Previous: Chapter 5. Adiabatic Flame Temperature
Ilya Prigogine’s quote from his 1977 Nobel lecture is a good starting point to complete our discussion of classical thermodynamics:
‘let us emphasize that one hundred fifty years after its formulation, the second law of thermodynamics still appears more as a program than a well defined theory in the usual sense, as nothing precise (except the sign) is said about the entropy production. Even the range of validity of this inequality is left unspecified. This is one of the main reasons why the applications of thermodynamics were essentially limited to equilibrium.’
Above Prigogine discusses thermodynamics of irreversible processes (non-equilibrium thermodynamics). He talks about a unified theory in which classical thermodynamics is a special case. However, such an ideal has not been reached yet – at present there are different variants of non-equilibrium thermodynamics. Therefore, I consider classical thermodynamics and non-equilibrium thermodynamics as distinct theories. The main difference is related to the absence of explicit time in classical thermodynamics. Time in the explicit form belongs to transport processes in continuum mechanics; non-equilibrium thermodynamics combines these equations with the Clausius inequality, leading to the entropy production.
The term ‘non-equilibrium state’ has a wide range of meanings; it is better to speak of a hierarchy of non-equilibrium states, and therefore it is important to more precisely define what is meant by a non-equilibrium state in a particulate case. Examples will be given that demonstrate the relativity of the terms ‘equilibrium’ and ‘non-equilibrium state’. The consideration of statistical mechanics in the following part of the book expands the hierarchy of non-equilibrium states, including non-equilibrium states beyond those found in continuum mechanics.
We start the discussion with entropy in classical thermodynamics, including non-equilibrium states in discontinuous systems. Classical thermodynamics does not explicitly include time, and thus there is no entropy production. The next step is to include time explicitly – transport processes in continuum mechanics are considered. This gives a transition to non-equilibrium thermodynamics. This way allow us to better understand and correctly position Prigogine’s quote above. The limits of applicability of classical thermodynamics will be examined at the end of the chapter.
- Entropy in classical thermodynamics
- Non-equilibrium states in discontinuous systems
- From temperature field to non-equilibrium thermodynamics
- Limits of applicability of classical thermodynamics
Entropy in classical thermodynamics
The new properties, internal energy and entropy, introduced by the first and second laws of thermodynamics apply to a substance, which in this part of the book is considered as a continuous medium. Therefore, the starting point is that entropy, internal energy, and the derived properties — enthalpy and Gibbs energy — are properties of substance. Let me remind you of the different thermodynamic phases; in classical thermodynamics, they are considered separately, and the entropy of a system consisting from several phases is the sum of the entropies of all phases. I would like to note that many metaphors of entropy in statistical mechanics detach entropy from the substance; the appearance of insubstantial entropy is a characteristic sign of going beyond thermodynamics.
The question of what entropy is, implies a desire to reduce new properties of substance to familiar properties. This, however, is impossible, since physics explains the familiar through the unfamiliar rather than vice versa. Understanding new concepts in physics requires working through examples to become accustomed to the unfamiliar, and entropy is no exception. Therefore, understanding the theory of physics requires working simple examples out using mathematical equations. A qualitative examination of physical theory without the use of mathematical equations is insufficient. A simple example: it is impossible to understand the difference between entropy and Gibbs energy without work with the equation G = H − TS.
In the case of entropy in classical thermodynamics, there are two levels of understanding. The first level concerns the problem of coordination, that is, finding the connection between the physical quantity entropy and the experiments being conducted. This requires at least an understanding of which experiments led to the appearance of the entropy column in the table of thermodynamic properties. It is important to remember that the entropy of a substance is a quantitative characteristic of the state of the substance. In this part of the book, two types of experiments for determining the entropy of a pure substance were discussed: measuring heat capacity in calorimetry and studying the thermal equation of state.
Let us be back to the question about heat capacity at constant pressure or constant volume — this helps to reflect on the question of what understanding is. Heat capacity is a property of a substance, directly determined in the experiment. Yet, one should ask oneself whether there is a clear understanding of what heat capacity is. For future discussions of entropy in statistical mechanics, it should be noted that entropy, as a property of a substance, is related to all other thermodynamic properties through derivatives. In this sense, entropy does not differ from other substance properties; therefore, any attempt to attribute properties related to ignorance or information to the entropy of a substance immediately leads to all thermodynamic properties being related to ignorance or information.
The difference between entropy and other properties arises in connection with the Clausius inequality as a criterion for a spontaneous process and a criterion for establishing global equilibrium in an isolated system. This is the second level of understanding entropy, and this step requires considering non-equilibrium states at the level of classical thermodynamics. Note that an increase in entropy and maximum entropy as a criterion for an equilibrium state only applies to isolated or adiabatically isolated systems. Under different external conditions, other criteria should be used, for example, the minimum Gibbs energy at constant external temperature and pressure. Under different external conditions, the entropy of a system in a spontaneous process may well decrease. A simple example is a glass of hot water on a table. In this case, the water spontaneously cools, thereby decreasing its entropy.
Non-equilibrium states in discontinuous systems
The analysis of non-equilibrium states in classical thermodynamics is the simplest in the case of discontinuous systems, where the system consists of subsystems, in each of which there are no gradients in temperature, pressure, and concentration. Two such examples are examined in Chapter 4 ‘Clausius Inequality as Equilibrium Criterion‘ and one example involving the calculation of the equilibrium composition was considered in Chapter 5 ‘Adiabatic Flame Temperature‘. A conceptual solution was proposed using the terms local and global equilibrium. Subsystems are in a state of local equilibrium; a non-equilibrium state for the whole system means the lack of global equilibrium. The Clausius inequality, in one form or another, is used to find global equilibrium for the entire system, when each subsystem is always in a local equilibrium state.
It is also important to note the relativity of the concepts of global and local equilibrium due to kinetic constraints. The thermodynamic conclusion about global equilibrium of the system says nothing about time needed to achieve it. Let us consider a candle in air. Thermal and mechanical equilibrium exist, the temperature and pressure are uniform, but there is no global chemical equilibrium. According to thermodynamics, the global equilibrium corresponds to the products of candle combustion, but without ignition, the combustion process does not begin by itself. Chemists in this case speak of an activation barrier. The reaction is energetically favorable, but to initiate it, the activation barrier must be overcome — a burning match is needed to light a candle.
Another extreme example of a thermodynamically unstable substance under normal conditions is diamond. Thermodynamic analysis shows that under normal conditions, the most stable modification of carbon is graphite. Therefore, according to thermodynamics, diamond should spontaneously be transformed into graphite. Moreover, in an air atmosphere, diamond, like graphite, thermodynamically speaking, should be transformed into carbon dioxide — the candle example applies to both graphite and diamond. However, the activation barrier in all reactions is too high, and therefore the conclusion that diamond is thermodynamically unstable does not affect its value. However, thermodynamic research has helped to identify the region of thermodynamic stability for diamond and thereby to find the conditions for artificial diamond synthesis.
From temperature field to non-equilibrium thermodynamics
In Chapter 1 ‘Temperature and Thermal Equation of State‘ we discussed the generalization of the concept of temperature to the case of a temperature field. Similarly, in continuum mechanics, a pressure field and a concentration field are introduced to describe the states of a system with gradients of intensive properties. In classical thermodynamics, the entropy of such states can be calculated by generalizing the principle of local equilibrium. A system with gradients is considered to be a system consisting of an infinitely large number of infinitely small subsystems. The entropy of such a system is the integral (sum) over all subsystems.
In classical thermodynamics, this approach is rarely used, since the search for global equilibrium generally employs the variational principle, which is implicitly based on the principle of local equilibrium. Ultimately, the variational principle yields an equilibrium state with uniform intensive variables without gradients, as in the case with discontinuous systems.
In the 19th century, parallel to the development of classical thermodynamics, equations for transport processes were developed, such as the Fourier heat equation, the Navier-Stokes equations, and the Fick diffusion equation. These equations explicitly include time and allow us to compute quantitative changes in a system with gradients without dealing with entropy. The goal of non-equilibrium thermodynamics was to unify the equations for transport processes with classical thermodynamics into a single theory — the Clausius inequality should be replaced by an entropy transport equation.
The first version of non-equilibrium thermodynamics (thermodynamics of irreversible processes) appeared toward the end of the first half of the 20th century. Notable works include those by Karl Eckart (1940), Josef Meixner (1943), and Ilya Prigogine (1947). In these works, transport equations were combined by introduction of energy and entropy balances. To achieve the entropy balance within the Clausius inequality, the introduction of a special term, entropy production, was required. This term was mentioned in Prigogine’s quote at the beginning of the chapter; after that Prigogine elaborates as follows:
‘To extend thermodynamics to non-equilibrium processes we need an explicit expression for the entropy production.’
In a number of cases, interesting results were obtained in non-equilibrium thermodynamics, but in general, entropy balance does not affect the course of transport processes. This means that the entropy balance calculation was performed in addition to the kinetics of the transport process and did not affect the process itself.
In extended irreversible thermodynamics new ideas have been developed – a different interpretation of the meaning of ‘state’ is proposed. In the case above, the temperature field was taken as a characteristic of the state, and on this basis, the principle of local equilibrium was introduced for a small region. In extended irreversible thermodynamics, the state is associated with a heat flux (temperature gradient), which leads to the rejection of the principle of local equilibrium.
In extended irreversible thermodynamics, local temperature is found as the derivative of local internal energy with respect to entropy, which introduces a ‘non-equilibrium temperature’ distinct from the ‘equilibrium’ temperature. However, I was unable to find a connection between the ‘non-equilibrium temperature’ of extended irreversible thermodynamics and metrology. As far as I understand, the problem of coordination in extended irreversible thermodynamics remains unresolved. Note that in non-equilibrium statistical mechanics, the term ‘non-equilibrium temperature’ has a different meaning.
In this light, Prigogine’s the quote cited at the beginning of the chapter is, in a certain sense, valid. There is still no a unified theory that combine the Clausius inequality with all possible processes in continuum mechanics. It is worth noting Prigogine’s work on dissipative systems far from equilibrium, under the slogan ‘Order from Chaos’ — Bénard cells, oscillatory chemical reactions, and others. It is unclear whether it is possible to rigorously prove that the Clausius inequality holds in all these processes.
Let us begin with the meaning of the expression ‘rigorously prove’; it basically concerns the question of the relationship between the theory of physics and the world. The first interpretation of ‘rigorously prove’ is associated with the level of proving mathematical theorems based on the mathematical formalism of the theory. Such a proof remains impossible.
On the other hand, nothing prevents us from considering the Clausius inequality as a successful research program. It is based on the assertion that a perpetual motion machine of the second kind is impossible. In the case of heat engines, the Carnot cycle under this assumption yields the Clausius inequality, which can then be generalized to all physical processes. The primary justification for the Clausius inequality is the practice of thermodynamic research and the practical applications developed from it. Classical thermodynamics has been actively used in practice for over a century and a half. A perpetual motion machine of the second kind has not yet been created, which serves as a good justification for the universality of the Clausius inequality.
Discussion of dissipative processes from this perspective comes down to the practical question of what happens to a dissipative process in an isolated system — for example, what happens to a dissipative process in a closed bunker. A dissipative system can exist in an energy flow only; the Clausius inequality predicts that in a closed bunker, energy flows will cease, which in turn would mean the disintegration of the dissipative process. Violation of the Clausius inequality would mean the possibility of a subsystem in the bunker, far from equilibrium, in which a perpetual motion machine of the second kind is realized and which therefore will never reach equilibrium.
In conclusion, let us consider Prigogine’s statement in the context of the distinction between an equilibrium and non-equilibrium state. Classical thermodynamics, as discussed above, allows us to estimate the entropy of non-equilibrium states within the framework of local equilibrium without time. In classical thermodynamics, all possible non-equilibrium states of continuum mechanics can be ranked by the distance from equilibrium. From this viewpoint, Prigogine’s statement appears too categorical. However, the introduction of time in the considerations requires entropy production in one form or another, so at this stage, the numerical value of the entropy of a non-equilibrium state depends on the theory of non-equilibrium thermodynamics. From this viewpoint, Prigogine’s statement is correct.
Limits of applicability of classical thermodynamics
Classical thermodynamics was developed to understand heat engines, and this imposes limitations that should be kept in mind. The development of statistical mechanics, which is discussed in the next part, demonstrated the possibility of fluctuations — spontaneous changes in a system at equilibrium. In other words, the equilibrium state should be formally considered as a dynamic state, where all quantities spontaneously oscillate around mean values. At a normal system size, fluctuations are small and can be neglected, but when we move to the study of mesosystems, fluctuations play a significant role. As already noted, at present many experiments are being conducted to study the behavior of mesosystems.
Another limitation is related to surface energy. The behavior of substance at a phase boundary differs from that of the substance within the phase. In this case, we speak of volumetric and surface energy densities, which differ from each other. In ordinary systems, the amount of the substance is big enough, and therefore the total surface energy can be neglected. When moving to dispersed systems, where the surface area increases, in classical thermodynamics there are some solutions available but not for all possible cases.
An additional problem arises with amorphous substances, such as glass. Thermodynamics is based on the concept of a state function, which presupposes the ability to characterize completely the state of a substance. Glass is formed by rapidly cooling a melt, which prevents crystallization, but the properties of the resulting glassy state depend on the cooling rate. Thus, with the same chemical composition, different states of glass can be obtained, which require additional characterization; it must be also included in the thermodynamic description of the substance.
As system sizes increase, we reach another limit, tied to the fact that classical thermodynamics as well as statistical mechanics relies on the assumption of additivity of extensive quantities. This means that classical thermodynamics and statistical mechanics do not include gravity. In Chapter 4 ‘Clausius Inequality as Equilibrium Criterion‘ it was mentioned Clausius’s extension of the Second Law to the entire Universe (heat death) — at that time, the existence of the additivity condition was not yet fully understood. At present, it should be clear that automatic extension of the Second Law to the entire Universe is impossible, since incorporating gravity requires the development of a new theory of thermodynamics.
This does not mean, however, that the conclusion about heat death is incorrect. Thus, in 1816, long before Clausius, Lord Byron vividly described the process of human extinction without the Sun’s energy in his poem ‘Darkness‘:
I had a dream, which was not all a dream.
The bright sun was extinguish’d, and the stars
Did wander darkling in the eternal space,
Rayless, and pathless, and the icy earth
Swung blind and blackening in the moonless air;
Morn came and went—and came, and brought no day …
No one knows what prompted Byron to write such a pessimistic poem: a bad mood, excessive reading of the Apocalypse, or the bad weather of 1816 (the year without a summer). Whatever the case, the poem provides a clear, common-sense picture of what would happen to humanity if sources of useful energy disappeared.
Let us return to cosmology. In this case, a generalization of thermodynamics to the case of general relativity is required. It is impossible in this case to use automatically conclusions of classical thermodynamics and statistical mechanics in respect to the behavior of an isolated system. By the way, there are even problems to include thermodynamics in the framework of special relativity — see ‘Relativistic Temperature: Ott-Planck Dissonance‘ in the section Additional Information.
Next: Part 2. Statistical Mechanics
References
Ilya Prigogine, Time, Structure and Fluctuations. Nobel Lecture, December 8, 1977.
Georgy Lebon, D. Jou. Early history of extended irreversible thermodynamics (1953–1983): An exploration beyond local equilibrium and classical transport theory. The European Physical Journal H 40, no. 2 (2015): 205-240.
Ingo Müller, Wolf Weiss. Thermodynamics of irreversible processes — past and present. The European Physical Journal H 37, no. 2 (2012): 139-236.
Barri J. Gold, ThermoPoetics: Energy in Victorian Literature and Science, 2010.
Additional information
In Russian. Relativistic Temperature: Ott-Planck Dissonance: Does the temperature of a moving body increase or decrease? Quotes are provided from Yu. G. Rudoy’s review of the history of relativistic thermodynamics and from Zhuang Liu’s paper on the correspondence between Einstein and von Laue.
Discussion