A term meaning flow chart for the etymology of the 1865 term entropy.
In thermodynamics, the etymology of term entropy, symbol S, derives
from the Greek word τροπη (tropos), meaning “transformation, coined by German physicist Rudolf Clausius in 1865. [1]

“Clausius coined the term entropy, from the Greek tropos, or transformation.”
— Eric Zencey (1983), “Entropy as Root Metaphor” [10]

The above quote by American scholar Eric Zencey, who did his PhD dissertation on entropy as a root metaphor, in the social sciences, gives a well-representative synopsis of the etymology of entropy; the Greek term "τροπη", however, to clarify, has the following alternative meanings, namely "turn" or "change": [11]

The exact meaning behind Clausius' use of the term "transformation" is in the sense of:

HeatWork

or the transformation of heat into work or work into heat as defined by the mechanical equivalent of heat.

There were, however, a number of precursory terms and phrases used by Clausius in the preceding fifteen years, such as diagrammed above, to define this quantity. In addition, there are to be found propagated in the literature a number synonyms and false etymologies. Both topic areas are discussed below.

Heat | History
The roots of the mathematical concept of entropy have its roots in the various historical models of heat: caloric, matter of fire, phlogiston, terra pinguis, sulphur model of heat, among others. To construct the quantity entropy, conceived as a state function, i.e. to transform the conception of a quantity of heat—conceived by Antoine Lavoisier previously as an indestructible fluid like particle called “caloric” (a theory disproved via Benjamin Thompson's 1798 cannon boring experiment)—Clausius employed the Euler reciprocity relation, whereby he used William Thomson’s absolute temperature (1854) scale T as the integrating factor for the differential (inexact differential) of a unit or quantity of heat Q, to make a new state variable function for heat: dQ/T (exact differential) which he came to call entropy.

Precursory etymologies
German physicist Rudolf Clausius, between 1850 and 1865, used the following phrasings and terminologies to define his heat divided by absolute temperature quantity in development:

 Date Conceptual Development of Entropy 1780 The matter of fire (indestructible) is called "caloric" (Antoine Lavoisier) and is disengaged from bodies during combustion. 1798 The synopsis conclusions of the famous caloric-questioning "cannon boring experiment" (Benjamin Thompson): “What is heat? Is there anything as igneous fluid? Is there anything that can with propriety be called caloric? That heat generated by friction [in the boring experiments] appeared, evidently, to be inexhaustible, [it] cannot possibly be a material substance; … it appears to me to me to be extremely difficult, if not quite impossible, to form any distinct idea of anything capable of being excited and communicated in the manner heat was excited and communicated in these experiments, except it be MOTION.” 1850 Clausius: stated that: an expression was needed to account for the experimental fact that "loss of heat occurs when work is done." (as Carnot had assumed did not occur). 1854(Jan 19) Rankine: introduced his so-called "thermodynamic function" of heat; which he later (Sep 1865) says is equivalent to Clausius' entropy function. 1854(Dec) Clausius: defined the ratio: Q/T and calls it "equivalence-value" (so to have relation to Joule's 1843 paper "Mechanical equivalent of heat"). 1856 Clausius: uses the phrase: "equivalence-value of all uncompensated transformations" (so as to have relation to Carnot's postulate: “we shall assume that the quantities of heat absorbed and emitted in these different transformations compensate each other exactly” involved in a cyclical process" (and gives it the symbol -N). 1862 Clausius: relates the integral of dQ/T to something he calls "disgregation" of the body having relation to arrangement of the molecules of the working body (to have relation to Carnot's 1824 paper On the Motive Power of Fire that characterized the "transformations" of "working substances" of an engine cycle, namely "mode of aggregation"). 1865(Apr 24) Clausius: lets dS = dQ/T and first calls S the "transformational content" of the working body, but then settles on entropy, so to have similarity to the word energy. 1877 Clausius: explains the origin of the term ‘entropie’ to James Maxwell in a Nov 8th letter (see below).

1865 coining
On April 24th, 1865, during a reading to the Philosophical Society of Zurich, Clausius gave the following statement, in what was his first mention of the term “entropy”, shown below, left in the original German and right in the first 1867 English translation by mathematical physicist Thomas Hirst: [1]

 Original German Hirst 1867 English translation “Sucht man für S einen bezeichnenden Namen, so könnte man, ähnlich wie von der Grosse U gesagt ist, sie sei der Wärmennd Werkinhalt des Körpers, von der Grosse S sagen , sie sei der Verwandlungsinhalt des Körpers. Da ich es aber für besser halte, die Namen derartiger für die Wissenschaft wichtiger Grossen aus den alten Sprachen zu entnehmen, damit sie unverändert in allen neuen Sprachen angewandt werden können, so schlage ich vor, die Grosse S nach dem griechischen Worte η τροπή, die Verwandlung, die Entropie des Körpers zu nennen. Das Wort Entropie habe ich absichtlich dem Worte Energie möglichst ähnlich gebildet, denn die beiden Grossen, welche durch diese Worte benannt werden sollen, sind ihren physikalischen Bedeutungen nach einander so nahe verwandt, dass eine gewisse Gleichartigkeit in der Benennung mir zweckmnssig zu sein scheint.” “We might call S the transformation content of the body, just as we termed the magnitude U its thermal and ergonal content. But as I hold it to be better terms for important magnitudes from the ancient languages, so that they may be adopted unchanged in all modern languages, I propose to call the magnitude S the entropy of the body, from the Greek word τροπή, transformation. I have intentionally formed the word entropy so as to be as similar as possible to the word energy; for the two magnitudes to be denoted by these words are so nearly allied their physical meanings, that a certain similarity in designation appears to be desirable.” Google German → English translation “If you are looking for specifically, S has a distinctive name, it might be similar to the size of U, it is the work Wärmennd content of the body, from the size of S say it is the transformation content of the body. Since I believe but for the better, such is the name for the science major in the Great found in ancient languages, so they can be used unchanged in all new languages, I propose that the size S of the Greek words η τροπή that Transformation to call the entropy of the body. The word entropy I have intentionally formed the word energy as similar as possible, because the two quantities, which are named by these words are related to their physical meanings to each other so close that a certain uniformity in the designation seems to me zweckmnssig.”

Rankine | thermodynamic function
In 1854, Scottish engineer-physicist William Rankine introduced what he called a "thermodynamic function", symbol Phi Φ, whose increase or diminution of which indicates whether heat is entering or leaving a system. [1] In 1873, Willard Gibbs, in two places, stated that Rankine’s thermodynamic function was the same and or synonymous with Clausius’ entropy; firstly, in his first thermodynamics article: [8]

“The term entropy, it will be observed, is here used in accordance with the original suggestion of Clausius, and not in the sense in which it has been employed by Professor Tait and others [Maxwell] after his suggestion. The same quantity has been called by Professor Rankine the thermo-dynamic function.”

Next, in his second 1873 thermodynamics article, Gibbs stated: [9]

“The entropy as defined by Clausius is synonymous with the thermodynamic function as defined by Rankine.”

Maxwell
Scottish physicist James Maxwell had a number of stumbling blocks in respect to his understanding of entropy, owing to his first rendering discerned from his reading of Peter Tait's 1868 Sketch of Thermodynamics, wherein the sign for entropy was "incorrectly" reversed, as follows (as explained (Ѻ) by Gibbs in 1873):

$dS = - \frac{dQ}{T}\,$

In a 1 Dec 1873 letter to Tait, after reading Gibbs clarifications, Maxwell stated the following: (Ѻ)

“It is only lately, under the conduct of Professor Willard Gibbs that I have been led to recant an error with I had imbibed from your θΔcs [thermodynamics] namely that the entropy of Clausius is unavailable energy while that of T' [Thomson's] is available energy. The entropy of Clausius is neither one nor the other it is only Rankine’s thermodynamic function.”

In 1877, Clausius, in a letter to Maxwell (Nov 8), explained the origin of the term ‘entropie’ (Ѻ), the letter, in the original German, from Maxwell’s collected scientific letters (Ѻ), is shown below:

In 1878, Maxwell, in discussion of Clausius' description of entropy in terms of "equivalence of transformations", referred to the theory of entropy as the theory of transformation-equivalents. [3]

Clausius | 1875 restatement
In the updated 1875 second edition to his Mechanical Theory of Heat textbook, Clausius reaffirmed his etymology as follows: [2]

“In another paper, after introducing a further development of the equivalence of transformations, the author proposed to call this quantity, after the Greek word τροπή, transformation, the entropy of the body [S].”

False etymologies
In the 1955 work Thermodynamics of Irreversible Processes, Belgian chemist Ilya Prigogine gave the following incorrect etymology: “The second principle of thermodynamics postulates the existence of a function of state, called entropy, from the Greek ενρωπη meaning “evolution”. [4] In any event, whatever his agenda was at this point, Prigogine and his co-author Dilip Kondepudi give the correct etymology in their 1998 textbook Modern Thermodynamics. [5] In the 1998 thermodynamics dictionary A to Z of Thermodynamics, French thermodynamicist Pierre Perrot gives the following incorrect etymology and incorrect date of coining: “[Entropy] a term proposed by Clausius in 1868, from the Greek root eντροπη, the act of turning round (τροπη, change of direction), implying the idea of reversibility.” [6] The Online Etymology Dictionary, since at least 2008, has been correctly claiming that entropy was conceived in the mind of Clausius it on the Greek term entropia, or "a turning toward," from en- "in" + trope "a turning"; whereas the correct sense Clausius uses the Greek term τροπή in the sense of 'transformation' or 'transformation content.' [7]

Entropy formualtions

References
1. (a) Clausius, Rudolf. (1865). “On Several Forms of the Fundamental Equations of the Mechanical Theory of Heat (Ninth Memoir).” Read at the Philosophical Society of Zurich on the 24th of April, 1865, published in the Vierteljahrsschrift of this society, Bd. x. S. 1.; Pogg. Ann. July, 1865, Bd. cxxv. S. 353; Journ. de Liouville, 2e ser. t. x. p. 361.
(b) Clausius, R. (1865). The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies (pg. 357) (Ninth Memoir). London: John van Voorst, 1 Paternoster Row. MDCCCLXVII.
2. (a) Clausius, Rudolf. Pogg. Ann. Vol. cxxv. p. 390.
(b) Clausius, Rudolf. (1879). The Mechanical Theory of Heat, (pg. 107). London: Macmillan & Co.
3. (a) Maxwell, James C. (1878). “Tait’s ‘Thermodynamics’ (I)”, (pgs. 257-59). Nature, Jan. 31.
(b) Maxwell, James C. (1878). “Tait’s ‘Thermodynamics’ (II)”, (pgs. 278-81). Nature, Feb. 07.
4. Prigogine, Ilya. (1955). Introduction to Thermodynamics of Irreversible Processes, (pg. 15). Charles C. Thomas.
5. Kondepudi, Dilip and Prigogine, Ilya. (1998). Modern Thermodynamics – from Heat Engines to Dissipative Structures, (pg. 80). New York: John Wiley & Sons.
6. Perrot, Pierre. (1998). A to Z of Thermodynamics (pg. 91). Oxford: Oxford University Press.
7. Entropy – Online Etymology Dictionary.
8. Gibbs, Willard. (1873). “Graphical Methods in the Thermodynamics of Fluids” (esp. 310n) (Ѻ), Transactions of the Connecticut Academy of Arts and Sciences, 2:309-42; in: Scientific Papers of J. Willard Gibbs, Volume One: Thermodynamics (pg. 2n). Oxbow Press, 1906.
9. Gibbs, Willard. (1873). “A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces”, Transactions of the Connecticut Academy, 2:382-404; in: Scientific Papers of J. Willard Gibbs, Volume One: Thermodynamics (pg. 53n). Oxbow Press, 1906.
10. Zencey, Eric. (1983). “Entropy as Root Metaphor”, Conference on Science, Technology, and Literature, Feb, Long Island University, New York; in: Beyond the Two Cultures: Essays on Science, Technology, and Literature (editors: Joseph Slade and Judith Lee) (§9:185-200, esp. pg. 189), Iowa State University Press, 1900.