A differential equation symbols cartoon by Nick Kim. [49] |

**symbols**or "notations" are means of representing measurable quantities or formulaic representations of these quantities by means of letters, modifications of letters, e.g. italics, bold, hats, cross-bars, etc., or other shapes, figurines, or operators.

“Philosophy is written in that great book that lies before our gaze—I mean the universe—but we cannot understand it if we do not first learn the language and grasp thesymbolsin which it is written.”— Galileo Galilei (1623),The Assayer[52]

“I know several men who see all nature insymbols, and express themselves conformably whether in Quintics or Quantics, Invariants or Congruents.”— James Maxwell (1863), letter (Ѻ) to (someone)

“The first time I heard about chemical thermodynamicswas when a second-year undergraduate brought me the news early in my freshman year. He told me a spine-chilling story of endless lectures with almost three-hundred numbered equations, all of which, it appeared, had to be committed to memory and reproduced in exactly the same form in subsequent examinations. Not only did these equations contain all the normalalgebraicsymbolsbut in addition they were liberally sprinkled withstars,daggers, andcirclesso as to stretch even the most powerful of minds.”

Table— Brian Smith (1973),Basic Chemical Thermodynamics[1]

The following is an (under-construction) table of the various symbols used in thermodynamics and its connective fields. Some symbols are linked to as separate article, as listed in the adjacent navigation box. Curiously, of note, it seems that smaller letter, e.g.

*p, t, v*, were preferred at the turn 19th century, whereas capital letters, e.g.

*P, T, V*, were preferred at the turn of the 20th century and onward, for the most part. In the meaning column (two), names shown in brackets are what people tend to call the symbol in the modern sense; if no brackets are shown, then it is the name used according to the reference given:

Symbol | Meaning | Formula | Date | Person |

Δ (link) | Fire or Heat | 450BC | Egyptians, Greeks, or Democritus [53] Empedocles. [14] | |

+ | (Plus sign) | 1489 1518 | Johannes Widmann (Ѻ) Henricus Grammateus [43] | |

– | (Minus sign) | 1489 1518 | Johannes Widmann (Ѻ) Henricus Grammateus [43] | |

= (link) | (Equals sign) | 1557 | Robert Recorde. [42] | |

> < | c.1600 | Thomas Harriot [51] | ||

Log. | Logarithm | 1616 | Appears as an abbreviation for logarithm in A Description of the Admirable Table of Logarithmes (1616), an English translation by Edward Wright of John Napier's work. [47] | |

∞ | Infinity | c.1655 | John Wallis [54] | |

≤ ≥ | 1670 | John Wallis [51] | ||

Integral | 1675 | Gottfried Leibniz: according to Leibniz's notebooks, a critical breakthrough occurred on 11 November 1675, when he employed integral calculus for the first time to find the area under a function y=ƒ(x). He introduced several notations used to this day, for instance the integral sign ∫ representing an elongated S, from the Latin word summa. [32] | ||

Derivative | 1675 | Gottfried Leibniz: in the same notebook, Leibniz employed the use of d for differentials, from the Latin word differentia. [32] | ||

ϕ (phi) | Phlogiston | 1704 | Georg Stahl: [24] | |

T | Temperature | ? | ||

V | Volume | ? | ||

P | Pressure or weight of the overlying atmosphere | 1738 | Daniel Bernoulli: “the weight P holding down the piston in [a given] position is the same as the weight of the overlying atmosphere, which we shall designate P in what follows.”[34] | |

n | Number of particles | 1738 | Daniel Bernoulli: [34] | |

f(x) | Function | 1755 | Leonhard Euler: was the first to write f(x) to denote the function f applied to the argument x. [44] | |

i | Imaginary number | 1755 | Leonhard Euler: [44] | |

e | Natural log base | 1755 | Leonhard Euler: [44] | |

∑ | Summation | 1755 | Leonhard Euler: [44] | |

→ | Dart (Reaction arrow) | 1757 | William Cullen: “the dart → between them expresses the elective attraction; when I put a dart with the tail to one substance and the point to another, I mean that the substance to which the tail is directed unites with the one to which the point is directed more strongly than it does with the one united to it in the crotchet {” . [9] | |

{ | Crotchet (Bonding bracket) | equates to: AB or A≡B (modern) | 1757 | William Cullen: “by the mark { I mean them united to another” [9] |

∂ | Partial differential | "curly d" | 1770 | Marquis de Condorcet |

AB | Chemical union | AB | 1775 | Torbern Bergman |

T | Vis viva (kinetic energy) | 1788 | Joseph Lagrange: [6] | |

Definite integral | 1822 | Joseph Fourier, in his Analytical Theory of Heat, introduced the symbol, and stated: “We refer in general by the sign (adjacent) as the Integral symbol with a and b as the Limits of the full integration that begins when the variable is equivalent to a, and is complete when the variable is equal to b.” [46] | ||

s (or e) | Heat | 1824 | Sadi Carnot: "[s is] the heat liberated (set free) or absorbed" when "a gas varies in volume without change in temperature."; "let e be the quantity of heat employed to maintain the temperature of the gas constant during its dilation." [21] | |

r (or u) | Motive power (work) | 1824 | Sadi Carnot: [21] | |

Q | Heat | 1834 | Emile Clapeyron: "Q is an absolute quantity of heat which the gas possesses." [21] The "Q" supposedly is short for small "quantity"; the usage may date earlier, possibly to Joseph Fourier, and his Analytical Theory of Heat (1822)? | |

R | Gas constant | 1834 | Emile Clapeyron: (possibly used earlier?) [21][23] | |

U | Force function | 1835 | William Hamilton: "the function which has been here called U may be named the force-function of a system." [7] | |

∇ | Quaternion operator Nabla (Maxwell, c.1873) Del (Gibbs and Wilson, 1901) | 1837 | William Hamilton Coined by Peter Tait, in dispute with Maxwell (link); or by Maxwell in honor of Tait, in his circa 1873/74 poem "Chief Musician upon Nabla" (link) (link) | |

T | Vis viva | ? | Rudolf Clausius: | |

J | Ergal | ? | Rudolf Clausius: [26] | |

U | Energy | T + J | ? | Rudolf Clausius: "[U is the] "energy of the system"; [according to the conservation of energy] the sum of the vis viva T and the ergal J [which] remains constant during its motion." [26] |

W | Work | F (xyz) + const. | ? | Rudolf Clausius: "[W is] the work done whenever a body moves under the influence of a force" and "the product of the force and distance moved through is the mechanical work which the force performs during the motion." |

S | Entropy | 1865 | Rudolf Clausius: [27] | |

U | (Internal energy) | 1865 | Francois Massieu: [3][12] | |

U' | (Heat content) | 1865 | Francois Massieu: [3][12] | |

Ψ (psi) | Characteristic function | or | 1869 | Francois Massieu: [12] |

Ψ' (psi prime) | or | 1869 | Francois Massieu: [12] | |

E | Entropy | 1872 | Ludwig Boltzmann: [36] | |

H | Heat | 1873 | Willard Gibbs: "heat received by the body in passing from one state to another". [5] | |

ε (epsilon) | Energy | 1873 | Willard Gibbs: "the energy of body in a given state". [5] | |

η (eta) | Entropy | 1873 | Willard Gibbs: "the entropy of a body in a given state". [5] | |

(d-hat) | Inexact differential | 1875 | Carl Neumann: [28] | |

χ (chi) | (Heat content) | ε + pv | 1876 | Willard Gibbs: "heat function at constant pressure" [3] |

Ψ (psi) | (Helmholtz function) | ε – tη | 1876 | Willard Gibbs: “the force function for constant temperature” [11] |

ζ (zeta) | (Gibbs function) | ε – tη + pv | 1876 | Willard Gibbs: |

μ (mu) | Potential (Chemical potential) | 1876 | Willard Gibbs: "if to any homogeneous mass in a state of hydrostatic stress we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous and it entropy and volume remaining unchanged, the increase of the energy of the mass divided by the quantity of the substance added is the potential for that substance in the mass considered." [8] | |

P | Potential difference (of a galvanic cell) or Pressure | 1877 | Hermann Helmholtz: [20] | |

ℰ | (Electromotive force) | 1877 | Hermann Helmholtz: "P is the potential difference (a and k referring to the anode and cathode, respectively) in a concentration cell." [20] | |

J | Mechanical equivalent of heat | 1882 | Hermann Helmholtz: [20] | |

θ | Absolute temperature | "temperature reckoned from –273° C" | 1882 | " " |

S | Entropy | 1882 | Hermann Helmholtz: [20] | |

U | Total energy | 1882 | Hermann Helmholtz: "the total energy (gesammt-energie) of the system [19] | |

, F | Free energy | U – JTS | 1882 | Hermann Helmholtz: "the free energy (frieie energie) of the system." [20] |

, B | Bound energy | JTS (J=1 in modern joule units) | 1882 | Hermann Helmholtz: "the latent energy (gebundene energie) of the system." [19] |

Reversible reaction | 1884 | Jacobus van't Hoff [45] | ||

A–B A=B | Uniting dashes (chemical bond) | represent the "force lines" associated with atomic valencies. | 1893 | Walther Nernst |

ln | Natural logarithm | 1893 | Used by Irving Stringham (1847-1909) in Uniplanar Algebra (Cajori vol. 2, page 107). [47] | |

U | Energy | 1897 | Max Planck: "the energy of a body or system of bodies." [19] | |

Φ (phi) | Entropy | 1897 | Max Planck: [19] | |

F | Free energy | U – TΦ | 1897 | Max Planck: “The function F, thus bearing the same relation to the external work that the energy U does to the sum of the external heat and work, has been called by Helmholtz the free energy (frieie energie) of the system (it should rather be called the ‘free energy for isothermal processes’).” [19] |

Ψ (psi) | (Gibbs energy) | or | 1897 | Max Planck: “multiplying Φ by – T, we get the thermodynamic potential at constant pressure U + pV – TΦ” [19][2] |

U | (Internal energy) | 1897 | Pierre Duhem: [18][3] | |

– | (Heat content) | 1897 | Pierre Duhem: [3] | |

(Helmholtz free energy) | 1897 | Pierre Duhem: [18][3] | ||

Φ (phi) | (Gibbs potential) | 1897 | Pierre Duhem: [18][3] | |

h | Universal constant (Planck constant) | 1900 | Max Planck: "universal constant"[38] | |

k | Universal constant (Boltzmann constant) | 1900 | Max Planck: "universal constant" [38] | |

A | Reaction energy | 1905 | Fritz Haber: [31] | |

Q | Reaction heat | 1905 | Fritz Haber: [31] | |

q | Latent heat of reaction | 1905 | Fritz Haber: "here we use minus q to designate the heat which is used up or in other words, becomes latent (or bound?)." [31] | |

S | Entropy | 1905 | Fritz Haber: [31] | |

U | Total energy | 1905 | Fritz Haber: [31] | |

N | Avogadro's constant | 1909 | Jean Perrin: “the invariable number N is a universal constant, which may appropriately be designated Avogadro’s constant.” [37] | |

H | Enthalpy | E + PV | 1909 | Heike Kamerlingh-Onnes: coined the name enthalpy from the Greek εν (en) ‘in’ and θαλπος (thalpos) ‘to heat’, to define the word enthalpos, to warm within. [4] |

U | Energy | 1912 | Otto Sackur: [17] | |

H | Heat content | U + pv | 1912 | Otto Sackur: “the heat of reaction at constant pressure is equal to the change in the function H. For this reason H is called the heat content of the system, or the heat function for constant pressure.” [17] |

Ψ (psi) | Free energy function | U – TS | 1912 | Otto Sackur: [17] |

ζ (zeta) | Thermodynamic potential | U – TS + pv | 1912 | Otto Sackur: “some authors use the term free energy for the function ζ, although this term was invented by Helmholtz for the function ψ (See G.N. Lewis, J. Am. Chem. Soc. 35: 14 (1913)).” [17] |

U | Energy content | 1917 | Walther Nernst: [33] | |

Q | Heat flow | 1917 | " " | |

A | Work | German “Arbeit”, meaning work, | 1917 | " " |

S | Entropy | 1917 | " " | |

Contour integral sign Line integral Path integral Curve integral | 1917 | Arnold Sommerfeld [50] | ||

E | Internal energy | 1923 | Lewis and Randall: "[E is] the energy contained within a system, or its internal energy, a property of the system." [10] | |

A | Helmholtz free energy | E – TS | 1923 | " " |

F | Free energy | (E + PV) – TS H – TS | 1923 | " " |

A | Work | 1924 | James Partington: [15] | |

d | Ordinary differential | " " | ||

θ (theta) or T | Temperature | 1924 | [" "]: “the symbol θ is used throughout to denote temperature measured on any scale, unless the scale is specified.” [15] | |

U | Intrinsic energy | 1924 | " " | |

H | Enthalpy; Total heat; Heat function | U + pV | 1924 | [" "]: “the function H was called by Gibbs the heat function at constant pressure, and denoted by χ; the name enthalpy has been proposed by Kamerling Onnes (τό θαλπος = heat). It is usually called by engineers the total heat, or heat content, but these names refer to an obsolete period in the theory of heat.” [15] |

F | Free energy | U – TS | 1924 | " " |

Z | Thermodynamic potential | U – TS + pV | 1924 | [" "]: “Z is called the thermodynamic potential by analogy with the potential function in dynamics: ϕ1 – ϕ2 = work” [15] |

Cycle integral ∮ (circle integral) closed path integral | 1924 | Used by Partington (1924) in a thermodynamic sense; may have been used by Arnold Sommerfeld in 1917, if not sooner? | ||

Δ | Change | "excess of final over initial value" (e.g. ) | 1933 | Edward Guggenheim - defined in formulaic terms [11]; symbols used previously by Gilbert Lewis (1923). |

H | Heat content | E + PV | 1933 | " " |

F | Helmholtz free energy | E – TS | 1933 | [" "]: “the function F is due to Helmholtz, and was named by him the ‘free energy’. It is sometimes called the ‘work function’. We shall call F the ‘Helmholtz free energy’.” [11] |

G | Gibbs free energy | H – TS | 1933 | Edward Guggenheim: “the function G is due to Gibbs, and is often referred to by modern writers as ‘free energy’. We shall call G the ‘Gibbs free energy’.” [11] |

U | Internal energy | 1934 | Theophile de Donder: [16] | |

E | Internal energy | 1936 | Theophile de Donder: [3] | |

A | Affinity | 1936 | " " | |

Ξ or B (?) | Exergy | 1956 | Zoran Rant: [39] | |

S | Entropy | 1968 | Clement Adkins: "[the rev subscript is used] to emphasize that the equality holds for reversible changes only." [41] | |

E | Total exergy | 2004 | Norio Sato: “we shall use the symbol E (capital epsilon) to express total exergy. [40] | |

ε | Molar exergy | 2004 | Norio Sato: “we shall use the symbol ε (small epsilon) to express molar (or specific exergy). [40] |

Undated

The following needed to be dated with reference to the actual person who first used the symbol (starting point references are shown in brackets):

Specific heat at constant pressure | [19] | |||

Specific heat at constant volume | [19] | |||

γ (gamma) | 1897 | Max Planck (?): [19] | ||

J | Massieu function | 1960 | Herbert Callen (?): [13] | |

Y | Planck function | 1960 | Herbert Callen (?): [13] | |

Discussion

The following is a discussion section on some of the symbols.

The Greek alphabet: used greatly in thermodynamics; capital delta Δ, for example, refers to heat (in chemistry) or change (in thermodynamics), as in before minus after of some variable, small delta δ represents an inexact differential, as in δW or δQ, as does đ (d-crossbar), and d refers to an exact differential, as in dU or dS. |

Some common symbol examples include, ∆ the symbol for heat or fire, θ∆ics the shorthand symbol for thermo-dynamics,

*U*for internal energy,

*G*for Gibbs free energy,

*A*of

*F*for Helmholtz free energy,

*S*for entropy,

*T*for temperature,

*P*for pressure,

*V*for volume, etc.

Etymologies

The etymologies of many symbols are very difficult to track down. Belgian chemist Theophile de Donder's 1936 book

*A Thermodynamic Theory of Affinity*contains one of the first symbol tables for the main thermodynamic functions used by various pioneering authors up to that year. [3] One of the few articles on symbol etymology history is the 1997 "A Brief History of Thermodynamics Notations" by American chemical thermodynamicist Rubin Battino who commented, in motivation prior to writing his article, that: [2]

“I was recently asked, ‘I understand the use ofHfor enthalpy because that isheatrelated, but where doesSfor entropy come from?’ I was stumped and decided to do some historical research to track down not only the origin ofSbut also the other principal thermodynamic terminology and notation.”

This semi error filled article (e.g. attributing enthalpy,

*H*, to Clapeyron, or the statement "Clausius in 1834 [age 12] was using

*Q*for heat"), however, fails in its essential mission to determine the origin of

*H*to enthalpy (which was assigned in 1909 by Dutch physicist Heike Kamerlingh-Onnes), and the origin of

*S*to entropy, assigned by German physicist Rudolf Clausius in 1865. The origin of S as entropy is arguably assigned either (a) on the following of Sadi Carnot's 1824 use of the small letter

“[

**represents] the heat liberated (set free) or absorbed”***s*during an isothermal expansion of a body of gas; or (b) in honor of S. Carnot's first name "Sadi". [21] Both of these tables are shown below:

De Donder's 1936 Symbol TableBattino's 2001 Symbol TableBelgian chemist Theophile de Donder’s 1936 thermodynamic potential notation table. [3] American chemical thermodynamicist Rubin Battino's 2001 thermodynamic symbol etymology table. [2]

Jensen's symbol research

A few interesting symbol etymology articles come from American chemistry historian William Jensen, who runs a semi-annual "Ask the Historian" column of the

*Journal of Chemical Education*, who takes email queries on "historical origins of symbols". In his 2003 “The Universal Gas Constant R”, a response to a reader's inquiry, which “traces the history of the gas constant

*R*and the probable reason for its representation by the letter

*R*", Jensen tracks the first use of the constant to French engineer Emile Clapeyron's use of R in his 1834

*Memoir on the Motive Power of Fire*. [22][23]

In his 2010 "Why Are

*q*and

*Q*Used to Symbolize Heat?", Jensen attempts to outline the historical usages of "q" and "Q" in the works of Clapeyron, Rudolf Clausius, August Horstmann, Hermann Helmholtz, Fritz Haber, up through Gilbert Lewis. [48]

See also

● Characteristic function

● Characteristic function table

IUPAC's 2007 Chemical Thermodynamics Symbol Table |

Table of common chemical thermodynamics symbols from the 2007 (3rd ed) of Quantities, Units and Symbols in Physical Chemistry. [30] |

References

1. Smith, Brian E. (1973).

*Basic Chemical Thermodynamics.*Imperial College Press.

2. (a) Battino, Rubin, Strong, L.E., and Wood, S.E. (1997). "A Brief History of Thermodynamics Notation",

*Journal of Chemical Education,*74: 304-305.

(b) Battino, Rubin. (2001). "Ch. 4: A Brief History of Thermodynamics Notation", in:

*Drug-Receptor Thermodynamics: Introduction and Application*by Robert Raffa. Wiley.

(c) Rubin Battino (homepage) – RubinBattino.com.

3. De Donder, T. (1936).

*Thermodynamic Theory of Affinity: A Book of Principles*(pg. xvi-xvii). Oxford: Oxford University Press.

4. Laidler, Keith J. (1993).

*The World of Physical Chemistry*(pg. 110). Oxford University Press.

5. Gibbs, J. Willard. (1873). "Graphical Methods in the Thermodynamics of Fluids",

*Transactions of the Connecticut Academy*(quantities).

*I. pp. 309-342, April-May.*

6. Lagrange, Joseph. (1788).

*Analytical Mechanics*. Publisher.

7. (a) Hamilton, W.R. (1834). “On a general method in dynamics by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function.”

*Philos. Trans. R. Soc. London,*124:247-308.

(b) Hamilton, W.R. (1835). “A second essay on a general method in dynamics.”

*Philos. Trans. R. Soc. London,*125:95-144.

8. Gibbs, Willard. (1876). "On the Equilibrium of Heterogeneous Substances",

*Transactions of the Connecticut Academy,*III. pp. 108-248, Oct., 1875-May, 1876, and pp. 343-524, may, 1877-July, 1878.

9. (a) Crosland, M. P. (1959). “The use of diagrams as chemical ‘equations’ in the lecture notes of William Cullen and Joseph Black.”

*Annals of Science*, Vol 15, Num 2, June.

(b) Thims, Libb. (2007).

*Human Chemistry (Volume Two)*, (preview), (pgs. 385-388). Morrisville, NC: LuLu.

10. Lewis, Gilbert N. and Randall, Merle. (1923).

*Thermodynamics and the Free Energy of Chemical Substances*(pgs. 155-59)

*.*McGraw-Hill Book Co., Inc.

11. Guggenheim, Edward, A. (1933).

*Modern Thermodynamics by the Methods of Willard Gibbs*(pg. 11)

*.*London: Methuen & Co.

12. (a) Massieu, Francois. (1869). “Sur les Fonctions Caracteristiques des Divers Fluids (On the Various Functions Characteristic of Fluids)”,

*Comptes Rendus*69, pgs. 858-62, 1057-61.

(b) Gibbs refers to Massieu’s notation in 1875 and 1876 (See:

*Sci. Pap.*1, pg. 86-87, 358).

(c) Aris, Rutherford, Davis, Howard T., and Stuewer, Roger H. (1983).

*Springs of Scientific Creativity*(ch. 5: The Scientific Style of Josiah Willard Gibbs, by Martin J. Klein, pgs 142-62, esp. pg. 161). University of Minnesota Press.

13. (a) Perrot, Pierre. (1998).

*A to Z of Thermodynamics*(Massieu function, pg. 190). Oxford University Press.

(b) Callen, Herbert B. (1960).

*Thermodynamics: an Introduction to Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics*(pg. 101). Wiley.

14. Greek philosopher Plato mentions the elements [fire (), earth (), air (), water ()] as of pre-Socratic origin, a list created by the Ionian philosopher Empedocles (ca. 450 BC).

15. Partington, James R. (1924).

*Chemical Thermodynamics: An Introduction to General Thermodynamics and its Applications to Chemistry.*D. Van Nostrand.

16. De Donder, Theophile. (1934).

*L’Affinitie*(Part I, Part II, Part III). Paris: Gauthier-Villars.

17. Sackur, Otto. (1912).

*Lehrbuch der Thermochemie und Thermodynamik*(

*A Textbook of Thermo-chemistry and Thermodynamics*(U, S: pg. 142, H: pgs. 171, 175, Ψ: pg. 172, ζ: 174, pg. 180)

*,*1917 Eng. trans. by George Gibson)

*.*MacMillan.

18. Duhem, P. (1897).

*Traite Elementaire de Mecanique Chimique*(

*Treatise on Elementary Chemical Mechanics*)

*.*Paris: Hermann.

19. Planck, Max. (1897).

*Treatise on Thermodynamics*(U: pg. 45, Φ pg. 89, F: pg. 113)

*.*Longmans, Green and Co.

20. Cahan, David (1993).

*Hermann von Helmholtz and the Foundations of Nineteenth-Century Science*(Contributors: Helge Kragh, pg. xvii; ch. 10: "Between Physics and Chemistry - Helmholtz's Route to a Theory of Chemical Thermodynamics" by Helge Kragh). University of California Press.

21. Carnot, Sadi. (1824).

*Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power*("s being the heat", pg. 82 (32)). Paris: Chez Bachelier, Libraire, Quai Des Augustins, No. 55.

22. Clapeyron, Emile. (1834). “Memoir on the Motive Power of Heat”,

*Journal de l’Ecole Polytechnique.*XIV, 153 (and Poggendorff's

*Annalender Physick,*LIX, [1843] 446, 566).

23. (a) Jensen, William B. (2003). “The Universal Gas Constant R” (abstract: “this column traces the history of the gas constant R and the probable reason for its representation by the letter R.),

*J. Chem. Edu.*80: 731.

(b) William B. Jensen (faculty) – Department of Chemistry, University of Cincinnati.

24. Henri-Victor Regnault – NNDB.

25. Partington, J.R. (1937).

*A Short History of Chemistry*(ϕ: pg. 148). Dover.

26. Clausius, Rudolf. (1879).

*The Mechanical Theory of Heat*

*(section: Mathematical introduction, pgs. 1-20).*

*London: Macmillan & Co.*

27. (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.*

(c) See: Entropy (etymology).

28. (a) Neumann, Carl. (1875).

*Lectures on the Mechanical Theory of Heat (Vorlesungen über die mechanische Theorie der Wärme)*, Germany.

(b) Laider, Keith, J. (1993).

*The World of Physical Chemistry*. Oxford University Press.

29. Mills I, Cvitas T, Homann K, et al. (1988).

*Quantities, and Units and Symbols in Physical Chemistry.*IUPAC. Oxford: Blackwell Scientific Publishing.

30. Cohen, E. Richard, Cvitas, Tomislav. (2007).

*Quantities, Units and Symbols in Physical Chemistry*,

*3rd ed. (Chemical Thermodynamics Symbol Table, pg. 56)*

*.*Royal Society of Chemistry.

31. Haber, Fritz. (1905).

*Thermodynamics of Technical Gas Reactions*, (Translator’s Preface, 1907, pg. vii).

*Longmans, Green, and Co.*

32. (a) This ingenious and suggestive notation (integral sign ∫ for integration and

*d*for differentiation) for calculus is probably his most enduring mathematical legacy. Leibniz did not publish anything about his calculus until 1684.

(b) For an English translation of Leibniz's paper, see Struik (1969: 271–84), who also translates parts of two other key papers by Leibniz on the calculus.

(c) Struik, D. J. (1969).

*A Source Book in Mathematics, 1200–1800*. Harvard University Press.

33. Nernst, Walther. (1918).

*The New Heat Theorem: Its Foundations in Theory and Experiment*(

*Die theoretischen und experimentellen Grundlagen des neuen Wärmesatzes*)

*,*[tr. 1926]. E.P. Dutton & Co.

34. (a) Bernoulli, Daniel. (1738). “On the Properties and Motions of Elastic Fluids, Especially Air” (

*Hydrodynamica*, Section 10) in:

*The Kinetic Theory of Gases of Gases*(pgs. 57-65), 2003, by Stephen G. Brush, Nancy S. Hall. Imperial College Press.

(b) Bernoulli, Daniel. (1738).

*Hydrodynamica, Sive Vivibus et Motimus Fluidorum Commentarii.*Sectio Decima: “De affectionibus atque botimus fluidorum elasticorum, praecipue autem aeris.” (pgs. 200-204). Argentorati, Sumptibus Johannes Reinholdi Dulseckeri.

35. Thomson, William. (1849). “An Account of Carnot’s Theory of the Motive Power of Heat – with Numerical Results Deduced from Regnault’s Experiments on Steam”, (127-203)

*Transactions of the Edinburgh Royal Society,*xiv.;

*Annales de Chime,*xxxv. 1852.

36. Boltzmann, Ludwig. (1872). "Further Studies on the Thermal Equilibrium of Gas Molecules" (“Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen”), in

*Sitzungsberichte der Akademie der Wissenschaften, Mathematische-Naturwissenschaftliche Klasse*(pgs. 275-370; term: "entropy" symbol "E", pgs. 263, 308, 346, etc), Bd. 66, Dritte Heft, Zweite Abteilung, Vienna: Gerold.

37. (a) Perrin, Jean. (1909). “Brownian Motion and Molecular Reality”

*(*"Mouvement brownien et réalité moléculaire"), Annales de Chimie et de Physique, 18: 1–114.

(b) Engl. Trans. by Frederick Soddy (London: Taylor and Francis, 1910) [Excerpt: sections 1-6 complete (from: David M. Knight, ed., Classical Scientific Papers: Chemistry (New York: American Elsevier, 1968) and the abridgment reprinted in Henry A. Boorse & Lloyd Motz, The World of the Atom, Vol. 1 (New York: Basic Books, 1966)].

[38] (a) Planck, Max. (1900). “On Irreversible Radiation Processes” (“Über Irreversible Strahlungsvorgänge”),

*Annalen der Physik.*Vol. 1, pg. 69, 99.

(b) Planck, Max. (1901). “On the Law of Distribution of Energy in the Normal Spectrum”,

*Annalen der Physik,*Vol. 4, pg. 553 ff.

(c) Hoffmann, Dieter. (2008).

*Max Planck: Annalen Papers*(Section: Max Planck’s Vita, pg. viii; section: Max Planck on Entropy and Irreversibility by Werner Ebeling, pgs 29-38, etc.). Wiley-VCH.

39. Rant, Zoran. (1956). "Exergie, ein neues Wort fur "Technische Arbeitsfahigkeit" (Exergy, a new word for "technical available work")". Forschung auf dem Gebiete des Ingenieurwesens, 22: 36–37.

40. Sato, Norio. (2004).

*Chemical Energy and Exergy: an Introduction to Chemical Thermodynamics for Engineers*(pg. 99). Elsevier.

41. Adkins, Clement J. (1967).

*Equilibrium Thermodynamics*(section 5.2: Entropy, pgs. 71-). McGraw-Hill.

42. Recorde, Robert. (1557).

*The Whetstone of Witte:*

*whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers.*Publisher.

43. (a) Grammateus, Henricus. (1518).

*A New Skill Book*(

*Ayn new Kunstlich Buech*). Vienna.

(b) Miller, Jeff. (2009). “Earliest Uses of Symbols of Calculus”, Jeff560.tripod.com.

(c) Plus and minus signs – Wikipedia.

44. Boyer, Carl B.; Uta C. Merzbach (1991).

*A History of Mathematics*(pp.439–445). John Wiley & Sons.

45. (a) Nernst, Walther. (1895).

*Theoretical Chemistry*

*: from the Standpoint of Avogadro’s Rule & Thermodynamics*(, pg. 358)

*.*MacMillan and Co.

(b) The exact publication of Van't Hoff likely being his 1884

*Studies in Chemical Dynamics.*

46. Miller, Jeff. (2009). “Earliest Uses of Symbols of Calculus”, Jeff560.tripod.com.

47. Cajori, Florian. (2008). “Earliest Uses of Function Symbols.”, Jeff560.tripod.com.

48. Jensen, William B. (2010). “Why Are q and Q Used to Symbolize Heat?”

*, Journal of Chemical Education*, 87, 1142.

49. Kim, Nick D. (c.2006). “Differential Equations”, NearingZero.net (WayBack).

50. (a) Table of mathematical symbols by introduction date – Wikipedia.

(b) Line integral – Wikipedia.

51. (a) Stedall, Jacqueline. (2003).

*The Greate Invention of Algebra: Thomas Harriot’s Treatise on Equations*(symbols: =, <, >, pg. 8; modern inequality signs, pg. 302). Oxford University Press.

(b) Stedall, Jacqueline. (2012).

*The History of Mathematics: A Very Short Introduction*. Oxford University Press.

(c) Table of mathematical symbols by introduction date – Wikipedia.

52. Crick, Francis. (1994).

*The Astonishing Hypothesis: the Scientific Search for the Soul*(pg. 203) (Ѻ). Simon and Schuster.

53. Thims, Libb. (2019).

*Human Chemical Thermodynamics: Chemical Thermodynamics Applied to the Humanities*

*–*Sociology, Economics, History, Philosophy, Ethics,*Government, Politics, Business, Religion, and Relationship*(pdf). Publisher.

54. Inwood, Stephen. (2003).

*The Man Who Knew Too Much: the Strange and Inventive Life of Robert Hooke 1653-1703*(pg. 374)

*.*Pan MacMillan.

Further reading

● Anon. (1936). “Symbols for Heat and Thermodynamics” (abstract),

*Nature*138: 281, Aug. 15.

● Chemical thermodynamics symbols – Old.IUPAC.org.

● Kjelstrup, Signe, Bedeaux, D., and Johannessen, E. (2008).

*Non-Equilibrium Thermodynamics for Engineers*(section: List of Symbols, pgs. 245-). World Scientific.

● Miller, Jeff. (2009).

*Earliest Uses of Various Mathematical Symbols.*Gulf High School.

External links

● Table of mathematical symbols by introduction date – Wikipedia.