Symbols

Symbols (cartoon)
A differential equation symbols cartoon by Nick Kim. [49]
In thermodynamics, 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 the symbols in which it is written.”
Galileo Galilei (1623), The Assayer [52]

“I know several men who see all nature in symbols, 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 thermodynamics was 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 normal algebraic symbols but in addition they were liberally sprinkled with stars, daggers, and circles so as to stretch even the most powerful of minds.”
Brian Smith (1973), Basic Chemical Thermodynamics [1]
Table
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:


SymbolMeaningFormulaDatePerson





Δ
(link)
Fire or Heat
450BCEgyptians, 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)
1557Robert Recorde. [42]
>
<


c.1600Thomas 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.1655John Wallis [54]



1670John Wallis [51]
\intIntegral
1675Gottfried 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]
 d \, Derivative
1675Gottfried Leibniz: in the same notebook, Leibniz employed the use of d for differentials, from the Latin word differentia. [32]
ϕ (phi)
Phlogiston
1704Georg Stahl: [24]
T
Temperature
?
VVolume
?
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]
nNumber of particles
1738Daniel Bernoulli: [34]
f(x)
Function
1755Leonhard Euler: was the first to write f(x) to denote the function f applied to the argument x. [44]
i
Imaginary number
1755Leonhard Euler: [44]
e
Natural log base
1755Leonhard Euler: [44]
Summation
1755Leonhard Euler: [44]
Dart
(Reaction arrow)
Cullen's reaction diagram (modern view)
1757William 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)
Bracket equates to: AB or A≡B (modern) 1757William Cullen: “by the mark { I mean them united to another” [9]
Partial differential"curly d"1770Marquis de Condorcet
ABChemical unionAB1775Torbern Bergman
TVis viva (kinetic energy)\tfrac{1}{2} mv^2 1788Joseph Lagrange: [6]
 \int_{a}^{b} \,Definite integral
1822Joseph 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
1824Sadi 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)
1824Sadi Carnot: [21]
QHeat
1834Emile 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)?
RGas constant
1834Emile Clapeyron: (possibly used earlier?) [21][23]
UForce function
1835William 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)
Quaternion operator1837William 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)
TVis viva\sum \frac{1}{2}mv^2?Rudolf Clausius:
JErgalJ_0 - \int_{t_0}^{t} \sum (Xdx +Ydy + Zdz)?Rudolf Clausius: [26]
UEnergyT + 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]
WWorkF (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."
SEntropy\frac{dQ}{T}1865Rudolf Clausius: [27]
U(Internal energy)
1865Francois Massieu: [3][12]
U'(Heat content)
1865Francois Massieu: [3][12]
Ψ (psi) Characteristic function~ \frac{-U+tS}{t} ~or \frac{-\psi}{t} \,1869 Francois Massieu: [12]
Ψ' (psi prime)
~ \frac{-U+tS-pv}{t} ~or \frac{-\zeta}{t} \,1869Francois Massieu: [12]
EEntropy
1872Ludwig Boltzmann: [36]
HHeat
1873Willard Gibbs: "heat received by the body in passing from one state to another". [5]
ε (epsilon) Energy
1873Willard Gibbs: "the energy of body in a given state". [5]
η (eta)Entropy
1873Willard Gibbs: "the entropy of a body in a given state". [5]
 \bar{d} \,(d-hat)
Inexact differential
1875Carl Neumann: [28]
χ (chi)(Heat content)ε + pv
1876Willard Gibbs: "heat function at constant pressure" [3]
Ψ (psi) (Helmholtz function) ε – tη 1876Willard Gibbs: “the force function for constant temperature” [11]
ζ (zeta)(Gibbs function)ε – tη + pv
1876Willard Gibbs:
μ (mu)Potential
(Chemical potential)

1876Willard 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]
PPotential difference (of a galvanic cell) or Pressure
1877Hermann Helmholtz: [20]
(Electromotive force)~ P_k - P_a ~1877Hermann Helmholtz: "P is the potential difference (a and k referring to the anode and cathode, respectively) in a concentration cell." [20]
JMechanical equivalent of heat
1882Hermann Helmholtz: [20]
θAbsolute temperature"temperature reckoned from –273° C"1882" "
SEntropy
1882Hermann Helmholtz: [20]
UTotal energy
1882Hermann Helmholtz: "the total energy (gesammt-energie) of the system [19]
Free energy symbol (Helmholtz), FFree energy U JTS1882Hermann Helmholtz: "the free energy (frieie energie) of the system." [20]
Bound energy (symbol), B

Bound energyJTS
(J=1 in modern joule units)
1882Hermann Helmholtz: "the latent energy (gebundene energie) of the system." [19]
 \leftrightarrows \,Reversible reaction
1884Jacobus van't Hoff [45]
A–B
A=B
Uniting dashes
(chemical bond)
represent the "force lines" associated with atomic valencies.1893Walther Nernst
lnNatural logarithm
1893Used by Irving Stringham (1847-1909) in Uniplanar Algebra (Cajori vol. 2, page 107). [47]
UEnergy
1897Max Planck: "the energy of a body or system of bodies." [19]
Φ (phi)Entropy
1897Max Planck: [19]
FFree energy U TΦ1897Max 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)~ \Phi  - \frac{U + pV}{T} ~or~ -\frac{G}{T} ~1897Max Planck: “multiplying Φ by – T, we get the thermodynamic potential at constant pressure U + pV – TΦ” [19][2]
U(Internal energy)
1897Pierre Duhem: [18][3]
(Heat content)
1897Pierre Duhem: [3]
~ \mathfrak{F} ~(Helmholtz free energy)
1897Pierre Duhem: [18][3]
Φ (phi)(Gibbs potential)
1897Pierre Duhem: [18][3]
hUniversal constant
(Planck constant)
 6.55 \cdot 10^{-27} erg \cdot sec \ 1900Max Planck: "universal constant"[38]
kUniversal constant
(Boltzmann constant)
 1.35 \cdot 10^{-16} \frac{erg}{deg} \ 1900Max Planck: "universal constant" [38]
AReaction energy
1905Fritz Haber: [31]
QReaction heat
1905Fritz Haber: [31]
qLatent heat of reaction
1905Fritz Haber: "here we use minus q to designate the heat which is used up or in other words, becomes latent (or bound?)." [31]
SEntropy\frac{q}{T}1905Fritz Haber: [31]
UTotal energy
1905Fritz Haber: [31]
NAvogadro's constant 70.5 \cdot 10^{22} \ 1909Jean Perrin: “the invariable number N is a universal constant, which may appropriately be designated Avogadro’s constant.” [37]
HEnthalpyE + PV1909 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]
UEnergy
1912Otto Sackur: [17]
HHeat contentU + pv
1912Otto 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 functionU TS1912Otto Sackur: [17]
ζ (zeta)Thermodynamic potentialU TS + pv1912Otto 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]
UEnergy content
1917Walther Nernst: [33]
QHeat flow
1917" "
AWorkGerman “Arbeit”, meaning work,1917" "
SEntropy
1917" "
\oint \ Contour integral sign
Line integral
Path integral
Curve integral

1917Arnold Sommerfeld [50]
EInternal energy
1923Lewis and Randall: "[E is] the energy contained within a system, or its internal energy, a property of the system." [10]
AHelmholtz free energyE TS1923" "
F
Free energy(E + PV) – TS
H
TS
1923" "
A
Work
1924James Partington: [15]
dOrdinary 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]
UIntrinsic energy
1924" "
H
Enthalpy; Total heat; Heat functionU + pV1924[" "]: “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 energyU TS1924" "
Z
Thermodynamic potentialU TS + pV1924[" "]: “Z is called the thermodynamic potential by analogy with the potential function in dynamics: ϕ1 – ϕ2 = work” [15]
circle integral (symbol)Cycle integral
(circle integral)
closed path integral
 \sum \frac{Q}{T} \,1924Used 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.  \Delta G = G_f - G_i \,)
1933Edward Guggenheim - defined in formulaic terms [11]; symbols used previously by Gilbert Lewis (1923).
H
Heat contentE + PV1933" "
FHelmholtz free energyE TS1933[" "]: “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 energyH TS1933Edward 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]
UInternal energy
1934Theophile de Donder: [16]
EInternal energy
1936Theophile de Donder: [3]
AAffinity~  - \left ( {\partial G\over \partial \xi} \right )_{p,T} ~1936" "
Ξ or B (?)
Exergy
1956Zoran Rant: [39]
SEntropy\frac{d Q_{rev}}{T}1968Clement Adkins: "[the rev subscript is used] to emphasize that the equality holds for reversible changes only." [41]
ETotal exergy
2004 Norio Sato: “we shall use the symbol E (capital epsilon) to express total exergy. [40]
εMolar exergy
2004Norio 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):

~ c_p ~Specific heat at constant pressure

[19]
~ c_v ~Specific heat at constant volume

[19]
γ (gamma)
~ \frac{c_p}{c_v} ~1897Max Planck (?): [19]
JMassieu function \frac{-F}{t} \,1960Herbert Callen (?): [13]
YPlanck function \frac{-G}{t} \,1960Herbert Callen (?): [13]











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

Greek symbols (no s)
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 of H for enthalpy because that is heat related, but where does S for entropy come from?’ I was stumped and decided to do some historical research to track down not only the origin of S but 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

“[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 Table
De Donder's symbol table
Battino's 2001 Symbol Table
Battino's symbol table
Belgian 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
IUPAC chemical thermodynamics symbols (2007)
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 (θ∆ics - Encyclopedia of Humanthermodynamics), earth (θ∆ics - Encyclopedia of Humanthermodynamics), air (θ∆ics - Encyclopedia of Humanthermodynamics), water (θ∆ics - Encyclopedia of Humanthermodynamics)] 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.
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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.
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(b) Miller, Jeff. (2009). “Earliest Uses of Symbols of Calculus”, Jeff560.tripod.com.
(c) Plus and minus signs – Wikipedia.
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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).
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(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.
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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.

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