Before continuing my journey into the thermodynamic world of J. Willard Gibbs, I wanted to take a step back and revisit Rudolf Clausius’s final sentence in his ninth and final memoir on the mechanical theory of heat (here is link to all nine memoirs that ran from 1850 to 1865).

Clausius’s Ninth Memoir

Realizing that his Ninth would be his final memoir, at least with regards to the mechanical theory of heat, Clausius must have felt the need for closure.  He brought the physics along as far as he could, even if he was not fully satisfied with where he stood.  While Clausius brought powerful focus to the quantity δQ/T that showed up in his work, his attempt to seek a deeper understanding of δQ/T didn’t really pan out. So how was he to conclude his fifteen years of labor on this topic? He needed to do so in a way that did justice to what he achieved and perhaps also to point the way towards future research.  It was in this context that he quietly introduced entropy.  The introduction of this property in and of itself would have made a fitting end to his story.  And if he had stopped there, his work would have remained powerful and groundbreaking.  But he didn’t stop here.

Instead, in a turn of historical significance and scientific confusion, he ended his Ninth Memoir with a final sentence that was a doozy, one that arguably was the single cause for the main scientific and philosophical impact of this publication.  In a sudden and unexpected flash for the reader, accompanied by a most minimal lead-in discussion, Clausius made a bold leap from Carnot to the cosmos.

From Carnot to cosmos

Clausius concluded his Ninth Memoir by stating two fundamental theorems of the Mechanical Theory of Heat:

1.  The energy of the universe is constant.  (Die Energie der Welt ist constant)

2.  The entropy of the universe tends to a maximum.  (Die Entropie der Welt strebt einem Maximum zu)

We’ll get to that rather large conceptual leap imbedded in the second and final sentence in a moment.  But first, let’s look at these final two statements together since they reflected a change of major proportions.  Clausius and his need for simplicity, generality and symmetry likely sought a grand way to bring his work to a close.  His shift from Carnot and the mechanical theory of heat to the fundamental laws of matter led him to effectively declare that the science of matter was based on two fundamental state properties, energy and entropy, that held much greater significance than the primitive properties on which they were based. Gibbs would later recognize the rightness of this navigational direction set by Clausius and base his own work on these properties as he made famously clear by his verbatim use of these two statements, in the original German, as a lead in to his groundbreaking paper, On the Equilibrium of Heterogeneous Substances (1875-78).  (One could argue that Clausius provided Gibbs his roadmap.)

It wasn’t just entropy that was important, but the declaration of an increasing entropy

But it was more than the declaration of energy and entropy that made this memoir so revolutionary.  It was the declaration of an increasing entropy in the universe that garnered the most interest and later fame for Clausius.  This must rank as one of the largest conceptual leaps in the history of science. 

By his Ninth Memoir, Clausius felt he had reached the peak of a mountain that only he saw, although I doubt he felt he had truly made it.  But his final two sentences were, to me, his attempt to declare the peak reached.  The first statement had been uttered before.  The second statement, on the other hand, was brand new to the field of thermodynamics, a whole new concept, one not only introducing a new property of matter but one also introducing a unique behavior of that property that brought quantification to thermodynamic irreversibility.  This was what was so revolutionary about this specific memoir.  It represented a whole new take on the 2^nd Law, one quite different than the early version originated by Carnot and modified by Clausius: “Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.” [1]   Clausius’ one short sentence launched a whole new field by upgrading the 2^nd Law from Carnot’s and other’s versions to a quantifiable entropy that truly brought the concept of direction into thermodynamics.  But where did this sentence come from?

Regarding the universe

By 1865, others had started to consider the universe as fair game in the analysis of whether or not the laws of thermodynamics remained valid.  As Clausius started publishing his theories on this subject, other scientists, rightly so, started to test their validity.  This was the Scientific Method at work.  Propose a hypothesis, then test it.  Does it survive?  As a true professional, Clausius respected this process and responded to each challenge with a good, open spirit.  

While such challenges provided motive for Clausius to extend his work beyond Carnot’s engine, it was really Thomson’s work that provided the defining motive.  In his Eighth Memoir, Clausius summarized key aspects of Thomson’s work on the universal tendency for energy to dissipate such that different temperatures move towards and not away from each other, a tendency that Clausius’s work with entropy would ultimately explain.  In this context, with precedent having already been established by Thomson and others, regarding the universe, it wasn’t a total surprise that Clausius looked to the universe in response, especially regarding his succinct capturing of the 1^st Law of Thermodynamics, this time in terms of his newly adopted concept of energy.

So the extension of the 1^st Law to the universe was not unreasonable; it had already been started by others.  But entropy, and especially the concept of an increasing entropy?  This was different.  Yes, it was a very bold and intuitive statement from an otherwise cautious scientist. But it was also a statement that was based on an assumption.

Regarding Clausius and transformations of heat

Clausius was driven to understand how heat transforms to work (per Joule and his concept of work-heat equivalence), on the one hand, and from high temperature to low temperature on the other, such as occurs, he believed, in thermal conduction. He devoted significant time studying these two transformations of heat and his subsequent writing on the topic was abstract, complicated, challenging to read and digest, and not entirely valid. I reviewed this material in Chapter 34 of my book as I felt it important to understand how Clausius arrived at an increasing entropy. While I don’t want to delve into this material here, I do want to focus on Clausius’s key central argument.

Regarding, once again, thermal conduction

I wrote about William Thomson’s challenge with thermal conduction in a previous post (here) and now shift the discussion to Clausius’s own challenge. Consider again the bottom half of this illustration from my book.

Consider the equation in the lower right. We know that entropy increases during thermal conduction because in the equation quantifying this, T_H > T_C and Q_H < O. But how do we know Q_H < O? It’s easy for us to know this today because, as dictated by the conservation of energy and the 1st Law of Thermodynamics, when thermal energy (Q) is exchanged between two bodies, 1) the two changes in energy are identical in number and opposite in sign, and 2) the signs themselves are not arbitrary.  When heat enters a body, the value of Q for that body must be positive since its dU is positive.  Likewise the value of Q for the other body must be negative since its dU is negative.  Recall per the 1st Law (as created by Clausius), dU = δQ (absent work).

But the above logic wasn’t easy for those in the mid-1800s. Folks were still coming out from behind the influence of the caloric theory and the new replacement concept of energy was still rather abstract for many, including, at least to some extent I suspect, Clausius.

How Clausius decided the sign on Q

Clausius’ challenge with the sign on Q was best reflected by the fact that he started with one convention in his Fourth Memoir and then changed to the correct convention in his Ninth, writing, “[In my Fourth Memoir] a thermal element given up by a changing body to a reservoir of heat is reckoned positive, an element withdrawn from a reservoir is reckoned negative… In the present memoir, however, a quantity of heat absorbed by a changing body is positive, and a quantity given off by it is negative.” [2]  In reading this one senses that he viewed the reservoir and the working substance differently and the heat that passed between them as an entity that existed on its own, with its own identity and sign, depending on which direction it was moving in. In this context, the decision about the sign seemed arbitrary. But he had to make this decision and so turned to a deeply held belief.

Consider Clausius’ leading paragraph to his famed final statement of his Ninth Memoir and note my bold italics:

“… all transformations occurring in nature may take place in a certain direction, which I have assumed as positive…  The application of this theorem to the Universe leads to a conclusion to which W. Thomson first drew attention, and of which I have spoke in the 8^th Memoir…. The entire condition of the universe must always continue to change in [a positive] direction.” [3]

Clausius believed that nature allowed certain transformations and built up his logic accordingly. He believed that heat flowed naturally from hot to cold, and not from cold to hot. So when he considered the equation in the lower right of the above illustration, he hypothesized that thermal conduction was a natural occurrence and so must represent a positive transformation, which meant that Q_H had to be negative for the equation to yield a positive number, which meant that the change in entropy had to be positive. In short, the natural tendency of entropy was to increase.

This logical construct was a hypothesis built on an emotional feeling that what happens in nature, e.g., conduction from hot to cold, must be positive.

But what happens to the entropy of an isolated body?

Clausius’ hypothesis that the change in entropy of the universe is positive held no substantive backing.  It was an intuition based largely on an assumption involving the conduction of heat.  We don’t really know what’s happening in the universe, but we do know that when two bodies meet and exchange energy due to the presence of gradients, the entropy of the equilibrated combined system is greater than the sum of the entropies of the two initial bodies.  It was this thinking that Gibbs and others picked up on and developed further, extending this beyond thermal gradients to include all forms of energy gradients. 

It’s interesting to me that Clausius himself did not really grasp the application of entropy to a given isolated body.  To him, the universe was comprised of a series of on-going transformations, the net of which, quantified by a change in entropy, was positive.  But the relevance of this logic to a closed system comprised of gradients didn’t seem to attract his attention.  In fact, once stated, it didn’t appear that Clausius saw any further use for entropy.  But fortunately, Gibbs did, to great effect.  Gibbs realized that one could consider a given non-equilibrated system to be comprised of component parts, each equilibrated internally and thus characterized by its own state properties such as U and S but none equilibrated with the others.  He also realized based on Clausius’ work and also Thomson’s that that transformations and dissipations would eventually cause the gradients—mechanical, thermal, chemical—to disappear or (per Thomson) “dissipate” and that the final fully equilibrated system would have an entropy greater than or equal to the sum of the components’ initial entropies.  Gibbs deeply understood the implications of what Clausius discovered much more so than Clausius himself did.

Am I being too harsh here by saying that Clausius ended his Ninth Memoir based on combined assumption and intuition as opposed to a sound, structured logic?  Perhaps.  Again, I certainly don’t mean to diminish the masterpiece of work that it was, as it clearly provided Gibbs the path forward on his own monumental work.  But evidence in support of this is provided by Clausius himself.  When he summarized all of his work on thermodynamics into his comprehensive The Mechanical Theory of Heat (1879),[4] he included no mention of a natural tendency for entropy to increase.

* * *

In a world embracing conservation, entropy must have come as a shock.  Mass is conserved.  Energy is conserved.  But entropy?  It’s not conserved.  The entropy of the equilibrated sum is greater than or equal to the combined entropies of the initial parts.  What an unusual property!  This is what gave Gibbs a powerful direction towards equilibrium.

Learn more!!!

Learn much, much more about Clausius and an increasing entropy and how Gibbs put these finds to great use in my book, Block by Block: The Historical and Theoretical Foundations of Thermodynamics.

References

[1] Clausius, R. 1867. The Mechanical Theory of Heat: With Its Applications to the Steam-Engine and to the Physical Properties of Bodies. Edited by Hirst, T. Archer. London: John Van Voorst. Fourth Memoir (1854), p. 117.

[2] Ibid, Ninth Memoir (1865), p. 329.

[3] Ibid, Ninth Memoir (1865), p. 364.

[4] Clausius, R. 1879. The Mechanical Theory of Heat (1879). Translated by Walter R. Browne. London: Macmillan.

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