
An example 1939 symbol table, compiled by Ralph Fowler and Edward Guggenheim, for the main six thermodynamic functions found commonly in modern textbooks of the day; the latter of which μ (mu) being the chemical potential. [2]

An example 1939 symbol table, compiled by Ralph Fowler and Edward Guggenheim, for the main six thermodynamic functions found commonly in modern textbooks of the day; the latter of which μ (mu) being the chemical potential. [2]

In thermodynamics, characteristic function notation table lists the various notation symbol schemes used in signifying the main characteristic functions of thermodynamics: the energy (U) of a body, the entropy (S) of the body, the enthalpy (H) or heat content of a body, the free energy (available energy of useful energy) of a body for isochoric-isobaric processes or Helmholtz free energy (F), and the free energy (available energy or useful energy) of a body for isobaric-isothermal processes or Gibbs free energy (G).

Symbol usage conflicts
The term "characteristic functions" is a term introduced in 1869 by French engineer Francois Massieu and for these functions, Massieu employed a Greek symbol notation scheme to represent the various thermodynamic functions characteristic of bodies. In 1873, this Greek symbol notation scheme was adopted by American engineer Willard Gibbs.

Prior to this, however, in 1865, the first two main thermodynamic functions, energy and entropy, were introduced using the English letter symbols U and S, respectively, by German physicist Rudolf Clausius. This was the start of a symbol use inconsistency that would last for at least a century, well into the early 21st century, with authors still inconsistent on their usage of the symbols A, short for arbeit (German for "work"), or F, short for frieie energie (German for "free energy"), to represent the Helmholtz free energy (U – TS).

Out of this conflict, into the 1930s, the symbol usage issue had become so chaotic and varied, that new thermodynamics textbooks were forced to compile notation tables and put this at the beginning of each book. The first of these notation tables seems to have been one compiled by Edward Guggenheim (1933), followed by a second made by Theophile de Donder (1936), and a third made by Ralph Fowler (1939).

In 1958, American geologist Hugh McKinstry, supposedly, wrote an extensive article on subject of the varieties of symbol use of the characteristic functions in thermodynamics; the article, supposedly, does not deal with thermodynamics directly, but is a commentary on the use of symbols. [4]

Function symbols table
An updated synthesis of these various notation tables (Guggenheim, de Donder, and Fowler) are regrouped together below, listed in chronological order:

			Energy U	Entropy S	Enthalpy U + PV	Helmholtz free energy U – TS	Gibbs free energy U + PV – TS
------------------------------------------------------------------		-------------------------------------------------------------------------------				------------------------------------------------------------	------------------------------------------------
					Heat content	Constant: V, P Free energy (Work function)	Constant: T, P Free energy (Available energy) (Useful energy)
Rudolf Clausius	1865	*Mechanical Theory of Heat*	U	S
Francois Massieu	1869	“On the Various Functions Characteristic of Fluids”	U		U’ (U prime)	– tψ (name)	– tψ’ (name)
James Maxwell	1871	Theory of Heat	e	φ (phi)
Willard Gibbs	1876	*On the Equilibrium of Heterogeneous Substances*	ε (epsilon)	η (eta)	χ (chi)	ψ (psi)	ζ (zeta)
Hermann Helmholtz	1882	"On the Thermodynamics of Chemical Processes"	U	S		, F German: ‘frieie energie’
Walther Nernst	1893	Theoretical Chemistry from the Standpoint of Avogadro’s Rule and Thermodynamics	U
Pierre Duhem	1897	Thermodynamique et Chimie	U			F	Φ (Phi)
Max Planck	1897	Treatise on Thermodynamics	U	Φ (Phi)	H	F	– Tψ (name)
Fritz Haber	1905	Technical Thermodynamics of Gas Phase Reactions	U	S		A German: ‘arbeit’ (work)
Otto Sackur	1912	Thermochemistry and Thermodynamics	U		H	ψ (psi)	ζ (zeta)
Percy Bridgman	1914	"A Complete Collection of Thermodynamic Formulas"	E	S	H	ψ (psi)	Z
Gilbert Lewis	1923	*Thermodynamics*	E	S	H	A	F
James Partington	1924	Chemical Thermodynamics	U	S	H	F	Z
Theophile de Donder	1926	L'Affinite	U	S	ψ (psi)	F	H
John Butler	1928	The Fundamentals of Chemical Thermodynamics	E	S	H		F
Walter Schottky	1929	Thermodynamik	U	S	H	F	G
Ralph Fowler	1929	Statistical Mechanics	E	S	H	– Tψ	– TΦ
Hermann Ulich	1930	Chemische Thermodynamik	U		W	F	G
Georges Lerberghe	1931	Calcul des Affinites Physico-Chimiques	U		I	F	H
Edward Guggenheim	1933	*Modern Thermodynamics*	E	S	H	F	G
Theophile de Donder	1936	Thermodynamic Theory of Affinity	E	S	H	F	F
Mark Zemansky	1937	*Heat and Thermodynamics*	U	S	H	F	G
Paul Epstein	1937	Textbook of Thermodynamics	U	S	χ (chi)	ψ (psi)	Φ (Phi)
Johannes Bronsted	1938	Physical Chemistry	E	S	H	F	G
Joseph Keenan	1941	Thermodynamics	E	S	H	ψ (psi)	Z
Frederick Rossini	1950	Chemical Thermodynamics	E	S	H		F, u
Juliana Goates and Bevan Ott	2000	Chemical Thermodynamics	U	S	H	A	G
IUPAC [3]	2007	Chemical Thermodynamics Symbol Table	U	S	H	A, F	G

An important function not listed in this table, owing to its consistency is ‘bound energy’, symbol ‘B’, equal to the function ‘TS’, which is defined, according to Herman Helmholtz (1882), the coiner of this term, as “expressing the mechanical equivalent of that quantity of heat which must be conveyed into a body at temperature T in order to raise its entropy to the value S.” Hence, total energy (or total energy plus pressure-volume work energy) less bound energy is what is called ‘free energy’ (Helmholtz or Gibbs, respectively):

References
1. (a) Guggenheim, Eduard, A. (1933). Modern Thermodynamics by the Methods of Willard Gibbs (pg. 28). London: Methuen & Co.
(b) De Donder, Theophile. (1936). Thermodynamic Theory of Affinity: A Book of Principles (pg. xvi). Stanford University Press.
2. Fowler, Ralph and Guggenheim, Eduard A. (1939). Statistical Thermodynamics: a Version of Statistical Mechanics for Students of Physics and Chemistry (pg. x). Cambridge University Press.
3. 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.
4. (a) Vanserg, Nicolas. (1958). “Mathmanship”, The American Scientist, 46: 94A-98A; in: Essays on Thinking and Writing in Science, Engineering, and Business, (pgs. 110-), W.C. Brown, 1963.
(b) Nicolas Vanserg (pen name), supposedly, is Hugh McKinstry, a professor of geology at Harvard for many years.
(c) McSween, Harry Y, Richardson, Steven M., Uhle, Maria. (2003). Geochemistry: Pathways and Processes (pg. 45). Columbia University Press.