DYNAMIC APPROACH TO THERMODYNAMIC SIMULATION OF CHEMICAL SYSTEMS:
FROM TRUE EQUILIBRIUM TO TRUE CHAOS.
B. Zilbergleyt,
System Dynamics Research Foundation, Chicago
E-mail: livent@ameritech.net
ABSTRACT. The paper presents new model of equilibrium in open chemical systems suggesting linear
dependence of the reaction shift from true equilibrium on the external thermodynamic force. Basic equation
of the model includes traditional logarithmic term and non-traditional parabolic term. Behavior of open
system is defined by relative contributions of both terms. If classical logarithmic term prevails open system
can be in equilibrium. Increased weight of non-classical chaotic term leads open system to bifurcations and
chaos. Area of open equilibrium serves as a water shed between true equilibrium and bifurcations and
eventual chaos. In simple isolated equilibrium the chaotic term equals to zero turning the equation to
traditional form of constant equation. Formally this term matches excessive thermodynamic function showing
linear relationship between reaction extent in open equilibrium and logarithm of coefficient of thermodynamic
activity. In form of chaotic term it reveals new behavioral features of open systems while method of
coefficients of thermodynamic activities hides them to keep open system disguised as isolated entity.
Discovered relationship prompts us to use the coefficient of linearity in combination with reaction shift rather
than activity coefficients. This coefficient in open equilibrium can be calculated in a simple way using basic
equation and reaction shift as independent variable. Numerical data obtained by various simulation techniques
have proved premise of the new theory and following from it method of chemical dynamics.
INTRODUCTION: BACK TO CHEMICAL DYNAMICS.
Nowadays we know that chemical self-organization happens in a vaguely defined area “far-fromequilibrium”[1], while classical thermodynamics defines what is frozen at the point of true equilibrium. What
occurs in between?
“True”, or “internal” thermodynamic equilibrium is defined by current thermodynamic paradigm only for
isolated systems. That’s why applications to real systems often lead to severe misinterpretation of their status,
bringing approximate rather than precise results. A few questions arise in this relation. Is it possible to expand
the idea of thermodynamic equilibrium to open systems? How to describe and simulate open equilibrium in
chemical systems? Is there any relationship between deviation of a chemical system from “true” equilibrium
and parameters of its non-ideality?
Current paradigm of chemical thermodynamics handles open chemical systems employing coefficients of
thermodynamics activities. The coefficients as well as the whole concept of equilibrium, based on GuldbergWaage’s equation, may be derived from the probability theory [2]. Indeed, chemical system with only one
type of collision is a simple chemical system: one type of collision – one reaction – one outcome, and chance
of the reaction to happen is proportional to P(A) - probability of reactants to collide, equal to the particles
mole fractions product. In complex systems with multiple interactions, according to the Bayes’ theorem,
instead of P(A) we consider P(A/B) – conditional probability of collision A given another collision, B, took
place. The ratio =P(A/B)/P(A) defines a coefficient, currently known as coefficient of thermodynamic
activity.Regardless of the initial idea, introduction of coefficients of thermodynamic activity was and still is a
great contribution to chemical thermodynamics mainly because it allowed scientists to keep expressions for
G and constant equation unchanged still holding open systems in a mask of isolated entities. The name of
the Law of Active Masses became more clear -  may be considered a fraction of a substance available to
participate in chemical reaction. Later on value of  was tied to excessive thermodynamic functions though
experiment is often the best way to obtain it.
The values to be introduced and discussed in this paper are sensitive to isolation or openness of the systems
where the chemical processes are to run. An isolated system will be referred to as system with possible true
thermodynamic equilibrium. Simple chemical system allows only one chemical reaction to run towards true
thermodynamic equilibrium. Thermodynamic state of such a system is defined by two thermodynamic (e.g.,
P, T) and one chemical (reaction coordinate parameters. For the sake of clearness, simplicity and to keep it
closer to reality, open system will be considered also allowing for one chemical reaction to run thus being a
simple subsystem – a part of bigger and more complex chemical system. The system contains a set of similar
entities as subsystems and constituted by chemical (or thermodynamic) interaction between them. For
1- 1
example, in some cases it is very convenient if reaction of formation of one chemical substance from elements
runs in each subsystem. This way number of subsystems will equal to the number of chemical species in the
system. We define thermodynamic equilibrium in open system as open equilibrium. It is the central idea of
the present paper, and we will get its basic equations, show where it leads the open system and how to bring
the ideas to numbers.
To do so we will use currently almost neglected method de Donder that introduced thermodynamic affinity
interpreting it as a thermodynamic force and considering the reaction extent a “chemical distance” [2]. It is
more convenient to use redefined reaction coordinate dj=dnkj/kj, instead of dj=dnkj/kj by de Donder, or
reaction extentj =nkj/kj in increments. Value of nkj equals to amount of moles, consumed or appeared in
j-reaction between its two arbitrary states, one of them may be the initial state. The ij value equals to a
number of moles of k-component, consumed or appeared in an isolated j-reaction on its way from initial state
to true equilibrium and may be considered a thermodynamic equivalent of chemical transformation. This
value is unambiguously related to standard change of free Gibbs’ energy or equilibrium constant of reaction
and represents natural and the only result of thermodynamic simulation of the reaction equilibrium. Above
redefined value of the reaction extent remains the same being calculated for any component of a simple
chemical reaction; the only (and easily achievable by appropriate choice of the basis of the chemical system)
condition for this is that each chemical element is involved in only one substance on each side of the reaction
equation like in reactions of formation from elements. Now, in our definition j is a dimensionless chemical
distance (“cd”) between initial and running states of j-reaction, 0≤ j≤1, and thermodynamic affinity A =
(G/)p,T turns into a classical force by definition, customary in physics and related sciences.
Chemical reaction in isolated system is driven only by internal force (eugenaffinity, Aij). True thermodynamic
equilibrium occurs at A´ij = 0 and at this point ´j = 1. Reactions in open system are driven by both internal
and external (Aej ) forces. The external force originates from chemical or, in general, thermodynamic (also
due to heat exchange, pressure, etc., affecting thermodynamic parameters of its state) interaction of the open
system with its environment. Linear constitutional equations of non-equilibrium thermodynamics at zero flow
give us a condition of open equilibrium with resultant affinity
A*ij + ie A*ej = 0,
(1)
where ie = aii /aie is a ratio of the Onsager/Kasimir coefficients [3]. The accent mark and asterisk relate
values to isolated (“true”) and open equilibrium correspondingly, and indices ‘i’ and ‘e’ define internal and
external variables and functions correspondingly.
In this work we will use only one assumption which in fact slightly extends the hypothesis of linearity. Given
a relation between the reaction shift from equilibrium j =1j and external thermodynamic force causing
this shift, we will suppose at the first approximation that the reaction shift in the vicinity of true
thermodynamic equilibrium is linearly related to the shifting force
j = ie Aej .
(2)
Recalling that Ai = (Gi /i ), or Ai = (Gi /i ) and substituting (2) into (1), we will get an intermediate
expression
– (G*ij /*j) + (ie/ie) *j = 0.
(3)
After a simple transformation, retaining in writing only j for j and j for j we turn it into
G*ij + bie *j *j = 0,
where bie = ie /ie). As usually, G*ij =
G0ij
(4)
+ RTln*j (, *j) , and corresponding constant equation is
 RTlnKj + RTln*j (, *j) + bie *j * j = 0,
(5)
where *j (, *j) is the product of mole fractions in open equilibrium expressed via j-reaction extent. Please
notice that equilibrium constant Kj = 'j (, ).
Obviously product *j *j is dimensionless, therefore bie has dimension of energy. To bring (5) to more
symmetric form we introduce a new value - the “alternative” temperature of the open system
Ta = bie / R,
(6)
R is universal gas constant. The value of Ta is introduced in this work for convenience and symmetry; we
cannot clearly explain its physical meaning at this moment. The logarithmic term contains traditional
thermodynamic temperature Tt , and (5) turns to
RTt ln`j + RTt ln*j + RTa *j*j= 0.
Now, dividing (7) by (RTt ), and denoting = Ta /Tt, we transform equation (7) into
ln [`kj,1)/ kj , *j)]  j*j*j = 0.
(8)
______________________________________________________________________________________

“Vicinity” in this case is certainly not less vague than “far-from-equilibrium”. Some discussion see below.
1- 2
So, as soon as chemical system becomes open, appropriate constant equation includes a non-linear, nonclassical term originated due to system’s interaction with its environment.
What opens up immediately is a similarity between the non-classical term of (5) and the well known product
rx(1x) from so called logistic map [4], which is one of the most convincing equations leading to
bifurcations and chaos. We refer alternative temperature to as chaotic temperature Tch and j as reduced
chaotic temperature.
Being divided by *j , this equation expresses linearity between the thermodynamic force and reaction shift
{ln [`kj,1)/ kj , *j)]}/*j =  j*j,
(9)
both parts of it are new expressions for the thermodynamic force. Equality (9) contains condition of open
equilibrium as balance of internal and external thermodynamic forces. Equation (8) can be written in more
general form as
*= *(10)
It is easy to see that in case of isolated system *= external thermodynamic force equals to zero, and (8)
turns to the normal constant equation. For better understanding of internal relations between (9) and constant
equation one should recall once again that serving as a parameter in(10), is the only output from the
solution to constant equation (because n` n0, where right side contains equilibrium and initial mole
amounts).
INVESTIGATION OF THE FORCE-SHIFT RELATIONSHIP.
First, consider the force expression from equation (9). Its numerator is a logarithm of a combination of molar
fraction products for a given stoichiometric equation. The expression under the logarithm is the molar fraction
product for ideal system divided by the same product where kj replaced by *jkj due to the system’s shift
from “true” equilibrium. Table 1 represents expressions for thermodynamic forces of some simple chemical
reactions with initial amounts of reactants A and B both equal to one mole. Graphs of the reaction shifts vs
thermodynamic forces are shown at Fig.1. One can see visually distinctive linearity (actually, quasi-linearity)
on shift-force curves. Extent of the linearity region depends on the value.
Going down to real objects, consider a model system containing a double oxide MeORO and an independent
reactant I (for instance, sulfur) such that I reacts only with MeO , while RO restricts reaction ability of MeO
Table 1. Thermodynamic forces {ln[`kj,1)/ kj , *j)]}/ (eq. 8) for some simple chemical reactions.
Initial amounts of reactants are taken equal to 1 mole and initial products to zero for simplicity.
Reaction equation.
A + B = C
A + 2B = C
2A + 2B = C
Thermodynamic force
{ln {(1/(2)/(2)][(1)/(1)]2}}/
{ln{(1/(1)/(1 [(12)/(12)]2}}/
{ln{(1/[(23233[(124}}/
1.0
1.0


0.5
0.5
0.0
1.0

0.5
0.0
0.0
0
25
F
50
0
25
F
50
0
25
F
50
Fig. 1. Shift of some simple chemical reactions from true equilibrium  vs. shifting force F, kJ/m·cd.
Reactions, left to right, values of in brackets(curves follow right to left) : A+B=AB (, 0.9),
A+2B=AB2 ( 2A+2B=A2B2 (. Also, linear areas on the curves
give an estimation of how far the “vicinity of equilibrium” extents.
1- 3
and frees in the reaction as far as MeO is consumed. Two competing processes are in equilibrium in the
isolated system - decomposition of MeORO, or restricting reaction: MeORO = MeO + RO, and leading
reaction: MeO + I = *L, the right side in the last case represents a sum of products. Resulting reaction in the
system obviously is MeORO + I = *L + R.
To obtain numbers for real species, we used thermodynamic simulation (HSC Chemistry for Windows) in the
model set of substances. The Is were S, C, H2, and MeORO were double oxides with symbol Me standing for
Co, Ni, Fe, Sr, Ca, Pb and Mn. As restricting parts RO were used oxides of Si, Ti, Cr, and some others.
Chosen double oxides had relatively high negative standard change of Gibbs’ potential to provide negligible
dissociation in absence of I. For more details see [5]. Some of the results for reactions (MeO RO+S) are
shown on Fig.2. In this group value of (G0C / *L)I plays role of external thermodynamic force regarding
the (MeO+S) reaction.
The most important is the fact that in both cases the data, showing the reality of linear relationship, have been
received using exclusively current formalism of chemical equilibrium where no such kind of relationship was
ever assumed. It is quite obvious that linear dependence took place in some cases up to essential deviations
from equilibrium. We call the method, described in the present work (also including original de Donder’s
approach), a force-shift method for explicit usage of chemical forces, originally introduced as thermodynamic
affinities, or a method of chemical dynamics (MCD). Results shown on Fig.1 and Fig.2 prove the premises
and some conclusions of the theory.

FeO*RO
0.5
CoO*RO
CaO*RO
0.0
0
300
F
600
Fig.2. Reaction shift * vs. force F= G0 MeO RO / *, kJ/m·cd, 298.15K, direct thermodynamic simulation.
Points on the graphs correspond to various RO. One can see a delay along x-axis for CaORO.
Within current paradigm of chemical thermodynamics, constant equation for non-ideal system with k 1 is
G0j =  RTt ln *k  RTtln*xk,
and xk are molar fractions, power values are omitted for simplicity. The non-linear term of the equation (8)
also belongs to a non-ideal system, and comparison of (8) and (15) leads to following equality in open
equilibrium with precision of the sign
j *j ln * k) /*j.
(16)
If * k<1, the minus sign should be placed on the left side. This result is quite understandable. For instance,
in case of MeORO the chemical bond between MeO and RO reduces reaction activity of MeO; the same
result will be obtained for reaction (MeO + I) in absence of RO and with reduced coefficient of
thermodynamic activity of MeO. Now, to avoid complexity and using only one common component MeO  in
both subsystems, the relationship between the shift of the (MeO +I) reaction and activity coefficient of MeO 
is very simple
* = (1/)(ln *)/*],
 (17)
where *, and* are related to the (MeO +I) reaction, and (ln *)/*] is external thermodynamic force
acting against it (divided by RTt). This expression for the force as well as the total equation (17) are new.
This equation connects values from the MCD with traditional values of chemical thermodynamics. Yet again,
at *L= 0 we have immediately *=1, and vice versa, a correlation, providing an explicit and instant
transition between open and isolated systems. In case of multiple interactions one should expect additivity of
the shift increments, caused by interaction with different reaction subsystems, which follows the additivity of 
appropriate logarithms of activity coefficients. This is also proven by simulation.
Data for Fig. 3 were obtained using two different methods of thermodynamic simulation. I-simulation relates
to an isolated (MeORO+I) system with real MeO and RO and MeO= 1 in all cases. In O-simulation a
_______________________________________________________________________________
1- 4

The numerator is free energy of formation of double oxides MeO RO from oxides MeO and RO.
combination of |MeO+Y2O3+I| represented the model of open system where RO was excluded and replaced
by yttrium oxide, neutral to MeO and I to keep the same total amount of moles in the system as in Isimulation and avoid interaction between MeO and RO, which role was played by Y 2O3. Binding of MeO into
double
compounds with RO, resulting in reduced reaction ability of MeO was simulated varying . I-simulation
provided a relationship in corresponding rows of the *L - * values, and O-simulation - with *L - G0MeO RO
1.0
SrO*RO
PbO*RO

0.5
CoO*RO
0.0
0
F
50
100
Fig. 3. *vs. (-ln *) (I-simulation, x) and vs. (G0MeORO/*) (O-simulation, o), (MeOR+S). PbO and
CoO at 298K, SrO at 798. Curve for SrO shows light delay along the x-axis.
correspondence. Standard change of Gibbs’ potentialG0MeO RO, determining strength of the MeORO bond,
was considered an excessive thermodynamic function to the reaction (MeO+I).
We have calculated some numeric values from the data used for plotting Fig.2, they are shown in Table 2. It
is worthy to mention that the range of activity coefficients, usable in equilibrium calculations, seems to be
extendible down to unusually low powers (see Fig.1).
Strong relation between reaction shifts and activity coefficients means automatically strong relation between
shifts and excessive thermodynamic functions, or external thermodynamic forces. Along with standard change
of Gibbs’ potential we also tried two others – the Q which was calculated by equation (17) with *, used in the
O- simulation, and another, G*MeORO, found as a difference between G0MeORO and equilibrium value of
RTtln*xk. Referring to the same *, all three should be equal or close in values. Almost ideal match,
illustrating this idea, was found in the CoORO - S system and is shown on Fig.5. In other systems all three
were less but still are close enough. Analysis of the values, which may be used as possible excessive
functions, shows that the open equilibrium may be defined using both external (like G0MeORO) and internal
(the bound affinity, see [4]) values as well as, say, a neutral, or general value like a function calculated by
(17) at given activity coefficient.
Table 2.
Reduced temperatures, standard deviations and coefficients of determination between * and (ln *) in
some MeORO-S systems. Initial reactants ratio S/MeO = 0.1.
CoO*RO
Tt , K

Standard deviation, %
Coefficient of determination
SrO*RO
PbO*RO
298.15
798.15
298.15
40.02
8.99
0.98
6.54
2.99
0.99
3.93
6.80
0.97
In principle all three may be used to calculate or evaluate L. This allows us to reword more explicitly the
problem set in the beginning of this work and explain the alternative temperature more clear. It is easy to see
that equation (17) represents another form of the shift-force linearity.
Recalling that  = (Tch /Tt), one can receive
* = [1/(RTch)]• (QE / *),
(18)
where QE is a general symbol for excessive thermodynamic function. It means that the shifting force is
unambiguously related to the excessive function, and the alternative temperature is just inverse to the
1- 5
coefficient of proportionality between the force and the shift it causes. The product RT ch has dimension of
energy while and  are dimensionless.
To compare values of received with different methods we put them together in one Table 3. Abbreviations
in the table means: R-simulation – thermodynamic simulation of homologous series of reactions with
MeORO, (ElRea)-simulation – abstract simulation of elemental reactions with corresponding reaction
equation and varied , and simulation stands for thermodynamic simulation with artificially varied
coefficients of thermodynamic activity. In some cases (like in reaction 2CoO+S=2Co+SO2 at 298K) results
match very well, but in some cases, like in the example below, the match is not very good.
All results depend upon precision of input thermochemical information used to calculate equilibrium
composition. It seems that in case of (ElRea)–simulation this dependence is less expressed, and this method
may be more preferable.
Table 3.
Reduced chaotic temperatures and average deviations. Reactions MeO+H2=Me+H2, 973K,
initial MeO:H2=1:1.

*scope
NiO
0.92
> 0.4
CdO
0.88
CoO
0.85
MeO
R-simulation

Avedev %%
24.68
1.00
~ 0.3
19.33
2.61
0.4 - 0.5
17.66
0.01
ElRea-simulation
simulation
 Avedev %% 
18.9
19.7
4
2.67
7
15.0
14.9
4
1.52
2
12.8
10.3
8
2.39
8
Avedev %%
1.93
1.74
1.75
CHEMICAL ANALOG OF THE HOOKE’S LAW.
Linearity of the shift vs. the force up to certain yield point and then sharp deviation from linearity (see Figs 1
and 4) bring to mind an idea that there should exist a kind of chemical analog to the Hooke’s law, well known
in mechanics of materials. We have investigated about a hundred of reactions between double oxides with
essential negative values of standard changes of Gibbs’ energy of formation and various reductants like S, H2,
1.0

0.5
0.0
6 ( G/RT)/ 12

0
Fig.4. Reaction shifts vs. external thermodynamic forces for reactions of 18 double
oxides with sulfur, Gibbs’ energies in kJ/(mole*K).
CO. In all cases relationship was similar. In addition to the mentioned pictures, typical reaction shifts
distribution in open equilibrium vs. external thermodynamic forces for reactions MeO RO+S at 298K is
shown in Fig.4.
According to theory of elasticity, deformation of a material, which in one-dimensional case equals to the ratio
of elongation to the initial length of sample, x/L, and tension in the deformed material, which is equal to the
ratio of deforming force to the area of the perpendicular to the force section, are related as
(x/L) = (1/E)(F/S),
(19)
where E  Young’s coefficient of elasticity.
In our case, replacing tension by the thermodynamic force, which has a dimension of force strength (because
it is a ratio of free energy change to the reaction extent, that is to “chemical distance” between initial and
1- 6
open-equilibrium states), juxtaposing deformation to the reaction shift *, and “true” equilibrium reaction
extent = 1 to the initial length of the sample, one can easily get
* = (1/)F,
(20)
Comparison between (17) and (20) unambiguously shows similarity between the reduced chaotic temperature
 and Young’s module E. As far as reaction shift * stays within the linearity region, elimination of
thermodynamic force will bring the system back to the state of isolated, “true” equilibrium, reaction shift
turns to zero. Reduced chaotic temperature plays role of the elasticity module, and the yield point corresponds
to the proportionality limit. In the mechanical law elastic potential, or potential energy of elastic deformation
is a quadratic function of deformation. The expression for the change of free Gibbs’ energy in open system it
is also a quadratic function of the reaction shift from equilibrium because It is quite obvious that
positive value of the reaction shift corresponds to “compression” while extension is analogous to the situation
with the negative reaction shift value (1). Suggestions of this chapter may lead to a new thermodynamic
approach to chemical hysteresis.
CHAOS OUT OF ORDER.
Previous parts of this paper were related to equilibrium states of open chemical systems while current part is
intended rather to analyze the features of basic equation. Let’s put down equation (8) in a short form
ln (`/ *)  ** = 0.
(21)
As it is shown on Fig.5, open equilibrium occurs at the crossing of the logarithmic term as function of j and
chaotic term jjj.
The origin of the term “chaotic” in application to some values of this work was explained above with regard
to logistic map, which in form o equation looks like
xn+1  xn(1 xn) = 0.
(22)
Values of *j in (21) and xn in (22) are supposed to stay within the range (0,1). Due to intensive study of the
chaotic processes for more than two decades, properties of (22) are well known while equation (21), to the
best of our knowledge, has never been investigated from this point of view. At the same time, very interesting
features of open chemical systems may be discovered through its study.
Value of as well as -value define the fate of iterations, which are important not only for calculating
algorithm but also for understanding and control processes in complex chemical systems. To refresh reader’s
memory, we will give a very brief summary of key points related to properties of the logistic equation
following [5]. Condition 0<<4 and initial choice x0(0,1) keeps all x values within the same range in the run
of iterations. If <1, the only steady solution is x=0. At 31first bifurcation occurs and solutions to
equation (22) split with period 2. At 3.5 next bifurcation takes place turning the period to 4, at 3.54
period doubles again and becomes equal to 8, and so forth. Further increase of beyond ~ 3.5699… leads to
non-repeating sequence of numbers referred to as chaotic.
It is obvious that in case of more complicated equation (21) (we will refer it to as OpEq equation or loglogistic map), the solutions behavior will depend upon relative contributions of both terms – classical
logarithmic, which tends toward classical pattern of isolated “true” thermodynamic equilibrium, and nonclassical chaotic, leading the open chemical system to bifurcations and chaos. This paper presents results of
preliminary investigation of the OpEq equation using as example elemental reaction A+B=C under the
influence of these two powers.
Classical paradigm of chemical thermodynamics admits only one state of equilibrium (Zel’dovich’s theorem,
[7]). This statement is valid only for simple isolated chemical system. There are two possibilities in case of
open equilibrium before bifurcations and then following chaos occur. Equation (21) has 2 solutions as
minimum – trivial at =0, where both curves have one joint point (as with =1 on Fig.5), and also non-trivial
with >0 which represents namely open equilibrium and has at least one crossing point with the second term
(upper parabola). If such point exists then open equilibrium has a non-trivial solution. Both logarithmic and
chaotic functions are continuous, differentiable and monotonous, and non-trivial solution of open equilibrium
exists if derivative of the logarithmic curve is less than derivative of the chaotic curve in the initial point =0
dln(`/*)/d < d()/d
(23)
Minimal value of , providing existence of non-trivial solution, can be easily received from (8). Its right side
equals merely to , while left side, taking into account that `=K, gives a product (1/K) d*/d, and
(1/K) d*/d.(24)
Minimal value of  is totally defined by K, or the thermodynamic equivalent of chemical transformation ,
and reaction equation (which defines expression for *). Though numbers used to plot graph on Fig.5 are
tajken for example, the result is quite simple - if condition (24) is reversely satisfied, the logarithmic term
essentially prevails and open system still has only one attractor - true thermodynamic equilibrium. All states
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below curve (1) in Fig.8 satisfy condition (24) and solution is =0. If a non-trivial solution to (21) exists,
possibility to find bifurcations due to the chaotic term becomes real.
In one-approach we investigated iterative behavior of the basic equation solutions at fixed values of just to
grasp the picture in general. Typical iterative graphs are shown on Fig.6. Period n means n possible solutions.
Fig.7 presents diagram of states of open chemical system with reaction A+B=C. This diagram perhaps is the
most important result of this work. One can distinctively see 3 areas on the diagram – true equilibrium where
1.0
ln(`*)
f (
=3
0.5
=1
0.0
0.0

0.5
1.0
Fig.5. Logarithmic and chaotic terms of the basic OpEq equation as functions of
the shift between open and true thermodynamic equilibrium *j.
curves are laying immediately on abscissa and all the way long have =0, open equilibrium from the points
of >0 to the split points, and bifurcations after the split points. At the tail-ends of the split curves period
doubles.
=5
1.4
=10, period 2
1.0



0.7
0.5
0.6
0.0
0.0
0
5
10
15
=20, period 4
1.2
0.0
0
5
10
15
0
5
10
15
Fig.6. Bifurcation of solutions to the basic equation. Reaction A+B=C, =0.1.
Abscissa – number of iterations.
Important is the fact that open system still stays in open equilibrium up to essential values of reduced chaotic
temperatures, and further increase of  must occur to move open system up to the split point.
1.0

0.5
1 2
0.0
0
3
6
12

Fig.7. State diagram of the open chemical system, reaction A+B=C,
corresponding values: 1 0.1, 2  0.5, 3 – 0.7.
Table 3.
Relation between parameter  and  on two equilibrium limits, TTDE – true equilibrium, OpEq – open
equilibrium, reaction A+B=C, initial reactant amounts equal to unity.

K
(G)/T
1- 8
TTDE
OpEq
0.1
0.3
0.5
0.7
0.9
0.24
1.04
3.00
10.11
99.00
12.06
-0.33
-9.13
-19.24
-38.21
1.2
1.6
2.7
5.2
13.1
7.4
8.6
10.3
12.5
18.5
Fig. 7 exemplifies explicitly the influence of the parameter 
asymbol of the reaction “classical
strength”. Numerical data is placed in Table 3 (initial amounts of reactants moles were equal to unity).
b
a
20
20

2

10
2
1
10
1
0
1
2
3
0
4  5
-40
-20
0
 G/T 20
Fig.8. Thresholds of trivial solution (upper limit to true equilibrium, 1) and first bifurcation (upper limit to
open equilibrium or equilibrium at all, 2) (with precision of  ~ +/-0.05): a) vs. TD-equivalent (points on
x-axis with numbers 1 to 5 correspond to =0.1, 0.3, 05, 0.7, 0.9), and b) vs. G/T, reaction A+B=C.
We have placed in Table 3 and on Fig.8b also more habitual values of equilibrium constant K and reduced
changes of Gibbs’ free energy G/Tt along with values of . Chaotic temperature in all cases as well as
change of free Gibbs’ energy is reduced by thermodynamic temperature, and both are counted in values per
Kelvin’s degree.
Due to above mentioned similarity with the Hooke's law, value of beyond the yield point is always different
than below it. This feature may influence the results. One should also account coordinate y of the yield point
itself, this brings the number of independent parameters to at least 4. To investigate the case with two values
of the calculation program was designed so that reaction shift was iterated first with initial value of 
corresponding to the left, ascending part of the graph on Fig.2. As soon as running value of  hit the yield
point, iterations automatically continued with second value of  always bigger than the first as it was already
mentioned above. It occurred that if y was within very reasonable limits (0.3  0.7; the values found earlier
for real systems) second  played major role in the system behavior. The most unexpected discovery was that
the two-calculation results didn’t have a big difference with one- approach - region of trivial solution
stayed almost untouched, and no matter how big were the values of the first  (if less than thresholds in Table
1), bifurcations eventually occurred when second  reached about the same bifurcation threshold numbers as
the only  in one- model. Because we couldn’t find any really distinguishable points for the two- model, our
major conclusion is that it features the same threshold values as one-approach.
We would like to focus reader’s attention once again on Fig.7 and Fig.8 containing graphical images of the
most important things discovered in quite simple reaction by the new theory. Area below curve 1 in Fig.8
corresponds to true thermodynamic equilibrium (=0) though value of already isn’t zero. Area between
curves 1 and 2 corresponds to open equilibria (0<where non-trivial solutions to (21) residerea above
curve 2 corresponds to bifurcations and no equilibrium at all. Therefore, zone of open equilibria with nonzero shifts separates true equilibrium and chaos, or classical and non-classical areas, serving as a water shed
between regular and strange attractors. It is noteworthy and quite natural that the more negative is standard
change of free Gibbs’ energy of reaction the higher are limit values of reduced chaotic temperatures.
A subtitle of this part of the paper could be “How simple is simple reaction A+B=C?”. As it follows from
here presented preliminary results, this reaction ceases to be simple as soon as it runs in open chemical
system.
PRACTICAL APPLICATIONS.
With current methods in chemical thermodynamics, we have to know appropriate coefficients of
thermodynamic activity in order to simulate and compute equilibrium composition of most complex chemical
systems. Their numeric values are not always available, and it is usually very expensive to get them when we
are in need. MCD offers an easier and involving much less efforts way to run that kind of simulation. Indeed,
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this new theory interprets equilibrium of complex chemical system as truly internal equilibrium constituted by
a set of subsystems’ equilibria, explicitly defined by appropriate set of basic equations. Now, having the kj
value from solution for the isolated state and j as a characteristic for subsystem response to external
thermodynamic perturbation (as it was above described in details), we can get equation (8) in form (21)
containing only variable and parameter  .
Current methods of simulation of complex equilibrium use the constant equations (or equivalent to it
minimization Gibbs’ potential of the system) in the same form as if subsystems are isolated, occasionally
using not always available coefficients of thermodynamic activity. In most cases their joint solution actually
only restricts consuming the common participants, watching material balance within the system and playing
role of accountant. Application of current methods to real systems leads to some definite errors in simulation
results originated due to misinterpretation of their status [6]. Method of Chemical Dynamics assumes states of
subsystems as open equilibria within complex equilibrium and should give more correct numerical output.
A principal feature of application following from our method consists in usage of reaction shift *j (as the
system’s response to external impact) multiplied by coefficient proportionality rather than activity coefficient
k. Providing with an easy way to obtain value of j in open equilibria within minutes, MCD brings new
opportunities into analysis and simulation of complex chemical systems.
Another new in principle opportunity consists in analysis of equilibrium and non-equilibrium areas of
solutions to the basic equation of the theory. This allows us to evaluate critical values of external impact
(values of to keep open system within desirable areas.
DISCUSSION.
The new basic equation derived in this work links equilibrium (corresponding to isolated systems) and nonequilibrium thermodynamics (making sense in open systems), and may be rewritten more generally as
GjG0j + RTt f t (*jRTch f ch (*j.
The method treats true, isolated thermodynamic equilibrium of a system as a reference state for its open
equilibrium when system becomes a part of a bigger system. This reference state is memorized in kj. Such an
approach matches well the interpretation of equilibrium at zero control parameters (area below line 1 in Fig.8)
as origin of the chaosity scale (the S-theorem, [8]). Based on a very simple and quite natural assumption, the
basic equation of the present work naturally and smoothly drags non-linearity into thermodynamics of open
systems thus bridging a gap between classical and non-classical thermodynamics. Offered in this paper
theory represents new, unified thermodynamics of chemical equilibria fitting open systems as well as isolated
chemical systems as a particular case.
Going down to the results, one can see that system behavior is essentially affected by parameter . Depending
upon the value of standard change of Gibbs’ energy (or ), system still will not deviate from true, classical
thermodynamic equilibrium if external impact doesn’t bring the ratio T ch/Tt beyond a certain value. The
bifurcation threshold features the same dependency. We can strongly declare that evolution of open chemical
system from thermodynamic “dead” order through bifurcations to “vivid” chaos, i.e. its transition from
kingdom of thermal energy to the point where it gives up to external power, is driven by ratio between
chaotic and thermodynamic temperatures.
We should confess so far that our actual understanding of physical meaning of the chaotic temperature and
how to estimate it independently is far from clarity. We know for sure that in open equilibrium value of
reduced chaotic temperature  can be easily found for any equilibrium value of reaction extent * directly
from the basic equation (e.g., 21) and may be immediately used in thermodynamic simulation. Also, the state
diagrams of open systems may be used to find correspondence between values of  and .This makes new
theory and following from it Method of Chemical Dynamics (MCD, [6]) immediately available for practice.
Practical advantage of the method of chemical dynamics is not restricted by opportunity to avoid usage of
coefficients of thermodynamic activity. The method, for instance, also leads to new in principle opportunity to
simulate internal equilibrium of a system with subsystems at different thermodynamic temperatures (like in
plasma).
To conclude we would like to mention that open chemical system is by definition coupled with another open
system, more exactly, with its compliment to a bigger system. Changing the control parameters may cause an
adjustment of the whole system to new equilibrium state through the bifurcations area, and one will observe it
as a system of coupled oscillators [5]. If they are convergent the system eventually may achieve internal
equilibrium.
Addressing to a skeptical reader, we’d like to underline that our non-traditional term of the basic equation
already existed in chemical thermodynamics in form of excessive thermodynamic function, having different
meaning and origin. This work gives alternative description in relation to an external impact and offers
unified concept of open chemical systems where true and open equilibrium, bifurcations and chaos are
1 - 10
logically tied together in unknown before combination. We just tried to find out what has been lost or hidden
when chemical system, the major object of chemical thermodynamics, has been idealized as an isolated entity.
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de Donder, T.; van Risselberge, T.
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Thermodynamic Theory of Affinity; Oxford University Press; Stanford, 1936.
[4]
Gyarmati, I.
[5]
Epstein, I., Pojman, J.
An Introduction to Non-linear Chemical Dynamics; Oxford University Press: New York, 1998.
Zilbergleyt, B.
Thermodynamic equilibrium in open chemical systems; // LANL Printed Archives,
Chem. Physics, 19.04/2000, http://arXiv.org/abs/physics/0004035.
Zel’dovich, Ya.
J. Phys. Chem.(USSR), 1938, 3, 385.
Non-Equilibrium Thermodynamics; Springer-Verlag; Berlin, 1970.
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Klimontovitch, Yu.
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