HUNGAR~JOURNAL 

OF INDUS1RIAL CHEMIS1RY 
VESZPREM 

Vol. 29. pp. 81- 86 (2001) 

COMPUTER CONSTRUCTION OF CHEMICAL REACTION MECHANISMS 

B. V. ALEXEEV, N. I. KOLTSOV and F. J. KElL 1 

(Department of Physical Chemistry of Chuvash State University, Moskovskii prospekt 15, 428015 Cheboksruy, RUSSIA 
1Department of Chemical Engineering, Technical University of Hrunburg-Harburg, Eissendorfer Strasse 38, D-21073 

Hrunburg, GERMANY) 

Received: Decernoer 28, 2000 

A computer progrrun was developed which permits the design of all possible mechanisms of a certain class of chemical 
reactions proceeding according to a step scheme via intermediate components and a brutto-reaction. The application of 
the progrrun for concrete reactions is described. 

Keywords: chemical reaction, reaction mechanism, algorithm, intermidiate substances, Maple progrrun 

Introduction 

It is of great importance to find a mechanism of a 
chemical reaction in accordance with measurements. 
The solution of this problem is based on the formulation 
of all possible step schemes describing the experimental 
peculiarities of the reaction. It is based also on choosing 
a mechanism consisting of intermediate components in 
the various steps which is confirmed by the 
corresponding physico-chemical methods. In the 
publications [1-3] the systematization of all possible 
three- and four-step mechanisms of catalytic reactions is 
carried out. For these mechanisms the relaxation time 
[1] and autooscillations [2,3] were investigated. As a 
base for generating these mechanisms, transient steps of 
intermediate components on a catalyst surface are taken 
without referring to special chemical reactants. In the 
present paper a program for the automatic generation of 
stoichiometric factors of a certain class of reactions will 
be presented. 

The algorithm and the program 

The computer program which allows the design of step 
schemes of chemical reactions like 

(1) 

(i = l, ... ,sA; l = l, ... ,nA), 

is described in this paper. They proceed via the 
following steps 

L,.a;x i = L,.aijX i , d,; 
j 

(i = l, ... ,sx, j == l, ... ,nx; r = l, ... ,q ), 

(2) 

where A1 , X j are reactants and intermediate substances; 

nA, nx are the numbers of reactants and intermediate 

substances; sA, sx are the number of brutto-reactions 

(1) and steps (2) in the mechanism, respectively; a; 2 0 
and c: 2 0 are stoichiometric coefficients; d,; > 0 are 
stoichiometric numbers of the i-th step in the r-th route 
(a route is a linear combination of elementary steps 
which lead to the final brutto reaction of the reactants); 
q is the number of routes. The goal of the program is to 
determine all possible mechanisms of step scheme (2) in 
the form 

2);A1 + L,.a;x j "'l:h;A1 + L,.a;;x j (3) 
l j l j 

taking into account brutto-reactions of eq. (1). It is 
worth noting that the scheme {2) is supposed to be given 
and the program provides the selection of basic 
substances for this scheme. The program searches not 

only for the stoichiometric coefficients b1i , but also 
guarantees the satisfaction of the laws of conservation 



82 

Table 1 The block diagram of algorithm of the program 

Defining of stage scheme a*X=a_ *X and brutto-scheme c* A=c_ *A by lists of stages, written in a 
form of linear equations, connecting the symbols of intermediates and basic substances, respectively . 

• Calculating the matrix of stoichiometric numbers a-a_ and the vector of intermediates X on a stage 
scheme given. 

• Calculating the number and basis of conservation laws alpha of stage scheme, defined from the 
property, that they belong to right kernel of the matrix of stoichiometric numbers . 

• Calculating the number and basis of routes d of stage scheme, defined from the property, that they 
belong to left kernel of the matrix of stoichiometric numbers . 

• Calculating the matrix of stoichiometric numbers c-c_ and the vector of basic substances A by the 
given brutto-scheme. 

• Calculating the number and basis of conservation laws beta of brutto-scheme, defined from the 
property, that they belong to right kernel of the matrix of stoichiometric numbers. 

t 
Calculating the number and basis of routes r of brutto-scheme, defined from the property, that they 
belong to left kernel of the matrix of stoichiometric numbers. 

t 
Finding the matrix of stoichiometric coefficients b-b_ from the matrix equation d*{b-b_)=u*(c-c_). 
Usually, the solution is not unique, and may possess some arbitrary parameters v in addition to 
parameters u. As separate determinion of matrices of stoichiometric numbers b and b_ is impossible 
under approach .exploited here, the coefficients of the matrix b could be considered as arbitrary. 
When matrices u, v and b are selected, matrix b_ could be calculated uniquely. 

t 
Calculating the molecularities matrix gamma from the matrix equation {a-a_)*gamma=-{b-b_)*beta, 
this constitutes the second condition mentioned above. The solution might be non-unique, wich may 
cause the appearing of another arbitrary matrix w. 

t 
Printing the problem solution in a form of stages b*A+a*X=b_ *A+a_ *X of a mechanism and 
chemical formulas of intermediates lnX=alpha *lnZ+gamma *lnP and the basic substances 
lnA=beta*lnP. 

t 

for reactants and intermediate substances and the 
chemical formulas of the species Xi and reactants A1 • (5) 

The algorithm is as follows. The chemical formulas 
of intermediates X 1 are written in the form 

x1 =fiz:"'*llP!", 

where ai* are numbers of adsorption sites of catalyst 

atoms of type Z" where the intermediates X. are 
J 

(4) 
k k 

where Zt and lk are the types of catalyst and types of 
reactant atoms. The chemical formulas of the reactants 
"'* have the following form 

adsorbed; f3 ik are numbers of adsorbing reactants of 
type lk on the adsorbing sites zk ; r lk are the numbers 
which connect of numbers a Jk , fJ J< and the 
stoichiometric coefficients by the following relations 



L(a1; -aij)aik =0; 
j 

:L<bt; -b~7)y,k =o, 
l 

Ldr/b; -bij)fJjl =0, 
i,j 

:Lee; -c~)Y1k +:Lea; -aij)/3jk = o. 
l j 

(6) 

The solution of these linear equations (unknown values 

are a ik , f3 ik , y1k_, b1; and b11 ) results in the mechanism 
(3) which is in accordance with the step scheme (2) and 
the brutto-reactions (l). 

This algorithm is realized in a program written in 
Maple programming language. 

The problem that this program solves is summarized 
below: 

Given: 

L Step scheme a*X==a_ *X 
2. Routes d*(a-a_}=O 
3. Conservation laws (a-a_)*alpha=O 
4. Chemical formulas InX=alpha*InZ 
5. Brutto-reaction c*A=c_*A 
6. Routes r*(c-c.J=O 
7. Conservation laws (c-c_)*beta=O 
8. Chemical formulas lnA=beta*JnP 

The following set of linear equations has to be solved to 
find alpha, beta, gamma, b and b_. 
Find b, b_, alpha, beta, gamma such that: 
1. Mechanism is b* A +a*X=b_ *A +a_ *X, 
2. Chemical formulas are 

InX=alpha*lnZ+gannna*InP, 
and the following conditions are fullfilled: 
a) on routes: d*(b-b_)*beta=O 
b) on conservation laws: 

(b-b_)*beta+(a-a_)*gamma=O 
Here a and a_ are matrices of stoichiometric 

coefficients of the intermediates in eq. (2); c, c_ and b, 
b_ are matrices of stoichiometric .coefficients of the 
reactants in eqs. (1) and (3), respectively; dis the matrix 
of stoichiometric numbers of steps in different routes 
(see eq.{2)); alpha, beta, gamma are matrices of 
coefficients aik, f3it, ylk (see eqs. (4),(5)); A, X, Z, P 

are vectors of A,, Xi, Zk, lk (see eqs.(4), (5)). Brief 
comments on the algorithm will be given. 

The first condition represents equality of all d-routes 
of the mechanism (i.e. routes, obtained using 
stoichiometric coefficients of matrix d) to the set of 
steps of the brutto-scheme. We mean an equality with 
respect to linear combinations of steps, i.e. equality of 
linear spans of vectors of stoichiometric numbers of 
steps. This condition could be written also in a form 
d*(b-b.J=n*{c-c_), where n is an arbitrary matrix of the 
corresponding size. It follows from the orthogonality of 
matrices d*(b·bJ and c-c_ to the same matrix beta. 

83 

The second condition describes the balance of each 
step of the mechanism of atoms P of reactants (the 
balance on atoms X of intermediates is satisfied 
automatically). The main steps of the program are given 
in a block scheme (see Table 1). 

Application of the program 

Let us analyze the applications of the program for 
concrete chemical reactions. 
For example, the reaction 2CO + 0 2 = 2C02 proceeds 
yia a four-step scheme 

1. 2X1 =2X2 , 

2. X 1 =X3 , 

3. X 2 +X3 ==2X1 , 

4. X 2 ==X1 • 

The initial data will look like 

StepScheme:=£2 *xi =2*x2,xl =x3,x2 +x3=2 *xl,x2==xl ]; 
BruttoScheme:=[2 *C0+02=2 *C02 ]; 

The step scheme in mechanism set (9) given below in 
the appendix becomes the simplest one if the parameter 
values are the following 

b12 =b2! =b4l =1, 

b11 = b13 = hzz = b23 = b3I = b32 = b33 = b42 = b43 ""0, 

u11 = v11 = w11 = 1, 

V31 = V32 = -1, U21 = Vzi = V12 = Vzz == Wzi = 0 · 
The following mechanism is the simplest: 

1. 0 2 +2K=2KO, 

2. CO+K=KCO, 
3. KO+KC0=2K +C02 , 

4. CO+KO=K+C02 , (7) 

which corresponds to a two-route scheme of carbon 
monoxide oxidation on platinum catalysts (4]. This 
model is important because mechanism (7) has two 
routes which are characterized by a similar brutto~ 
reaction ( 2CO + 0 2 = 2C02 ). 

If a three-step mechanism 

1. 2X1 =X2 +X3 , 

2. X 2 =X1 , 

3. X 3 =X2 , 

is supposed to be realized for the reaction 
2CO + 2NO = 2C02 + N 2 , the initial data for the 
program will be as follows: 

StepScheme:=[2*xl=x2+x3,x2=xl,x3=x2]; 
BruttoScheme:=[2*C0+2*N0=2*C02+N2]. 

The program gives not one, but a set of mechanisms 
(10) which is characterized by the parameters 



84 

b;p wii, vii, uii. The selection of fixed values of these 

parameters like 

b13 = bzl = b33 = 1 ' 

b11 = h12 = b14 = bzz = bz3 = bz4 = h31 = h32 = b34 = 0 ' 

uu = vu = v32 = 1, Vzt = V4z = -1, 
V31 =V41 =v12 =vzz =wn =wzl =w31 =0 

allows one to write down a mechanism and chemical 
formulas of reactants and intermediate substances in the 
following form 

N0+2X 1 = X 2 +X3 , CO+X 2 = 
= C02 + X 1 , NO+X 3 =N2 +X 2 , 

X 1 =Zp X 2 =Z1P2P3
1

, X 3 =Z1 Jl-Pz-
1~, 

CO=P3 ,C02 =P2,NO=~,N2 =~
2P2-

2~2 • 

This is a mechanism 

l.N0+2K =KO+KN, 
2. CO+ KO = K + C02 , 
3.NO+KN=KO+N2 • (8) 

It coincides with the known mechanism [5] of carbon 
monoxide interaction with nitrogen oxide on platinum 
catalysts. 

Conclusion 

It is known that the structure of a mechanism to 
determine the dependence of the reaction rate as a 
function of concentration of reactants or time [6]. 
Reactions proceeding via linear mechanisms are 
characterized by monotonous dependences of stationary 
rates vs. initially taken substance concentrations. 
Reactions proceeding via nonlinear mechanisms are 
characterized by critical phenomena like multiplicity of 
stationary states (MSS) and autoosciUations. There are 
papers in which step schemes describing autooscillations 
[3] and MSS [6] are published. Thus, if a given reaction 
proceeds in an autooscillation regime, its mechanism 
can be established by the step schemes which are given 
in the paper [3]. For each scheme it is possible with the 
aid of our program to determine the chemical structure 
of intermediate components and the amount of reactants 
in each step. Some other schemes are formed in the 
same manner. As a result, a set of mechanisms is 
formed. In the end that mechanism containing 
intermediate components must be chosen, which is 
confirmed by experimental methods. 

Thus~ the developed program enables one to 
construct mechanisms of chemical reactions with the aid 
of a computer. This program gives the ability to search 
for chemical formulas of reactants and intermediate 
components in each step of a reaction mechanism. The 
program can be used for the solution of the step scheme 
identification problem, which is connected with a unique 
mechanism determination of a reaction which is 
characterized by a definite kinetic beha'<1ior. 

REFERENCES 

1. ALEXEEV B. V., FEDOTOV V. KH. and KOLTSOV N. 
I.: Chern. Phys., 1982, 1, 776-779 (in Russian) 

2. FEDOTOV V. KH., ALEXEEV B. V. and KOLTSOV N. 
I.: Izv. Vuzov. Chemistry and Chern. Technology, 
1985,28, 66-68 (in Russian) 

3. KOLTSOV N. I., ALEXEEV B. V. and FEDOTOV V. 
KH.: Dokl. Akad. Nauk SSSR, 1994, 337, 761-764 
(in Russian) 

4. IVANOV V. P., EWKHIN V.I., YABLONSKI! G. S., 
SAVCHENKO V. I. and TATAUROV V. L.: Kinet. 
Katal., 1981, 22, 1040-1042 (in Russian) 

5. DRUZHININ D. L., IVANOVA A. N. and FuRMAN G. 
A.: Chern. Phys., 1986, 5, 1410-1412 (in Russian) 

6. KOLTSOV N. I., FEDOTOV V. KH. and ALEXEEV B. 
V.: Kinet. Katal., 1995, 36,42-46 (in Russian) 

APPENDIX 

The program written in MAPLE and examples 

Program 

with(linalg): 
# Stepl Step scheme a*X=a_ *X & brutto-scheme c* A=c_ *A 
# StepScheme:=[xl=x2,x2=x3,2*xl +x3=3*xl]; 
# StepScheme:=[2*xl=2*x2,xl=x3,x2+x3=2*xl,x2=xl]; 
# BruttoScheme:=[2*C0+02=2*C02]; 
# 02+2K=2KO, KO+KC0=2K+C02, CO+K=KCO, 

CO+KO=K+C02 
# StepScheme:=[2*xl=x2+x3,x2=xl,x3=x2]; 

BruttoScheme:=[2*C0+2*N0=2*C02+N2]; 
# N0+2K=KO+KN, CO+KO=K+C02, NO+KN=KO+N2 
StepScheme:=[xl=x2,x2=x3,x3=xl]; 
BruttoScheme:=[2*C0+2*N0=2*C02+N2]; 
#N0+2K=KO+KN,CO+KO=K+C02,~0+KN=KO+N2 
#Step 2 Stoichiometric matrix a-a_ & vector X 
Steps:=nups(StepScheme): lprint('Intermediate reaction step 
scheme given' ,StepScheme ); 
Intermediates Vector:=transpose(matrix([[ op(indets(StepSche 
me))]])): Intermediates:=rowdim(Intermediates Vector): 
lprint('Steps founded' ,Steps); 
if Steps=O then ERROR(' At least one step must be given') fi; 
!print(' Intermediates founded' ,Intermediates): 
I print(' Intermediates vector'): print(Intermediates Vector): 
if Intermediates=() then ERROR(' At least one intermediate 
must be founded') fi; 
a:=matrix(Steps,Intermediates,['['coeff( op(l,StepScheme[i]}, 
intermediates Vector[j, 1 ]}'$'j'= ! . .Intermediates ]'$'i'=l .. Steps ]): 
a....:=matrix(Steps,Intermediates,['['coeff( op(2,StepScheme[i]), 
IntermediatesVector[j,l])'$T=l .. Intermediates]'$'i'=l .. Steps]): 
'a-a_ ":=matadd(a,a_, 1,-1 ): 
Iprint('Stoichiometric matrices of intermediate reaction step 
scheme,a,a_,a-a_ "}; 
print(a,a_.'a-a_ ')~ 
# Step 3 Conservation laws alpha 
alpha:=transpose(matrix([op{kernel('a-a_ ')) ])): 
l...aws:=coldim(alpha): lprint('Intermediates conservation laws 
founded' ,Laws); 
if Laws=O then ERROR(' At least one conservation law must 



exist') else lprint('Intermediates conservation laws matrix, 
alpha'); print( alpha); 
# Step 4 Routes d 
d:=matrix([op(kemel(transpose('a-a_ ')))]): 
Routes:=rowdim( d): lprint('Routes founded' ,Routes); 
ifRoutes>O then lprint('Routes matrix, d'); print(d); fi fi: 
# Step 5 Matrix c-c_ & vector A 
lprint('Brutto reaction step scheme given',BruttoScheme); 
Brutos:=nops(BruttoScheme ): I print(' Steps founded',Brutos); 
Species V ector:=transpose(matrix([[ op(indets(BruttoScheme))] 
])): Species:=rowdim(SpeciesVector): lprint('Species 
founded',Species): lprint('Species vector'): 
print( Species Vector): if ( { op(Intermediates Vector)} intersect 
{ op(SpeciesVector) })<>{} then ERROR('Intermediates 
vector' ,Intermediates Vector, 'of step scheme', 
StepScheme, 'and species vector',Species Vector,' ofbrutto 
reaction' ,Brutto, 'must have zero intersection', 
{ op(Intermediates Vector)} intersect { op(Species Vector)}) fi; 
'c-c_':=njatrix(Brutos,Species,[seq([seq(coeff(op(l, 
BruttoScheme[i])-op(2,BruttoScheme[i]), 
SpeciesVector[j,l]),j=l .. Species)],i=l .. Brutos)]): 
lprint('Stoichiometric matrix ofbrutto reaction step scheme, 
c-c_ '); print('c-c_ '); 
# Step 6 Conservation laws beta 
beta:=transpose(matrix([op(kernel('c-c_'))])): 
BruttoLaws:=coldim(beta): lprint('Brutto conservation laws 
founded' ,BruttoLaws ); 
if BruttoLaws>O then !print(' Conservation laws matrix, beta'); 
print(beta) fi: 
# Step 7 Routes r 
r:=matrix([op(kernel(transpose('c-c_ ')))]): 
BruttoRoutes:=rowdim(r): lprint('Brutto routes founded', 
BruttoRoutes); if BruttoRoutes>O then lprint('Brutto routes 
matrix'); print(r) fi; 
# Step 8 Stoichiometric matrix b-b_ 
RowSp:=matrix([ op(rowspan(' c-c_ ')) ]): 
u:=array(l .. Routes, l .. rowdim(RowSp) ): 
mu:=multiply(u,RowSp )~ 'b-b_ ':=linsolve( d,mu,'R','v'): 
lprint('Matrix b-b_ '): print('b-b_ '): !print(' Verification: 
matrix expression d*(b-b_)*beta'): 'd*(b-b_)*beta':=evalm(d 
&* 'b-b_' &* beta); print(' d*(b-b_)*beta '): I print(' must be 
zero matrix'): 
ifiszero('d*(b-b_)*beta') then lprint('Verification passed') 
else ERROR(' Matrix expression d*(b-b_)*beta must be zero 
matrix') fi; b:=array(l..Steps,l..Species): 
lprint('Matrix b'): print(b): b_:=matadd(b,'b-b_ ',1,-1): 
lprint('Matrix b_ '): print(b_): 
#Step 9 Matrix gamma 
gama:=linsolve('a-a_ ',evalm(-'b-b_" & * beta),'r2' ,'w'): 
I print(' Stoichiometric matrix, gama '); print(gama); 
Iprint('Verification: matrix expression (a-a_)*gama+(b-
b_)*beta'); '(a-a_)*gama+(b-b_)*beta':=evalm('a-a_' &* 
gama+'b-b_' &*beta): print('(a-a_)*gama+(b-b_)*beta'); 
lprint('must be zero matrix'); 
if iszero('(a-a_}*gama+(b-b_)*beta ') then lprint('Verification 
passed') else ERROR('Matrix expression (a-a_)*gama+(b-
b_)*beta must be zero matrix') fi; 
# Step 10 Print the solution 
lprint("Intennediates step scheme'): print(StepScheme): 
lprint('Brutto step scheme'): print(BruttoScheme ): 
Z:=['Z.i'$'i'=l..Laws]: P:=['P .i'$'i'= I .. BruttoLaws ]: 
m:=evalm(b &*Species Vector+ a&* Intermediates Vector): 
m_:=evalm(b_ &*Species Vector+ a_&* 
Intermediates Vector): 
eqs:=array(['['m[i.j]=m_[i,j]'$'i'= l .. rowdim(m) ]'$)'=I .. coidim( 
m_)J): lprint('Reaction mechanism'):print(eqs): 
m3:=evalm(alpha &* Z + gama &* P): 
ln:=[seq(IntermediatesVector[i,l]=m3[i],i=l . .Intennediates)]: 

lprint('Chemical formulas of intermediates~): print(In): 
m4:=evalm(beta &* P): 
Sp:=[seq(SpeciesVector[i,l]=m4[i],i=l .. Species)]: 
lprint('Chemical fonnulas of species'); print(Sp): 

Example 1. 

85 

For the reaction 2CO + 0 2 = 2C02 , proceeding in a 
four-step scheme 

1. 2X1 = 2X 2 , 
2. X 1 = X 3 , 

3. X2 +X3 =2X 1 , 

4. X2 =X1 , 

initial data was entered in a form 
StepScheme:=[2*x1=2 *x2,xl=x3,x2+x3=2*xl,x2=x 1]; 
BruttoScheme:=[2*C0+02=2 *C02]; 
The set of mechanisms of a reaction of carbon monoxide 
oxidation, which is found by the program, has the 
following form 

1. b 11 CO + b 12 0 2 + bi3C02 + 2X1 = 

=(b 11 -4u 21 -2u 11 +2v 11 )CO+ 

+(blz -2uzl-ul! +2Vzr)02 + 

+(b 13 +2u 11 +4u 21 +2v31 )C02 +2X 2, 

2. b 21 CO + b 22 0 2 + h 23C0 2 + xl = 

= (b 21 +2u 21 +v 12 -v 11 )C0+ 

+(b2I2 +u21 +Vzz -Vzi)Oz + 

+(b23 -2Uzt +v32 -v31)C0z +X3, 

3. b:11Co + b 32 0 2 + b 33C0 2 + X 2 + x 3 = 
= (b31 -vtz)CO+(b32 -Vzz)Oz + 

+(b33 -v3z)COz +2Xl, 
4. b 41CO+b 42 0 2 +b 43 C02 +X2 = 

=(b41 -vn)CO+(b42 -Vzt)Oz + 
+ (b43- V31) COz + Xp 

(9) 

where bu, Uu, v u are free parameters. The chemical 
formulas of intermediate substances are determined by 
the relations 

where Z1 are free centres on a catalyst surface, ~ and 

P2 are atoms or atom complexes out of which the basic 

substances are fonned. w11 , w21 are free parameters. 

The chemical formulas of basic substances are 
determined by the program as 

CO= f;, 0 2 = F;-
2 P2

2
, C02 = P-:.. 



86 

li and ~ complexes in this model form oxide and Brutto step scheme 
carbon dioxide respectively. 2CO + 2NO = 2C02 + N 2 

Reaction mechanism 
Example 2. 1. b11 CO +b12 C02 +b13 NO +b14 N 2 + 2X 1 = 

Three-step mechanism 

1. 2X1 = X 2 + X3 , 
2. X 2 =X1 , 

is supposed to be realized for the reaction 
2CO + 2NO = 2C02 + N 2 . The initial data for the 

program will be 
StepScheme:=[2*xl=x2+x3,x2=xl ,x3=x2]; 
BruttoScheme:=[2*C0+2*N0=2*C02+N2]; 
The part of the program output is given below. 
Intermediates step scheme 

2X1 = X 2 +X3, X 2 = X1, X3 = X2 

= (b 11 - 2u11 + 2v11 + v12 )CO+ 

+ (b12 + 2u11 + 2v21 + v22 )C02 + 

+ (b13 - 2u11 + 2v31 + v32 )NO+ 

+ (b14 + Uu + 2v 41 + v 42 )N 2 + X 2 + X 3' 
2. h21 CO +h22 C02 +h23 NO +b24 N 2 + X 2 = 

= (b21 - v11 )CO + (b22 - v21 )C02 + (10) 
+ (b23 -v31 )NO + (b24 - v41 )N2 + Xp 

3. b31 CO+b32 C02 +b33 NO+b34 Nz +X3 = 
= (b31 - v12 )CO+ (b32 - v22 )C02 + 

+ (b33 -v3z)NO + (b34- v42)N2 + Xz 

Chemical formulas of intermediates 

X - Z P. Wn P. w21 P. w31 
I- 1 I 2 3 ' 

X _ z P.-v31-2v41+wn p_-v2I+2v4I+w2t R-vu-2v4I +w3I 
2- I 1 2 3 ' 

X -z v-v32 -2v42 -2v4I-v3I +wu p -vzz-v2t+2v4t+2v42+w2t 
3- tlJ ? 

.p-vt2-2v42 -vu-2v4I +w3I 
3 

Chemical formulas of species 

CO=?:,, C02 = P2 , NO= .Fi, N 2 = li
2 
P2-

2 
P3

2
• 


	Page 85 
	Page 86 
	Page 87 
	Page 88 
	Page 89 
	Page 90