Differential equations as a general treatment in kinetics

In linear systems of reactions, the differentiations of the degrees of advancement with respect to time t depend linearly on the concentrations. According to the principles mentioned in Sections 2.1.3.1 and 2.1.4.2, the last column of the stoichiometric matrix representation (see Example 2.7) becomes for the kth of the r different partial reactions (eq. (2.15)) 

This linear system of equations can be written as a matrix equation  

As defined in Table 2.2, in this equation x symbolises the vector of ail r degrees of advancement differentiated with respect to time, a is the vector of the concentrations of all the reactants. Kq represents the matrix of all the rate constants. This matrix will have n columns and r rows. It can be directly derived from the scheme discussed in Section 2.1.1.1. This differential equation is valid for all concentrations. By combination of eqs. (2.3) and (2.29) one obtains the differential equation 

It has been mentioned that linear dependencies between partial reactions have to be avoided (see Section 2.1.4) and the conservation of mass between the concentrations has to be used. Under these conditions according to eq. (2.6) and the definition of P (defined by eq. (2.7) the following relationship is obtained  

The reduced matrix v has to be used, otherwise v cannot be formed, since a quadratic matrix is necessary for inversion Therefore the change in concentration with time is given according to eq. (2.30) by 

---