The Reaction Coordinate
Consider again the general chemical reaction depicted. During the progress of the reaction, at each point the extent of depletion of the reactants, and the enhancement in the amount of product is exactly in proportion to their respective stoichiometric coefficients. Thus for any change dni in the number of moles of the ith species for a differential progress of the reaction one may write: Since all terms are equal, they can be all set equal to a single quantity , representing the extent of reaction as follows : The general relation between a differential change dni in the number of moles of a reacting species and is therefore: (i = 1,2, ...N)
This new variable , called the reaction coordinate, describe the extent of conversion of reactants to products for a reaction. Thus, it follows that the value of is zero at the start of the reaction. On the other hand when , it follows that the reaction has progressed to an extent at which point each reactant has depleted by an amount equal to its stoichiometric number of moles while each product has formed also in an amount equal to its stoichiometric number of moles. For dimensional consistency one designates such a degree of reaction as corresponding to Now, considering that at the point where the reaction has proceeded to an arbitrary extent characterized by (such that ), the number of moles of ith species is ni we obtain the following relation: Thus: ;(i = 1,2,...,N)

Thus the total number of moles of all species corresponding to extent of reaction: Or: Where:  Thus, The foregoing approach may be easily extended to develop the corresponding relations for a set of multiple, independent reactions which may occur in a thermodynamic system. In such a case each reaction is assigned an autonomous reaction co-ordinate (to represent the jth reaction). Further the stoichiometric coefficient of the ith species as it appears in the jth reaction is designated by Since a species may participate in more a single reaction, the change in the total number of moles of the species at any point of time would be the sum of the change due each independent reaction; thus, in general: (i= 1,2,...N)
On integrating the above equation starting from the initial number of moles to corresponding to the reaction coordinate of each reaction: (i = 1,2,...,N)
Or: Summing over all species gives: (a) Now: and, We may interchange the order of the summation on the right side of eqn. (a); thus:   Using eqns. 8.32 and 8.38 one obtains: (i = 1,2,....,N)