HUNGARIAN JOURNAL 
OF INDUSTRIAL CHEMISTRY 

VESZPREM 
Vol. 29. pp. 87-90 (2001) 

ESTIMATION OF STEADY STATES NUMBER OF ONE-ROUTE CATALYTIC 
REACTIONS 

E. S. PATMAR, B. V. ALEXEEV, N. I. KOLTSOV and F. J. KEIL1 

(~epartment of Physic~ Chemi~try o! Chuvash State University, Moskovskii prospekt 15, 428015 Cheboksary, RUSSIA 
Department of Chemical Engmeenng, Technical University ofHamburg-Harburg, Eissendorfer Strasse 38, D-21073 

Hamburg, GERMANY) 

Received: December 28, 2000 

For one-route catalytic reactions the relationship between the number of reversible steps and the number of internal 
steady states (ISS) is i~vestigated. ~or r~actions having trimolecular steps, among which there is only one reversible 
stage, the number ISS mcreases to mfimty together with the increase of the number of intermediate substances. The 
~o~el.of a reaction with 2n intermediate substances for which the ISS number is equal to n-1 is described. If there are no 
hmitatwns. in the number of reversible ~teps in ~ reaction mechanism, then the number of ISS grows exponentially 
toget~er with the growth ~f ~umber of mtermed1ate substances. The model of a bimolecular homogeous reaction (a 
reactiOn whose steps have Similar molecularity) with 3n intermediate substances is given. It can have 211 ISS. 

Keywords: catalytic reaction, internal steady states, intermediate substances, mechanism 

Introduction 

One of the important problems . of catalytic reaction 
kinetics is to find out the reasons for the occurrence and 
the number of steady states (SS). If there are some SS 
under the same conditions of reaction, then the reaction 
rates in these SS are different. Therefore, the 
investigation of the possibility of existance of SS is a 
topical problem. Determining a SS which has the highest 
rate of reaction allows one to solve optimization 
probems. The classification of types of SS is given in 
[ lJ. According to this classification both internal (ISS) 
and boundary (BSS) steady states are determined. In 
these steady states the number of intermediate 
substances with zero concentrations is equal or not equal 
to zero, respectively. Some papers [2-41 were devoted to 
the investigation problems of multiplicity (MSS) of SS. 
The necessary conditions of MSS is the existence of 
interaction steps of different intermediate substances in 
a reaction mechanism [2]. The sufficient conditions of 
unique occurrence of ISS in catalytic reactions are 
presented in [3]. However~ the results described in [2] 
and [3] do not have both necessary and sufficient 
conditions of multiplicity, that is they are not criteria for 
MSS. For the first time, the multiplicity criterion was 
described in [4]. Later on this criterion was presented in 
an algorithmic form and a computer program [51. 

However, this criterion does not allow to determine the 
number of ISS. Some estimates of the number of ISS for 
several classes of catalytic reactions were obtained [6-
8]. For one-route reactions of which the steps are 
irreversible and have arbitrary molecularity the ISS 
number does not exceed two [6]. Some reaction models 
with two intermediate substances of which ISS number 
can be infinite are presented in [7]. However, these 
reactions have a high molecularity of steps. For obvious 
reasons one may ask the question: are there any classes 
of reactions with low molecularity allowing an unlimited 
increase of the number of ISS? The answer is given in 
[8]. The reaction models with n intermediate substances 
of which the number of ISS is not less than n+ 1 are 
given in this paper. This means that with the help of 
these reaction models steady regimes can be obtained, 
which are characterized by a large number of ISS. The 
reactions investigated in [8] are multi-route, therefore, 
the possibility of the unlimited increase of the number of 
ISS for one-route reactions with a low molecularity of 
each of their steps has not yet been studied. This paper 
gives new results obtained by the authors on the 
investigation of ISS for different classes of catalytic 
reactions. For one-route reactions proceeding according 
to inhomogeneous schemes and having trimolecular 
steps we have proved the possibility of an unlimited 
increase of ISS accompanied by an increase of the 



88 

number of steps. This means that the existence of one 
reversible step in reaction schemes having a great 
number of ISS is not only a necessary [6] but also a 
sufficient condition. Additionally, the exponential 
increase of the number of ISS under the condition of 
increasing the number of steps for one-route reactions 
proceeding via homogeneous bimolecular schemes is 
demonstrated. 

Results and discussion 

Let us describe the reaction A=> B proceeding via the 
scheme which contains 2n inhomogeneous (different 
molecularity) steps, one of which is reversible: 

t.A+X1 =X2 
2.2X1 +X3 -7X4 +2X2 
3.X~ +X5 --=>X6 +X2 

(n+l).X1 +X2 -72X3 ; 

(n+2).X3 +X4 +X2 -72X5 +X1 
(n+3).X5 +X6 +X2 -72X1 +X4 

(n-2).X2,_.. + X2•-~' -7 X:bo-4 +X2 (2n-2).X2._5 + X2 ...... + X2 -72X2._3 + Xz...,; 
(n-1). Xz.-1 + Xz-3 -7 2Xz._2 (2n -1). X2._3 + X2,_2 + X2 -7 2X2._1 + X2.-4 

n. Xz._2 + Xz...1 -7 2X2• 2n. Xz._1 + 2X2• -7 3X1 + B 

(1) 

where A and Bare reactants, X 1 are catalytic centers, Xi 

are intermediate substances (i=2, ... ,2n). This reaction is 
one-route because it proceeds via 2n steps in which 2n-1 
independent intermediate substances take part. 

According to the theory of stationary reactions [9] 
under isothermal conditions of a gradientless differential 
reactor for scheme (1) the stationary equations have the 
following form: 

k2x~x3 = r 
k3x4 x5 = r 

kn+zXzX3X4 = r 

kn+3XzXsX6 = r 

kn-2X2n-6X2n-5 = r kzn-zXzXzn-sXzn-4 = r 
kn-tX2n-4Xzn-3 = r kzn-!XzX2n-3Xzn-2 = T 
knXzn-zXzn-1 :;:;: r kznXzn-lxin == T 

(2) 

where ki = const are stage rate constants, CA is the 
concentration of A, X; are concentrations of intermediate 

substances Xi ~ r is the reaction rate. 
Let us introduce new variables x = c1x 2 ,y = c2x 2 1 x1 ~ 

where cpc;: must be in such a form that the first two 
equations of scheme (2) after substituting r. lead to the 
equation x + y = l . Further, let us mark aU xi by x and 
\' .;• 

X3 = P3Y; 

Xs = PsY·X; 

Xz = PzX 

x4 = p4xl y2 
x6 = p6/ yz 

x2n-3 = P2n-3Y. xn-
3

; x2n-Z = P2nx-n+4 I Y2 

Xzn-1 = P2n-1Y. xn-Z; x2n = P2nx-nl 2+2 I y 

where the Pi are determined by the equations 

k_l kl 
Pt=k,p2=k, 

n+l n+l 

- kn+t ( kt )j-tTij-t km+n . 2 
Pzj-1 --- --, ]= , ... ,n, 

k_l kn+l m=l km+l 

(3) 

P2i = (~)2( kn+I )j-1 IT~. j = 2, ... ,n -1 ' 
kn+l kl m=Z km+n 

Pzn = (!sJ...)2( kn+I )n-2 IT~ 
kn+l kl m=Z km+n 

The numbers Pi can have any positive values by 

changing the rate constants ki . Look at the relations 

P1Pz 3 -~:....:::...-kn+l' j = , ... ,n, 
P2i-4P2i-3 

k _ PtPz k 2n- 2 n+l • 
Pzn-tPzn 

This note is necessary in order to work with such values 

of numbers Pi. Thus, let us mark all xi by x and y, 
and reduce scheme (2) to one equation: 

P3Y + PsY·X+ ... + P2n.-rY·xn-2 -1+ 

+ Ptxly+ P2X+ P4XI Y
2 

+ P61Y
2 

+ ... , (4) 

where y=1-x. 

Let us consider the function 

<p(x) = P3Y+ PsY·x+ ... + P2n_1y·xn-
2 -1 and show 

that the number of different roots qJ(x) lying in the 

interval (0.1) is equal to n-1 for the corresponding 
selection of numbers p 3 , p 5 , ••• , p 211_1 • The function l{J(x) 

can be represented in the form: 

1- P3 + (p3 - Ps )x + · ·· + (Pzn-3 - P2n-1 )x"-2 + 
n-l 

+ Pzn-lxll-i = Pzn-ITI(x-xi) 
i=l 

Hence we obtain 

(5) 

Jr-i 

Pz • .-~n Cr-x,)= Pz~~-l(x"-1 -s1x"-2 + ... +(-lt1s1d) (6) 
t4 



where the si are symmetrical root functions in xi , 

whereby the parameters P;, according to relations (5), 
n-1 

can be written in the form p 21_1 == 1- L ( -1)i si , 
i=n-j+l 

j = 2, . .. ,n. By choosing different small positive roots 

xi the symmetrical functions will be also small positive 

numbers and the p21 _1 values will be close to 1. 

Selecting the corresponding ppp2 ,p4 , ••••• ,p2n 

constants, the number of roots of equations (4) lying in 
the interval (0,1) is also equal to n-1. In fact, if we make 
the constants p 1 , p 2 , p 4 , ... , Pzn small, then this results 

from the theorem on the continuous dependence of the 
roots on coefficients [10]. Thus, scheme (1) has n-1 ISS 
at some sets of rate constants. 

In order to show the exponential increase of the 
number of ISS for an increasing number of steps, let us 
consider the reaction A=>B, proceeding via the 
following steps 

l.A+.:t2 +~ =:bi 1'.4 +.:t2 -?2xs 
2.Xs +4; =2x4 2'.x4 +Xs-?2xs 

r".2Xj_ =2.x.,; 

:t'.2x4 =2x6; (7) 

The stationary equations are written in the following 
form: 

Is· xi - k_t•.xi = dl.r 
k2.x; - k_2.~ = d2-r 

k.X:m-t.:13.. -k-nxin-Z =d.r k.·.:l3..-zAJn-l =d •• r k.·xi.-z -k_ •• c8 :x;,. =d •• r 
}:x;=l 

i=l . .JIJ 

(8) 

Dividing the first three equations of system (8) in the 

first line by x[ and substituting r we get two equations 
with two unknown values y 2 =x2 /x1,y3 =~lx1 • We 
continue the same procedure with three equations, then 
with four ones and so on. Further, let us select rate 
constants for these equations in such a way that the 
equations obtained could have two different solutions 
(for example, use the ISS multiplicity criterion for one-
route reactions [5]). Then for these systems the solutions 
have the following form 

~j-l = uiiX3j-Z ;X:J1 = U~X3J-l i1 E [1,2], j = l..n (9) 

Using these solutions we can exclude x3i-PX:Ji, 

mark then X3i-Z, and reduce 3j-l equations of systems 

(8) to the following form: 

cix-;1_2 = dlr i1 E [1,2], j = l..n (10) 

Further, let us substitute r and mark x3i_2 by x1 • 
Using relations (9), let us denote all x

1 
by x1 • On the 

basis of the last equation of system {8) we calculate x1 
and then determine xi • Knowing the data for uj

1
, u;; • 

the solutions of system (9) can be simply found. This 

89 

r, $-1 

0.415 

0.3!5 0.40 0.45 0.50 
CA • aibitraryunits 

Fig.] Dependence of rate on concentration o A for reaction 
proceeding via scheme (7) at k 1 = lUU, k 2 == 4:L, k 3 = :>U, 

k_1 =k_2 =k_3 =11.0, kr =k2, =ky =:ll, _1 
k1• = k2• = k3• = 106, k_1• = k_2• = k_3• = 169 ( s ) 

means that the unique solution of system (9) 
corresponds to just one vector out of a set of 2n vectors. 
We want to show that all the solutions are different. If it 

is not so, then all numbers coincide for two different u/ 
sets. This is impossible because these sets differ at least 
in one of the coordinates. Scheme (7) is constructed on 
the base of scheme (1) in Table 1 of paper [11]. If we 
take any other scheme in Tables 1 and 2 of paper [11] or 
their combination, we can construct schemes having the 
exponential increase of the number of ISS. The kinetic 
dependence in Fig.l illustrates the number of ISS for 
schemes (7). This dependence has the following 
interesting peculiarity: all the roots appear 
simultaneously (in a general case two solutions appear). 
This peculiarity is connected with system (8) by splitting 
the system into independent equation subsystems. This 
property of system (8) leads to this unusual type of 
kinetic dependence. Schemes (7) can be used as test for 
the efficiency of computer programs which determine 

the number of ISS. The absent of solutions at some C A 
(Fig.l) is connected with the presence of boundary 
steady states [1] for scheme (7). 

·conclusion 

The presence of reversible steps in mechanisms of one-
route catalytic reactions is a necessary and sufficient 
condition of an unlimited ISS number. If there are no 
limitations in the number of reversible steps. the number 
of ISS increases exponentially together with the number 
of intermediate substances. 



90 

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