Constraints due to the combined first and second laws

In this section, we analyze a system exchanging heat and chemical substances with its environment. We avoid a full mathematical treatment here and keep equations to the bare minimum necessary. The mathematically inclined can consult the literature (Roels (1983, 2010) and the references therein). In the system, transformation of material substances in chemical reactions takes place. In that case, introduction of a new concept that considers the combined effect of the energy and the entropy content facilitates identifying the restrictions posed by the first and second laws of thermodynamics, we refer to the concept of free energy that we discuss later. For simplicity’s sake, we consider a system where two transformations involving two substances appear. This simple system reflects all the features of the general case involving a multitude of compounds and chemical reactions. Fig. 3.2 reflects the system.

In this figure )₁and )₂ are the flows of substances 1 and 2 to the system, )_Qis the heat flow to the environment. The N’s are the amount of the two substances in the system. The R’s are the net rates of production of the two substances in the transformations in the system. E₁ and E2 are the energy contents per unit substance 1 and 2 respectively. S₁ and S₂ are the entropy contents per unit of those substances.

We write the first and second law based balance equations in the usual way:

First law:

E Q

dt E

dE )_{1 1})_{2 2}) (3.14) Rates of exchange

)Q

) )₁, ₂,

2 1 2 1,E ,S,S E

State variables: E,S,N1,N2

Rates of net production: R₁,R₂

Fig. 3.2. System involving chemical transformation of material substances.

31 For the amounts of compounds 1 and 2, balance equations of the same structure apply, for compound 1:

We now invoke a fundamental result of thermodynamics, the so-called Gibbs equation of state: In eqn. 3.18 new state variables appear.G₁and G₂are the free energies of compounds 1 and 2 respectively. We return to the nature and definition of free energy below.

Combining eqns. 3.14-3.18 results in (see Roels (1983, 2010) for the mathematical detail):

) (R₁G₁ R₂G₂

T3_S (3.19)

Combing this result with the second law that requires entropy production positive if processes take place in the system, i.e. if it is not at equilibrium, the following restriction results:

2 0

Eqn. 3.20 states that free energy decreases because of the transformations in the system. We discuss the significance of this inequality shortly.

A similar restriction applies to a system in a steady state, i.e. a state in which the changes in the state variables become zero. That implies by virtue of eqns. 3.16 and 3.17 that the flow of each substance from the system equals its net rate of production, i.e. R1 )1 and R₂ )₂. Substitution of these equalities in eqn. 3.20 results in:

2 0

2 1

1 ) !

)G G (3.21) For a system in a steady state, the net effect of flows of energy bearing substances to the system must result in a positive net flow of free energy to the system if the system is not at equilibrium. In the words of the Nobel laureate Schrödinger (Schrödinger (1945)) when referring to living systems: The system “feeds on negentropy”. With this statement, he indicates that transport of free energy bearing substances to the system, casually called

32

negentropy (“negative” entropy), serves to keep the system in a non-equilibrium steady state by compensating the entropy production in the processes that take place in the system.

These restrictions are the counterpart of the ones we derived for system exchanging heat and work with the environment and prove to be important in analyzing complex patterns of transformations such as in an organism. Later on in Chapter 5, we argue that such restrictions have counterparts in systems where complex patterns of economic transactions take place. So much for mathematics that I hope remained sufficiently elementary.

The system we analyze here involves only two transformations and two compounds. What we find if we do the math for more complex systems (e.g. Roels (1983, 2010)), are the following statements that apply to any transformation pattern involving an arbitrary large number of compounds and reactions:

In any system, the net effect of a pattern of transformations can only be the destruction of free energy.

In addition, again with perfect generality:

In such systems, we can put no restriction to the direction of individual transformations.

Transformations in which entropy decreases, i.e. reactions against the natural direction defined by the second law, are perfectly possible if supported by transformations in the natural direction. What we can say is that at least one transformation must proceed in the natural direction and that the effect of the total pattern of transformations must result in the production of entropy.

The phenomenon we phrase above, henceforth called coupling, i.e. combining a process in the natural direction with one against this natural direction, shows of crucial importance to the understanding of evolutionary phenomena in chemistry, biology and economics.

The restriction to the reaction pattern applies to any system be it isolated or open. Hence, it also applies to the universe. The direction of its evolution involves an irreversible destruction of free energy that sets the time arrow of evolution.

For a system in steady state, we formulate a restriction that does not depend on the complexity of the transformations in the system. We analyze the constraint based on a black box approach. We police the boundaries of the system and simply apply free energy accounting to the flows going in and out. This usually leads to a vast reduction of the complexity of the analysis. The present author (Roels (1980, 1983)) applies this principle to functioning microorganism that engage in very complex patterns of chemical reactions to build up their structure and to produce metabolic energy to support their maintenance and growth. By the very nature of their successful functioning, the significant exchange flows with the environment are vastly smaller in number than the myriad of compounds supporting their metabolism. This observation leads to powerful tools for the analysis and optimization of processes in which microorganisms are used.

For systems in a steady state the restriction due to the combined first and second laws reads:

For a system in steady state, the perfectly general conclusion is that the net effect of the exchange flows with the environment must be transport of free energy to the system. Again, we can say nothing about individual flows.

For a complete understanding of the nature of the restrictions introduced above the concept of free energy that we left unexplained, needs further analysis. This is the subject of the next section.

33 3.5. Free energy.

Free energy is a powerful concept. It arises from combining of the first law, the second law and the concept of temperature. The definition of free energy is relatively straightforward.

Firstly, it is a macroscopic function of state and it thus solely depends on the present state of the system. The trajectory the system follows in arriving in that state, its history, is immaterial. Let us assume that the system is originally in the reference state for energy, i.e.

the arbitrary datum level for energy, and let us additionally assume that the entropy is initially also zero. We further assume that the system travels from the initial to the final state by reversibly exchanging heat with the environment. In that case, following the reasoning in the Section 3.3 that leads to the identification of the relation between entropy increase and heat transferred to the system, we write:

T dt dS )Q

(3.22) When we phrase this a little bit casually, we can identify the product of temperature and the change of entropy, and hence the product of temperature and entropy as we started from a reference state in which the entropy was zero, as the low quality, heat related, portion of the system’s total energy. Thus, it seems reasonable to diminish the system energy with the low quality portion defined by the product of temperature and entropy. In this way, we define free energy G as the difference between internal energy and the product of temperature and entropy. The author realizes that the reasoning above is far from being rigorous but it conforms to the result of a full mathematical analysis and may be of use to the less mathematically inclined reader. Anyhow, we arrive at free energy using the following expression:

G=E-TS (3.23) Formally, eqn. 3.23 is the definition of the Helmholtz and not the Gibbs flavor of free energy but that is immaterial to the treatment here. We now suspect that G is an expression of the portion of the available energy that we can transform into useful work, i.e. is interesting from the perspective of “relevant” energy. Total internal energy thus not directly relates to the real available capacity to obtain work. The restrictions we derive in the previous section put a limitation to transformation and transfer of relevant energy or free energy. In Chapter 4 when discussing the relation between the macroscopic and microscopic approaches, we provide a clearer explanation of the difference between energy and free energy.

We summarize the root cause for the difference between potentially available internal energy and the fact that only a portion is accessible to do useful work as follows. We resort to a macroscopic description, as the microscopic picture of reality is far too complex to analyze in detail. Therefore, we seek refuge to an approach that involves far less state variables by lumping together many microstates in one macroscopic state. In the associated averaging process, information gets lost. Entropy is the state variable that quantifies that loss of information. Entropy is proportional to the amount of information lost in the averaging process (an amount of information can be quantified using information theory as we show in Chapter 4). The proportionality constant is the famous Boltzmann constant indicated with the symbol k. It is a mishap of the history of thermodynamics to use the Celsius degree as the unit for temperature. On choosing the product of the Boltzmann constant and our present-day temperature as the temperature scale, equality results between entropy and lack of information on using the macroscopic approach. This reveals that the product of temperature and lacking

34

amount of information represents the energy penalty resulting from the macroscopic approach. That penalty results in a portion of the internal energy becoming “hidden” by the lack of information in the macroscopic approach to reality and hence that portion is unavailable to do useful work.

Entropy is less a property of the system than a property of the model of reality we can realistically develop. Our picture is always of a reduced information nature and we cannot avoid paying the penalty that this involves.

3.6. Conclusion.

This chapter summarizes the basic framework of macroscopic thermodynamics. It introduces the concepts of internal energy, entropy and free energy, the first and second laws of thermodynamics and the nature of temperature as a so-called integrating factor for the heat flow. We discuss the relation of temperature to the cost of the information that is lacking in the macroscopic approach. These principles result in restrictions to transformations and the associated flows of work, heat and energy bearing substances to and from the systems’

environment.

35 CHAPTER 4. MACROSCOPIC AND MICROSCOPIC MODELS: STATISTICAL THERMODYNAMICS.

4.1. Introduction.

We discussed the vast complexity of almost all physically and socioeconomically relevant systems. A detailed modeling of the microscopic picture of reality is not practical. That is why physicists resort to macroscopic models containing vastly less information than needed to specify the microscopic richness of the details of a system. These methods are one of the hallmarks of modern science and an important driver of scientific and technological progress.

This chapter unveils the relation between the macroscopic and the microscopic approaches by the presentation of a statistical analysis. This allows the reader to obtain a better understanding of the power and the limitations of the macroscopic approach. To this effect, we need a background in statistics and information theory. This chapter treats these formalisms as well as the foundations of a statistical theory of energy transformations.