses Mayer’s work, how it was confirmed by Joule’s 1844 work, and how Joule and Thomson’s further study of the effect of density on gas properties (the Joule-Thomson effect, 1854) helped establish the Kelvin scale [11].

**Joule.**In late 1840, Joule published his observation of Joule heating, that wires heat up at a rate proportional to their electrical resistance and to the current squared [61,62]. Around 1842 Joule began a series of independent experiments that gave accurate values for the mechanical equivalent of heat. He first studied what is known as Joule heating due to electric current driven by chemical energy from a battery, and next studied what we now know as “Joule heating” due to electric current driven by electromagnetic induction from mechanical energy. In 1844, Joule wrote of some of his experiments on air, that “the heat evolved by compressing the air was found to be the equivalent of the mechanical power employed, and, vice versa, the heat absorbed in rarefaction was found to be the equivalent of the mechanical power developed.” As Hutchison notes, although in this paper Joule did not study the conversion of heat to work (all other things held constant), he did show that (all other things not held constant) work converts to heat and that (some) heat converts to work, in equivalent amounts [11]. In this same passage, Joule quotes Mayer as having in 1842 proposed such an equivalence, “without, however, attempting an experimental demonstration of its accuracy.” One cannot fail to recognize the methodical, thorough, and careful approach taken by Joule [63].

**Colding.**Colding, a Danish civil engineer, is largely ignored in textbook discussions of the history of energy conservation. Working at Oersted’s institute, he published in Scandinavian journals, which were largely unread out of the Scandinavian countries. In 1863, he wrote, in English, a summary of his work [65]. There he noted that in 1843 he gave many examples of heating due to compression of both liquids and solids, and developed an apparatus to study heating due to friction, in part to establish that the heating was proportional to the ‘mechanical power lost’. He published his results in 1847, with a value for the mechanical equivalent not far from that of Mayer.

**Holtzmann.**In 1845, Holtzmann (“a schoolteacher and competent mathematician” [66]), aware of work on the mechanical equivalence of heat but, apparently, not of energy conservation, considered Clapeyron’s extension of Carnot. Using the same method as Mayer he obtained a value for the mechanical equivalence of heat that substantially agreed with that of Joule. However, Holtzmann did not take the step of arguing that something is conserved, nor did he redefine the temperature scale via ${T}_{g}=a+t$, where Holtzmann took the temperature offset $a=267$ [7].

**Helmholtz.**In 1847, Helmholtz wrote a paper mathematically enunciating energy conservation (he called it ‘force conservation’) [30]. First, he considered mechanical energy, including both what we would call kinetic energy and potential energy, showing that for central forces their sum is conserved. He then discussed heat, arguing that Joule’s many experiments show that heat and work are both quantities of the same type, what we would now call energy, and thus that total energy (for Helmholtz, ‘total force’) is conserved [30].

## 10. Clausius 1850

#### 10.1. Ideal Gas and Specific Heats

#### 10.2. Ideal Gas and Adiabatic Condition

#### 10.3. Thermodynamic Efficiency

#### 10.4. Steam

- (1)
- $s,\sigma $ as the volumes of a unit mass of vapor (steam) and of fluid (water);
- (2)
- $m,\mu $ as the masses of vapor and fluid;
- (3)
- $h,c$ as the specific heats per mass of vapor and fluid.
- (4)
- r as the latent heat per vapor mass (rather than Clapeyron’s latent heat per vapor volume k).

#### 10.5. Steam and Clausius-Clapeyron Equation

## 11. Thomson 1851

#### 11.1. Part I. Fundamental Principles

#### 11.2. Part II. Finite Temperature Differences

#### 11.3. Part III. Applications to Physical Properties of all Substances

#### 11.4. Part IV. Measurement of, for a Compressed Fluid, the Mechanical Work and the Heat Produced

#### 11.5. Part VI. Thermo-Electric Currents

## 12. Clausius 1854

#### 12.1. Clausius’s Two Types of “Equivalences of Transformation”

- (1)
- One type, the equivalence value function $f(t)$, applies when work W is completely converted into unit heat $Q=1$ at a reservoir of temperature t, with $f(t)>0$. For more general Q, $f(t)$ appears in the form $Qf(t)$. Conversely, if heat Q at t completely converts to work, then its equivalence value is $-Qf(t)$:$$\mathrm{equivalence}\phantom{\rule{4.pt}{0ex}}\mathrm{value}-Qf(t)\phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}Q\phantom{\rule{4.pt}{0ex}}\mathrm{at}\phantom{\rule{4.pt}{0ex}}t\phantom{\rule{4.pt}{0ex}}\mathrm{converts}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{4.pt}{0ex}}W$$
- (2)
- The other type, the equivalence value function $F({t}_{1},{t}_{2})$, applies when unit heat $\overline{Q}=1$ spontaneously flows from reservoir ${t}_{1}$ to reservoir ${t}_{2}$, as occurs if ${t}_{1}>{t}_{2}$, with $F({t}_{1},{t}_{2})>0$. For more general $\overline{Q}$, $F({t}_{1},{t}_{2})$ appears in the form $\overline{Q}F({t}_{1},{t}_{2})$:$$\mathrm{equivalence}\phantom{\rule{4.pt}{0ex}}\mathrm{value}\phantom{\rule{4.pt}{0ex}}\overline{Q}F({t}_{1},{t}_{2})\phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}\overline{Q}\phantom{\rule{4.pt}{0ex}}\mathrm{flows}\phantom{\rule{4.pt}{0ex}}\mathrm{from}\phantom{\rule{4.pt}{0ex}}{t}_{1}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{4.pt}{0ex}}{t}_{2}$$

#### 12.2. The Reversibility Condition: An Early Version of Entropy

#### 12.3. Finding the Absolute Temperature

## 13. Thomson-Joule 1854

## 14. Priority Disputes

- I think it fair to say that Mayer is the first to publish the idea of energy conservation and to obtain a value for the mechanical equivalent of heat. However, without further experimental support, as in the independent and extensive and careful work of Joule, it would not have served as a foundation for others to build upon. On the other hand, the underlying implication, not stated by Mayer, is that the constituents of the gases are so weakly interacting that volume changes due to expansion or compression do not affect their interactions.
- Thomson expended great effort in the laboratory to test his friend Joule’s “conjecture” that $\mu =J/{T}_{g}$. But Thomson strategically labeled and downgraded Mayer’s more well-founded but implicit assumption, that the constituents of an ideal gas interact only weakly, to the level of a “hypothesis”. In my opinion, this is a loss to the physics community. Mayer’s unstated but well-founded hypothesis about the lack of interactions in the ideal gas should be taught at the outset of all thermodynamics courses, in part because it so readily yields a value for the mechanical equivalent of heat. Further, as argued above, the relationship $\mu =J/{T}_{g}$ did not define temperature uniquely; uniqueness was first shown to follow from the integration factor argument of Clausius. Modern arguments, depending on the analysis of a Carnot cycle, and its generalization to arbitrary reversible cycles, are simpler than the argument of Clausius.
- S. Thompson, in his biography of Kelvin, condemns Clausius as arriving at a “vague statement” that heat tends to pass from hot to cold [52]. However, a similar comment can also be made about Kelvin’s initial statement of his version of the Second Law of thermodynamics. Moreover, Thompson also seems to appreciate neither the origin of Thomson’s specification of the reversibility condition in a “conjecture” rather than a theoretical deduction, nor Clausius’s extension of it in integral form to develop the entropy [78].
- Although I have read comments attributing to nationalism the priority jockeying between Thomson, Tait and Joule, on the one hand, and Clausius and Mayer on the other hand, I think it is more likely that it was a matter of personal loyalty in the staking of claims. To my knowledge, of all of the participants during the development of thermodynamics, only Thomson and Joule were friends. On the contrary, in 1862 the English Tyndall supported the German Clausius, likely simply out of a sense of fairness; after this Kelvin’s friend Tait entered the fray. I believe that Thomson had it within his ability to call off the polemical Tait; if Thomson had done so, various disagreements would have been more amicable, but perhaps in that case neither Joule nor Thomson would have had their eponymous units of energy and temperature. Not until somewhat later did Thomson and the German Helmholtz (a distant relative of the American William Penn) become good friends.
- By 1860 Maxwell, and by 1873 Gibbs, had entered the picture. These two had a calming influence. Gibbs wrote:“in the memoir of Clausius ...the science of thermodynamics came into existence”. However, he goes on with “In the development of the various consequences of the fundamental propositions of thermodynamics, as applied to all kinds of physical phenomena, Clausius was rivaled, perhaps surpassed, in activity and versatility by Sir William Thomson” [78]. Here Gibbs implicitly refers to Thomson’s far-sighted applications of thermodynamics to elasticity and to thermoelectricity.

## 15. Clausius and Entropy 1865

## 16. Rankine

## 17. On Entropy Production and Wasted Energy

## 18. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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A History of Thermodynamics: The Missing Manual. *Entropy* **2020**, *22*, 77.
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A History of Thermodynamics: The Missing Manual. *Entropy*. 2020; 22(1):77.
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