Number of independent chemical reactions

For a PVnr system of uniform T and P containing N species and 7T phases at thermodynamic equiUbrium, the intensive state of the system is fully deterrnined by the values of T, P, and the (N — 1) independent mole fractions for each of the equiUbrium phases. The total number of these variables is then 2 + 7t N — 1). The independent equations defining or constraining the equiUbrium state are of three types equations 218 or 219 of phase-equiUbrium, N 7t — 1) in number equation 245 of chemical reaction equiUbrium, r in number and equations of special constraint, s in number. The total number of these equations is A(7t — 1) + r -H 5. The number of equations of reaction equiUbrium r is the number of independent chemical reactions, and may be deterrnined by a systematic procedure (6). Special constraints arise when conditions are imposed, such as forming the system from particular species, which allow one or more additional equations to be written connecting the phase-rule variables (6). 

The number of independent chemical reactions / can be determined as follows ... 

Gibbs phase rule phys chem A relationship used to determine the number of state variables F, usually chosen from among temperature, pressure, and species composition in each phase, which must be specified to fix the thermodynamic state of a system in equilibrium F = C - P - M+2, where C is the number of chemical species presented at equilibrium, P is the number of phases, and M is the number of independent chemical reactions. Also known as Gibbs rule phase rule. gibz faz, rijl I... 

The mole numbers in Equation (5.63) could represent the species present in the system, rather than the components. However, in such a case they are not all independent. For each independent chemical reaction taking place within the system, a relation given by Equation (5.46) must be satisfied. There would thus be (S — R) independent mole numbers for the system if S is the number of species and R is the number of independent chemical reactions. In fact, this relation may be used to define the number of components in the system [6]. If the species are ions, an equation expressing the electroneutrality of the system is another condition equation relating the mole numbers of the species. The total number of components in this case is C = S — R — 1. 

R number of independent chemical reactions s solid state s displacement S entropy... 

The governing equations for each of the three ideal reactors are material balances for reactants and products of the reaction. In general, one material balance equation must be written for each independent reaction taking place in the reactor. For a group of Nsp species (i.e., reactants and products) in the reactor consisting of iVei elements, the number of independent chemical reactions typically is equal to (/Vsp - Nei). 

When the number of independent chemical reactions equals C - p, where p is the rank of the atom matrix (mjk), Gibbs free energy is minimized subject to atom balance constraints ... 

Identify the number of independent chemical reactions associated with each reaction step in the manufacture of the desired product. Include intermediates that can be separated and recycled. At each step indicates the phase(s), as well as the range of feasible temperatures and pressures. 

Because the vector m is constrained by the mass conservation requirement = const, the space of possible m values has — 1 dimensions. If the number of independent chemical reactions, R, is less than - 1, then some vectors m are not accessible at some assigned M this, as will be seen, has important consequences in the consideration of heterogeneous chemical equilibria. Now consider the special case where R = N — 1, so that indeed all admissible m s are accessible. Because the kernel of a contains only the zero vector, there exists an M X N matrix A such that... 

We now move to stoichiometry in infinite-dimensional space. The number of independent chemical reactions can, of course, be infinitely large, and one needs to introduce a reaction label u that plays for reactions the same role as the com-... 

Finally, when you are using either molecular species balances or extents of reaction to analyze a reactive system, the degree-of-freedom analysis must account for the number of independent chemical reactions among the species entering and leaving the system. Chemical reactions are independent if the stoichiometric equation of any one of them cannot be obtained by adding and subtracting multiples of the stoichiometric equations of the others. 

Number of independent chemical reactions, phase rale Molar or specific entropy Partial entropy, species i in solution Excess entropy = S —... 

The fact that chemical reactions are expressed as linear homogeneous equations allows us to exploit the properties of such equations and to use the associated algebraic tools. Specifically, we use elementary row operations to reduce the stoichiometric matrix to a reduced form, using Gaussian elimination. A reduced matrix is defined as a matrix where all the elements below the diagonal (elements 1,1 2,2 3,3 etc.) are zero. The number of nonzero rows in the reduced matrix indicates the number of independent chemical reactions. (A zero row is defined as a row in which all elements are zero.) The nonzero rows in the reduced matrix represent one set of independent chemical reactions (i.e., stoichiometric relations) for the system. 

In the preceding section, we discussed how to determine the number of independent chemical reactions and how to select a set of independent reactions. The number of independent reactions indicates the number of equations that we should solve to determine the composition of the reactor. To solve these equations. 

A method to determine the number of independent chemical reactions and how to select a set of independent reactions... 

Number of independent chemical reactions Number of phases... 

Now, let us choose n — h linearly independent covariant vectors gP as basis in the reaction subspace V and show that n - /j is the number of independent chemical reactions in the mixture, cf. below (4.45) and Rem. 4. These vectors can be written in the basis ofW as... 

Regular (quadratic) matrix means that its determinant is non-zero. Assertions in conditions 2,3 about ranks n —h (number of independent chemical reactions, see Sect. 4.2) foUow with the use of Lemma product of quadratic regular matrix with rectangultir matrix of maximal rank has also this maximal rank (this follows from Sylvester s inequeilities for rtmk of matrix product, see [134, 13.2.7]). 

The sets of mole numbers that satisfy (11.2.3) are sets of stoichiometric coefficients, Vy, and the subspace of N that satisfies (11.2.4) is called the nullspace of A. The dimension of the nullspace is the number of independent vectors Vy (basis vectors) that satisfy (11.2.3) that is, the nullspace has dimension H, which is the number of independent chemical reactions. 

Here A, U, W, and V are each square of dimension (C x C). In addition, W is a diagonal matrix, and since A is singular, some diagonal elements in W are zero. The number of zero elements equals the dimension of the nullspace, which for our problem is % the number of independent chemical reactions. 

The number of independent equations of relation (see 1.30), respectively, the number of independent chemical reactions is equal to a rank rk(v) of the stoichiometric matrix that is why a system of linear homogeneous equations (1.31) has m-rk(v) fundamental solutions for un-known elements of the vector fx. So, in order for a vector /t to be determined synonymously, it is necessary and sufficient to assign only m-rk(v), its elements giving the remainder from equation of relation (1.31). 

---