## The work done by a system imbedded into an environment

One of the founders of modern thermodynamics, Ostwald, did not like the word "entropy". He generally tried to avoid the use of both the word and the concept of entropy. He tried to replace the latter by the concept of work. Since the main interaction in thermodynamics is the interaction between the system and its environment (keeping in mind that the interaction is non-symmetric), there are two sorts of work, namely, work done by the system on its environment and work done by the environment on the system embedded within it (see also Chapter 2).

This work can be used in order to bring the system into the state of macroscopic motion or to displace the chemical equilibrium of the system or to displace the state of a system in some external "field" defined by those "potentials" about which we were speaking above.

We shall count work A done by the external forces (which are associated with the environment) on the system as positive. Conversely, the negative work, A < 0 will be associated with the work lAl done by the system on its environment. A supersystem "the system + its environment" can be considered as a closed system, but its components are not in thermodynamic equilibrium with each other. In the course of the equilibrium becoming established between the system and its environment the former can perform a work on the latter. The supersystem can arrive at a number of different final states (in particular as regards its energy and entropy), because the transition to equilibrium can also differ. Therefore, the total work, which can be performed by some non-equilibrium system, depends also on the way along which thermodynamic equilibrium is reached. Naturally, the following question arises: along which way will this work be maximal? Note that since we are interested in the work performed by a non-equilibrium system, we have not taken into account the work which might be performed during a general expansion (or compression) of a supersystem. The work could also be done by a system, which itself is in equilibrium. This means that neither the total volume nor the number of particles of each sort contained within the supersystem change, but remain constant. It is interesting that the conservation of the total number of particles of each sort is a necessary

condition for the existence of the corresponding global biogeochemical cycle. Let us consider the global carbon cycle (GCC) as an example.

Let us have one local ecosystem as the system and another part of the biosphere as its environment. Naturally, the whole biosphere is our supersystem. Living matter is a combination of carbon atoms, which have been drawn from the surrounding biosphere (in particular, from atmospheric carbon dioxide) and "packed" into different "boxes" called genes, proteins, organisms, species, etc. upstairs in the hierarchy of the organisation of living matter. A certain amount of carbon atoms are continually released into the atmosphere while at the same time other atoms are returned in the process of the decomposition of living matter. If these amounts are permanently changed, tending to constant when the ecosystem tends to equilibrium with another part of the biosphere (all this is, namely, the GCC), then the total amount of carbon (and, as a consequence, the total number of atoms of carbon) remain constant (Fig. 5.1).

We shall now consider the problem of maximal work in a special situation. We follow here, in general, the Landau and Lifshitz book (1995). Suppose the system is in an external environment whose temperature T0, pressure p0 and (chemical) potentials m0 = (m?, Mn} differ from those of the system (T, p and m = (m?, • mil). The internal energy U0, volume V0 and number of particles N0 = (N0,N°} of the environment are so large that changes in them due to processes taking place in the system do not produce any significant change in the temperature, pressure or potentials of the environment; these may, therefore, be considered to be constant.

Let at its initial state the system be in thermodynamic equilibrium with its environment. In order to change the initial state "0" and to "move" the system to another state, "1", in which it would not be in thermodynamic equilibrium (although always in a dynamic equilibrium), i.e. to realise the transition 0 ! 1, the environment has to do a certain work 8A01. If the system is a thermally isolated object then the work done by the environment on the system for a given change in its state (i.e. from the given initial to the final state) would be a uniquely defined quantity equal to the change in the internal energy of the system. The coupling of two systems, when one of them becomes the environment of another and the environment also is involved in the process, makes the result no longer unique, and the question arises as to what is the minimal work an environment can do for a given change in state.

Fig. 5.1. The system and its environment. Ecodemon is working on the system (left), and the system is working on the ecodemon (right). Ecodemon is working as Sisyphus.

Fig. 5.1. The system and its environment. Ecodemon is working on the system (left), and the system is working on the ecodemon (right). Ecodemon is working as Sisyphus.

If, in a transition from one state to another, the environment does work on the system, then in an inverse transition from the state "1" to the initial state "0", the system must do work on the environment. To a direct transition, which requires the external source to supply the minimum work min(SA)01 = SAmin, and the transition is forced for the system, there is a corresponding inverse transition, in which the system does the maximum work maxl(SA)10l = l8Amaxl. The latter transition is started when the action which supports it stops and the system begins spontaneously to move towards its thermodynamic equilibrium. It is evident that the amounts SAmin and lSAmaxl are identical. During these transitions, the system can exchange heat, matter and work with the environment.

Thus, the total change DU in the internal energy of the system for some (not necessarily small) change of state is made up of three parts: the "useful" work SA done on the body by the environment, the work of compression done by the environment and the heat received from the environment. As was pointed out, owing to the great size of the environment, its temperature, pressure and chemical potentials may be considered to be constant; hence the work done by it on the system is p°DV°, the work done by environmental particles is 5j=i iM°DNI, and the heat given out by it is — T°DS° (letters with sub- and superscripts refer to the environment and those without them to the system). Thus we have n

Since the volume of the environment and the system together remains constant DV° = —DV. From the conditions of conserving the number of particles and the invariability of chemical potentials we have for each sort of particle DN° = — AN,-; i = 1,..., n. Furthermore, in accordance with the law of increase of entropy we have AS + DS° > 0; thus DS° > —AS. Hence, from Eq. (1.1) we obtain:

Equality is attained for a reversible process. Thus we again conclude that the transition occurs with the minimum expenditure of work (and hence the inverse transition with the maximum work done) if it is reversible. The value of the minimum work is given by the formula n

(T°, P° and being constant, can be taken under the operator A), i.e. this work is equal to the change in the quantity U — T°S + p°V — Y!n=i M°Ni. For maximum work the formula must obviously be rewritten with the opposite sign, SAmax = — SAmin, since the initial and final states are interchanged. Here it is necessary to note that the operator A is the difference between values at the first and zero states, and not the opposite. For instance, SAmin = F1 — F° = DF (here F embraces all terms that are within brackets in Eq. (1.3)). Then SAmax = F° — F1 = —DF.

If, at each instant during the process, the system is in a dynamic equilibrium (but not, of course, in thermodynamic equilibrium with the environment), then for an infinitesimal change of state, Eq. (1.3) may be written in a different form. Substituting the expression for the full differential of internal energy (Gibbs' equation: dU = T dS - p dV + JJ^ fii dNi) into the differential form of Eq. (1.3), dAmn = dU - T0 dS + p0V - X"=1 mO dNi, we obtain:

n dAmin = IdAmaxI = (T - To)dS - (p - Po)dV + ^ (Mi - M^. (1.4)

There are two special cases. If the volume and the temperature of the system remain constant and the latter is equal to that of the environment, then from Eq. (1.3) we have

i.e. the maximal work which can be performed by the system tending to an equilibrium with its environment is equal to the decrease in the free energy of the system. If the temperature and pressure are constant, and T = T0, p = p0 we have

i.e. the system doing the work on the environment decreases its corresponding thermodynamic potential. Note that both conditions are typical for biochemical reactions and the ecological system.

Since SAmax # 0, A(U - T0S + p0V - Yj=\ mON) # 0. This means that as a result of processes undergone by the system the quantity U - T0S + p0V - ^n=i MM0Ni will decrease so that at equilibrium it will be a minimum.

In particular, during spontaneous transition "1"!"0" at constant temperature T = T0 and constant pressure p = p0 the thermodynamic potential G of the system decreases, and during transitions at constant temperature T = T0 and volume, its free energy F also decreases. Note that the deduction given here does not assume that the temperature and the volume (or pressure) of the system remain constant during the whole of the transition: it can be asserted that the thermodynamic potential (or free energy) of the system decreases in every process in which the temperature and pressure (or volume) are the same (and are equal to those of the environment) at the beginning and the end, even if they change at some stage of the process. For this it is sufficient that the process (transition) is quasi-stationary (as discussed above).

Yet, another thermodynamic meaning can be given to the maximum work. Let St be the total entropy of the supersystem (the system together with its environment). If the system is in equilibrium with the environment then St is a function of their total internal energy Ut: St = St(Ut). If the system is not in equilibrium with the environment then their total entropy (for the same value of their total energy Ut) differs from the value St = St(Ut) by some quantity SSt < 0.

In Fig. 5.2 the solid line represents the function St = St(Ut) and the vertical line ab the quantity - SSt. The horizontal line bc is the change in the total energy during the transition from the state of thermodynamic equilibrium with the environment to the state represented by the point b. In other words, this line represents the minimum work an environment must do to bring the system from a state of equilibrium with the environment to the given state.

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