Differential equations of macrothermodynamics
The systems and the processes
As a rule, modern thermodynamics studies simple or complex systems with
similar processes taking place in a single fixed time scale. (Usually,
the processes are localized in one or several hierarchies.) However, the
processes in real systems usually imply complicated transformations involving
structures of various hierarchies and relating to different time scales.
Let us consider some differential equations and characteristic functions
that are important for macrothermodynamics - thermodynamics of complex
hierarchic natural systems.
We use the commonly known equation combining the first and the second
laws of thermodynamics for complex closed systems where physical-chemical
processes take place:
Here T denotes temperature, S the entropy, U the internal
energy, p pressure, V volume, Xk any generalized
force except pressure, xk any generalized coordinate
except volume, m k chemical
potential, mk the mass of the k-th substance,
which can be replaced by the number of moles. The equality sign relates
to the case of reversible changes, the inequality describes the irreversible
ones. The work performed by the system is negative. (In some literature
it is considered to be positive; for instance, see the monographs by K.Denbigh
Suppose that in the system under study not only chemical reactions take
place but also transformations between the structure elements of other
j-th hierarchic levels (i-th partial evolution). Moreover,
let the reagents of each i- th evolution condense into the particles
of the i-th evolution phase - a structure of higher level of substance
organization. In their turn, the latter are reagents of the next, (i+1)
-th, partial evolution. (The order of the partial evolution, i and
of the structure hierarchy, j corresponds to the hierarchy order
of the reagents.) Transformation of similar reagents of some hierarchic
level (j) into similar reagents of the next hierarchic levels (j+1,
j+2,...) can be represented as follows:
Here n, m, r >>1, and to each j-th structure hierarchy
the i-th process of phase-transition type is put into correspondence.
Each partial evolution (i, i+1, i+2, ...) takes place
in its own time scale t (ti, ti+1,
ti+2, ...), which is comparable with the life-span of
the corresponding system or its subsystem (j, j+1, j+2,
...) of the given structure level (for instance, organells, cells, organisms).
At times t the natural subsystems (j) are partly kinetically
quasi-closed, i.e., non-stationary. Since a hierarchic system is kinetically
closed only with respect to separate components of the flow coming out,
it is open as a whole, and therefore, its volume and composition vary.
At the same time, in the subsystems of the j-th, (j+1), (j+2)
levels local quasi-equilibrium are achieved at small times
t i, t
i+2, ... . These quasi-equilibrium
are “drifting” ones, they are achieved at every “moment” during the evolution
of the corresponding system. (Different-size arrows in the equalities (2)
show that in the process of thermodynamic self-organization - self-assembly
we consider namely quasi-equilibrium “drifting to the right”; the large
upper arrow shows the direction where the equilibrium is shifted.) Therefore,
each partial evolution is a process of self-assembly (i) for the
substance of variable composition in an open non-stationary system (j).
At any moment of the i-th partial evolution, the subsystem j
(for instance, the subsystem of supramolecular structures) can be characterized
by the value of Gibbs function of the system formation, .
One can also use the variation of the specific value of other state function,
see Chapter II. The value relates
to the process (i) of self-assembly (condensation) of the elements
of j-th structure level, which are initially in the standard state:
for instance, the ideal gas or “ideal solution”. As a result of such self-assembly,
the products of partial evolution i are formed, their composition
varying in time. These products are the reagents of the partial evolution
(i+1). One of the partial evolution with the participation of different
reagents of the k-th type can be presented as
where t i,k do not differ
much, and, therefore, the total process of fast relaxation takes place
in a single time scale.
We stress that the schemes (2) and (3) describe a real evolution process
consisting of partial evolution. These partial evolution are characterized
by different mean times of fast relaxation t
i, t i+1,
... and go on in the corresponding time scales ti, ti+1,
... . To a certain approximation, these schemes reflect the hierarchy of
structures (j) and the corresponding hierarchy of processes (i)
in Nature. The evolution of natural biostructures can be characterized
as avalanche-condensation structure formation. From the viewpoint of its
mechanism, it is opposite to avalanche or branched processes, like electron
avalanches accompanying the electric discharge in gases or branched chemical
Taking into account the schemes (2) and (3), one can write the generalized
equation of the first and the second laws in the form:
where the subscript i relating, as before, to the partial evolution
and k to the component of the i-th evolution. The upper index
j indicates that we consider the behaviour of the j-th system.
Relation (4) introduces the potentials of the components of various
partial evolution. It is the most general equation in the thermodynamics
of hierarchic structures, for it accounts for any kind of interaction (work),
transformations of the substance in the system (variation of its amount
in different subsystems), and also the entropy variation in the course
of all partial evolution. Note that this relation can be used, to a certain
approximation, within any given time scale.
It follows from the relation (4) that the full differentials of the
characteristic functions of corresponding characteristic (independent)
variables given below can be written for the case of complex system as
where the indices j are omitted for convenience and the superscript
* means that we study the behaviour of complex systems. The functions H*
and G*, which do not correspond with their analogues for simple
systems, H and G, can be written as H*=H-Xx, G*=G-Xx.
The evolution potentials in the equations (5)-(8) are defined as
where all corresponding variables of the type m are chosen constant,
with the exception for those used as the differentiation variables. Since in
the equation (9) relates to a complex system, it can be denoted by .
For a complex hierarchic, say, ecological, system, contributions of
different terms in (4)-(9) are determined by the choice of the time scale
and the parameters i,k. For instance, for a real system where
higher evolution take place involving rather large structures (organisms
and so on), the contribution of the terms TidSi,
pidVi and some others, is extremely small.
However, the values and can
be rather large in this case. Indeed, in the case of small particles the
energy of thermal motion kBT>>En. (En
is the energy of natural magnetic, electric, and gravity fields acting
on the particle.) Still, for sufficiently large structures the inverse
is valid: En >>kBT. Hence, the relative contribution
of the corresponding terms is large. While passing from lower evolution
to higher ones, one can also observe variations of the relative contributions
of the terms characterizing surface, photochemical, magnetic-hydrodynamic,
deformational, and other kinds of work.
While studying real processes with fixed time scales, one can simplify
considerably the equation (4) by neglecting the small terms. Certainly,
the thermodynamics (thermostatic) of systems is not interested whether
the transition from one state into another one is equilibrium or not. Still
it should be noted that there are many real chemical, biological, and other
processes studied experimentally that go practically in equilibrium
regimes. (Such processes are studied by the thermodynamics of equilibrium
processes.) When fast chemical reactions take place in the case
of a stationary flow (the situation is close to the one of the Van’t -Hoff
box), the actual useful work of the system (Wx)T,p,...
approaches the maximal work of the process, i.e., the relation -(Wx)T,p,...@
- D G*
holds true. This relation is also satisfied with high accuracy in many
natural chemical and biological systems where stationary processes take
place. As an example, recall that for the processes in fuel cells the rational
efficiency h R = -(Wx)T,p/-D
G* is close to unity. (In the industrial hydrogen-oxygen fuel cells,
the value of h R can reach
0,9 at low currents.) For electrochemical reactions carried out in laboratory
conditions, one can easily achieve h R
= 0,99. Several equilibrium (quasi-equilibrium) chemical transformations
in living organisms are described in literature. Thus, we come to the conclusion
that many natural processes can be characterized by low irreversibility:
to a certain good approximation they can be considered as quasi-static
(quasi-equilibrium) ones and investigated by means of the classical thermodynamics
of drifting equilibrium. As to the natural chemical and biological systems,
in the majority of cases they can be characterized with high accuracy by
the corresponding functions of the states (see the Chapter I). Besides,
from the principle of structure stabilization and the statements following
from macrothermodynamics one can conclude that the rational efficiencies
of spontaneous evolutionary processes tend to the unity in the course
of the evolution (h R ®
1). This conclusion is in good agreement with the general statements
of modern ecology.
Certainly, the results presented do not neglect the existence of dissipative
structures and the effects of dynamic self-organization in natural systems
(I.Prigogine), for instance, when such systems undergo "revolutionary changes"
due to the influence of external factors, and their state is far from the
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