Linear independent reaction

These quantities are preserved like atoms in the given reactions and hence are called reaction invariants (ref. 16). In this example we found 4 linearly independent reaction invariants. It does not mean, however, that the species Ml, 2,. .. M6 are built necessarily from 4 atoms. In fact, introducing the species M = CH4, 2 — O2, 3 = M4 = 5 = 2 9 and M — 2 2... 

We consider a chemical system consisting of the following species methane (CH4), water (H20), carbon monoxid (CO), carbon dioxide (CO2), and hydrogen (H2). There are two linearly independent reactions among these species, e.g.,... 

Such a definition can, evidently, be extended to any number of routes. It is clear that if A(1), A(2), A<3) are routes of a given reaction, then any linear combination of these routes will also be a route of the reaction (i.e., will produce the cancellation of intermediates). Obviously, any number of such combinations can be formed. Speaking in terms of linear algebra, the reaction routes form a vector space. If, in a set of reaction routes, none can be represented as a linear combination of others, then the routes of this set are linearly independent. A set of linearly independent reaction routes such that any route of the reaction is a linear combination of these routes of the set will be called the basis of routes. It follows from the theorems of linear algebra that although the basis of routes can be chosen in different ways, the number of basis routes for a given reaction mechanism is determined uniquely, being the dimension of the space of the routes. Any set of routes is a basis if the routes of the set are linearly independent and if their number is equal to the dimension of the space of routes. 

The number of complexes minus that of connected components of the graph for their conversions equals the number of linearly independent reactions (stoichiometric vectors). A second Horn and Jackson condition for quasi-thermodynamic behaviour is the weak reversibility of the graph for complex conversions. This graph is called weakly reversible if any of its connected components contain a route to get from any node to any other moving in the direction of its arrows. For example, the scheme... 

As already discussed, at equilibrium only the reaction rates wr of the R linearly independent reactions can initially be assumed to vanish. To handle the correspondence between the linearly dependent and independent reactions we relate the A, and Ar affinities through linear equations of the form... 

However, there is one fly in the ointment here It may not be possible to determine the rate laws for each of the reactions. In this case it may be necessary to work with the minimum number of reactions and hope that a rate law can be found for each reaction. That is, you need to find the number of linearly independent reactions m your reaction set. In Example 6-8 just discussed, there are four reactions given [(E6-8.5) through (E6-8.8)]. However, only three of these reactions are independent, as the fourth can be formed from a linear combinafion of the other three, Tcchttiques for determining the number of independent reactions are given by Aris. ... 

Consider two reactions of the type A B and B 5 C. If the third process A C is not feasible the number of elementary reactions is the same as the number of linearly independent reaction equations the reactions are said to be uncoupled. If A 4= C represents a feasible reaction the three processes are said to be coupled generally, coupling occurs whenever there is a redundancy in the number of reaction steps. 

Consider the following case of three coupled reactions denoted by A B (I), B C (II) and C A (III), and compare this situation to the linearly independent reactions A B (1) and B C (2). The rate of disappearance of the various reagents is related to the reaction rates, co, by... 

To appreciate the special role of a conical intersection as a transition point between the excited and the ground state in a photochemical reaction, it is useful to draw an analogy with a transition state associated with the barrier in a potential energy surface in a thermally activated reaction (Figure 6.6). In the latter, one characterizes the transition state with a single vector that corresponds to the reaction path through the saddle point. The transition structure is a minimum in all coordinates except the one that corresponds to the reaction path. In contrast, a conical intersection provides two possible linearly independent reaction path directions. 

In the case of mechanisms whose elementary steps incorporate one intermediate to the left and right of the reaction equality (called by Temkin linear mechanisms ), each edge in the cyclic graph stands for an elementary step of the reaction mechanism, i.e. for a pair of mutually reversed elementary reactions. Each vertex of the kinetic graph corresponds to a certain intermediate while the linearly independent reaction routes are represented by graph cycles. For example, the mechanism of the water vapour methane conversion over Ni incorporates two independent routes, five intermediates, and six steps it is depicted by kinetic graph 1. 

Even if the actual chemical reaction pathways are not known, a set of linear independent reactions can be generated from some given set of components, thereby allowing Equation (4.541). This approach is known as the stoichiometric formnlation and is considered in some detail in Section 4.5.5. We mnst emphasize that the independent reactions generated by this method do not necessarily describe the actnal chemical reactions that occur in the system. These independent reactions relate the mole changes of each species in each reaction to changes in the extent of reaction via a stoichiometric coefficient, i.e., Eqnation (4.540). Since the final equilibrinm state will be independent of the particnlar pathway taken, the final eqnilibrinm compositions of the given species will stiU be correctly described by Eqnation (4.541). 

Once the matrix D is determined, the smallest nnmber of linear independent reactions can be generated from the known set of components. Let the rank of the matrix D be denoted by c. With... 

Another example for the check of the stoichiometric consistency is the catalytic decomposition of nitrogen oxide (NO) with hydrogen (H2) over alumina supported Rli monolith. Besides the main reaction, the formation of nitrous oxide (N2O) and self-decomposition also take place. The overall reactions are given below 2N0+2H2=>N2+H20, 2NO+H2=>N20+H20 2N20=>2N2+02. The system consists of three linearly independent reactions. The vector of chemical symbols is a = [ A NO 02 N20 H2 ] and consequently, the stoichiometric matrix becomes... 

Notice that one must consider linearly independent reactions for the statement in the example to be true. If we considered an arbitrary network for example,... 

Notice we have zeros below the diagonal of the first four rows and columns. We know by inspection that these reactions are linearly independent, which is why we wrote them in the first place, So the rank of V is four. See Exercise 2.9 for a proof that there are no other valid linearly independent reactions among these species and elements. 

For a given reaction nemork with nt linearly independent reactions, any steady state that is achievable by any reactor-separator design with total reactor volume V is achievable by a design with not more than nt + I CSTRs, also of total, reactor volume V. Moreover the concentrations, temperatures and pressures in the CSTRs are arbitrarily close to those occurring in the reactors of the original design. 

I Say we would like to find the number of linearly independent reactions, and then compute the species production rates from given reaction rates, [n Y2 = [12 3J. A typical set of Octave or Matlab commands to perform these operations would appear as follows... 

The second equation tells usRb n f2 = Ra + Rc)- In other words, using the linearly independent reactions, we can compute both reaction rates and we find a restriction on the possible production rates. We explore subsequently what happens when this restriction is violated by the production-rate measurements. 

A set of reactions is linearly independent if no reaction in the set can be written as a linear combination of the other reactions in the set. linear independence of reactions is equivalent to.linear independence of the rows of v. The rank of a matrix is the number of linearly independent rows (equivalently co.lumns) of the matrix., so the rank of v is the number of. linearly Independent reactions in the network. 

Armed with this result, consider Equation 2.67 and establish that the number of linearly independent reactions that satisfy the element balances is... 

Notice w e cannot state a phase rule with chemical, reaction if we have not defined linearly mdependent reactions, which is another motivation for defining linearly independent reactions in (Chapter 2. 

There exist some other possibilities to reduce the number of linear independent reactions, degrees of advancement, or concentrations at conditions specifically obtained in kinetics. Further information is given in Section 2.3. 

The program calculates y for. r + 1 equidistant supporting points from r = 0 to r = T. A parabola of the order of the number of linear independent reaction steps s is fitted to the supporting points. The values for the s intersections of zero represent the required eigenvalues. They can be obtained by use of common numerical procedures such as the Newton method, nest of intervals, and so on. The program checks whether all s intersections of zero are found. The procedure continues with the following steps ... 

---