Structure of the analytical theories of heat

de Oliveira, Mário J.

doi:10.1590/1806-9126-RBEF-2022-0008

Mário J. de Oliveira ^** Correspondence email address: oliveira@if.usp.br

Universidade de São Paulo, Instituto de Física, Rua do Matão, 1371, 05508-090 São Paulo, SP, Brasil.

http://orcid.org/0000-0002-4139-1445

* Correspondence email address: oliveira@if.usp.br

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Figure 1
A small Carnot cycle in the pressure-volume diagram. The lines ab and dc represent isothermal processes whereas the lines ad and bc represent process without the intervention of heat. The figure is based on figure 3 of the Clapeyron paper of 1834 [²⁷[27] E. Clapeyron, Journal de l’École Royale Polytechnique 14, 153 (1834).] and on the similar figure 2 of the Clausius paper of 1850 [³³[33] R. Clausius, Annalen der Physik und Chemie 79, 368, 500 (1850).].

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Figure 2
Isotherms of carbon dioxide according to experiments carried by Andrews [⁴⁵[45] T. Andrews, Philosophical Transactions of the Royal Society of London 159, 575 (1869).] for various temperatures indicated in degrees Celsius, and pressure in atmospheres. The figure is adapted from the figure 15 of Maxwell book on Theory of Heat [²⁰[20] J.C. Maxwell, Theory of Heat (Longmans, London, 1871).]. The dashed line is an addition of Maxwell and represents the boundart of the region where the liquid and its vapor coexist.

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Figure 3
According to James Thomson, the isotherm of a fluid below the critical temperature may have an S shape given by the line ABCDEFGHK, which means that the liquid transforms continuously into vapor. The straight line CG connecting the liquid and vapor states is proposed by Maxwell and describes the coexistence of liquid and vapor. Adapted from figure 16 of Maxwell book on Theory of Heat [²⁰[20] J.C. Maxwell, Theory of Heat (Longmans, London, 1871).].

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Figure 4
A small Carnot cycle ABCD in the pressure-volume diagram. BC and DA are isothermal and AB and CD are adiabatic. The areas of the rectangles AKPk, ALQl, AMRm, and ANSn are equal to each other and equal to the parallelogram ABDC. The figure is based on figure 24 of the Maxwell book [²⁰[20] J.C. Maxwell, Theory of Heat (Longmans, London, 1871).].

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Figure 5
The curve AB represents a process connecting the states A and B, and the lines A

α

and B

β

are two adiabatics. The area of ABba represents the work performed and the area AB

β α

represents the heat absorbed by the body undergoing the process AB. The line CBD is an isotherm. The figure is based on figure 26 of the Maxwell book [²⁰[20] J.C. Maxwell, Theory of Heat (Longmans, London, 1871).].

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Figure 6
An isotherm at a temperature below the critical temperature according to the van der Waals equation (). Along the S shaped line CDEFG, the liquid transforms continuously into vapor. But this cannot be realized experimentally because along DEF the body is unstable. Based on the figure 6 of the van der Waals thesis [⁴⁹[49] J.D. van der Waals, Over de Continuitet van den Gas- en Vloeistoftoestand (Sijthoff, Leiden, 1873).].

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Figure 7
The thermodynamic surface ABCD, which gives the energy U as a function of the entropy S and the volume V. The straight lines FEG and HEK belong to the plane tangent to the thermodynamics surface at the point E. The inclinations of the lines FEG and HEK are related to the temperature T and to the pressure p, respectively. The thermodynamic surface remains above the plane tangent to any point of the surface, which is equivalent to say that it has the convexity property.

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Figure 8
Thermodynamic surface displaying the coexistence of three phases, solid, liquid, and vapor, represented by the plane triangle SLV. Any point of the triangle correspond to the same temperature and the same pressure. It also displays the coexistence of two phases, represented by the dashed straight line segments. Any point of a certain line correspond to the same temperature and pressure. The point C is the critical point where two phases become identical.

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Figure 9
Top panel. Massieu handwriting [¹³⁶[136] F. Massieu, Exposé des principes fondamentaux de la théorie mécanique de la chaleur, 1873; https://iris.univ-lille.fr/handle/1908/4877.
https://iris.univ-lille.fr/handle/1908/4... ] of 1873 for the first law of thermodynamics in differential form,

d Q = d U + A d

T_e. The symbol T for work is a French cursive capital letter ‘T’, and A is the heat equivalent of work. Bottom panel. Illustration of the first law in a secondary school book of 1965 [¹³⁷[137] T.N.O. Folmer-Johnson, Elementos de Termologia (Nobel, São Paulo, 1965).].

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Figure 10
Illustration of a Legendre transformation. (a) Energy U versus entropy S, (b) temperature

T = \partial U / \partial S

versus S, (c) entropy

S = - \partial F / \partial T

versus T, (d) free energy F versus T. The curves in (a) and (d) are Legendre transforms of each other, and both hold the convexity property. The functions shown in (b) and (c) are inverse of each other. The full circle in (d) indicates that the left and right derivatives are distinct. The dot lines are guides to the eyes.

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Figure 10
Illustration of a Legendre transformation. (a) Energy U versus entropy S, (b) temperature

T = \partial U / \partial S

versus S, (c) entropy

S = - \partial F / \partial T

versus T, (d) free energy F versus T. The curves in (a) and (d) are Legendre transforms of each other, and both hold the convexity property. The functions shown in (b) and (c) are inverse of each other. The full circle in (d) indicates that the left and right derivatives are distinct. The dot lines are guides to the eyes.

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Table 1
Notations for temperature θ, absolute temperature T, heat Q, work W, entropy S, energy U, Helmholtz free energy F, enthalpy H, Gibbs free energy G. An asterisk* indicates that the author mentions the Legendre transformation.

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Table 2
The author and the abbreviated title of the work that we have analyzed together the year of its publication and references.

(1)

(2)

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author	θ	T	Q	W	S	U	F	H	G	year	ref.
Poisson	θ		q							1823	[²²[22] S.D. Poisson, Annales de Chimie et de Physique 23, 337 (1823).]
Clapeyron	t		Q							1834	[²⁷[27] E. Clapeyron, Journal de l’École Royale Polytechnique 14, 153 (1834).]
Clausius	t		Q			U				1850	[³³[33] R. Clausius, Annalen der Physik und Chemie 79, 368, 500 (1850).]
Clausius	t	T	Q	W		U				1854	[³⁸[38] R. Clausius, Annalen der Physik und Chemie 93, 481 (1854).]
Clausius		T	Q	W	S	U				1865	[³⁹[39] R. Clausius, Annalen der Physik und Chemie 125, 353 (1865).]
Massieu	t	T	Q		S	U		$U^{'}$		1869	[⁴¹[41] F. Massieu, Comptes Rendus 69, 858, 1057 (1869).]
Maxwell	t	t	h	w	$ϕ$	e				1871	[²⁰[20] J.C. Maxwell, Theory of Heat (Longmans, London, 1871).]
Gibbs		t	H	W	$η$	$ϵ$				1873	[⁵⁸[58] J.W. Gibbs, Transactions of the Connecticut Academy of Arts and Sciences 2, 309 (1873).]
Massieu	t	T	Q	T	S	U	$- H$	$U^{'}$	$- H^{'}$	1876	[⁴²[42] F. Massieu, Mémoires Présentés par Divers Savants a l’Académie de Sciences de l’Institut Nationale de France 22, 2 (1876).]
Gibbs		t	Q	W	$η$	$ϵ$	$ψ$	$χ$	$ζ$	1876	[⁶⁰[60] J.W. Gibbs, Transactions of the Connecticut Academy of Arts and Sciences 3, 108 (1876).]
Helmholtz		$ϑ$	Q	W	S	U	$𝔉$			1882	[⁹³[93] H. von Helmholtz, Sitzungsberichte der Königlich preussischen Akademie der Wissenschaften zu Berlin XXVI, 1 (1882), p. 72.]
Duhem		T	Q	T	S	U	F		$Φ$	1886	[⁸⁷[87] P. Duhem, Le Potentiel Thermodynamique et ses Applications à la Mécanique Chimique et à l’Étude des Phénomènes Électriques (Hermann, Paris, 1886).]
Bertrand	t	T	Q		S	U	$- H$		$- H^{'}$	1887	[¹¹⁷[117] J. Bertrand, Thermodynamique (Gauthier-Villars, Paris, 1887).]
Poincare	t	T	Q	$τ$	S	U	$- H$		$- H^{'}$	1892	[¹¹⁸[118] H. Poincaré, Thermodynamique (Georges Carré, Paris, 1892).]
Nernst		T	Q		S	U	A			1893	[¹⁰¹[101] W. Nernst, Theoretische Chemie vom Standpunkte der Avogadro’schen Regel und der Thermodynamik (Ferdinand Enke, Stuttgart, 1893).]
Duhem	$ϑ$	T	Q	T	S	U	F		H, $Φ$	1897	[⁹⁰[90] P. Duhem, Traité Élémentaire de Mécanique Chimique Fondée sur la Thermodynamique (Hermann, Paris, 1897-1899), 4 vols.]
Planck	t	$ϑ$	Q	A	S	U	F			1897	[⁷⁸[78] M. Planck, Vorlesungen über Thermodynamik (Von Veit, Leipzig, 1897).]
Planck	t	T	Q	A	S	U	F	W		1911	[⁷⁹[79] M. Planck, Vorlesungen über Thermodynamic (von Veit, Leipzig, 1911), dritten Auflage.]
Partington	θ	T	Q	A	S	U	$Ψ$	W	$ϕ$	1913	[¹²⁶[126] J.R. Partington, A Text-Book of Thermodynamics (Constable, London, 1913).]
Planck	t	T	Q	A	S	U	F	W		1922	[⁸⁰[80] M. Planck, Vorlesungen über Thermodynamic (Gruyter, Berlin, 1922), siebente Auflage.]
Lewis
and Randall		T	q	w	S	E	A	H	F	1923	[¹⁰⁶[106] G.N. Lewis and M. Randall, Thermodynamics and the Free Energy of Chemical Substances (McGraw-Hill, New York, 1923).]
Partington	θ	T	Q	A	S	U	F	H	Z	1924	[¹²⁰[120] J.R. Partington, Chemical Thermodynamics (Constable, London, 1924).]
De Donder		T	Q	W	S	U	F	$Ψ$	H	1927	[¹¹⁰[110] T. De Donder, L’Affinité (Gauthier-Villars, Paris, 1927).]
Schottky		T	Q	A	S	U	F	W	G	1929	[¹²¹[121] W. Schottky, Thermodynamik (Springer, Berlin, 1929).]
van Lerberghe		T	Q	T	S	U	F	I	H	1931	[¹²²[122] G. van Lerberghe, Calcul des Affinités Physico-Chimiques (Gauthier-Villars, Paris, 1931).]
Guggenheim		T	q	w	S	E	F	H	G	1933	[¹²⁷[127] E.A. Guggenheim, Modern Thermodynamics by the method of Willard Gibbs (Methuen, London, 1933).]
De Donder		T	Q		S	E	F	H	G	1936	[¹¹³[113] T. De Donder, L’Affinité, première partie (Gauthier-Villars, Paris, 1936), 2 ed.]
Epstein*		T	Q	W	S	U	$Ψ$	X	$Φ$	1937	[¹³¹[131] P.S. Epstein, Textbook of Thermodynamics (Wiley, New York, 1937).]
Fermi		T	Q	L	S	U	F		$Φ$	1937	[¹²³[123] E. Fermi, Thermodynamics (Prentice-Hall, New York, 1937).]
Zemansky		T	Q	W	S	U	F	H	G	1937	[¹³²[132] M.K. Zemansky, Heat and Thermodynamics (McGraw-Hill, New York, 1937).]
Guggenheim		T	q	w	S	U	F	H	G	1949	[¹³⁰[130] E.A. Guggenheim, Thermodynamics (North-Holland, Amsterdam, 1949).]
Sommerfeld*		T	Q	W	S	U	F	H	G	1952	[¹²⁴[124] A. Sommerfeld, Vorlesungen über theoretische Physik, Thermodynamik und Statistik (Dieterische, Wisbaden, 1952), band V.]
Landau
and Lifshitz		T	Q	R	S	E	F	W	$Φ$	1958	[¹²⁵[125] L.D. Landau and E.M. Lifshitz, Statistical Physics (Pergamon Press, London, 1958).]
Callen*		T	Q	W	S	U	F	H	G	1960	[¹³³[133] H.B. Callen, Thermodynamics (Wiley, New York, 1960).]
ter Haar
and Wergeland*		T	Q	L	S	U	F	H	G	1966	[¹²⁸[128] D. ter Haar and H. Wergeland, Elements of Thermodynamics (Addison-Wesley, Reading, 1966).]
Kubo*		T	Q	W	S	U	F	H	G	1968	[¹²⁹[129] R. Kubo, Thermodynamics (North-Holland, Amsterdam, 1968).]

author	work	year	ref.
Laplace	Theory of elastic fluids	1822	[²¹[21] P.S. Laplace, Connaissance des Tems, ou des Mouvements Célestes, a l’Usage des Astronomes et des Navigateurs, pour l’an 1825 (Forgotten Books, Paris, 1825), p. 219, 302.]
Poisson	Heat of gases and vapors	1823	[²²[22] S.D. Poisson, Annales de Chimie et de Physique 23, 337 (1823).]
Carnot	Motive Power of Fire	1824	[²⁶[26] S. Carnot, Réflexions sur la Puissance Motrice du Feu (Bachelier, Paris, 1824).]
Clapeyron	Motive power of heat	1834	[²⁷[27] E. Clapeyron, Journal de l’École Royale Polytechnique 14, 153 (1834).]
Clausius	Mechanical theory of heat	1850	[³³[33] R. Clausius, Annalen der Physik und Chemie 79, 368, 500 (1850).]
Maxwell	Theory of Heat	1871	[²⁰[20] J.C. Maxwell, Theory of Heat (Longmans, London, 1871).]
van der Waals	Continuity of the Gaseous
	and Liquid States	1873	[⁴⁹[49] J.D. van der Waals, Over de Continuitet van den Gas- en Vloeistoftoestand (Sijthoff, Leiden, 1873).]
Gibbs	Equilibrium of
	heterogeneous substances	1876	[⁶⁰[60] J.W. Gibbs, Transactions of the Connecticut Academy of Arts and Sciences 3, 108 (1876).]
Planck	Treatise on Thermodynamics	1897	[⁷⁸[78] M. Planck, Vorlesungen über Thermodynamik (Von Veit, Leipzig, 1897).]
Duhem	Chemical Mechanics	1897	[⁹⁰[90] P. Duhem, Traité Élémentaire de Mécanique Chimique Fondée sur la Thermodynamique (Hermann, Paris, 1897-1899), 4 vols.]
Nernst	New Heat Theorem	1906	[¹⁰³[103] W. Nernst, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen Jahre 1906, 1 (1906).]
De Donder	Affinity	1927	[¹¹⁰[110] T. De Donder, L’Affinité (Gauthier-Villars, Paris, 1927).]
Prigogine	Irreversible Phenomena	1945	[¹¹⁴[114] I. Prigogine, Étude Thermodynamique des Phénomènes Irreversibles (Desoer, Liège, 1947).]

F = \sum_{i} m_{i} [ε_{i} + T (c_{i} - h_{i} - c_{i} \ln T + a_{i} \ln \frac{m_{i}}{V})] .

[1] * Correspondence email address: oliveira@if.usp.br