Entropy

The basic model of "entropy", symbol S, and "entropy increase", symbol N, shown in respect to one cycle of the Papin engine, as per the Lavoisier-Carnot cycle (1824), wherein Lavoisier-Carnot quantity of heat , shown by the dot " • ", heat believed here as a particle, is defined as "caloric", and the Clausius cycle (1854), wherein the Thomson-Clausius quantity of heat , heat here conceptualized as vibration, rotation, and motion, is defined as an an inexact ${\displaystyle \delta Q}$ amount, equal to the bound energy: , where T is the absolute temperature, and is a differential change in entropy.

In thermodynamics, entropy (CR:2,075) (LH:77) (TL:2,152|#3), from Greek en- (η), meaning "in" + -tropy (τροπή), meaning in "transformation", or verwandlung in German, aka "equivalence value" or "equivalence value of the transformation" (Clausius, 1854)[1] or "transformation content" (Clausius, 1865), or "transformation equivalent" (Maxwell, 1878)[2], symbol S or , is the mechanical equivalent of heat (Joule, 1843) based state function re-formulation of the Lavoisier-Carnot model of heat as caloric (Lavoisier, 1783; Carnot, 1824); defined formulically as:[3]

${\displaystyle {\frac {\delta Q}{T}}}$

where is an inexact differential, the symbol meaning "inexact", of a quantity Q of heat, and T is the absolute temperature (Thomson, 1848), measured at the point where the unit of heat enters or leaves the boundary of the body (Thomson, May 1854; Clausius, Dec 1854).

The entropy function is based on the reciprocity relation (Euler, 1739).[4] The fraction is the integrating factor that converts the inexact quantity of heat into an exact differential . This is exactly the same has how is the integrating fact that converts the inexact quantity of work ${\displaystyle \delta W}$ into an exact differential , as pioneered by Clapeyron (1834) graphically, via indicator diagram (Southern, 1796), based on Carnot (1824), wherein the work of the engine could be measured via the calculus of integrals, then defined as the method of fluents and fluxions (Newton, 1665).

Overview

The entropy change ${\displaystyle dS}$ , associated with the transformation of a work body, in one Clausius cycle, is defined as the sum of the equivalence values of all uncompensated transformations:[5]

${\displaystyle N=-\int {\frac {dQ}{T}}}$

Since, the value of N cannot be known exactly, per reason that it involves measuring the work that the molecules of the system or working body do on each other, in one cycle, the inequality sign ">" or "<", depending, was introduced as a patch solution, according to which the magnitude of the value of N, i.e. |N|, will tend to increase in numerical value, until the cyclical transformation process stops, at which time N will be at a maximum (Clausius, 1855). Clausius later renamed replaced N with S, the term "equivalence value" with entropy, and the phrase "sum of the equivalence values of all uncompensated transformations" with "entropy increase" or "entropy tends to a maximum".

Entropy, as a differential, is defined by the symbol dS, which is an exact differential (the "d" symbol meaning "exact"), representative of a differential amount, aka small amount, of the entropy:

${\displaystyle dS={\frac {\delta Q}{T}}}$

entering or leaving the working body.

Etymology

The heat cycle model of Lavoisier (1873), wherein adding caloric particles   to bodies increases volume, and removing caloric particles, decreases volume, these "caloric" particles are indestructible, according to which at the end of the expansion and contraction cycle, NO change occurs in the "working body", i.e. the blue volume above (e.g. a body of water in a piston and cylinder). In 1854, Lavoisier's caloric   became the Thomson-Clausius " " (1854), aka "entropy" (Clausius, 1865) model.

Precursor models to entropy are: sulfur principle, "matter of heat", “phlogiston”, symbol: "phi" ϕ (Stahl, 1703), note phi- being the root of the phoenix, “caloric”, symbol:   (Lavoisier, 1783)[6] or "Lavoisier heat quantity" (as Lavoisier did not employ a symbol for caloric, but just spoke of it as a "quantity" Q of heat.[7]

In 1824, Sadi Carnot, in his On the Motive Power of Fire, building on Lavoisier, and his caloric-based volume change cycle model, shown adjacent, gave the following formula, in general verbalized outline:

${\displaystyle Q_{L_{H1}}+Q_{L_{H2}}+Q_{L_{H1}}\dots -Q_{L_{C1}}-Q_{L_{C2}}-Q_{L_{C3}}\dots =0}$

where,  H1 is one caloric particle coming from the hot body into the working body,  H2 is second caloric particle coming from the hot body into the working body, etc.,  C1 is one caloric particle going from the working body into the cold body,  C2 is second caloric particle going from the working body into the cold body, etc., which all sum to zero, at the completion of the heat cycle, the working body returning to its original state. The Lavoisier-Carnot cycle (or Carnot cycle) is reversible, in this sense, i.e. in the sense of the working body returning to its "original state", which no "change" in nature occurring.

Thomson-Clausius

In May 1854, William Thomson, in his "On the Dynamical Theory of Heat, Part 5: Thermoelectric Currents"[8], gave the following formula:

${\displaystyle {\frac {H_{t}}{t}}+{\frac {H_{t}'}{t'}}+{\frac {H_{t}''}{t''}}+\dots =0}$

where H is a quantity of heat, going into our out of the boundary of the system, at the absolute temperatures t, t', t'', etc, at those boundary locations, or "localities, respectively", as Thomson phrased things.

In Dec 1854, Rudolf Clausius, in his "On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat", building, presumably, on Thomson, as historians (Caldwell, 1871; Brush, 1975) have conjectured, gave the following expression:

${\displaystyle N={\frac {Q_{1}}{T_{2}}}+{\frac {Q_{2}}{T_{2}}}+{\frac {Q_{3}}{T_{3}}}\dots }$

or:

${\displaystyle N=\sum {\frac {Q}{T}}}$

This expression, in turn, following discussion, assumes the form:

${\displaystyle N=\int {\frac {dQ}{T}}}$

where the integral extends "over all the quantities of heat received by several bodies", as he says. Clausius at this point was referring to N as the "total value of all the transformations" and in 1856 as "equivalence value of all the uncompensated transformations" (see: characteristic function, row: 12). This combined logic, of Thomson (May 1854) and Clausius (Dec 1854), as surmised well by Donald Caldwell (1971)[9] and by Stephen Brush (1975), quoted below, is where the basic model of entropy arose.

“It would seem that one should date the discovery or invention of the entropy concept from this 1854 paper, since the change in terminology from ‘equivalence-value of a transformation’ to ‘entropy’ can have no effect on the physical meaning of the concept itself.”
Stephen Brush (1975), The Kind of Motion We Call Heat (pg. 576)[10]

In short, in these seven months, Thomson and Clausius usurped and upgraded the previous "caloric model" of heat cycles of Lavoisier (1783) and Carnot (1824), therein replacing the invalid Carnot cycle with the corrected to reality Clausius cycle.

In 1865, Clausius, to simplify his previous names for   as a sum, and   as a "transformation equivalent" phrases, introduced the term "entropy" based on the Greek τροπή, meaning transformation, which he describes as follows:

“We might call S the ‘transformation content’ of the body, just as we termed the magnitude U [internal energy] 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 words words η [in or the] + τροπή, meaning in ‘transformation’ (or change) [Verwandlung]. 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.”
Rudolf Clausius (1865), Mechanical Theory of Heat (pg. 357)

Here, we note that the isopsephy value of τροπή is 558, which translates directly as: "change" or "solstice", but also in secret name numerical equivalent as: τεληεις (NE:558), meaning: "perfect, complete", μητις (NE:558), meaning: "wisdom, counsel", and: μη-τις (NE:558), meaning: "none, nothing" (Barry, 1999).[11] Hence, entropy, in short, is the thermodynamical measure of "change", give or take pre-Clausius secret name meaning.

Thus, the cryptic motto about how the "entropy of the universe tends to a maximum" is decoded to mean that the "universe tends towards change".[12]

Confusables

Entropy, itself, as a single unit or quantity of heat,  , entering or leaving a body, to note, is NOT to be confused with "entropy change", aka "transformation equivalents" (Maxwell, 1878)[2] or integral sum of the "equivalence values of all uncompensated transformations" (Clausius, 1856), at the at the end of a Clausius cycle of system transformations. The former, i.e. entropy, has no increase or decrease associated with it, other than the indication as to whether heat is entering or leaving the body, quantified by a ± sign. The latter, i.e. entropy change or transformation equivalent change, does have an increase associated with it, which is associated with the work the molecules of the system do on each other, during the cyclical transformation, quantified by the value "N", the sum of the uncompensated transformations, of heat into work, and work into heat, at the end of the Clausius cycle. Great confusion abounds, in respect to the term "entropy increase" (which people incorrectly equate with "disorder increase"), which references the latter, but not the former.

End matter

References

1. Equivalence value – Hmolpedia 2020.
2. Transformation equivalents – Hmolpedia 2020.
3. Clausius, Rudolf. (1865). The Mechanical Theory of Heat (translator: Thomas Hirst) (entropy, 10-pgs; uncertainty, 3-pgs; disorder, 0-pgs; randomness, 0-pgs). Macmillan & Co, 1867.
4. Euler reciprocity relation – Hmolpedia 2020.
5. Equivalence value of all uncompensated transformations – Hmolpedia 2020.
6. Lavoisier, Antoine; Laplace, Pierre. (1783). Memoir on Heat (translator: Henry Guerlac). Neale, 1982.
7. Entropy formulations – Hmolpedia 2020.
8. Thomson, William. (1854). “On the Dynamical Theory of Heat. Part V: Thermo-electric Currents”, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science (pgs. 214-25; quote, pg. 232), May 1; in: Transactions of the Royal Society of Edinburgh, 21:123-#. Royal Society, 1857.
9. Cardwell, Donald. (1971). From Watt to Clausius: the Rise of Thermodynamics in the Early Industrial Age (pgs. 258-60). Cornell.
10. Brush, Stephen. (1975). The Kind of Motion We Call Heat: Statistical Physics and Irreversible Processes (pg. 576). Holland.
11. Barry, Kieren. (1999). The Greek Qabalah: Alphabetic Mysticism and Numerology in the Ancient World (pdf) (§:Dictionary of Isopsephy, #558, pg. 239). Weiser.
12. Note: this is a strong correction to the Planck entropy (1909) model, which eludes to the conclusion that the "universe tends towards disorder", which is so confusedly bandied about in pop culture.