# Characteristic function

Here we see the "root" functions of what we now refer to as "entropy" and "entropy increase", developed by Clausius, building on Thomson, in 1854 to 1862, which form the "core" the so-called "characteristic functions" of thermodynamics (Massieu, 1869), which form the basis of the famously cryptic statement that the "entropy of the universe tends to a maximum" (Clausius, 1865).

In functions, characteristic function (TR:17) (LH:7) (TL:24) refers to a "single" thermodynamical function that is "characterizes" a system or body (Massieu, 1869).[1]

## Overview

### Modern functions

The following are the characteristic functions of thermodynamics, shown in their modern notion; the three "root" formulations behind what we now call "entropy" are shown in green background; the key seven characteristic functions, relevant to the characteristic functions of humans are shown bolded:[2]

# Formula Names / Definitions
A ${\displaystyle -{\frac {\partial G}{\partial \xi }}_{p,T}}$
• The effect of likes tending towards likes (Plato, 370BC)
• Qualities which cause attraction or repulsion between things (Magnus, 1255)
• Attraction; certain powers, virtues, or forces operating at the chemical level; any force by which small particles of bodies attract towards each other (Newton, 1717)[3]
• Affinity (Donder, 1936)
P ${\displaystyle {\frac {F}{A}}}$
• Compression, owing to weight, of the air or water surrounding a body, increasing with depth (Beeckman, (1620)
• Weight of the atmosphere pushing on a surface (Torricelli, 1644)
• Force F exerted on a surface per unit are A (Bernoulli, 1738)
• Pressure of a given body in any state (Gibbs, 1873)

Alternative symbols: p (Gibbs, 1873)

F
• That which is impressed upon a body to cause a change in its state of motion (Newton, 1687)
• The product of the mass of a body and its acceleration (Newton, 1687)
• An attractive action proportional G to the ratio of the product of the mass of two bodies, m1 and m2, and the square of their distance of separation r2 (Newton, 1687)
T ${\displaystyle {\frac {1}{C}}}$
• [Description] (Thomson, 1849)[4]
• Function of the temperature which the changing matter has at the moment when it receives the element of heat δQ (Clausius, 1856)
• Absolute temperature (Clausius, 1856)
• Temperature reckoned from absolute zero; the reciprocal of the the Carnot function C (Maxwell, 1871)
• Absolute temperature of a given body in any state (Gibbs, 1873)

Alternative symbols: t (Massieu, 1869), θ (Maxwell, 1871), t (Gibbs, 1873)

V ${\displaystyle dV}$
• Volume
• Volume of a given body in any state (Gibbs, 1873)

Alternative symbols: v (Gibbs, 1873)

Q ${\displaystyle \delta Q=TdS}$
• Quantity of heat (Lavoisier, 1783)
• Heat imparted to the changeable body during the cyclical process (Clausius, date)
• Heat received by the body in passing from one state to another; heat given out by the body is considered as a negative quantity of heat received by it (Gibbs, 1873)

Alternative symbols: H (Maxwell, 1871; Gibbs, 1873)

W
• Product of the force acting on a body and the distance through which the body is moved (Coriolis, 1829)[5][6]
• External work preformed during the cyclical process (Clausius, 1856)
• Work done by the body in passing from one state to another; work spent upon the body is considered as a negative quantity of the work done by the body (Gibbs, 1873)
J ${\displaystyle {\frac {Q}{W}}}$
• Ratio of "heat expended" to "work produced" (Clausius, 1850)
• Thermal equivalent of a unit of work (Clausius, 1856)
• Mechanical equivalent of heat (Partington, 1924)

Alternative symbols: A (Clausius, 1850), (Helmholtz, 1882) [7]

U ${\displaystyle T+J}$
• Arbitrary function of V an T; comprising sensible heat and the heat necessary for interior work (Clausius, 1850)
• Mechanical energy of a body in a given state (Thomson, 1851)
• One function completely determined by the initial and final states of the body (Clausius, 1854)
• Interior heat of the body (Zeuner, 1860)
• Internal work of the body (Zeuner, 1860)
• Function of activity (Kirchhoff, c.1862)
• Quantity of action (Kirchhoff, c.1862)[8]
• Energy of the body (Clausius, 1864)
• Thermal content plus the ergonal content of the body (Clausius, 1864)
• Energy (Clausius, 1866)
• Energy of a given body in any state (Gibbs, 1873)
• Internal energy (Clausius, 1875)
• Sum of the vis viva T[9] and the ergal J (Clausius, 1875)
• Total store of energy contained in the body (Helmholtz, 1882)

Alternative symbols: e (Maxwell, 1871), ε and E (Gibbs, 1873; Lorentz, 1922),

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

#### Entropy | Origin

• Equivalence value (Clausius, Dec 1854)[10]; based on Thomson's function (May 1854)
• Equivalence value of the transformation
N
• Total value of all the transformations (Clausius, 1854)[11]
• Equivalence value of all the uncompensated transformations (Clausius, 1856)
${\displaystyle \int {\frac {\delta Q}{T}}\geqq 0}$
S ${\displaystyle dS={\frac {\delta Q}{T}}}$
• "Transformational content" of the body (Clausius, 1865)[11]
• Entropy from the Greek word τροπη meaning "transformation" (Clausius, 1865)
• Entropy of a given body in any state (Gibbs, 1873)
• Transformation equivalent (Maxwell, 1878)[16]

Alternative symbols: φ (Maxwell, 1871), η and H (Gibbs, 1873), Φ (Planck, 1897)

Z ${\displaystyle dZ}$
• Disgregation of the body; depends on the arrangement of the particles of the body (Clausius, 1865)
• Transformation value of the existing arrangement of the particles of the body (Clausius, 1865)
[17]
• Bound energy (Helmholtz, 1882)[8]
• Bound work (Helmholtz, 1882)
• Transformation content energy (Thims, 2011)[18]

Alternative symbols: B (Thims, 2012)[19]

H ${\displaystyle U+PV}$
• Heat function at constant pressure (Gibbs, c.1876)[20]
• Enthalpy (Onnes, 1909)
• Absolute heat function (Schottky, 1929)[21]

Alternative symbols: U’ (Massieu, 1869), χ (Gibbs, c.1873; Epstein, 1936), W (Planck, 1897; Ulich, 1930), ψ (Donder, 1926), I (Lerberghe, 1931)

F ${\displaystyle U-TS}$

Alternative symbols: – tψ (Massieu, 1869), ψ (Gibbs, 1875; Sackur, 1912; Bridgman, 1914; Lorentz, 1922; Epstein, 1936; Keenan, 1941), (Duhem, 1897), A (Haber, 1905; Lewis, 1923; Goates, 2000), – Tψ (Fowler, 1929)

G

Alternative symbols: – tψ’(Massieu, 1869), ζ (Gibbs, 1875; Sackur, 1912; Lorentz, 1922), Φ (Duhem, 1897; Epstein, 1936), – Tψ (Planck, 1897), Z (Bridgman, 1914; Partington, 1924; Keenan, 1941), H (Donder, 1926; Lerberghe, 1931), F (Lewis, 1923; Butler, 1928), – TΦ (Fowler, 1929), u (Rossini, 1950)

μ
• Differential coefficient of energy taken with respect to a specific mass of a given substance in homogeneous mass of matter of various kinds enclosed in a rigid and fixed envelope, which is impermeable to and unalterable by any of the substances enclosed, and perfectly non-conducting to heat (Gibbs, 1875)
• Gibbsian chemical potential (Schottky, 1929)[21]

### Six main magnitudes | Clausius 1865

In 1865, Clausius, at the close of his Mechanical Theory of Heat, gave the following summary of what he considered to be the six main magnitudes of thermodynamics:

“Before proceeding further, let us collect together, for the sake of reference, the ‘magnitudes’ which have been discussed in the course of this memoir, and which have either been introduced into science by the mechanical theory of heat, or have obtained thereby a different meaning. They are six in number, and possess in common the property of being defined by the present condition of the body, without the necessity of our knowing the mode in which the body came into this condition:
1. the thermal content
2. the ergonal content
3. the sum of the two foregoing, that is to say the thermal and ergonal content, or the energy
4. the transformation-value of the thermal content
5. the disgregation, which is to be considered as the transformation-value of the existing arrangement of particles
6. the sum of the last two, that is to say, the transformational content, or the entropy.”
— Rudolf Clausius (1865), The Mechanical Theory of Heat (pgs. 357-58)

Clausius follows this, with a discussion of how to calculate the energy and entropy of a "homogeneous" body. Eight years later, in 1873, Gibbs, of course, expanded on this model, as applied to equilibriums of "heterogeneous" substances.

Extending this logic to the human formation scale, we can calculate the characteristic functions of a human, bound state of humans, or system of humans by the "property" of being able to defined the present condition or state of a given human, according to the energies and entropies involved in forming the body, without having to know the mode in which the body came into this condition or state of existence. This problem was first broached by Dolloff (1975).

### Terminology | Overlap

Here, as listed in the "alternative symbols" used section, we see a good deal of terminological discontinuity, which did not become homogenized or uniform until the 1940s, give or take a decade or two. Namely, we see the letters used for more than one thing:

• A used 4 different ways: affinity, area, mechanical equivalent of heat, and isothermal isochoric potential.
• H used 4 different ways: heat, enthalpy, entropy, and isothermal isobaric potential.
• F used 3 different ways: force, isothermal isochoric potential, and isothermal isobaric potential.
• W used 2 different ways: work and enthalpy.
• Φ used 2 different ways: entropy and isothermal isobaric potential.
• ψ used 2 different ways: enthalpy and isothermal isochoric potential.
• Z used 2 different ways: disgregation and isothermal isobaric potential.
• J used 2 different ways: ergal and the mechanical equivalent of heat.
• T used 2 different ways: temperature and vis viva.

Not to mention a number of mixed formulations, e.g. – Tψ and – TΦ both being used for isothermal isobaric potential.

### Entropy increase | Maximum entropy

In rows 11 to 12, in respect to equivalence values, positive transformations, negative transformations, uncompensated transformations, sum of the equivalence values of all uncompensated transformations, etc., all of which Clausius, in 1865, tried to condense into the catch phrase "entropy of the universe tends to a maximum", resulting in terms, such as: entropy increase or maximum entropy, we see the source of the origin of a large amount of ongoing confusion in modern thermodynamics. In short, modern people think that all of this means "disorder tends to a maximum", whereas, in fact, Clausius never used the term "disorder" once!

### Formation energy

In 66AE, Thims, owing to on-going confusion of namesakes related to the "isothermal isobaric potential" (row 18), began employ "formation energy" as the new namesake so accurately explain meaning, in respect to the energy behind the formation of a human. The term "formation energy" being short for Lewis' 1923 phrase: "free energy of formation of a chemical substance".[23] The substance in focus herein being a "human".

## Historical tables

In 1929, Walter Schottky, student of Max Planck, in his Thermodynamics: the Study of Cyclical Processes, Physical and Chemical Changes and Equilibria, co-authored with: Hermann Ulich and Carl Wagner, was the first to use the letter "G", for the isothermal isobaric thermodynamic potential, in honor of Willard Gibbs, who he credits with being the main founder of chemical thermodynamics. Specifically, in his section "Chemical Potential", Schottky derives the following function:[21]

### Guggenheim | 1933

In 1933, Edward Guggenheim, in his Thermodynamics: by the Methods of Willard Gibbs, gave the following so-called table of "notation used by various authors for the four characteristic functions", as he says:[24]

Guggenheim says that the notation he used was a compromise between Lewis (1923)[23], on one hand, and Walter Schottky, Hermann Ulich, and Carl Wagner (1929), on the other hand.

### Donder | 1936

In 1936, Theophile Donder, in his Thermodynamics Theory of Affinity, gave the following so-called table of "notations used for the thermodynamic potentials in some important texts on thermodynamics", as he calls it:[25]

### Guggenheim | 1939

In 1939, Guggenheim, in his Statistical Thermodynamics: Statistical Mechanics for Students of Physics and Chemistry, co-authored by Ralph Fowler, gives the following so-called table of "correlated symbols used for the chief thermodynamic functions" in other modern textbooks:[26]

## Quotes

The following are quotes:

Massieu has shown how all the properties of a fluid ‘which are considered in thermodynamics’ may be deduced from a single function, which he calls a characteristic function of the fluid considered; he introduces two different functions of this kind, vis a function of the temperature and volume, which he denotes by Ψ, and a function of the temperature and pressure, which he denotes by Ψ’; in both cases he considers a constant quantity (one kilogram) of the fluid, which is regarded as invariable in composition.”
Willard Gibbs (1876), On the Equilibrium of Heterogeneous Substances (pgs. 86, 358)[27]
“A stupid man's report of what a clever man says is never accurate, because he unconsciously translates what he hears into something that he can understand.”
Bertrand Russell (1945), History of Western Philosophy (pg. 45)[28]

## End matter

### References

1. Massieu, Francois. (1869). “On the Various Functions Characteristic of Fluids and on the Theory of Vapors” (Sur les Functions Caracteristiques des Divers Fluides et Sur la Theorie des Vapeurs), Comptes Rendus, 69: 858-62, 1057-61.
2. Characteristic function notation table – Hmolpedia 2020.
3. See: Query 31 (Newton, 1717).
4. Carnot function – Hmolpedia 2020.
5. Principle of the transmission of work – Hmolpedia 2020.
6. Calculation of the Effect of Machines – Hmolpedia 2020.
7. Note: Clausius (who uses the symbol "A") says: "A denotes a constant which expresses the equivalent of heat for he unit of work". The symbol "J", in honor of James Joule, later came to be employed, at was set equal to 1, a decade or so after Helmholtz (1882), who uses this symbol: for the mechanical equivalent of heat.
8. Helmholtz, Hermann. (1882). “On the Thermodynamics of Chemical Processes” (pdf), in: Physical Memoirs Selected and Translated from Foreign Sources, 1: 43-97. Physical Society of London, Taylor and Francis, 1888.
9. Note: here Clausius, in his "Mathematical Introduction" (1875), uses the symbol T for ergal, whereas in his previous memoirs had used the symbol T for absolute temperature; hence, we have inherent symbol confusion, at this point (a problem problem seen in the half-century to follow)
10. Note: Clausius uses "Q", the symbol began to be used after Clausius, to signify that Q is an "inexact differential"
11. Clausius, Rudolf. (1865). The Mechanical Theory of Heat (translator: Thomas Hirst). Macmillan, 1867.
12. Positive transformation  - Hmolpedia 2020.
13. Negative transformation – Hmolpedia 2020.
14. Uncompensated transformation – Hmolpedia 2020.
15. Entropy increase – Hmolpedia 2020.
16. Transformation-equivalents – Hmolpedia 2020.
17. Note: this is JTS with J, i.e. the mechanical equivalent of heat, set equal to one.
18. Transformation content energy – Hmolpedia 2020.
19. Note: the symbol actually used is: , according to the English translation; and he doesn't use "J" as shown but: .
20. Planck, Max. (1897). Treatise on Thermodynamics (translator: Alexander Ogg) (pgs. 74-75). Longmans, 1903.
21. Schottky, Walter; Ulich, Hermann; Wagner, Carl. (1929). Thermodynamics: the Study of Cyclical Processes, Physical and Chemical Changes and Equilibria (Thermodynamik: die Lehre von den Kreisprozessen, den physikalischen und chemischen Veränderungen und Gleichgewichten) (Amz) (Gibbsian chemical potential (Gibbschen chemischen potential), pg. xiii; §: The Chemical Potential (Die chemischen Potentiale), pgs. 213-; Gibbs, 5+ pgs). Springer.
22. Note: this term a new truncated shorthand of the all the former; generally themed on Lewis' free energy tables (1923) of the "free energies of formation of chemical substances".
23. Lewis, Gilbert. (1923). Thermodynamics: and the Free Energy of Chemical Substances (amanuensis: Merle Randall). McGraw-Hill.
24. Guggenheim, Eduard. (1933). Modern Thermodynamics: by the Methods of Willard Gibbs (pg. 28). Methuen.
25. Donder, Theophile. (1936). Thermodynamic Theory of Affinity: A Book of Principles (pg. xvi). Stanford University Press.
26. Fowler, Ralph; Guggenheim, Eduard. (1939). Statistical Thermodynamics: a Version of Statistical Mechanics for Students of Physics and Chemistry (pg. x). Cambridge.
27. Gibbs, Willard. (1876). On the Equilibrium of Heterogeneous Substances; in: Transactions of the Connecticut Academy, III. pp. 108-248, Oct., 1875-May, 1876, and pp. 343-524, may, 1877-July, 1878; in: The Scientific Papers of Willard Gibbs, Volume One: Thermodynamics (§3:55-371). Longmans, 1906.
28. Russell, Bertrand. (1945). History of Western Philosophy (pg. 45). Simon & Schuster.