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10.3: Appendix C - Matrices and Stoichiometric Schemata

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    44515
  • In Appendix C1 we review concepts associated with matrix algebra. In Appendix C2 we illustrate how one can transform an equation to a picture in a rigorous manner, and this leads to a schema for a single independent stoichiometric reaction. This schema can also be obtained by “counting atoms” and “balancing a chemical equation”. For multiple independent stoichiometric reactions, a schema cannot be constructed by “counting atoms” and in Appendix C3 we show how the schemata can be developed in a rigorous manner.

    C1: Matrix Methods and Partitioning

    In order to support the results obtained for the atomic matrix studied in Chapter 6 and for the mechanistic matrix studied in Chapter 9, we need to consider that matter of partitioning matrices. All the information necessary for our studies of stoichiometry is contained in Eq. \((6.2.10)\); however, that information can be presented in different forms depending on how the atomic matrix and the column matrix of net rates of production are partitioned. In our studies of reaction kinetics, all the information that we need is contained in the mechanistic matrix; however, that information can also be presented in different forms depending on presence or absence of Bodenstein products. In this appendix we review the methods required to develop the desired different forms.

    Matrix addition

    We begin our study of partitioning with the process of addition (or subtraction) as illustrated by the following matrix equation

    \[\begin{bmatrix} {a_{11} } & {a_{12} } & {a_{13} } & {a_{14} } \\ {a_{21} } & {a_{22} } & {a_{23} } & {a_{24} } \\ {a_{31} } & {a_{32} } & {a_{33} } & {a_{34} } \\ {a_{41} } & {a_{42} } & {a_{43} } & {a_{44} }\end{bmatrix} + \begin{bmatrix} {b_{11} } & {b_{12} } & {b_{13} } & {b_{14} } \\ {b_{21} } & {b_{22} } & {b_{23} } & {b_{24} } \\ {b_{31} } & {b_{32} } & {b_{33} } & {b_{34} } \\ {b_{41} } & {b_{42} } & {b_{43} } & {b_{44} }\end{bmatrix} = \begin{bmatrix} {c_{11} } & {c_{12} } & {c_{13} } & {c_{14} } \\ {c_{21} } & {c_{22} } & {c_{23} } & {c_{24} } \\ {c_{31} } & {c_{32} } & {c_{33} } & {c_{34} } \\ {c_{41} } & {c_{42} } & {c_{43} } & {c_{44} }\end{bmatrix} \label{1}\]

    This can be expressed in more compact nomenclature according to

    \[\mathbf{A} + \mathbf{B} = \mathbf{C} \label{2}\]

    The fundamental meaning of Eqs. \ref{1} and \ref{2} is given by the following 16 equations:

    \[\begin{align} a_{11} + b_{11} = c_{11} && a_{21} + b_{21} = c_{21} \nonumber\\ a_{12} + b_{12} = c_{12} && a_{22} + b_{22} = c_{22} \nonumber\\ a_{13} + b_{13} = c_{13} && a_{23} + b_{23} = c_{23} \nonumber\\ a_{14} + b_{14} = c_{14} && a_{24} + b_{24} = c_{24} \nonumber\\ {}\label{3} \\ a_{31} + b_{31} = c_{31} && a_{41} + b_{41} = c_{41} \nonumber\\ a_{32} + b_{32} = c_{32} && a_{42} + b_{42} = c_{42} \nonumber\\ a_{33} + b_{33} = c_{33} && a_{43} + b_{43} = c_{43} \nonumber\\ a_{34} + b_{34} = c_{34} && a_{44} + b_{44} = c_{44} \nonumber\end{align}\]

    These equations represent a complete partitioning of the matrix equation given by Equation \ref{1}, and we can also represent this complete partitioning in the form

    \[\begin{bmatrix} \color{red}{a_{11} } & \vdots & {a_{12} } & \vdots & {a_{13} } & \vdots & {a_{14} } \\ \hdashline {a_{21} } & \vdots & {a_{22} } & \vdots & {a_{23} } & \vdots & {a_{24} } \\ \hdashline {a_{31} } & \vdots & {a_{32} } & \vdots & {a_{33} } & \vdots & {a_{34} } \\ \hdashline {a_{41} } & \vdots & {a_{42} } & \vdots & {a_{43} } & \vdots & {a_{44} }\end{bmatrix} + \begin{bmatrix} \color{red}{b_{11} } & \vdots & {b_{12} } & \vdots & {b_{13} } & \vdots & {b_{14} } \\ \hdashline {b_{21} } & \vdots & {b_{22} } & \vdots & {b_{23} } & \vdots & {b_{24} } \\ \hdashline {b_{31} } & \vdots & {b_{32} } & \vdots & {b_{33} } & \vdots & {b_{34} } \\ \hdashline {b_{41} } & \vdots & {b_{42} } & \vdots & {b_{43} } & \vdots & {b_{44} }\end{bmatrix} = \begin{bmatrix} \color{red}{c_{11} } & \vdots & {c_{12} } & \vdots & {c_{13} } & \vdots & {c_{14} } \\ \hdashline {c_{21} } & \vdots & {c_{22} } & \vdots & {c_{23} } & \vdots & {c_{24} } \\ \hdashline {c_{31} } & \vdots & {c_{32} } & \vdots & {c_{33} } & \vdots & {c_{34} } \\ {c_{41} } & \vdots & {c_{42} } & \vdots & {c_{43} } & \vdots & {c_{44} }\end{bmatrix} \label{4}\]

    Here we have colored the particular partition that represents the first of Eqs. \ref{3}. The complete partitioning illustrated by Equation \ref{4} is not particularly useful; however, there are other possibilities that we will find to be very useful and one example is the row/column partition given by

    \[\begin{bmatrix} {a_{11} } & {a_{12} } & \vdots & {a_{13} } & {a_{14} } \\ {a_{21} } & {a_{22} } & \vdots & {a_{23} } & {a_{24} } \\ \hdashline {a_{31} } & {a_{32} } & \vdots & {a_{33} } & {a_{34} } \\ {a_{41} } & {a_{42} } & \vdots & {a_{43} } & {a_{44} }\end{bmatrix} + \begin{bmatrix} {b_{11} } & {b_{12} } & \vdots & {b_{13} } & {b_{14} } \\ {b_{21} } & {b_{22} } & \vdots & {b_{23} } & {b_{24} } \\ \hdashline {b_{31} } & {b_{32} } & \vdots & {b_{33} } & {b_{34} } \\ {b_{41} } & {b_{42} } & \vdots & {b_{43} } & {b_{44} }\end{bmatrix} = \begin{bmatrix} {c_{11} } & {c_{12} } & \vdots & {c_{13} } & {c_{14} } \\ {c_{21} } & {c_{22} } & \vdots & {c_{23} } & {c_{24} } \\ \hdashline {c_{31} } & {c_{32} } & \vdots & {c_{33} } & {c_{34} } \\ {c_{41} } & {c_{42} } & \vdots & {c_{43} } & {c_{44} }\end{bmatrix} \label{5}\]

    Each partitioned matrix can be expressed in the form

    \[\mathbf{A} = \begin{bmatrix} \mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{21} & \mathbf{A}_{22} \end{bmatrix} = \begin{bmatrix} {a_{11} } & {a_{12} } & \vdots & {a_{13} } & {a_{14} } \\ {a_{21} } & {a_{22} } & \vdots & {a_{23} } & {a_{24} } \\ \hdashline {a_{31} } & {a_{32} } & \vdots & {a_{33} } & {a_{34} } \\ {a_{41} } & {a_{42} } & \vdots & {a_{43} } & {a_{44} }\end{bmatrix} \label{6}\]

    and the partitioned matrix equation is given by

    \[\begin{bmatrix} \mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{21} & \mathbf{A}_{22} \end{bmatrix} + \begin{bmatrix} \mathbf{B}_{11} & \mathbf{B}_{12} \\ \mathbf{B}_{21} & \mathbf{B}_{22} \end{bmatrix} = \begin{bmatrix} \mathbf{C}_{11} & \mathbf{C}_{12} \\ \mathbf{C}_{21} & \mathbf{C}_{22} \end{bmatrix} \label{7}\]

    We usually think of the elements of a matrix as numbers such as \(a_{11}\), \(a_{12}\), etc.; however, the elements of a matrix can also be matrices as indicated in Equation \ref{7}. The usual rules for matrix addition lead to

    \[\mathbf{A}_{11} + \mathbf{B}_{11} = \mathbf{C}_{11} \label{8a}\]

    \[\mathbf{A}_{12} + \mathbf{B}_{12} = \mathbf{C}_{12} \label{8b}\]

    \[\mathbf{A}_{21} + \mathbf{B}_{21} = \mathbf{C}_{21} \label{8c}\]

    \[\mathbf{A}_{22} + \mathbf{B}_{22} = \mathbf{C}_{22} \label{8d}\]

    and the details associated with Equation \ref{8a} are given by

    \[\begin{bmatrix} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} }\end{bmatrix} + \begin{bmatrix} {b_{11} } & {b_{12} } \\ {b_{21} } & {b_{22} }\end{bmatrix} = \begin{bmatrix} {c_{11} } & {c_{12} } \\ {c_{21} } & {c_{22} }\end{bmatrix} \label{9} \]

    A little thought will indicate that this matrix equation represents the first four equations given in Eqs. \ref{3}. Other partitions of Equation \ref{1} are obviously available and will be encountered in the following paragraphs.

    Matrix multiplication

    Multiplication of matrices can also be represented in terms of submatrices, provided that one is careful to follow the rules of matrix multiplication. As an example, we consider the following matrix equation

    \[\begin{bmatrix} {a_{11} } & {a_{12} } & {a_{13} } & {a_{14} } \\ {a_{21} } & {a_{22} } & {a_{23} } & {a_{24} } \\ {a_{31} } & {a_{32} } & {a_{33} } & {a_{34} } \\ {a_{41} } & {a_{42} } & {a_{43} } & {a_{44} }\end{bmatrix} \begin{bmatrix} {b_{11} } & {b_{12} } \\ {b_{21} } & {b_{22} } \\ {b_{31} } & {b_{32} } \\ {b_{41} } & {b_{42} }\end{bmatrix} = \begin{bmatrix} {c_{11} } & {c_{12} } \\ {c_{21} } & {c_{22} } \\ {c_{31} } & {c_{32} } \\ {c_{41} } & {c_{42} }\end{bmatrix} \label{10}\]

    which conforms to the rule that the number of columns in the first matrix is equal to the number of rows in the second matrix. Equation \ref{10} represents the 8 individual equations given by

    \[a_{11} b_{11} + a_{12} b_{21} + a_{13} b_{31} + a_{14} b_{41} = c_{11} \label{11a}\]

    \[a_{11} b_{12} + a_{12} b_{22} + a_{13} b_{32} + a_{14} b_{42} = c_{12} \label{11b}\]

    \[a_{21} b_{11} + a_{22} b_{21} + a_{23} b_{31} + a_{24} b_{4} = c_{21} \label{11c}\]

    \[a_{21} b_{12} + a_{22} b_{22} + a_{23} b_{32} + a_{24} b_{42} = c_{22} \label{11d}\]

    \[a_{31} b_{11} + a_{32} b_{21} + a_{33} b_{31} + a_{34} b_{41} = c_{31} \label{11e}\]

    \[a_{31} b_{12} + a_{32} b_{22} + a_{33} b_{32} + a_{34} b_{42} = c_{32} \label{11f}\]

    \[a_{41} b_{11} + a_{42} b_{21} + a_{43} b_{31} + a_{44} b_{41} = c_{41} \label{11g}\]

    \[a_{41} b_{12} + a_{42} b_{22} + a_{43} b_{32} + a_{44} b_{42} = c_{42} \label{11h}\]

    which can also be expressed in compact form according to

    \[ \mathbf{AB} = \mathbf{C} \label{12}\]

    Here the matrices \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\) are defined explicitly by

    \[\mathbf{A} = \begin{bmatrix} {a_{11} } & {a_{12} } & {a_{13} } & {a_{14} } \\ {a_{21} } & {a_{22} } & {a_{23} } & {a_{24} } \\ {a_{31} } & {a_{32} } & {a_{33} } & {a_{34} } \\ {a_{41} } & {a_{42} } & {a_{43} } & {a_{44} }\end{bmatrix} \quad \mathbf{B} = \begin{bmatrix} {b_{11} } & {b_{12} } \\ {b_{21} } & {b_{22} } \\ {b_{31} } & {b_{32} } \\ {b_{41} } & {b_{42} }\end{bmatrix} \quad \mathbf{C} = \begin{bmatrix} {c_{11} } & {c_{12} } \\ {c_{21} } & {c_{22} } \\ {c_{31} } & {c_{32} } \\ {c_{41} } & {c_{42} }\end{bmatrix} \label{13}\]

    In Eqs. \ref{1} through \ref{9} we have illustrated that the process of addition and subtraction can be carried out in terms of partitioned matrices. Matrix multiplication can also be carried out in terms of partitioned matrices; however, in order to conform to the rules of matrix multiplication, we must partition the matrices properly. For example, a proper row partition of Equation \ref{10} can be expressed as

    \[\begin{bmatrix} {a_{11} } & {a_{12} } & {a_{13} } & {a_{14} } \\ {a_{21} } & {a_{22} } & {a_{23} } & {a_{24} } \\ \hdashline {a_{31} } & {a_{32} } & {a_{33} } & {a_{34} } \\ {a_{41} } & {a_{42} } & {a_{43} } & {a_{44} }\end{bmatrix} \begin{bmatrix} {b_{11} } & {b_{12} } \\ {b_{21} } & {b_{22} } \\ {b_{31} } & {b_{32} } \\ {b_{41} } & {b_{42} }\end{bmatrix} = \begin{bmatrix} {c_{11} } & {c_{12} } \\ {c_{21} } & {c_{22} } \\ \hdashline {c_{31} } & {c_{32} } \\ {c_{41} } & {c_{42} }\end{bmatrix} \label{14}\]

    In terms of the submatrices defined by

    \[\mathbf{A}_{11} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \end{bmatrix} , \quad \mathbf{A}_{21} = \begin{bmatrix} a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{bmatrix} \\ \mathbf{C}_{11} = \begin{bmatrix} {c_{11} } & {c_{12} } \\ {c_{21} } & {c_{22} } \end{bmatrix} \quad \mathbf{C}_{21} = \begin{bmatrix} {c_{31} } & {c_{32} } \\ {c_{41} } & {c_{42} }\end{bmatrix} \label{15}\]

    we can represent Equation \ref{14} in the form

    \[\begin{bmatrix} \mathbf{A}_{11} \\ \mathbf{A}_{21} \end{bmatrix} \quad \mathbf{B} = \begin{bmatrix} \mathbf{A}_{11} \mathbf{B} \\ \mathbf{A}_{21} \mathbf{B}\end{bmatrix} = \begin{bmatrix} \mathbf{C}_{11} \\ \mathbf{C}_{21} \end{bmatrix} \label{16}\]

    Often it is useful to work with the separate matrix equations that we have created by the partition, and these are given by

    \[\mathbf{A}_{11} \mathbf{B} = \mathbf{C}_{11} \label{17}\]

    \[\mathbf{A}_{21} \mathbf{B} = \mathbf{C}_{21} \label{18}\]

    The details of the first of these can be expressed as

    \[ \mathbf{A}_{11} \mathbf{B} = \begin{bmatrix} {a_{11} } & {a_{12} } & {a_{13} } & {a_{14} } \\ {a_{21} } & {a_{22} } & {a_{23} } & {a_{24} } \end{bmatrix} \begin{bmatrix} {b_{11} } & {b_{12} } \\ {b_{21} } & {b_{22} } \\ {b_{31} } & {b_{32} } \\ {b_{41} } & {b_{42} } \end{bmatrix}, \quad \mathbf{C}_{11} = \begin{bmatrix} {c_{11} } & {c_{12} } \\ {c_{21} } & {c_{22} }\end{bmatrix} \label{19a}\]

    Multiplication can be carried out to obtain

    \[ \begin{bmatrix} a_{11} b_{11} +a_{12} b_{21} +a_{13} b_{31} +a_{14} b_{41} & a_{11} b_{12} +a_{12} b_{22} +a_{13} b_{32} +a_{14} b_{42} \\ a_{21} b_{11} +a_{22} b_{21} +a_{23} b_{31} +a_{24} b_{41} & a_{21} b_{12} +a_{22} b_{22} +a_{23} b_{32} +a_{24} b_{42} \end{bmatrix} = \begin{bmatrix} {c_{11} } & {c_{12} } \\ {c_{21} } & {c_{22} } \end{bmatrix} \label{19b}\]

    and equating the four elements of each matrix leads to

    \[{a_{11} b_{11} +a_{12} b_{21} +a_{13} b_{31} +a_{14} b_{41} } = c_{11} \\ {a_{11} b_{12} +a_{12} b_{22} +a_{13} b_{32} +a_{14} b_{42} } = c_{12} \\ {a_{21} b_{11} +a_{22} b_{21} +a_{23} b_{31} +a_{24} b_{41} } = c_{21} \\ {a_{21} b_{12} +a_{22} b_{22} +a_{23} b_{32} +a_{24} b_{42} } = c_{22} \label{19c}\]

    Here we see that these four individual equations (associated with the partitioned matrix equation) are those given originally by Eqs. \ref{11a} through \ref{11d}. A little thought will indicate that the matrix equation represented by Equation \ref{18} contains the four individual equations represented by Eqs. \ref{11e} through \ref{11h}. All of the information available in Equation \ref{10} is given explicitly in Eqs. \ref{11a}-\ref{11h} and partitioning of the original matrix equation does nothing more than arrange the information in a different form.

    If we wish to obtain a column partition of the matrix \(\mathbf{A}\) in Equation \ref{10}, we must also create a row partition of matrix \(\mathbf{B}\) in order to conform to the rules of matrix multiplication. This column/row partition takes the form

    \[\begin{bmatrix} {a_{11} } & {a_{12} } & \vdots & {a_{13} } & {a_{14} } \\ {a_{21} } & {a_{22} } & \vdots & {a_{23} } & {a_{24} } \\ {a_{31} } & {a_{32} }& \vdots & {a_{33} } & {a_{34} } \\ {a_{41} } & {a_{42} } & \vdots & {a_{43} } & {a_{44} } \end{bmatrix} \begin{bmatrix} {b_{11} } & {b_{12} } \\ {b_{21} } & {b_{22} } \\ \hdashline {b_{31} } & {b_{32} } \\ {b_{41} } & {b_{42} }\end{bmatrix} = \begin{bmatrix} {c_{11} } & {c_{12} } \\ {c_{21} } & {c_{22} } \\ {c_{31} } & {c_{32} } \\ {c_{41} } & {c_{42} }\end{bmatrix} \label{20}\]

    and the submatrices are identified explicitly according to

    \[\mathbf{A}_{11} = \begin{bmatrix} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ {a_{31} } & {a_{32} } \\ {a_{41} } & {a_{42} }\end{bmatrix} \quad \mathbf{A}_{12} = \begin{bmatrix} {a_{13} } & {a_{14} } \\ {a_{23} } & {a_{24} } \\ {a_{33} } & {a_{34} } \\ {a_{43} } & {a_{44} }\end{bmatrix} \quad \mathbf{B}_{11} = \begin{bmatrix} {b_{11} } & {b_{12} } \\ {b_{21} } & {b_{22} }\end{bmatrix} \quad \mathbf{B}_{21} = \begin{bmatrix} {b_{31} } & {b_{32} } \\ {b_{41} } & {b_{42} }\end{bmatrix} \label{21}\]

    Use of these representations in Equation \ref{20} leads to

    \[\begin{bmatrix}\mathbf{A}_{11} & \mathbf{A}_{12} \end{bmatrix} \begin{bmatrix} \mathbf{B}_{11} \\ \mathbf{B}_{21} \end{bmatrix} = \mathbf{C} \label{22}\]

    and matrix multiplication in terms of the submatrices provides

    \[\mathbf{A}_{11} \mathbf{B}_{11} + \mathbf{A}_{12} \mathbf{B}_{21} = \mathbf{C} \label{23}\]

    In come cases, we will make use of a complete column partition of the matrix \(\mathbf{A}\) which requires a complete row partition of the matrix \(\mathbf{B}\). This partition is illustrated by

    \[\begin{bmatrix} {a_{11} } & \vdots & {a_{12} } & \vdots & {a_{13} } & \vdots & {a_{14} } \\ {a_{21} } & \vdots & {a_{22} } & \vdots & {a_{23} } & \vdots & {a_{24} } \\ {a_{31} } & \vdots & {a_{32} }& \vdots & {a_{33} } & \vdots & {a_{34} } \\ {a_{41} }& \vdots & {a_{42} } & \vdots & {a_{43} } & \vdots & {a_{44} } \end{bmatrix} \begin{bmatrix} {b_{11} } & {b_{12} } \\ \hdashline {b_{21} } & {b_{22} } \\ \hdashline {b_{31} } & {b_{32} } \\ \hdashline {b_{41} } & {b_{42} }\end{bmatrix} = \begin{bmatrix} {c_{11} } & {c_{12} } \\ {c_{21} } & {c_{22} } \\ {c_{31} } & {c_{32} } \\ {c_{41} } & {c_{42} }\end{bmatrix} \label{24}\]

    and in terms of the submatrices it can be expressed as

    \[\begin{bmatrix}\mathbf{A}_{11} & \mathbf{A}_{12} & \mathbf{A}_{13} & \mathbf{A}_{14} \end{bmatrix} \begin{bmatrix} {\mathbf{B}_{11} } \\ {\mathbf{B}_{21} } \\ {\mathbf{B}_{31} } \\ {\mathbf{B}_{41} }\end{bmatrix} = \mathbf{C} \label{25}\]

    Matrix multiplication leads to

    \[\mathbf{A}_{11} \mathbf{B}_{11} + \mathbf{A}_{12} \mathbf{B}_{21} + \mathbf{A}_{13} \mathbf{B}_{31} + \mathbf{A}_{14} \mathbf{B}_{41} = \mathbf{C} \label{26}\]

    and we will find this form of Equation \ref{10} to be especially useful in our discussion of stoichiometric schemata.

    C2: Single Independent Stoichiometric Reaction

    When the rank of the atomic matrix is \(N -1\), we can use Eq. \((6.2.10)\) to obtain Eqs. \((6.2.28)-(6.2.31)\) that can be expressed as

    \[\frac{R_{A} }{R_{N} } = \nu_{A} \label{27a}\]

    \[\frac{R_{B} }{R_{N} } = \nu_{B} \label{27b}\]

    \[\frac{R_{C} }{R_{N} } = \nu_{C} \label{27c}\]

    .

    \[\frac{R_{N-1} }{R_{N} } = \nu_{N-1} \label{27m}\]

    \[\frac{R_{N} }{R_{N} } = \nu_{N} \label{27n}\]

    Here we have identified the ratios of reactions rates as the stoichiometric coefficients, \(\nu_{A}\), \(\nu_{B}\), etc. These stoichiometric coefficients are not necessary to determine reaction rates, as we indicated in Examples \(6.2.1\), \(6.2.3\) and \(6.2.4\); however, they can be used to develop a picture of the stoichiometry of the reaction. To see how this is done, it is convenient to express Eqs. \ref{27a}-\ref{27n} in the form

    \[R_{A} = \nu_{A} R_{N} , \quad A = 1,2,3,....N \label{28}\]

    which can be used with Equation \((6.2.8)\) to obtain

    \[\sum_{A = 1}^{A = N}N_{JA} \nu_{A} R_{N} = 0 , \quad J = 1,2,3,....T \label{29}\]

    In matrix form this can be expressed as

    \[\begin{bmatrix} {N_{1A} } & {N_{1B} } & {N_{1C} } & {...} & {N_{1N} } \\ {N_{2A} } & {N_{2B} } & {N_{2C} } & {...} & {N_{2N} } \\ {.} & {.} & {.} & {...} & {.} \\ {N_{TA} } & {N_{TB} } & {N_{TC} } & {...} & {N_{TN} }\end{bmatrix} \begin{bmatrix} {\nu_{A} } \\ {\nu_{B} } \\ {\nu_{C} } \\ {.} \\ {\nu_{N} }\end{bmatrix}R_{N} = \begin{bmatrix} {0} \\ {0} \\ {.} \\ {0}\end{bmatrix} \label{30}\]

    and the single unknown reaction rate, \(R_{N}\), can be cancelled to obtain

    \[\begin{bmatrix} {N_{1A} } & {N_{1B} } & {N_{1C} } & {...} & {N_{1N} } \\ {N_{2A} } & {N_{2B} } & {N_{2C} } & {...} & {N_{2N} } \\ {.} & {.} & {.} & {...} & {.} \\ {N_{TA} } & {N_{TB} } & {N_{TC} } & {...} & {N_{TN} }\end{bmatrix} \begin{bmatrix} {\nu_{A} } \\ {\nu_{B} } \\ {\nu_{C} } \\ {.} \\ {\nu_{N} }\end{bmatrix} = \begin{bmatrix} {0} \\ {0} \\ {.} \\ {0}\end{bmatrix} \label{31}\]

    Here it is understood that \(\nu_{N}\) is equal to one.

    Each column in the chemical composition matrix identifies the structure of a molecule. For example, the first column in the chemical composition matrix indicates the atoms associated with molecular species \(A\), and the second column indicates the atoms associated with species \(B\). We can partition the chemical composition matrix into \(N\) molecular species submatrices to obtain

    \[\begin{bmatrix} {N_{1A} } & \vdots & {N_{1B} } & \vdots & {N_{1C} } & \vdots & {...} & \vdots & {N_{1N} } \\ {N_{2A} } & \vdots & {N_{2B} } & \vdots & {N_{2C} } & \vdots & {...} & \vdots & {N_{2N} } \\ {.} & \vdots & {.} & \vdots & {.} & \vdots & {...} & \vdots & {.} \\ {N_{TA} } & \vdots & {N_{TB} } & \vdots & {N_{TC} } & \vdots & {...} & \vdots & {N_{TN} }\end{bmatrix} \begin{bmatrix} {\nu_{A} } \\ \hdashline {\nu_{B} } \\ \hdashline {\nu_{C} } \\ \hdashline {.} \\ \hdashline {\nu_{N} }\end{bmatrix} = \begin{bmatrix} {0} \\ {0} \\ {.} \\ {0}\end{bmatrix} \label{32}\]

    and this can be expanded (see Eqs. \ref{24} through \ref{26}) in the form

    \[\begin{bmatrix} {N_{1A} } \\ {N_{2A} } \\ {.} \\ {N_{TA} }\end{bmatrix} \nu_{A} + \begin{bmatrix} {N_{1B} } \\ {N_{2B} } \\ {.} \\ {N_{TB} }\end{bmatrix} \nu_{B} + \begin{bmatrix} {N_{1C} } \\ {N_{2C} } \\ {.} \\ {N_{TC} }\end{bmatrix} \nu_{C} + ... + \begin{bmatrix} {N_{1N} } \\ {N_{2N} } \\ {.} \\ {N_{TN} }\end{bmatrix} \nu_{N} = \begin{bmatrix} {0} \\ {0} \\ {.} \\ {0}\end{bmatrix} \label{33}\]

    We now note that some of the stoichiometric coefficients will be negative and some will be positive, i.e., the reactants will have negative coefficients and the products will have positive coefficients. If species \(A\) and \(B\) are reactants and species \(C\) through \(N\) are products, we represent this idea as

    \[\nu_{A} = -\left|\nu_{A} \right| , \quad \nu_{B} = -\left|\nu_{B} \right| , \quad \nu_{J} \ge 0 , \quad J \Rightarrow C,D, ..., N \label{34}\]

    This allows us to express Equation \ref{33} in the form

    \[\begin{bmatrix} {N_{1A} } \\ {N_{2A} } \\ {.} \\ {N_{TA} }\end{bmatrix} \left|\nu_{A} \right| + \begin{bmatrix} {N_{1B} } \\ {N_{2B} } \\ {.} \\ {N_{TB} }\end{bmatrix} \left|\nu_{B} \right| = \begin{bmatrix} {N_{1C} } \\ {N_{2C} } \\ {.} \\ {N_{TC} }\end{bmatrix} \nu_{C} + \begin{bmatrix} {N_{1D} } \\ {N_{2D} } \\ {.} \\ {N_{TD} }\end{bmatrix} \nu_{D} + ... + \begin{bmatrix} {N_{1N} } \\ {N_{2N} } \\ {.} \\ {N_{TN} }\end{bmatrix} \nu_{N} \label{35}\]

    This matrix equation represents the concept that atomic species are neither created nor destroyed by chemical reactions. For example, the first equation in the set of equations represented by Equation \ref{35} is given by

    Atomic Species #1: \[N_{1A} \left|\nu_{A} \right| + N_{1B} \left|\nu_{B} \right| = N_{1C} \nu_{C} + N_{1D} \nu_{D} + ... + N_{1N} \nu_{N} \label{36}\]

    which is simply a statement of Axiom II for atomic species #1. In order to use Equation \ref{35} to construct a stoichiometric schema, we make use of the following two transformations:

    Transformation I. The pictures associated with the molecular species submatrices are constructed according to the transformations indicated by

    \[\begin{bmatrix} {N_{1A} } \\ {N_{2A} } \\ {.} \\ {N_{TA} }\end{bmatrix} \Rightarrow A , \quad \begin{bmatrix} {N_{1B} } \\ {N_{2B} } \\ {.} \\ {N_{TB} }\end{bmatrix} \Rightarrow B , \quad \begin{bmatrix} {N_{1C} } \\ {N_{2C} } \\ {.} \\ {N_{TC} }\end{bmatrix} \Rightarrow C , \quad {etc.,} \label{37}\]

    Transformation II. The equal sign in Equation \ref{35} is transformed to arrows depending on the sign of \(R_{N}\). These transformations are given by

    \[\begin{align} && = && \Rightarrow && \to && \text{when} && R_{N} > 0 \nonumber\\ && = && \Rightarrow && \leftarrow && \text{when} && R_{N} < 0 \label{38}\\ && = &&\Rightarrow && \mathop{\longleftarrow}\limits^{\displaystyle\longrightarrow} && \text{when} && {R_{N} = \pm \left|R_{N} \right|} \nonumber\end{align}\]

    For the first condition given by Equation \ref{38}, we can use these transformations to express Equation \ref{35} in terms of the picture given by

    \[\left|\nu_{A} \right|A + \left|\nu_{B} \right|B\to \nu_{C} C + \nu_{D} D + ... + \nu_{N} N \label{39}\]

    Here one must understand that this represents a picture of the stoichiometry of a reacting system in which the molecular species, \(A\) and \(B\), react to form the molecular species represented by \(C\), \(D\), ..., and \(N\). While we have assigned an equation number to this picture, it is not an equation.

    Example \(\PageIndex{1}\): Schema for complete oxidation of ethane

    In this example we want to apply the ideas given in the previous paragraphs to develop the stoichiometric schema for the complete combustion of ethane. For that reaction, the molecular species under consideration are

    Molecular species: \[\ce{C2H6} , \quad \ce{O2} , \quad \ce{H2O} , \quad \ce{CO2} \label{a1}\tag{1}\]

    and the chemical composition matrix can be illustrated as

    \[\text{Molecular Species}\to \ce{C2H6} \quad \ce{O2} \quad \ce{H2O} \quad \ce{CO} \\ \begin{matrix} {carbon} \\ { hydrogen} \\ {oxygen} \end{matrix} \begin{bmatrix} \color{\red}{ 2} & { 0} & {0} & {1} \\ \color{\red}{ 6} & { 0} & {2} & {0} \\ \color{\red}{ 0} & { 2} & {1} & {2} \end{bmatrix} \label{a2}\tag{2}\]

    Here we have colored the column that represents the molecular species submatrix for ethane, and it should be clear that the numbers in that colored column are associated with the ethane molecule, \(\ce{C2H6}\). For the complete combustion of ethane, Equation \ref{33} takes the form

    \[\begin{matrix} {carbon} \\ { hydrogen} \\ {oxygen} \end{matrix} \begin{bmatrix} {2} \\ {6} \\ {0}\end{bmatrix} \nu_{\ce{C2H6}} + \begin{bmatrix} {0} \\ {0} \\ {2} \end{bmatrix} \nu_{\ce{O2}} + \begin{bmatrix} {0} \\ {2} \\ {1} \end{bmatrix} \nu_{\ce{H2O}} + \begin{bmatrix} {1} \\ {0} \\ {2} \end{bmatrix} \nu_{\ce{CO2}} = \begin{bmatrix} {0} \\ {0} \\ {0} \end{bmatrix} \label{a3}\tag{3}\]

    in which carbon dioxide has been chosen as the pivot species. For this reaction one can follow the development in Example \(6.2.1\) (see Eqs. 5) to show that the stoichiometric coefficients are given by

    \[\nu_{\ce{C2H6}} = -\frac{1}{2} \label{a4a}\tag{4a}\]

    \[\nu_{\ce{O2}} = -\frac{7}{4} \label{a4b}\tag{4b}\]

    \[\nu_{\ce{H2O}} = +\frac{3}{4} \label{a4c}\tag{4c}\]

    \[\nu_{\ce{CO2}} = +1 \label{a4d}\tag{4d}\]

    When these values for the stoichiometric coefficients are used in Equation \ref{a3} we obtain the matrix equation given by

    \[\begin{matrix} {carbon} \\ { hydrogen} \\ {oxygen} \end{matrix} \frac{1}{2} \begin{bmatrix} {2} \\ {6} \\ {0}\end{bmatrix}+\frac{7}{4} \begin{bmatrix} {0} \\ {0} \\ {2}\end{bmatrix} = \frac{3}{2} \begin{bmatrix} {0} \\ {2} \\ {1}\end{bmatrix}+ \begin{bmatrix} {1} \\ {0} \\ {2} \end{bmatrix} \label{a5}\tag{5}\]

    To construct a picture, or stoichiometric schema, on the basis of Equation \ref{a5} we make use of the transformations represented by Eqs. \ref{37} and \ref{38}.

    Transformation I. The pictures associated with molecular species submatrices are extracted directly from the atomic matrix according to

    \[\begin{matrix} {carbon} \\ { hydrogen} \\ {oxygen} \end{matrix} \begin{bmatrix} {2} \\ {6} \\ {0}\end{bmatrix} \Rightarrow {\ce{C2H6}} , \quad \begin{bmatrix} {0} \\ {0} \\ {2} \end{bmatrix} \Rightarrow {\ce{O2}} , \quad \begin{bmatrix} {0} \\ {2} \\ {1} \end{bmatrix} \Rightarrow {\ce{H2O}} , \quad \begin{bmatrix} {1} \\ {0} \\ {2} \end{bmatrix} \Rightarrow {\ce{CO2}}\label{a7}\tag{6}\]

    Transformation II. The equal sign in Equation \ref{5} is transformed to arrows depending on the sign of . These transformations are given by

    \[\begin{align} && = && \Rightarrow && \to && \text{when} && R_{\ce{CO2}} > 0 \nonumber\\ && = && \Rightarrow && \leftarrow && \text{when} && R_{\ce{CO2}} < 0 \label{a8}\tag{7} \\ && = &&\Rightarrow && \mathop{\longleftarrow}\limits^{\displaystyle\longrightarrow} && \text{when} && {R_{\ce{CO2}} = \pm \left|R_{\ce{CO2}} \right|} \nonumber\end{align}\]

    When we follow these transformation rules, the matrix equation given by Equation \ref{5} becomes a stoichiometric schema having the following possibilities:

    \[\frac{1}{2} \ce{C2H6} + \frac{7}{4} \ce{O2} \longrightarrow \frac{3}{2} \ce{H2O} + \ce{CO2} , \quad R_{\ce{CO2}} > 0 \label{a9}\tag{8}\]

    \[\frac{1}{2} \ce{C2H6} + \frac{7}{4} \ce{O2} \longleftarrow \frac{3}{2} \ce{H2O} + \ce{CO2} , \quad R_{\ce{CO2}} < 0 \label{a10}\tag{9}\]

    \[\frac{1}{2} \ce{C2H6} + \frac{7}{4} \ce{O2} \mathop{\longleftarrow}\limits^{\displaystyle\longrightarrow} \frac{3}{2} \ce{H2O} + \ce{CO2} , \quad R_{\ce{CO2}} = \pm \left|R_{\ce{CO2}} \right| \label{a11}\tag{10}\]

    For a single independent stoichiometric reaction, such as the complete combustion of ethane, these stoichiometric schemata are easily developed by counting atoms; however, it is important to have a general methodology for creating these pictures from Eqs. \((6.2.8)\).

    It is important to remember that Axiom II given by Eqs. \((6.2.8)\) can be used to determine ratios of reaction rates, or stoichiometric coefficients, as indicated by Eqs. \ref{27a}-\ref{27n}. In addition, Eqs. \((6.2.8)\) can be used to derive the matrix equation given by Equation \ref{35}. Equation \ref{35} is not necessary for solving problems, but it can be transformed to Equation \ref{39} which is a useful pictorial representation. We find it convenient to discuss chemical reactions using stoichiometric schemata such as those given in Example A.1; however, they should not be confused with the mathematical equations represented by Eqs. \((6.2.8)\).

    C3: Multiple Independent Stoichiometric Reactions

    At this point we have shown that Eqs. \((6.2.8)\) can be used to constrain stoichiometric reaction rates in a manner that depends on the number of atomic species and the number of molecular species involved in the process. If there are \(N-1\) independent equations in the set of equations illustrated explicitly by Eqs. \((6.2.11)\), we can use Axiom II to determine all the rates of reaction in terms of a single pivot species. This is referred to as the case of a single independent reaction. The number of independent stoichiometric reaction rates associated with Eqs. \((6.2.11)\) is given by

    \[\begin{Bmatrix} \text{number of} \\ \text{independent} \\ \text{reactions} \end{Bmatrix} = N - r \label{40}\]

    in which \(r = rank[N_{JA} ]\). By rank we mean explicitly the row rank which represents the number of linearly independent equations contained in Eqs. \((6.2.11)\). In Example \(6.2.2\) the chemical composition matrix provided \(N = 3\) and \(rank = 2\), thus we had an example of a single independent stoichiometric reaction. This meant that a single reaction rate had to be measured in order to determine all the other reaction rates. In Example \(6.2.3\) the chemical composition matrix gave \(N = 4\) and \(rank = 2\), and we were confronted with two independent stoichiometric reactions. In this case two reaction rates had to be measured in order to determine all the other reaction rates. The determination of reaction rates for systems having multiple independent reactions is straightforward and requires only the direct application of Eqs. \((6.2.8)\). In Example A.1 we showed how to develop the stoichiometric schema for a single independent stoichiometric reaction rate, and in the following paragraphs we extend that development to the case of multiple independent stoichiometric reactions.

    In order to avoid the complex algebra associated with the development of schemata for the general case, we direct our attention to the specific example of the partial combustion of ethane that was studied in Example \(6.3.1\). There we showed that the reaction rates for \(\ce{C2H6}\), \(\ce{O2}\) and \(\ce{H2O}\) could be expressed in terms of the reaction rates for \({CO}\), \(\ce{CO2}\), and \(\ce{C2H4O}\), and our analysis took the form

    Step I: \[\underbrace{ \begin{bmatrix} R_{\ce{C2H6}} \\ R_{\ce{O2}} \\ R_{\ce{H2O}} \end{bmatrix} }_{non-pivots} = \underbrace{\begin{bmatrix} - \frac{1}{2} & - \frac{1}{2} & {- 1} \\ - \frac{5}{4} & - \frac{7}{4} & {- 1} \\ + \frac{3}{2} & + \frac{3}{2} & + 1 \end{bmatrix}}_{\text{pivot matrix}} \underbrace{ \begin{bmatrix} R_{\ce{CO}}^{I} \\ R_{\ce{CO2} }^{II} \\ R_{\ce{C2H4O}}^{III} \end{bmatrix} }_{pivots} \label{41}\]

    Here we have identified the reaction rates for the pivots as \(R_{\ce{CO}}^{I}\), \(R_{\ce{CO2}}^{II}\) and \(R_{\ce{C2H4O}}^{III}\) with the idea that these are the three independent rates of production which must be determined experimentally. In Chapter 6 we identified the net rates of production for the pivot species as \(R_{ \ce{CO}} \), \(R_{ \ce{CO2} }\) and \(R_{\ce{C2H4O}}\); however, in this case we have added the superscripts I, II, and III as additional identifiers. For the mathematical computation required to analyze material balance problems with chemical reaction, this type of nomenclature is unnecessary and could be considered cumbersome. However, our objective here is to transform equations to pictures and this is a more difficult task than using Axiom II to simply compute stoichiometric net rates of production.

    From Equation \ref{41} we see that the reaction rate for ethane takes the form

    \[R_{\ce{C2H6}} = - \frac{1}{2} R_{\ce{CO}}^{I} - \frac{1}{2} R_{\ce{CO2} }^{II} - R_{\ce{C2H4O}}^{III} \label{42}\]

    and a little thought will indicated that the coefficients, \(-{\frac{1}{2}}\), \(-{\frac{1}{2}}\) and \(-1\) represent the stoichiometric coefficients for ethane in terms of the three independent rates of production for carbon monoxide, carbon dioxide and ethylene oxide. We identify the matrix of coefficients in Equation \ref{41} as the stoichiometric coefficients associated with the three independent net rates of production in order to express the net rates of production for the non-pivot species as

    \[R_{\ce{C2H6}} = \nu_{\ce{C2H6}}^{I} R_{\ce{CO}}^{I} + \nu_{\ce{C2H6}}^{II} R_{\ce{CO2}}^{II} + \nu_{\ce{C2H6}}^{III} R_{\ce{C2H4O}}^{III} \label{43a}\]

    \[R_{\ce{O2}} = \nu_{\ce{O2}}^{I} R_{\ce{CO}}^{I} + \nu_{\ce{O2}}^{II} R_{\ce{CO2}}^{II} + \nu_{\ce{O2}}^{III} R_{\ce{C2H4O}}^{III} \label{43b}\]

    \[R_{\ce{H2O}} = \nu_{\ce{H2O}}^{I} R_{\ce{CO}}^{I} + \nu_{\ce{H2O}}^{II} R_{\ce{CO2} }^{II} + \nu_{\ce{H2O}}^{III} R_{\ce{C2H4O}}^{III} \label{43c}\]

    In terms of these stoichiometric coefficients, Equation \ref{41} can be expressed as

    Step II: \[\underbrace{ \begin{bmatrix} R_{\ce{C2H6}} \\ R_{\ce{O2}} \\ R_{\ce{H2O}} \end{bmatrix} }_{non-pivots} = \underbrace{\begin{bmatrix} \nu_{\ce{C2H6}}^{I} & {\nu_{\ce{C2H6}}^{II} } & {\nu_{\ce{C2H6}}^{III} } \\ {\nu_{\ce{O2}}^{I} } & {\nu_{\ce{O2}}^{II} } & {\nu_{\ce{O2}}^{III} } \\ {\nu_{\ce{H2O}}^{I} } & {\nu_{\ce{H2O}}^{II} } & {\nu_{\ce{H2O}}^{III} } \end{bmatrix}}_{\text{pivot matrix}} \underbrace{ \begin{bmatrix} {R_{\ce{CO}}^{I} } \\ {R_{\ce{CO2} }^{II} } \\ {R_{\ce{C2H4O}}^{III} } \end{bmatrix} }_{pivots} \label{44}\]

    where the numbers equivalent to these stoichiometric coefficients are available in Equation \ref{41} and are listed here as

    \[\begin{bmatrix} \nu_{\ce{C2H6}}^{I} & \nu_{\ce{C2H6}}^{II} & \nu_{\ce{C2H6}}^{III} \\ \nu_{\ce{O2}}^{I} & \nu_{\ce{O2}}^{II} & \nu_{\ce{O2}}^{III} \\ \nu_{\ce{H2O}}^{I} & \nu_{\ce{H2O}}^{II} & \nu_{\ce{H2O}}^{III} \end{bmatrix} = \begin{bmatrix} - \frac{1}{2} & - \frac{1}{2} & {- 1} \\ - \frac{5}{4} & - \frac{7}{4} & {- 1} \\ + \frac{3}{2} & + \frac{3}{2} & {+ 1} \end{bmatrix} \label{45}\]

    We now make use of an identity transformation for the pivot species

    \[\begin{bmatrix} {R_{\ce{CO}}^{I} } \\ {R_{\ce{CO2} }^{II} } \\ {R_{\ce{C2H4O}}^{III} }\end{bmatrix} = \begin{bmatrix} {1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{bmatrix} \begin{bmatrix} {R_{\ce{CO}}^{I} } \\ {R_{\ce{CO2} }^{II} } \\ {R_{\ce{C2H4O}}^{III} }\end{bmatrix} = \begin{bmatrix} {\nu_{\ce{CO}}^{I} } & {0} & {0} \\ {0} & {\nu_{\ce{CO2} }^{II} } & {0} \\ {0} & {0} & {\nu_{\ce{C2H4O}}^{III} }\end{bmatrix} \begin{bmatrix} {R_{\ce{CO}}^{I} } \\ {R_{\ce{CO2} }^{II} } \\ {R_{\ce{C2H4O}}^{III} }\end{bmatrix} \label{46}\]

    along with the partition described by Eqs. \ref{14} through \ref{18} to obtain a solution for the complete column matrix of production given by

    Step III: \[\begin{bmatrix} R_{\ce{C2H6}} \\ R_{\ce{O2}} \\ R_{\ce{H2O}} \\ R_{\ce{CO} } \\ R_{\ce{CO2} } \\ R_{\ce{C2H4O}} \end{bmatrix} = \begin{bmatrix} \nu_{\ce{C2H6}}^{I} & \nu_{\ce{C2H6}}^{II} & \nu_{\ce{C2H6}}^{III} \\ \nu_{\ce{O2}}^{I} & \nu_{\ce{O2}}^{II} & \nu_{\ce{O2}}^{III} \\ \nu_{\ce{H2O}}^{I} & \nu_{\ce{H2O}}^{II} & \nu_{\ce{H2O}}^{III} \\ \nu_{\ce{CO}}^{I} & {0} & {0} \\ {0} & \nu_{\ce{CO2} }^{II} & {0} \\ {0} & {0} & \nu_{\ce{C2H4O}}^{III} \end{bmatrix} \underbrace{ \begin{bmatrix} R_{\ce{CO}}^{I} \\ R_{\ce{CO2} }^{II} \\ R_{\ce{C2H4O}}^{III} \end{bmatrix} }_{pivots} \label{47}\]

    Here one must remember that the stoichiometric coefficients for the pivot species are all equal to one, i.e.,

    \[\nu_{\ce{CO}}^{I} = 1 , \quad \nu_{\ce{CO2} }^{II} = 1 , \quad \nu_{\ce{C2H4O}}^{III} = 1 \label{48}\]

    In addition, one must keep in mind that \(R_{\ce{C2H4O}}^{III}\) on the right hand side of Equation \ref{47} has exactly the same physical significance as \(R_{\ce{C2H4O}}\) on the left hand side of Equation \ref{47}.

    The column matrix of reaction rates on the left hand side of Equation \ref{47} is the column matrix that appears in Axiom II, and for this special case we express that axiom as

    Axiom II \[\sum_{A = 1}^{A = 6}N_{JA} R_{A} = 0 , \quad J\Rightarrow \ce{C} , \ce{H} , \ce{O} \label{49}\]

    In this analysis, we set up the chemical composition matrix to explicitly represent the pivot species as \(\ce{CO}\), \(\ce{CO2}\), and \(\ce{C2H4O}\). This leads to the visual representation given by

    \[\text{Molecular Species}\to \ce{C2H6} \quad \ce{O2} \quad \ce{H2O} \quad \ce{CO} \quad \ce{CO2} \quad \ce{C2H4O} \\ \begin{matrix} {carbon} \\ { hydrogen} \\ {oxygen} \end{matrix} \begin{bmatrix} { 2} & { 0} & {0} & {1} & { 1} & {2 } \\ { 6} & { 0} & {2} & {0} & { 2} & {4 } \\ { 0} & { 2} & {1} & {1} & { 0} & {1 }\end{bmatrix} \label{50}\]

    while the matrix representation of Equation \ref{49} takes the form

    \[\begin{bmatrix} {2} & {0} & {0} & {1} & {1} & {2} \\ {6} & {0} & {2} & {0} & {0} & {4} \\ {0} & {2} & {1} & {1} & {2} & {1}\end{bmatrix} \begin{bmatrix} R_{\ce{C2H6}} \\ R_{\ce{O2}} \\ R_{\ce{H2O}} \\ R_{\ce{CO}} \\ R_{\ce{CO2} } \\ R_{\ce{C2H4O}} \end{bmatrix} = \begin{bmatrix} {0} \\ {0} \\ {0}\end{bmatrix} \label{51}\]

    Substitution of Equation \ref{47} into this equation yields

    Step IV: \[\begin{bmatrix} {2} & {0} & {0} & {1} & {1} & {2} \\ {6} & {0} & {2} & {0} & {0} & {4} \\ {0} & {2} & {1} & {1} & {2} & {1}\end{bmatrix} \begin{bmatrix} {\nu_{\ce{C2H6}}^{I} } & {\nu_{\ce{C2H6}}^{II} } & {\nu_{\ce{C2H6}}^{III} } \\ {\nu_{\ce{O2}}^{I} } & {\nu_{\ce{O2}}^{II} } & {\nu_{\ce{O2}}^{III} } \\ {\nu_{\ce{H2O}}^{I} } & {\nu_{\ce{H2O}}^{II} } &{\nu_{\ce{H2O}}^{III} } \\ {\nu_{\ce{CO}}^{I} } & {0} & {0} \\ {0} & {\nu_{\ce{CO2} }^{II} } & {0} \\ {0} & {0} & {\nu_{\ce{C2H4O}}^{III} } \end{bmatrix} \underbrace{ \begin{bmatrix} {R_{\ce{CO}}^{I} } \\ {R_{\ce{CO2} }^{II} } \\ {R_{\ce{C2H4O}}^{III} }\end{bmatrix} }_{pivots} = \begin{bmatrix} {0} \\ {0} \\ {0}\end{bmatrix} \label{52}\]

    and at this point we construct a complete column/row partition of the second and third matrices to obtain

    Step V: \[\begin{bmatrix} {2} & {0} & {0} & {1} & {1} & {2} \\ {6} & {0} & {2} & {0} & {0} & {4} \\ {0} & {2} & {1} & {1} & {2} & {1}\end{bmatrix} \begin{bmatrix} {\nu_{\ce{C2H6}}^{I} } & \vdots & {\nu_{\ce{C2H6}}^{II} } & \vdots & {\nu_{\ce{C2H6}}^{III} } \\ {\nu_{\ce{O2}}^{I} } & \vdots & {\nu_{\ce{O2}}^{II} } & \vdots & {\nu_{\ce{O2}}^{III} } \\ {\nu_{\ce{H2O}}^{I} } & \vdots & {\nu_{\ce{H2O}}^{II} } & \vdots & {\nu_{\ce{H2O}}^{III} } \\ {\nu_{\ce{CO}}^{I} } & \vdots & {0} & \vdots & {0} \\ {0} & \vdots & {\nu_{\ce{CO2} }^{II} } & \vdots & {0} \\ {0} & \vdots & {0} & \vdots & {\nu_{\ce{C2H4O}}^{III} } \end{bmatrix} \begin{bmatrix} R_{\ce{CO}}^{I} \\ \hdashline R_{\ce{CO2} }^{II} \\ \hdashline R_{\ce{C2H4O}}^{III} \end{bmatrix} = \begin{bmatrix} {0} \\ {0} \\ {0}\end{bmatrix} \label{53}\]

    Following the analysis given in Sec. B.1, this column/row partition leads to

    Step VI: \[\begin{bmatrix} {2} & {0} & {0} & {1} & {1} & {2} \\ {6} & {0} & {2} & {0} & {0} & {4} \\ {0} & {2} & {1} & {1} & {2} & {1}\end{bmatrix} \left\{\begin{bmatrix} \nu_{\ce{C2H6}}^{I} \\ \nu_{\ce{O2}}^{I} \\ \nu_{\ce{H2O}}^{I} \\ \nu_{\ce{CO}}^{I} \\ {0} \\ {0} \end{bmatrix} R_{\ce{CO}}^{I} + \begin{bmatrix} \nu_{\ce{C2H6}}^{II} \\ \nu_{\ce{O2}}^{II} \\ \nu_{\ce{H2O}}^{II} \\ {0} \\ \nu_{\ce{CO2} }^{II} \\ {0} \end{bmatrix} R_{\ce{CO2} }^{II} + \begin{bmatrix} \nu_{\ce{C2H6} }^{III} \\ \nu_{\ce{O2}}^{III} \\ \nu_{\ce{H2O}}^{III} \\ {0} \\ {0} \\ \nu_{\ce{C2H4O}}^{III} \end{bmatrix} R_{\ce{C2H4O}}^{III} \right\} = \begin{bmatrix} {0} \\ {0} \\ {0}\end{bmatrix} \label{54}\]

    We now make use of the fact that the stoichiometric reaction rates of the pivot species are independent, and this provides the following three equations:

    Step VII: \[\begin{bmatrix} {2} & {0} & {0} & {1} & {1} & {2} \\ {6} & {0} & {2} & {0} & {0} & {4} \\ {0} & {2} & {1} & {1} & {2} & {1}\end{bmatrix} \begin{bmatrix} \nu_{\ce{C2H6}}^{I} \\ \nu_{\ce{O2}}^{I} \\ \nu_{\ce{H2O}}^{I} \\ \nu_{\ce{CO}}^{I} \\ {0} \\ {0} \end{bmatrix} R_{\ce{CO}}^{I} = \begin{bmatrix} {0} \\ {0} \\ {0} \end{bmatrix} \label{55}\]

    Step VII: \[\begin{bmatrix} {2} & {0} & {0} & {1} & {1} & {2} \\ {6} & {0} & {2} & {0} & {0} & {4} \\ {0} & {2} & {1} & {1} & {2} & {1} \end{bmatrix} \begin{bmatrix} \nu_{\ce{C2H6}}^{II} \\ \nu_{\ce{O2}}^{II} \\ \nu_{\ce{H2O}}^{II} \\ {0} \\ \nu_{\ce{CO2} }^{II} \\ {0} \\\end{bmatrix} R_{\ce{CO2}}^{II} = \begin{bmatrix} {0} \\ {0} \\ {0} \end{bmatrix} \label{56}\]

    Step VII: \[\begin{bmatrix} {2} & {0} & {0} & {1} & {1} & {2} \\ {6} & {0} & {2} & {0} & {0} & {4} \\ {0} & {2} & {1} & {1} & {2} & {1} \end{bmatrix} \begin{bmatrix} \nu_{\ce{C2H6}}^{III} \\ \nu_{\ce{O2}}^{III} \\ \nu_{\ce{H2O}}^{III} \\ {0} \\ {0} \\ \nu_{\ce{C2H4O}}^{III} \end{bmatrix} R_{\ce{C2H4O}}^{III} = \begin{bmatrix} {0} \\ {0} \\ {0} \end{bmatrix} \label{57}\]

    Each of these three equations has exactly the same form as Equation \ref{30}, thus we can repeat the procedure developed for a single independent stoichiometric reaction rate to determine the stoichiometric schemata for the three independent stoichiometric reaction rates. For example, the complete column/row partition of Equation \ref{55} leads to the matrix equation given by

    Step VIII: \[ \begin{matrix} {carbon} \\ { hydrogen} \\ {oxygen} \end{matrix} \begin{bmatrix} {2} \\ {6} \\ {0}\end{bmatrix} \nu_{\ce{C2H6}}^{I} + \begin{bmatrix} {0} \\ {0} \\ {2}\end{bmatrix} \nu_{\ce{O2}}^{I} + \begin{bmatrix} {0} \\ {2} \\ {1}\end{bmatrix} \nu_{\ce{H2O}}^{I} + \begin{bmatrix} {1} \\ {0} \\ {1}\end{bmatrix} \nu_{\ce{CO}}^{I} = \begin{bmatrix} {0} \\ {0} \\ {0}\end{bmatrix} \label{58}\]

    while the use of the numerical values of the stoichiometric coefficients given in Equation \ref{45} provides

    \[\begin{matrix} {carbon} \\ { hydrogen} \\ {oxygen} \end{matrix} \frac{1}{2} \begin{bmatrix} {2} \\ {6} \\ {0}\end{bmatrix}+ \frac{5}{4} \begin{bmatrix} {0} \\ {0} \\ {2}\end{bmatrix} = \frac{3}{2} \begin{bmatrix} {0} \\ {2} \\ {1}\end{bmatrix}+\begin{bmatrix} {1} \\ {0} \\ {1}\end{bmatrix} \label{59}\]

    This represents a bona fide equation that has no use other than to create a stoichiometric schema. To do so, we make use of Transformation I (see Equation \ref{37}) to obtain

    \[\begin{matrix} {carbon} \\ { hydrogen} \\ {oxygen} \end{matrix} \begin{bmatrix} {2} \\ {6} \\ {0}\end{bmatrix} \Rightarrow \ce{C2H6} , \quad \begin{bmatrix} {0} \\ {0} \\ {2} \end{bmatrix} \Rightarrow \ce{O2} , \quad \begin{bmatrix} {0} \\ {2} \\ {1}\end{bmatrix} = \ce{H2O} , \quad \begin{bmatrix} {1} \\ {0} \\ {1}\end{bmatrix} \Rightarrow \ce{CO} \label{60}\]

    and then employ Transformation II (see Equation \ref{38}) so that Equation \ref{59} provides the stoichiometric schema for the first independent reaction.

    Stoichiometric schema I: \[\frac{1}{2} \ce{C2H6} + \frac{5}{4} \ce{O2} \mathop{\longleftarrow}\limits^{\displaystyle\longrightarrow} \frac{3}{2} \ce{H2O} + \ce{CO} , \quad R_{\ce{CO}}^{I} = \pm \left|R_{\ce{CO}}^{I} \right| \label{61a}\]

    The schemata for the second and third independent reactions are developed in the same manner leading to

    Stoichiometric schema II: \[\frac{1}{2} \ce{C2H6} + \frac{7}{4} \ce{O2} \mathop{\longleftarrow}\limits^{\displaystyle\longrightarrow} \frac{3}{2} \ce{H2O} + \ce{CO2} , \quad R_{\ce{CO2}}^{II} = \pm \left|R_{\ce{CO2}}^{II} \right|\label{61b}\]

    Stoichiometric schema III: \[\ce{C2H6} + \ce{O2} \mathop{\longleftarrow}\limits^{\displaystyle\longrightarrow} \ce{H2O} + \ce{C2H4O} , \quad R_{\ce{C2H4O}}^{III} = \pm \left|R_{\ce{C2H4O}}^{III} \right|\label{61c}\]

    While these stoichiometric schemata are unnecessary for problem solving, they are quite traditional and in this section we have provided a methodical route for their development. In terms of solving problems, one must remember that the net rates of production for the non-pivot species are given in terms of the net rates of production of all the pivot species, and this idea is illustrated by Eqs. \ref{41} through \ref{44}.

    In Chapter 6 we developed a rigorous mathematical statement of the concept that atomic species are neither created nor destroyed by chemical reaction. This concept is identified as Axiom II and it is represented by Eqs. \((6.2.8)\) or by Eq. \((6.2.10)\). Axiom II was used to identify the number of independent net rates of production that are required for any given set of molecular species. In this appendix we have shown how Axiom II can be used to develop the stoichiometric schemata for the independent rates of production. These pictures are convenient for the discussion of independent stoichiometric reactions but they are not necessary for solving material balance problems.

    C4 Problems

    Section C1

    1. Given a matrix equation of the form \(\mathbf{c} = \mathbf{Ab}\) having an explicit representation of the form,

    \[\begin{bmatrix} {c_{1} } \\ {c_{2} } \\ {c_{3} } \\ {c_{4} } \\ {c_{5} }\end{bmatrix} = \begin{bmatrix} {a_{11} } & {a_{12} } & {a_{13} } & {a_{14} } \\ {a_{21} } & {a_{22} } & {a_{23} } & {a_{24} } \\ {a_{31} } & {a_{32} } & {a_{33} } & {a_{34} } \\ {a_{41} } & {a_{42} } & {a_{43} } & {a_{44} } \\ {a_{51} } & {a_{52} } & {a_{53} } & {a_{54} }\end{bmatrix} \begin{bmatrix} {b_{1} } \\ {b_{2} } \\ {b_{3} } \\ {b_{4} }\end{bmatrix} \label{1.1}\]

    develop a partition that will lead to an equation for the column vector represented by

    \[\begin{bmatrix} {c_{1} } \\ {c_{2} } \\ {c_{3} }\end{bmatrix} = ? \label{1.2}\]

    If the elements of c are related to the elements of b according to

    \[\begin{bmatrix} {c_{4} } \\ {c_{5} }\end{bmatrix} = \begin{bmatrix} {b_{3} } \\ {b_{4} }\end{bmatrix} \label{1.3}\]

    what are the elements of the matrix normally identified as \(\mathbf{A}_{ 21}\)?

    2. Construct the complete column/row partition of the matrix equation given by

    \[\begin{bmatrix} {a_{11} } & {a_{12} } & {a_{13} } & {a_{14} } \\ {a_{21} } & {a_{22} } & {a_{23} } & {a_{24} } \\ {a_{31} } & {a_{32} } & {a_{33} } & {a_{34} } \\ {a_{41} } & {a_{42} } & {a_{43} } & {a_{44} }\end{bmatrix} \begin{bmatrix} {b_{1} } \\ {b_{2} } \\ {b_{3} } \\ {b_{4} }\end{bmatrix} = \begin{bmatrix} {c_{1} } \\ {c_{2} } \\ {c_{3} } \\ {c_{4} }\end{bmatrix} \label{2.1}\]

    and show how it can be represented in the form

    \[\begin{bmatrix} {} \\ {} \\ {}\end{bmatrix}b_{1} + \begin{bmatrix} {} \\ {} \\ {}\end{bmatrix}b_{2} + \begin{bmatrix} {} \\ {} \\ {}\end{bmatrix}b_{3} + \begin{bmatrix} {} \\ {} \\ {}\end{bmatrix}b_{4} = \begin{bmatrix} {} \\ {} \\ {}\end{bmatrix} \label{2.2}\]

    Section C2

    3. Given a reacting system containing \(\ce{C3H6}\), \(\ce{NH3}\), \(\ce{O2}\), \(\ce{C3H3N}\) and \(\ce{H2O}\), construct the chemical composition matrix and determine the rank of that matrix. Use the development presented in Sec. 6.4 to develop the stoichiometric schema for the single independent reaction involving propylene, ammonia, oxygen, acrylonitrile and water.

    4. Construct the stoichiometric schema for the reaction described in Problem 6-13 for which the stoichiometric coefficients are given by

    \[\frac{R_{\ce{NaOH}} }{R_{\ce{NaBr}} } = \nu_{\ce{NaOH}} = -1 , \quad \frac{R_{\ce{CH3Br}} }{R_{\ce{NaBr}} } = \nu_{\ce{CH3Br}} = -1 , \quad \frac{R_{\ce{CH3OH}} }{R_{\ce{NaBr}} } = \nu_{\ce{CH3OH}} = 1 , \quad \frac{R_{\ce{NaBr}} }{R_{\ce{NaBr}} } = \nu_{\ce{NaBr}} = 1 \nonumber\]

    Section C3

    5. Fogler2 has proposed the following gas phase kinetic schemata and kinetic rate equations involving the oxidation of ammonia and the reduction of nitric oxide:

    I. \[\ce{4NH3} + 5\ce{O2} \to 4\ce{NO} + 6\ce{H2O} , \quad R_{\ce{NH3}} = -k_{1} c_{\ce{NH3}} (c_{\ce{O2}} )^{2} \nonumber\]

    II. \[\ce{2NH3} + {\frac{3}{2}} \ce{O2} \to \ce{N2} + 3\ce{H2O} , \quad R_{\ce{NH3}} = -k_{2} c_{\ce{NH3}} c_{\ce{O2}} \nonumber\]

    III. \[\ce{2NO} + \ce{O2} \to 2\ce{NO2} , \quad R_{\ce{O2}} = -k_{3} (c_{\ce{NO}} )^{2} c_{\ce{O2}} \nonumber\]

    IV. \[\ce{4NH3} + 6\ce{NO}\to 5\ce{N2} + 6\ce{H2O} , \quad R_{\ce{NO}} = -k_{4} c_{\ce{NO}} (c_{\ce{NH3}} )^{2 / 3} \nonumber\]

    Analyze a system containing these six molecular species and these three atomic species to determine the number of independent reactions. Are there any restrictions on the choice of pivot and non-pivot species? Do you have any ideas about how one would measure the rate of consumption of ammonia in Reaction I independently from the rate of consumption of ammonia in Reaction II in order to determine the rate constants \(k_{1}\) and \(k_{2}\)?

    6. When methane is partially combusted with oxygen, one finds the following molecular species: \(\ce{CH4}\), \(\ce{O2}\), \(\ce{CO}\), \(\ce{CO2}\), \(\ce{H2O}\)and \(\ce{H2}\). Determine the number of independent reactions and develop the stoichiometric schemata for this system.


    1. Fogler, S.H. 1992, Elements of Chemical Reaction Engineering, Second Edition, Prentice Hall, Englewood Cliffs, New Jersey.↩
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