# From Steam Engines to Chemical Reactions: Gibbs’ Contribution to the Extension of the Second Law

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## Abstract

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## 1. Introduction

“Thermodynamics gives me two strong impressions: first of a subject not yet complete or at least of one whose ultimate possibilities have not yet been explored, so that perhaps there may still be further generalizations awaiting discovery; and secondly and even more strongly as a subject whose fundamental and elementary operations have never been subject to an adequate analysis.”[1] (p. 6)

“C’est au professeur W. Gibbs que revient l'honneur d'avoir, par l’emploi systématique des méthodes thermodynamiques, créé une nouvelle branche de la science chimique dont l'importance, tous les jours croissante, devient aujourd'hui comparable à celle de la chimie pondérale créée par Lavoisier.”[2] (p. 6)

“This matter is still of relevant importance nowadays and its interest is not only historical. Only a small part of the almost immense quantity of results—herewith discussed and suggested—has been exploited yet. There are still hidden treasures of the greatest variety and importance for either theoretical or experimental researchers.”[3] (p. 6) (Translated by the authors)

“Gibbs succinct and abstract style and unwillingness to include examples and applications to particular experimental situations made his work very difficult to read. Famous scientists such as Helmholtz and Planck developed their own thermodynamic methods in an independent fashion and remained quite unaware of the treasures buried in the third volume of Transactions of the Connecticut Academy of Arts and Sciences.”[4] (p. 339)

## 2. Historical Hints

## 3. Geometrical Foundations of Gibbs’ Thermodynamic Model

“This equation evidently signifies that if ε be expressed as function of v and η, the partial differential coefficients of this function taken with respect to v and to η will be equal to −p and to t respectively.”[6] (p. 2)

“An equation giving ε in terms of η and v, or more generally any finite equation between ε, η and v for a definite quantity of any fluid, may be considered as the fundamental thermodynamic equation of that fluid, as from it by aid of Equations (1)–(3) (numberings according to the present paper) may be derived all the thermodynamic properties of the fluid (so far as reversible processes are concerned).”[6] (p. 2)

“Now the relation between the volume, entropy, and energy may be represented by a surface, most simply if the rectangular coordinates of the various points of the surface are made equal to the volume, entropy, and energy of the body in its various states. It may be interesting to examine the properties of such a surface, which we will call the thermodynamic surface of the body for which it is formed.To fix our ideas, let the axes of v, η and ε have the directions usually given to the axes of X, Y, and Z (v increasing to the right, η forward, and ε upward).”[7] (p. 34)

“When the body is not in a state of thermodynamic equilibrium, its state is not one of those, which are represented by our surface.”[7] (p. 35)

“The section of the surface by a vertical plane parallel to the meridian (i.e., the plane parallel to (Z, Y) in Figure 2) is a curve of constant volume. In this curve the temperature is represented by the rate at which the energy increases as the entropy increases, that is to say, by the tangent of the slope of the curve.”[13] (p. 197)

## 4. Thermodynamic Equilibrium

#### 4.1. Thermodynamic Equilibrium “à la Gibbs”

“As the discussion is to apply to cases in which the parts of the body are in (sensible) motion, it is necessary to define the sense in which the word energy is to be used. We will use the word as including the vis viva (i.e., kinetic energy in pre–modern language) of sensible motions [15].”[7] (p. 39)

“If all parts of the body are at rest, the point representing its volume, entropy, and energy will be the center of gravity of a number of points upon the primitive (thermodynamic) surface. The effect of motion in the parts of the body will be to move the corresponding points parallel to the axis of ε, a distance equal in each case to the vis viva of the whole body.”[7] (p. 39)

“…the body (is) separated from the medium by an envelop which will yield to the smallest differences of pressure between the two, but which can only yield very gradually, and which is also a very poor conductor of heat. It will be convenient and allowable for the purposes of reasoning to limit its properties to those mentioned, and to suppose that it does not occupy any space, or absorb any heat except what it transmits, i.e., to make its volume and its specific heat 0. By the intervention of such an envelop, we may suppose the action of the body upon the medium to be so retarded as not sensibly to disturb the uniformity of pressure and temperature in the latter.”[7] (p. 39)

“Now let us suppose that the body having the initial volume, entropy, and energy, ${v}^{\prime},{\eta}^{\prime}\text{and}{\epsilon}^{\prime}$ is placed (…) in a medium having the constant pressure P and temperature T, and by the action of the medium and the interaction of its own parts comes to a final state of rest in which its volume, etc., are ${v}^{\u2033},{\eta}^{\u2033}\text{and}{\epsilon}^{\u2033}$; we wish to find a relation between these quantities.”[7] (p. 40)

#### 4.2. Energy Variation in a Dissipative Process

- Kinetic energy is dissipated inside the system itself; hence internal energy increases accordingly. In such case, the total energy of the system does not change since the internal energy gain compensates the kinetic energy loss. Being ${\epsilon}_{k}^{\prime}$ the kinetic energy of non-equilibrium initial state and ${\epsilon}_{i}^{\prime \prime}\text{and}{\epsilon}_{i}^{\prime}$ the internal energies in the final and initial states, respectively, the previous energy balance translates as:$${\epsilon}_{i}^{\prime \prime}={\epsilon}_{i}^{\prime}+{\epsilon}_{k}^{\prime}$$Hence, from Equation (4):$${\epsilon}^{\u2033}={\epsilon}^{\prime}$$
- Kinetic energy is dissipated outside the system (i.e., towards the medium through the envelope). Hence the total energy of the system decreases and the following inequality holds valid:$${\epsilon}^{\u2033}<{\epsilon}^{\prime}.$$

“As the sum of the energies of the body and the surrounding medium may become less, but cannot become greater (this arises from the nature of the envelope supposed), we have ${E}^{\u2033}+{\epsilon}^{\u2033}\le {E}^{\prime}+{\epsilon}^{\prime}.$”[7] (p. 40)

#### 4.3. Entropy Variation in a Dissipative Process

#### 4.4. Volume Invariance

#### 4.5. Getting to a “Geometrical” Synthesis

- Equation (6) is arranged to give:$$-{E}^{\u2033}+T{H}^{\u2033}-P{V}^{\u2033}=-{E}^{\prime}+T{H}^{\prime}-P{V}^{\prime};$$
- Equation (12) is used as it is:$${E}^{\u2033}+{\epsilon}^{\u2033}\le {E}^{\prime}+\epsilon \prime ;$$
- both members of Equation (14) are multiplied by $\u2013T$$$-T{H}^{\u2033}-T{\eta}^{\u2033}\le -T{H}^{\prime}-T{\eta}^{\prime};$$
- both members of Equation (15) are multiplied by $P$$$P{V}^{\u2033}+P{v}^{\u2033}=P{V}^{\prime}+P{v}^{\prime},$$

“Now the two members of this equation (i.e.,Equation (19)) evidently denote the vertical distances of the points $\left({v}^{\u2033},{\eta}^{\u2033},{\epsilon}^{\u2033}\right)$ and $\left({v}^{\prime},{\eta}^{\prime},{\epsilon}^{\prime}\right)$ above the plane passing through the origin and representing the pressure P and temperature T. And the equation expresses that the ultimate distance is less or at most equal to the initial. It is evidently immaterial whether the distances be measured vertically or normally, or that the fixed plane representing P and T should pass through the origin; but distances must be considered negative when measured from a point below the plane.”[7] (p. 40)

#### 4.6. Stability Criteria

- (1)
- The thermodynamic surface is all above the tangent plane (Figure 4). In this case the point common to the surface and the plane corresponds to a stable state. In Gibbs’ own words:“Now, if the form of the surface be such that it falls above the tangent plane except at the single point of contact, the equilibrium is necessarily stable.”[7] (p. 42)

“… for if the condition of the body be slightly altered, either by imparting sensible motion to any part of the body, or by slightly changing the state of any part, or by bringing any small part into any other thermodynamic state whatever, or in all of these ways, the point representing the volume, entropy, and energy of the whole body will then occupy a position above the original tangent plane, and the proposition above enunciated (Equation (19)) shows that processes will ensue which will diminish the distance of this point from that plane, and that such processes cannot cease until the body is brought back into its original condition, when they will necessarily cease on account of the form supposed of the surface.”[7] (p. 42)

- (2)
- If the surface is all underneath the tangent plane, the common point to the surface and the plane corresponds to an unstable state.“On the other hand, if the surface have such a form that any part of it falls below the fixed tangent plane, the equilibrium will be unstable. For it will evidently be possible by a slight change in the original condition of the body (that of equilibrium with the surrounding medium and represented by the point or points of contact) to bring the point representing the volume, entropy, and energy of the body into a position below the fixed tangent plane, in which case we see by the above proposition that processes will occur which will carry the point still farther from the plane, and that such processes cannot cease until all the body has passed into some state entirely different from its original state.”[7] (p. 42)

“Von diesem Gesichtspunkt aus betrachtet, befindet sich z. B. ein Knallgasgemenge nur im labilen Gleichgewicht, weil die Hinzufügung einer geeigneten, aber beliebig kleinen Menge Energie (z. B. eines elektrischen Funkens, dessen Energie im Vergleich zu der des Gemenges verschwindend klein sein kann) eine endliche Zustandsänderung, also einen Uebergang zu einem Zustand grösserer Entropie, zur Folge hat.”[18] (p. 474)

- (3)
- The third circumstance occurs when the surface does not “anywhere fall below the fixed tangent plane, nevertheless meets the plane in more than one point”. Gibbs names this instance neutral equilibrium [7] (p. 42) due to its intermediate character between the cases already considered. He further specifies this definition by remarking “if any part of the body be changed from its original state into that represented by another point in the thermodynamic surface lying in the same tangent plane, equilibrium will still subsist” [7] (p. 42).

## 5. Dealing with Isolated Systems

“The business of ‘isolating’ a system perhaps requires more discussion. What shall we say to one who claims that no system can ever be truly ‘isolated’, for he may maintain that the velocity of electromagnetic disturbances within it or the value of the gravitational constant of the attractive force between its parts is determined by all the rest of the matter in the universe? To this all that can be answered is that this is not the sort of thing contemplated; even if an observer stationed on the surface enclosing the system is able to see other objects outside the surface and at a distance from it, the system may nevertheless still be pronounced isolated. It would probably be impossible under these conditions to give a comprehensive definition of what is meant in thermodynamics by ‘isolation’, but the meaning would have to be conveyed by an exhaustive cataloguing of all the sorts of things which might conceivably occur to an observer stationed on the enclosing surface, with an explicit statement that this or that sort of thing does or does not mean isolation.”[1] (p. 81)

“By homogeneous is meant that the part in question is uniform throughout, not only in chemical composition, but also in physical state.”[8] (p. 63)

“Le mot hétérogène a été traduit par chimique parce qu'en français le mot hétérogène sert habituellement indiquer une différence d'état physique tandis qu’il exprime ici une différence d’état chimique.”[2] (p. 1)

## 6. Gibbs’ principle Applied to Homogeneous Systems

- For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations of the state of the system, which do not alter its energy, the variation of its entropy shall either vanish or be negative. If ε denote the energy, and η the entropy of the system, and we use a subscript letter after a variation to indicate a quantity of which the value is not to be varied, the condition of equilibrium may be written:$${(\delta \eta )}^{}{}_{\epsilon}\le 0,$$
- For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations in the state of the system, which do not alter its entropy, the variation of its energy shall either vanish or be positive. This condition may be written:(δε)
_{η}≥ 0”. [8] (p. 56)

“But to distinguish the different kinds of equilibrium in respect to stability, we must have regard to the absolute values of the variations. We will use ∆ as the sign of variation in those equations, which are to be construed strictly, i.e., in which infinitesimals of the higher orders are not to be neglected.

_{ε}≤ 0, i.e., (δε)

_{η}≥ 0 ”. [8] (p. 57)

“Actually, Gibbs did not claim that this statement presents a formulation of the second law. But, intuitively speaking, the Gibbs principle, often referred to as principle of maximal entropy, does suggest a strong association with the second law. Gibbs corroborates this suggestion by placing Clausius’ famous words (‘Die Entropie der Welt strebt ein Maximum zu’) as a slogan above his article. Indeed, many later authors do regard the Gibbs principle as a formulation of the second law.Gibbs claims that his principle can be seen as ‘an inference naturally suggested by the general increase of entropy which accompanies the changes occurring in any isolated material system’. He gives a rather obscure argument for this inference.”[5] (p. 52)

- (1)
- The system does not change its volume (i.e., it is mechanically isolated) and the transformation does not alter energy value. These two constraints mathematically correspond to $\delta v=0$ and $\delta \epsilon =0$, implying that Equation (29) transforms into:$$\delta {\eta}^{}\le 0.$$

- (2)
- The system does not change its volume (i.e., it is mechanically isolated) and the transformation does not alter entropy value. These two constraints mathematically correspond to $\delta v=0$ and $\delta \eta =0$ implying that Equation (29) transforms into:$$\delta {\epsilon}^{}\ge 0.$$

## 7. Towards Chemical Equilibrium

“The substances ${S}_{1},{S}_{2}\dots {S}_{n}$, of which we consider the mass composed, must of course be such that the values of the differentials shall be independent, and $d{m}_{1},d{m}_{2}\dots d{m}_{n}$ shall express every possible variation in the composition of the homogeneous mass considered, including those produced by the absorption of substances different from any initially present. It may therefore be necessary to have terms in the equation relating to component substances which do not initially occur in the homogeneous mass considered, provided, of course, that these substances, or their components, are to be found in some part of the whole given mass.”[8] (p. 63)

“the conditions of equilibrium for a mass thus enclosed are the general conditions which must always be satisfied in case of equilibrium.”[8] (p. 62)

“If we call a quantity μ_{x}, as defined by such an Equation as (21) (numberings according to the present paper), the potential for the substance S_{x}in the homogeneous mass considered, these conditions (i.e., chemical equilibrium conditions) may be expressed as follows:The potential for each component substance must be constant throughout the whole mass.”[8] (p. 65)

“Equations (39) and (40) [numberings according to the present paper] express the conditions of thermal and mechanical equilibrium, viz., that the temperature and the pressure must be constant throughout the whole mass. In Equation (41) we have the conditions characteristic of chemical equilibrium.”[8] (p. 65)

For the sake of clarity, we would like to apply this treatment to the specific case—mentioned by Gibbs in his text—of “equilibrium in a vessel containing water and free hydrogen and oxygen.”[8] (p. 63)

“This will make n + 3 independent known relations between the n + 5 variables, $\epsilon ,\eta ,v,{m}_{1},{m}_{2},\dots ,{m}_{n},t,p,{\mu}_{1},{\mu}_{2},\dots ,{\mu}_{n}$. These are all that exist, for of these variables, n + 2 are evidently independent. Now upon these relations depend a very large class of the properties of the compound considered, we may say in general, all its thermal, mechanical, and chemical properties, so far as active tendencies are concerned, in cases in which the form of the mass does not require consideration. A single equation from which all these relations may be deduced we will call a fundamental equation for the substance in question.”[8] (p. 86)

## 8. Discussion

#### 8.1. Where’s the Second Law Hidden in Gibbs’ Treatment of Equilibrium?

#### 8.1.1. Isolated and Homogeneous System with Constant Chemical Composition

#### 8.1.2. Isolated and Homogeneous System with Variable Chemical Composition

#### 8.1.3. Isolated and Heterogeneous Systems with Variable Chemical Composition

#### 8.2. The Introduction of Potential: Gibbs’ Strategy for Extending the Domain of Thermodynamics

#### 8.3. The Mathematical Structure of Gibbs’ Formal Apparatus

“Gibbs was weaving the plot of a more general mechanics: he followed the track of Analytical Mechanics, but aimed at a wider–scope mechanics, which encompassed mechanics, thermodynamics and chemistry. He did not try to describe complex thermodynamic systems by means of mechanical models: on the contrary, purely mechanical systems were looked upon as specific instances of thermodynamic ones. The relationship between Mechanics and Thermodynamics consisted of a formal analogy: the mathematical structure of Mechanics offered a formal framework for the mathematical structure of Thermodynamics.”[21] (p. 48)

## 9. Conclusions

## Author Contributions

## Conflicts of Interest

## References and Notes

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“If S represents the entropy of the system in such a state that can be named maximum state, the variation of entropy—consistent with infinitely small state change—must be δS = 0. Maximum conditions and the state of absolute maximum (i.e., of stable equilibrium) can be found when the system state satisfies the latter condition.”(p. 37) Translated by the authors
- This notation follows Gibbs’ convention used throughout his papers to express finite variations. In this particular case it holds:
δE = E″ − E′δH = H″ − H′δV = V″ − V′
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**Figure 2.**Representation of quantities t and p within Gibbs’ geometrical model: they are the slopes of two tangent lines lying on the planes (Z, Y) (i.e., (energy, entropy) at $\left(v=constant\right)$) and (Z, X) (i.e., (energy, volume) at ($\eta =constant$)), respectively. Each point of the thermodynamic surface is then associated with a couple of (t, p) values, expressed by a plane tangent to the surface at that very point.

**Figure 3.**Geometrical interpretation of a process leading a system—placed in a medium at constant temperature T and pressure P—towards equilibrium. Points A and B represent the initial and final states, respectively. A′ and B′ are their projection on a plane, passing through the origin, whose slope is defined by the couple (t, p). The length of BB′, i.e., the distance of B from the fixed plane—is shorter than AA′, i.e., the distance of A from the same plane ($BB\prime <AA\prime $).

**Figure 4.**Geometrical representation of a stable equilibrium state: the thermodynamic surface is all above the tangent plane that represents pressure and temperature of the medium wherein the system is placed. In this case, the point common to the surface and the plane corresponds to a stable equilibrium state.

Gibbs’ Scripts | Current Notation | Variable Name |
---|---|---|

$\epsilon ,E$ ^{1} | $U$ | Energy |

$\eta ,H$ ^{1} | $S$ | Entropy |

$v,V$ | $V$ | Volume |

$t$ | $T$ | Absolute Temperature |

$p$ | $P$ | Pressure |

$W$ | $W$ | Work |

$H$ ^{2} | $Q$ | Heat |

^{1}E and H are Greek scripts for capital epsilon and eta;

^{2}H is Latin scripts for capital aitch.

Type of Equilibrium | Equilibrium Conditions | Equation Number |
---|---|---|

Thermal | ${t}^{\prime}={t}^{\u2033}={t}^{\u2034}=\dots $ | (39) |

Mechanical | ${p}^{\prime}={p}^{\u2033}={p}^{\u2034}=\dots $ | (40) |

Chemical | ${{\mu}^{\prime}}_{1}={{\mu}^{\u2033}}_{1}={{\mu}^{\u2034}}_{1}=\dots $ | (41) |

${{\mu}^{\prime}}_{2}={{\mu}^{\u2033}}_{2}={{\mu}^{\u2034}}_{2}=\dots $ | ||

${{\mu}^{\prime}}_{3}={{\mu}^{\u2033}}_{3}={{\mu}^{\u2034}}_{3}=\dots $ |

**Table 3.**Correspondence between the types used in Gibbs’ writings and the current thermodynamic notation.

Gibbs’ Greek Scripts | Original Gibbs’ Definition | Current Symbols | Current Definition | Current Variable Name |
---|---|---|---|---|

$\chi $ | $\chi =\epsilon +pv$ | $H$ | $H=U+pV$ | Enthalpy |

$\psi $ | $\psi =\epsilon -t\eta $ | $A$ | $A=U-TS$ | Helmholtz’s function |

$\zeta $ | $\zeta =\epsilon -t\eta +pv$ or $\zeta =\chi -t\eta $ | $G$ | $G=H-TS$ | Gibbs’ function |

**Table 4.**Summary of Gibbs’ fundamental equations and conditions of equilibrium, quoted from Gibbs’ third paper [8] (p. 87–88).

Thermodynamic Variables | Fundamental Equations | Equilibrium Conditions |
---|---|---|

$\epsilon ,\eta ,v,{m}_{1,}{m}_{2},\dots ,{m}_{n}$ | $d\epsilon =+td\eta -pdv+{\mu}_{1}d{m}_{1}+\dots {\mu}_{n}d{m}_{n}$ | ${\left(\delta \epsilon \right)}_{\eta}\ge 0$ or ${\left(\delta \eta \right)}_{\epsilon}\le 0$ |

$\chi ,\eta ,p,{m}_{1,}{m}_{2},\dots ,{m}_{n}$ | $d\chi =+td\eta +vdp+{\mu}_{1}d{m}_{1}+\dots {\mu}_{n}d{m}_{n}$ | |

$\psi ,t,v,{m}_{1,}{m}_{2},\dots ,{m}_{n}$ | $d\psi =-\eta dt-pdv+{\mu}_{1}d{m}_{1}+\dots {\mu}_{n}d{m}_{n}$ | ${\left(\delta \psi \right)}_{t}\ge 0$ |

$\zeta ,t,p,{m}_{1,}{m}_{2},\dots ,{m}_{n}$ | $d\zeta =-\eta dt+vdp+{\mu}_{1}d{m}_{1}+\dots {\mu}_{n}d{m}_{n}$ | ${\left(\delta \zeta \right)}_{p,t}\ge 0$ |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Pellegrino, E.M.; Cerruti, L.; Ghibaudi, E.
From Steam Engines to Chemical Reactions: Gibbs’ Contribution to the Extension of the Second Law. *Entropy* **2016**, *18*, 162.
https://doi.org/10.3390/e18050162

**AMA Style**

Pellegrino EM, Cerruti L, Ghibaudi E.
From Steam Engines to Chemical Reactions: Gibbs’ Contribution to the Extension of the Second Law. *Entropy*. 2016; 18(5):162.
https://doi.org/10.3390/e18050162

**Chicago/Turabian Style**

Pellegrino, Emilio Marco, Luigi Cerruti, and Elena Ghibaudi.
2016. "From Steam Engines to Chemical Reactions: Gibbs’ Contribution to the Extension of the Second Law" *Entropy* 18, no. 5: 162.
https://doi.org/10.3390/e18050162