# 6.4: Axioms and Theorems

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To summarize our studies of stoichiometry, we note that atomic species are neither created nor destroyed by chemical reactions. In terms of the *atomic matrix* and the *column matrix of net rates of production*, this concept can be expressed as

**Axiom II:** \[ \mathbf{AR} = 0 \label{74}\]

As indicated in the previous section, the *row reduced echelon form* of the atomic matrix can always be developed, thus we can express Equation \ref{74} as

**Row Reduced Echelon Form:** \[\mathbf{A}^*\mathbf{R} = 0 \label{75}\]

This product of the atomic matrix times the column matrix of net rates of production can be partitioned according to (see Sec. 6.2.6)

**Column/Row Partition:** \[ \begin{bmatrix} \mathbf{I} & \mathbf{W} \end{bmatrix} \begin{bmatrix} \mathbf{R}_{NP} \\ \mathbf{R}_{P} \end{bmatrix} =0\label{76}\]

Here the *column partition* of \(\mathbf{A}^*\) provides the *non-pivot submatrix* \(\mathbf{I}\) and the *pivot submatrix* \(\mathbf{W}\), while the *row partition* of \(\mathbf{R}\) provides the *non-pivot column submatrix* \(\mathbf{R}_{NP}\) and the *pivot column submatrix* \(\mathbf{R}_P\). Carrying out the matrix multiplication indicated by Equation \ref{76} leads to

\[ \mathbf{IR}_{NP} + \mathbf{WR}_{P} = 0 \label{77}\]

and operation of the unit matrix on \(\mathbf{R}_{NP}\) provides the obvious result given by

\[ \mathbf{IR}_{NP} = \mathbf{R}_{NP}\label{78}\]

At this point we define the *pivot matrix* \(\mathbf{P}\) according to

**Pivot Matrix:** \[ \mathbf{P} = -\mathbf{W} \label{79}\]

and we use this result, along with Equation \ref{78}, in Equation \ref{77} to obtain the *pivot theorem*

**Pivot Theorem:** \[ \mathbf{R}_{NP} = \mathbf{PR}_P \label{80}\]

These five concepts represent the foundations of stoichiometry, and they appear in various special forms throughout this chapter and in subsequent chapters. When ionic species are involved, *conservation of charge* must be taken into account as indicated in Appendix E. Heterogeneous reactions can be analyzed using the framework presented in this chapter and the details are discussed in Appendix F. Reactions involving *optical isomers* require some care that is illustrated in Section 6.5, Problems 32 and 33.

In Chapter 7 we will make repeated use of the *global form* of the pivot theorem. To develop the global form we integrate Equation \ref{80} over the volume \(\mathscr{V}\) to obtain

**Global Pivot Theorem:** \[ \pmb{\mathscr{R}}_{NP} = \mathbf{P}\pmb{\mathscr{R}}_P \label{81}\]

The elements of the column matrices are given explicitly by the following version of this theorem

\[ \mathscr{R}_A = \sum^{B=N_P}_{B=1} P_{AB}\mathscr{R}_{B}, \quad A = N_P + 1, N_P+2,\dots.. N \label{82}\]

in which the *global* net rate of production is related to the *local* net rate of production by Eq. \((6.2.18)\). A summary of the matrices presented in this chapter is given in Sec. 9.4 along with a discussion of the several matrices used in the study of chemical reaction kinetics.