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

“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.

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,

Etymologies

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

“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,

“[*s* represents] the heat liberated (set free) or absorbed”

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

In his 2010 "Why Are

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).

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

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

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

3. De Donder, T. (1936).

4. Laidler, Keith J. (1993).

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

6. Lagrange, Joseph. (1788).

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.”

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

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

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

(b) Thims, Libb. (2007).

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

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

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

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

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

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

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

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).

16. De Donder, Theophile. (1934).

17. Sackur, Otto. (1912).

18. Duhem, P. (1897).

19. Planck, Max. (1897).

20. Cahan, David (1993).

21. Carnot, Sadi. (1824).

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

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.),

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

24. Henri-Victor Regnault – NNDB.

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

26. Clausius, Rudolf. (1879).

27. (a) Clausius, Rudolf. (1865). “On Several Forms of the Fundamental Equations of the Mechanical Theory of Heat (Ninth Memoir).”

(b) Clausius, R. (1865).

(c) See: Entropy (etymology).

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

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

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

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

31. Haber, Fritz. (1905).

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

(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).

33. Nernst, Walther. (1918).

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

(b) Bernoulli, Daniel. (1738).

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)

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

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

(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”),

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

(c) Hoffmann, Dieter. (2008).

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).

41. Adkins, Clement J. (1967).

42. Recorde, Robert. (1557).

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

(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).

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

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

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?”

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).

(b) Stedall, Jacqueline. (2012).

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

52. Crick, Francis. (1994).

53. Thims, Libb. (2019).

54. Inwood, Stephen. (2003).

Further reading

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

● Chemical thermodynamics symbols – Old.IUPAC.org.

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

● Miller, Jeff. (2009).

External links

● Table of mathematical symbols by introduction date – Wikipedia.