|The etymology of the term entropy (see: entropy (etymology)), comprised of or embodying the terms: en- (inside) + -trope (transformation) + equivalence-value (1854) + transformation content (1856/1865). |
In 1865, Clausius, using differential notation, redefined entropy dS as follows:
or in modern Greek delta δQ notation (Neumann, 1875), to explicitly defined the differential unit of heat as an inexact differential, we have:
Clausius, accordingly, made the unit of heat the derivative of heat a complete differential or path independent state function, the logic of which became embodied in what Clausius defined as the "second main principle" of the mechanical theory of heat. The mathematical expression of entropy was conceived by Clausius to quantify the effect of irreversibility (or the irreversible change of state of a body), in the working body, e.g. a body of steam in a steam engine, during an engine cycle; an effect that French physicist Sadi Carnot assumed, in 1824, did not occur due to his view that heat was form of caloric particles.
|A depiction of the different equation formulations of entropy (using an equation overlay method), shown over a rabbit smelling a flower (indicative of natural governing nature of the second law); the function (δQ/T) on the right hand side of the equation in lower right hand corner being the 1854 "equivalence-value" formulation of heat; the equation in the upper left hand corner being the 1856 "equivalence-value of all uncompensated transformations" formulation of entropy; the second row equation a version of the probability-based Boltzmann entropy (1872) / Planck entropy (1901) / Gibbs entropy (1902); the equation in upper right hand corner being the partial of entropy with respect to some variable X at constant energy. |
In general, according to Clausius, when a body (working body) changes its state, work is performed externally and internally at the same time, the exterior work having reference to the forces which extraneous bodies exert upon the body under consideration, and the interior work to the forces exerted by the constituent molecules of the body in question upon each other. The interior work is for the most part so little known, and connected with another equally unknown quantity (in fact with the increase of heat actually present in the body) in such a way, that in treating of it we are obliged in some measure to trust to probabilities; whereas the exterior work is immediately accessible to the observation and measurement, and thus admits of more strict treatment.
As such, to avoid everything hypothetical, we can exclude the interior work, by confining heat operations to the consideration of cyclical process— that is to say, operations in which the modifications that the body undergoes are so arranged that the body finally returns to its original condition. In such operations the interior work which is performed during the several modifications, partly in a positive sense and partly in a negative sense, neutralizes itself, so that nothing but exterior work remains, for which the theorem of the equivalence of transformations can then be demonstrated with mathematical strictness.
The theorem of the equivalence of transformations argues that when a body goes through a cyclical process, a certain amount of exterior work may be produced, in which case a certain quantity of heat must be simultaneously expended; or, conversely, work my be expended and a corresponding quantity of heat may by gained. This may be expressed by saying: Heat can be transformed into work, or work into heat, by a cyclical process.
There may also be another effect of a cyclical process: heat may be transferred form one body to another, by the body which is undergoing modification absorbing heat form the one body and giving it out again to the other. In this case the bodies between which the transfer of heat takes place are to be viewed merely as heat reservoirs, of which we are not concerned to know anything except the temperatures. If the temperatures of the two bodies differ, heat passes, either from a warmer to a colder body, or from a colder to a warmer body, according to the direction in which the transference of heat takes place. Such a transfer of heat may also be designated, for the sake of uniformity, a transformation, inasmuch as it may be said that heat of one temperature is transformed into heat of another temperature.
The two kinds of transformations that have been mentioned are related in such a way that one presupposes the other, and that they can mutually replace each other. If we call transformations which can replace each other equivalent, and seek the mathematical expressions which determine the amount of the transformations in such a manner that the equivalent transformations become equal in magnitude, we arrive at the following expression: If the quantity of heat Q of the temperature t is produced from work, the equivalence-value of this transformation is:
In addition, if the quantity of heat Q passes from a body whose temperature is T1 into another whose temperature is T2, the equivalence-value of this transformation is:
|A tattoo of the principle of the equivalence of transformations (1856), using the 1875 inexact differential notation δ of German physicist Carl Neumann, on the forearm of Ivanka, a newly graduated philosophy student.  |
here T is a function of the temperature which is independent of the kind of process by means of which the transformation is effected, and T1 and T2 denote the values of this function which correspond to the temperatures of bodies one and two. By separate considerations, according to Clausius, T is in all probability the absolute temperature. These two expressions further enable us to recognize the positive or negative sense of the transformations. In the first, Q is taken as positive when work is transformed into heat, and as negative when heat is transformed into work. In the second, we may always take Q as positive, since the opposite senses of the transformations are indicated by the possibility of the difference:
being either positive or negative. It will thus be seen that the passage of heat from a higher to a lower temperature is to be looked upon as a "positive transformation", and its passage form a lower to a higher temperature as a "negative transformation".
If we represent the transformations which occur in a cyclical process by these expressions, the relation existing between them can be stated in a simple and definite manner. If the cyclical process is reversible, the transformations which occur therein must be partly positive and partly negative, and the equivalence-values of the positive transformations must be together equal to those of the negative transformations, so that the algebraic sum of all the equivalence-values become = 0. If the cyclical process is not reversible, the equivalence values of the positive and negative transformations are not necessarily equal, but they can only differ in such a way that the positive transformations predominate.
The theorem respecting the equivalence-values of the transformations may accordingly be stated thus: The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to nothing. The mathematical expression for this theorem is as follows. Let dQ be an element of the heat given up by the body to any reservoir of heat during its own changes, heat which it may absorb from a reservoir being here reckoned as negative, and T the absolute temperature of the body at the moment of giving up this heat, then the equation:
must be true for every reversible cyclical process, and the relation:
must hold good for every cyclical process which is in any way possible. The value of dQ/T is called entropy. 
See main: Entropy (etymology)Between 1850 and 1865, Clausius published a series of nine memoirs, which in 1865 were collected in the textbook Mechanical Theory of Heat. The outline of the theoretical development of the concept and terminology of entropy, went through a number of name changes: "an expression was needed" (1850), equivalence-value (1854), "equivalence-value of all uncompensated transformations" (1856), "disgregation" (1862), "transformation-content" and then finally arriving at the word entropy (1865). 
Entropy of the universe tends to a maximum
American physicist Michael Guillen argues that the following formulation of the second law:
which states that the entropy of the final state of the universe will be greater than or ‘maximal’ as compared to the initial state entropy of the universe, is one the of the five equations that most changed the world; along with Isaac Newton’s law of universal gravitation, Daniel bernoulli’s law of hydrostatic pressure, Michael Faraday’s law of electromagnetic induction, and Albert Einstein’s mass-energy equivalence relation. 
|A basic thermodynamic data table, alluding to the methodology according to which one would measure and list the entropy of a given human or rather human molecule, the human molecular formula of which is shown in the bottom row (next to an unknown value of positive entropy).|
Entropy of a human molecule
See main: Human entropyThe conception of the possession of an entropy value of individual person or human molecule or species of human molecules at a specific reference point in time is what is referred to as "human entropy".
The first person, historically, to make the suggestion that each person has a different value of entropy was American engineer William Fairburn in mentioned in his 1914 book Human Chemistry.
In theory, each individual person can be assigned an entropy value, in reference to a base value, similar to smaller molecules. Shown adjacent, for instance, are standard measures of entropy for four different molecules.
A table such as this is similar to the "material entropy" postulate, but with reference on the measure of entropy per species.
Human system entropy
See main: Economic entropy, Social entropy, etc.In human systems, the definition of entropy is the same with the translation that the "working body" is defined such that instead of water molecules, confined to the internal regions of a steam engine, put in alternating contact with a hot body (a fire) and a cold body (cool water), driven to do mechanical work (push a piston), we have human molecules, confined to the internal regions of various regions of social systems, put in alternating contact with a hot body (the day sun) and a cold body (the cool night sky), driven to do the daily work of life, e.g. economic work, social work, volunteer work, household work, parenting work, territorial expansion work, interpersonal work, relationship work, etc. 
From a reaction point of view, i.e. human chemistry point of view, boundaries to "working bodies" of human systems, i.e. interactive collections of human molecules confined to economic systems, are defined as being the the 90 percent probability regions in which a specific number of socially interactive or energetically-coupled humans are found. In this point of view, entropy is defined as the internal system energy (internal work) dissipated as humans act on each other, energy that does not find conversion into system external work.
|A listing of mostly incorrect misinterpretation definitions of entropy in a letter to The Electrician (London) from Sydney Evershed, January 09, 1903, supposedly in connection to the great “what is entropy debate” (1902-1904) started by British electrical engineer James Swinburne. |
See main: Entropy misinterpretationsHistorically, entropy has been subject to much confusion, misinterpretation, misapplication, a subject about which is prolonged and involved.
The first dominant person to confuse entropy was Peter Tait who in his 1868 Sketch of Thermodynamics presented an incorrect view of entropy. Tait's book was read by his associate James Maxwell, who included this incorrect view of entropy in his 1871 Theory of Heat. The error went unnoticed until the fourth edition, during which time it was corrected.
Other famous misinterpretations of entropy were dug out during the great 1902 to 1904 "what is entropy debate", such as are listed adjacent 1903 letter by Syndney Evershed. 
● Entropy (quotes)
1. Clausius, Rudolf. (1862). "On the Application of the Theorem of the Equivalence of Transformations to Interior Work", (pp. 215-250). Communicated to the Naturforschende Gesellschaft of Zurich, Jan. 27th, 1862; published in the Viertaljahrschrift of this Society, vol. vii. P. 48; in Poggendorff’s Annalen, May 1862, vol. cxvi. p. 73; in the Philosophical Magazine, S. 4. vol. xxiv. pp. 81, 201; and in the Journal des Mathematiques of Paris, S. 2. vol. vii. P. 209.(b) Thims, Libb. (2007). Human Chemistry (Volume Two), (preview). Morrisville, NC: LuLu.
2. (a) Thims, Libb. (2007). Human Chemistry (Volume One), (preview). Morrisville, NC: LuLu.
2. (a) Thims, Libb. (2007). Human Chemistry (Volume One), (preview). Morrisville, NC: LuLu.
3. Clausius, R. (1865). The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies. London: John van Voorst, 1 Paternoster Row. MDCCCLXVII.
4. Reeve, Sidney. (1907). “The Question of Entropy”, Harvard Engineering Journal (pgs. 138-54), Vol. 6.
5. Guillen, Michael. (1996). Five Equations that Changed the World: the Power and Poetry of Mathematics (ch. 4: An Unprofitable Experience: Rudolf Clausius and the Second Law of Thermodynamics, pgs. 165-214). Hyperion.
6. Adeline, Chloe. (2010). “Make Your Life Simple and Focused: With Science!”, SimpleRabbit.com, Apr 24.
● Fast, J.D. (1962). Entropy - the Significance of the Concept of Entropy and its Applications in Science in Technology. New York: McGraw-Hill Book Co. Inc.
● Arnheiim, Rudolf. (1974). Entropy and Art: an Essay on Disorder and Order. University of California Press.
● McIntyre, Vonda N. (1981). The Entropy Effect. Publisher: Star Trek.
● Brooks, Daniel R. and Wiley E.O. (1988). Evolution and Entropy (2nd ed.). Chicago: University of Chicago Press.
● Dugdale, J.S. (1998). Entropy and its Physical Meaning. London: Taylor and Francis.
● Greven, Andreas, Keller, Gerhard, and Warnecke, Gerald. (2003). Entropy (Princeton Series in Applied Mathematics). Princeton University Press.
● Ben-Naim, Arieh. (2007). Entropy Demystified - the Second Law Reduced to Plain Common Sense. London: World Scientific.
● Grady, Walter T. (2008). Entropy and the Time Evolution of Macroscopic Systems. Oxford University Press.
● Entropy – Wikipedia.