Introduction

Following a classic biosphere and ecological tradition (Vernadsky, 1926; Lindeman, 1942), we can say that the structure, functioning and evolution of the biosphere, as a whole, and ecosystems in particular, are mainly determined by the flows of matter, heat and radiation. A physical study of their interaction is possible in the framework of thermodynamics. The subject of classic thermodynamics is the consideration of conformities in energy transformations when they are transferred between different physical bodies (which we shall call "systems") in the form of heat and work. Thermodynamics methods are applicable to so-called macroscopic systems consisting of a great number of particles. In physics, the particles are molecules.

Thermodynamics is now one of the most complete and elegant chapters of theoretical physics. However, the domain of the applications of its general concept and methods is much larger than purely physical and chemical systems. Generally speaking, a thermodynamics approach allows us to describe general (macroscopic, systemic) properties of the systems consisting of the large number of interacting elementary (microscopic) objects ( particles). Therefore, when we talk about a thermodynamic approach in such a semi-empirical and phenomenological natural science as ecology, we hope implicitly that these methods help us to overcome an ecological "perdition of dimension". In other words, we could reduce a huge number of the individual descriptions of specimens, populations, their interactions, concentrations of different chemical substances, etc. to a few macroscopic variables and parameters, which will determine some generalised states of the ecosystem.

Towards a Thermodynamic Theory for Ecological Systems, pp. 13-40 © 2004 Elsevier Ltd. All Rights Reserved.

Before applying the thermodynamic concepts and methods to ecological systems we have to tell our readers about them. But before that, we shall try to answer the question: "What, strictly speaking, do we understand by the notions heat, energy and entropy?" We also have to define: "What is meant by a 'system'?" The latter is a main word in the thermodynamic thesaurus. When we talk about a system, naturally we imply that we know what a "system border" and "system environment" are. The system exchanges matter and energy with its environment across its borders. A type of exchange generates the systems classification.

In an isolated system neither energy nor matter exchanges with its environment. In a closed system only energy exchanges with its environment, but it cannot exchange matter. An open system is the system which exchanges energy and matter with its environment. From a thermodynamic point of view, any ecosystem (except the biosphere, which is the biggest ecosystem of our planet) is an open system, which exchanges energy (incoming solar radiation and outgoing heat irradiation) and matter (water, carbon dioxide, nutrients, organic matter, etc.) with the environment. The biosphere is a typical closed system, since it exchanges only energy with space (matter exchange can be neglected with a good approximation). Certainly, all these classes are theoretical models of real systems. In reality, there are not purely isolated, closed and open systems. The same system can be isolated during some time intervals, but be open during the others. For instance, the biosphere is a closed system with respect to thousands of years, and it is an open system with respect to geological times, when we cannot neglect the matter exchange between the core and the mantle. Generally speaking, the successful setting of a system gives a more than 50% guarantee of solving the problem.

The main bodies in thermodynamics are the system and its environment. As a rule, this pair forms an absolutely isolated supersystem, while the system and its environment are exchanged by matter and energy through a border of the system (Fig. 2.1). There are two sorts of equilibriums in thermodynamics. The first, thermodynamic equilibrium, takes place when there is no exchange of matter and energy between the system and its environment, and its state does not change. The second, dynamic equilibrium (steady state), is maintained by non-zero flows of energy, matter (and entropy) across borders of the system.

The world of the classic Newton's mechanics is isotropic, i.e. there are not some special preferable directions with respect to both the space and the time. For instance, all equations of Newton's mechanics do not change (they are invariant) when the positive time is replaced by the negative one. Another situation applies in thermodynamics, when the time is strictly oriented from the past towards the future. What could generate this anisotropy of time? In standard textbooks on thermodynamics, the reason is shyly hidden, but for us it is very important, since one of the main concepts of our book, namely, the concept of exergy (see Chapter 5), is based on this reason.

The basic hypothesis is very simple: the environment is much more than the system, so that the latter is a very small part of the first. A state of the system depends on a state of the environment, and any change in the environment causes some change in the system. However, the inverse is not true: the system does not influence its environment. Any processes can take place within the system but they cause no effects on the state of the environment.

Supersystem

Supersystem

Fig. 2.1. System, its environment and their superposition into a supersystem.

It is obvious that acceptance of the hypothesis immediately provides the anisotropy of action: the action is not equal to the counteraction. In other words, we have to depart from the Third Newton's Law: action is equal to counteraction. But this immediately allows us to depart from the temporal symmetry of Newton's mechanics. Note that, generally speaking, thermodynamics does not need this mechanics.

A state of the system is determined by a collection of thermodynamic macroscopic parameters (variables) that in turn are divided between extensive and intensive. Each extensive parameter for the system is equal to the sum of the corresponding extensive parameters of the macroscopic subsystems, which are components of the original system. It is easy to see that the energy, mass, number of particles and volume are typical extensive parameters. In other words, the extensivity of parameters reflects such a fundamental property of the system as its additivity. But such parameters as the temperature, pressure and density are intensive ones. They are some mean characteristics of the system. In order to clarify the concept of intensitivity, we consider the following example. Let us have two volumes V1 and V2, the number of particles in which are equal to N1 and N2, respectively. Then the densities will be equal to n1 = N1/V1 and n2 = N2/V2. By joining these subsystems into a single system we have the following expression for density of particles in this joined system:

N + N2 n v + n2 V2 n = VTTV, = V1 + V2 = n1p1 + n2p2'

where p1 = V1 /(V1 + V2), p2 = V2 /(V1 + V2) are the volume fractions of each subsystem in the total system. We can see that the total density is a mean of subsystem densities. Generally speaking, we can say that the system state (in relation to intensive parameters)

Fig. 2.2. Cyclic process: x1 and x2 are state variables, "0" = "4" are the initial and final states, "1", "2" and "3" are the intermediate states.

is described by mean parameters that are obtained by averaging corresponding parameters on the entire ensemble of subsystems.

Processes which take place in the system may be either equilibrium (reversible) or non-equilibrium (irreversible). Changes in the system caused by reversible processes can be neglected as a result of inverse consequence of local transitions without any changes in the environment. On the contrary, when the system moves inversely to its initial state in an irreversible process then the movement is accompanied by residual changes in the environment. Note that mainly reversible equilibrium processes are considered in classic thermodynamics.

We introduce also one important auxiliary concept, viz. the concept of cyclic process. Let the system start from some initial state, which is described by state variables. The system moves along the sequence of intermediate states and finishes where the movement started (Fig. 2.2). This is a cyclic process.

2.2. Matter and energy in mechanics and thermodynamics. Energy conservation as the First Law of Thermodynamics. Fundamental Gibbs equation

The conservation law of matter had probably already been accepted in ancient Greece and by the alchemists in medieval times, although it has also been presumed violated when the reproduction of microorganisms was observed. Lavoisier was the first to formulate the mass conservation principle when he stated, in 1785, that the total mass in a system remained constant after both chemical and physical processes.

The mass conservation principle is applied widely in chemical stoichiometric calculations, as the elements of the reactants are equal to the elements of the products. The mass conservation principle may also be formulated in everyday language: nothing comes from nothing. If we keep in mind the definition of an isolated system, then we can formulate the mass conservation principle in the following form: in the isolated system the total amount of matter is constant.

Along with matter, the ancient Grecians and medieval alchemists also considered a "very special matter" such as "fire" or "flogiston", i.e. heat. Now, we know that heat is not matter (in the proper sense) but it is one of the forms of energy. Heat can be transformed into other forms (mechanical energy, chemical, electrical, etc.), and the other forms of energy can be transformed into heat. Energy cannot be created or destroyed. This is the energy conservation principle called the First Law of Thermodynamics. It is obvious that in the isolated system the total amount of energy is constant. If the system is open then both matter and energy may be either accumulated or spent. This means that for both matter and energy the following equation, that could be called the "book keeping equation" since it can be also used in book keeping, is valid:

Change of mass or energy = inputs — outputs. (2.1)

The change of energy dE can be represented as the sum of two items: the item which corresponds to internal processes within the system (i), and the item which corresponds to processes of exchange between the system and its environment (e):

Since the energy cannot be created or destroyed within the system then diE = 0. The term deE can be represented in the form of the sum of work done on the system by its environment, SA, and heat SQ gained by the system from its environment:

This is a quantitative form of the First Law, initiated with James Joule's work (published in 1843 and 1849). He has shown that mechanical and electrical energy could be converted into heat, and demonstrated that the same amount of heat was always produced from a given amount of mechanical or electrical work, and that both the amount of heat and work as its equivalent are measured in the same units, calories or joules (1 cal = 4.18 J). Note that here we encounter the new concept of work, which is closely connected with transformations of energy and accompanies them.

An external force applied to a body, i.e. an action of the environment on a system, does work on it. In accordance with the general laws of mechanics, the work is the product of the force and the shift distance produced by it. The work can be used to change the kinetic and potential energy of the system, to change its chemical composition, etc. However, here and later on we shall exclude all the processes that lead to the changes in the kinetic and potential energy of the system. Then the total energy can be represented as U = E — Ekinetic — Epotential, where U is the so-called internal energy of the system.

It is necessary to distinguish the work SA done by the environment on the system, which counts as positive, and the work SA' done by the system on its environment (and, certainly, changes it), which is negative. Analogously, we shall count the heat received by the system from its environment as positive, and vice versa.

Let us consider the following example: there is a cylindrical tube with piston of cross-section area s filled by some "ideal" gas (system). The piston is densely adjacent to the interior lateral area of the tube. We press on the piston with force ps, where p is dE = diE + deE.

the pressure, and compress the gas. The piston is shifted the distance dx. We consider this as a transition from the initial point "0" to the final point "1". Since in mechanics the work is defined as the product of force by distance then we (i.e. the environment) do the work D4 = —pa dx = —p dV, where the change of volume dV < 0, since gas is compressed. Then D4 > 0. In addition, we can heat the gas, gaining as a result some amount of heat DQ. As a result we also increase the internal energy of the gas so that the change of internal energy d^01 = DQ + D4. After this the system is capable to do some work (on the environment), but does not do it until we (the environment) set free the piston, and the system starts to move backwards towards its initial state, i.e. "free" equilibrium. In the process of the movement it does the useful work D4', gives back the environment some heat SeQ' and produces some heat SiQ/ within the system. The latter is a consequence of irreversible processes within the system, for instance the friction of piston on the walls of the tube. As a result, the change of internal energy will be equal to d^10 = DQ' + DA', where DQ' = DeQ' + SiQ/. In accordance with the energy conservation law applied to the sum of these two transitions (this is a typical cyclic process) d^010 = d^01 + d^10 = 0, but it is not necessary that 8A = — 8A' and DQ = — DQ'. Therefore, the infinitesimal change in internal energy is the full differential but the changes in heat and work are not the full differentials. In other words, U is a state variable but A and Q are not the same, and the system energy cannot be split into heat energy and, for instance, mechanical or chemical energy.

The work done and the capability to do work are not equivalent. Even for a very simple system (such as an ideal gas in the tube with a piston), the system is capable to do the work only after the environment has done some work on the system. Although the two events are subdivided by a temporal interval in the dynamic case, this difference disappears in the static case. Thus, the fact that 8A' and 8A become equivalent (but not equal!) gives birth to a lot of misunderstanding. But let us not forget about their original non-equivalence!

Both conservation laws (matter and energy) are independent of each other, but according to Einstein's famous equation E = mc2, matter can be transferred to matter and energy to matter and the equivalent amount of mass and energy can be found by using this equation. E symbolises the energy, m the mass and c the light velocity (c = 3 X 108 m/s). Therefore, these laws are not independent but, since the transfers from matter to energy and vice versa have interest for nuclear and plasma processes, in biology we can consider them as independent. For practical use of the conservation laws in ecosystems, it can be stated that matter and energy are conserved. However, Einstein's equation is very important in this theory since it provides the energy positiveness (E0 = m0c2 > 0 where m0 is the "rest-mass").

The energy conservation law is a fundamental of mechanics (and electrodynamics), but it is insufficient for the complete dynamical description of a mechanical system. We need also some relation connecting three main mechanical variables, mass m, force F and velocity v. The second Newton law gives the relation: d(mv)/dt = F, while the other two laws define the structure of "mechanical" space and time. However, in ecology (and, for instance, in chemical kinetics also) there are not any analogies of Newton's equations; using for the deduction of Lotka-Volterra equations the so-called principle of collisions is rather a phenomenological one.

Strictly speaking, the laws of mechanics (and electrodynamics) are already sufficient in order to acquire the statement about the conservation of energy, but there is not any kind of general theoretical proof of the First Law: this is an empirical generalisation of our experience. In particular, despite the huge number of attempts to create perpetuum mobile, up to now it has not been constructed.

Let us consider a macroscopic system that can be described by such extensive variables as the internal energy U, apresently unknown variable, entropy, S, volume Vand numbers of different virtual particles n1,..., nm,..., and also such intensive variables as the temperature T, pressure P, etc. As some unit of quantity of matter, either a single particle or 1 mol will be used. Note that this is not important since in our general formulas the choice of unit is not essential. So, by having done the work SA on the system and gaining to it the external heat SQ, we transfer it (quasi-steadily, without the disturbance of equilibrium) from the initial equilibrium "0" to some new state "1". This is a forced transition, as a result of which the internal energy increases in the value of dU01 = dU > 0.

In accordance with the First Law, deE = dU = SQ + SA. Since energy cannot be created or destroyed within the system, then the latter changes only by means of exchange processes or external work. The work in turn changes the state of the system, i.e. changes its extensive variables (only small, infinitesimal differentials are considered). However, in reality, some part of the work is spent in order to overcome different "resistances" (such as friction, resistance of conductor by electrical current, etc.). Therefore, work can be represented as SA = SAus + SAirrev where the item SAus is "useful" work and the item SAirrev is a contribution of real irreversible processes into the increase of internal energy. If the "0" state is a thermodynamic equilibrium, then in the course of inverse spontaneous transition to the equilibrium the internal energy dU is spent in the processes of performance of the different work and released in the form of heat. The main Gibbs idea was that dU can be represented as a bilinear form of intensive variables and differentials of conjugated extensive variables, so that SA = ^£Xk dxk + ^Nm denm and SQ + SAirrev — Y.Nm dinm = T dS. Here, Xk and xk correspondingly form the pair [Xk, xk} of conjugate (by them) variables of work. For instance, the pair corresponding to the work done for compression is the pressure — p and the volume V, so that the corresponding bilinear term is — p dV. If nm corresponds to the number of the mth sort of virtual particles then dnm = denm + dinm are the sum of change of their numbers as a result of matter exchange between the system and its environment, and their birth and death within the system (for instance, as a result of biological "birth and death" processes or irreversible chemical reactions). Then the expression for the full differential of internal energy is written as dU =YXk dxk + YNm denm + YNm dinm + T dS

This is a general formal representation of the famous Gibbs equation.

Let di nm = 0, then the values of Nm are potentials that maintain the exchange of matter. In thermodynamics, the system is separated from its environment by potential barriers. In order for any virtual particle to penetrate from the environment into the system it has to overcome these barriers. For this, it has to perform certain work moving

Table 2.1

Different forms of energy and their intensive and extensive variables (potential and kinetic energies are denoted as mechanical energy)

Table 2.1

Different forms of energy and their intensive and extensive variables (potential and kinetic energies are denoted as mechanical energy)

Energy form

Extensive variable

Intensive variable

Heat

Entropy (J/K)

Temperature (K)

Expansion

Volume (m3)

Pressure (Pa = kg/s m)

Chemical

Moles (M)

Chemical potential (J/mol)

Electrical

Charge (A s)

Voltage (V)

Potential

Mass (kg)

(Gravity)(height) (m2/s2)

Kinetic

Mass (kg)

0.5(velocity)2 (m2/s2)

in a field of the thermodynamic potential. A value of the specific work (per single particle) is equal to Nm. In the case when the particles are chemical molecules, then Nm = Mm are chemical potentials. The value SAirrev is a contribution of real irreversible processes (such as friction, heating of conductor by electrical current, etc.) into the increase of internal energy. Note that D4irrev can also be represented as a product of an intensive variable and differential of a conjugated extensive variable. For instance, in the case of friction, D4irrev = ffr ds where f denotes the force of friction and ds is the infinitesimal distance. In the case of electrical current, D4irrev = U de where U denotes the voltage and de is the change of charge. It is interesting that the first D4irrev can be represented as SAirrev = ffrv dt, and the second as D4irrev = UI dt, where v is the velocity of movement, I is the amperage and di is the time differential. In both the cases the products ffrv and UI are powers.

We are keeping in mind that in thermodynamics there is such an intensive variable as the temperature T. What kind of conjugate extensive variable corresponds to it? Gibbs has postulated that this extensive variable S is entropy.

Finally, if we select the term connected with work of expansion or compression and consider only "chemical" particles, then the expression for the full differential of internal energy is represented in the form:

This is a standard form of the fundamental Gibbs equation. Certainly, along with "thermal", "mechanical" and "chemical" products T ds, — p dV and Mm dnm, we can also consider other pairs of intensive and extensive variables connected with two other forms of energy (see Table 2.1) and other sorts of virtual particles. Potential and kinetic energy is denoted here as mechanical energy.

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