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6.2: Conservation of Atomic Species

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    44495
  • To be precise about the role of atomic species in chemical reactions, we first need to replace the word statement given by Equation \((6.1.8)\) with a word equation that we write as

    \[\left\{\begin{array}{c} \text{ the molar rate of production} \\ \text{ per unit volume of }J\text{-type atoms} \\ \text{ owing to chemical reactions} \end{array}\right\}=0 , \quad J=1,2,...,T \label{13}\]

    Here the nomenclature is intended to suggest aTomic species. From this we need to extract a mathematical equation, and in order to do this we define the number \(N_{JA}\) as

    \[N_{JA} =\left\{\begin{array}{c} \text{ number of moles of} \\ J \text{-type atoms per mole} \\\text{ of molecular species }A \end{array}\right\} , \quad J=1, 2,...,T, \text{ and }A=1, 2,...N \label{14}\]

    We refer to \(N_{JA}\) as the atomic species indicator and we identify the array of coefficients associated with \(N_{JA}\) as the atomic matrix 3. To illustrate the structure of the atomic matrix, we consider the complete oxidation of ethane illustrated in Figure \(6.1.1\). That process provides the basis for the following visual representation of the atomic matrix:

    \[\text{ Molecular Species} \to \ce{C2H6} \quad \ce{O2} \quad \ce{H2O} \quad \ce{CO2} \\ \begin{matrix} {carbon} \\ { hydrogen} \\ {oxygen} \end{matrix} \begin{bmatrix} {2} & { 0} & { 0} & {1 } \\ {6} & { 0} & { 2} & {0 } \\ {0} & { 2} & { 1} & {2 } \end{bmatrix} \label{15}\]

    This representation connects atoms with molecules in a convenient manner, and it is exactly what one uses to count atoms and balance chemical equations. There are two symbols that are useful for representing the atomic matrix. The first of these is given by \(\left[N_{JA} \right]\) which has the obvious connection to Equation \ref{14}, while the second is given by A which has the obvious connection to the name of this matrix. In this text we will encounter both representations for the atomic matrix as indicated by

    \[\left[N_{JA} \right]=\begin{bmatrix} {2} & {0} & {0} & {1} \\ {6} & {0} & {2} & {0} \\ {0} & {2} & {1} & {2} \end{bmatrix}, \quad \text{ or } \quad \mathbf{A} = \begin{bmatrix} {2} & {0} & {0} & {1} \\ {6} & {0} & {2} & {0} \\ {0} & {2} & {1} & {2} \end{bmatrix} \label{16}\]

    In order to use the chemical composition indicator, \(N_{JA}\), to construct an equation representing the concept that atoms are neither created nor destroyed by chemical reaction, we first recall the definition of \(R_{A}\)

    \[R_{A} =\left\{\begin{array}{c} net \text{ molar rate of production} \\ \text{ per unit volume of species }A \\ \text{ owing to chemical reactions} \end{array}\right\} \label{17}\]

    which is consistent with the pictorial representation of \(R_{\ce{CO2}}\) given in Figure \(4.1.1\). Next we form the product of the chemical composition indicator with \(R_{A}\) to obtain

    \[N_{JA} R_{A} =\left\{\begin{array}{c} \text{ number of moles of} \\ J \text{-type atoms per mole} \\ \text{ of molecular species }A \end{array}\right\} \left\{\begin{array}{c} net \text{ molar rate of production} \\ \text{ per unit volume of species }A \\ \text{ owing to chemical reactions} \end{array}\right\} \label{18}\]

    A little thought will indicate that the product of \(N_{JA}\) and \(R_{A}\) can be described as

    \[N_{JA} R_{A} =\left\{\begin{array}{c} \text{ net molar rate of production per unit} \\ \text{ volume of }J\text{-type atoms owing to the } \\ \text{ molar rate of production of species }A \end{array}\right\} \label{19}\]

    and the axiomatic statement given by Equation \ref{13} takes the form4

    Axiom II: \[\sum_{A = 1}^{A = N}N_{JA} R_{A} =0 , \quad J=1,2,...,T \label{20}\]

    This equation represents a precise mathematical statement that atomic species are neither created nor destroyed by chemical reactions and it provides a set of \(T\) equations that constrain the \(N\) net rates of production, \(R_{A}\), \(A=1,2,...,N\). While Axiom II provides \(T\) equations, the equations are not necessarily independent. The number of independent equations is given by the rank of the atomic matrix and we will be careful to indicate that rank when specific processes are examined. If ions are involved in the reactions, one must impose the condition of conservation of charge as described in Appendix E. Some comments concerning heterogeneous reactions are given in Appendix F.

    The net rate of production of species \(A\) indicated by \(R_{A}\) can also be expressed in terms of the creation and consumption of species \(A\) according to

    \[R_{A} =\left\{\begin{array}{c} \text{ molar rate of }creation \text{ of} \\ \text{ species }A \text{ per unit volume} \\ \text{ owing to chemical reactions} \end{array}\right\} - \left\{\begin{array}{c} \text{ molar rate of }consumption \text{ of} \\ \text{ species }A \text{ per unit volume} \\ \text{ owing to chemical reactions} \end{array}\right\} \label{21}\]

    Here we need to think carefully about the description of \(R_{A}\) given by Equation \ref{17} where we have used the word net to represent the sum of the creation of species \(A\) and the consumption of species \(A\). This means that Eqs. \ref{17} and \ref{21} are equivalent descriptions of \(R_{A}\) and the reader is free to chose which ever set of words is most appealing.

    If we make use of the atomic matrix and the column matrix of the net rates of production we can express Axiom II as

    Axiom II: \[\begin{bmatrix} {N_{11} } & {N_{12} } & {N_{13} } & {......} & {N_{1, N-1,} } & {N_{1N} } \\ {N_{21} } & {N_{22} } & . & {......} & {N_{2, N-1} } & {N_{2N} } \\ {N_{31} } & {N_{32} } & . & {......} & . & . \\ {.} & . & . & {......} & . & . \\ {.} & . & . & {......} & . & . \\ {N_{T1} } & {N_{T2} } & . & {......} & {N_{T, N-1} } & {N_{TN} } \end{bmatrix} \begin{bmatrix} {R_{1} } \\ {R_{2} } \\ {R_{3} } \\ {.} \\ {.} \\ {R_{N-1} } \\ {R_{N} } \end{bmatrix} = \begin{bmatrix} {0} \\ {0} \\ {0} \\ {.} \\ {0} \end{bmatrix} \label{22}\]

    Everything we need to know about the conservation of atomic species is contained in this linear matrix equation; however, we need this information in different forms that will be developed in this chapter. In our development we will find patterns associated with the atomic matrix and these patterns will be connected to the physical problems under consideration.

    Axioms and theorems

    In dealing with axioms and proved theorems, it is important to accept the idea that the choice is not necessarily unique. From the authors’ perspective, Equation \ref{20} and Equation \ref{22} represent the preferred form of the axiom indicating that atoms are neither created nor destroyed by chemical reactions. We have identified both as Axiom II; however, we have also identified Eq. \((6.1.1)\) as Axiom II. Equation \((6.1.1)\) indicates that mass is conserved during chemical reactions while Equation \ref{20} indicates that atoms are conserved during chemical reactions. Surely both of these are not independent axioms, thus we should be able to derive one from the other. In the following paragraphs we show how the axiom given by Eqs. \ref{20} can be used to prove Eq. \((6.1.1)\) as a theorem.

    To carry out this proof, we first multiply Eqs. \ref{20} by the atomic mass of the \(J^{th}\) atomic species leading to

    \[AW_{J} \sum_{A = 1}^{A = N} N_{JA} R_{A} =0 , \quad J=1, 2,...,T \label{23}\]

    Here we have used \(AW_{J}\) to represent the atomic mass of species \(J\) in the same manner that we have used \(MW_{A}\) to represent the molecular mass of species \(A\). We now sum Equation \ref{23} over all atomic species to obtain

    \[\sum_{J = 1}^{J = T} AW_{J} \sum_{A = 1}^{A = N} N_{JA} R_{A} =0 \label{24}\]

    Since the sum over \(J\) is independent of the sum over \(A\), we can place the sum over \(J\) inside the sum over \(A\) leading to the form

    \[\sum_{A = 1}^{A = N}\sum_{J = 1}^{J = T}AW_{J} N_{JA} R_{A} =0 \label{25}\]

    We now note that \(R_{A}\) is independent of the process of summing over all \(J\), thus we can take \(R_{A}\) outside of the first sum and express Equation \ref{25} as

    \[\sum_{A = 1}^{A = N}R_{A} \sum_{J = 1}^{J = T}AW_{J} N_{JA} =0 \label{26}\]

    At this point we need only recognize that the molecular mass of species \(A\) is defined by

    \[MW_{A} =\sum_{J = 1}^{J = T}AW_{J} N_{JA} \label{27}\]

    in order to express Equation \ref{26} as

    \[\sum_{A = 1}^{A = N} R_{A} MW_{A} =0 \label{28}\]

    Use of the definition given by Equation \((6.1.4)\) leads to the following proved theorem

    Proved Theorem: \[\sum_{A = 1}^{A = N}r_{A} =0 \label{29}\]

    This result was identified as Axiom II by Equation \((6.1.1)\) and by Equation \((6.1.10)\) in our initial exploration of the axioms for the mass of multicomponent systems. It should be clear from this development that one person’s axiom might be another person’s proved theorem. For example, if Equation \ref{29} is taken as Axiom II, one can prove Equation \ref{20} as a theorem. The proof is the object of Section 6.5, Problem 4, and it requires the constraint that the rates of production, \(R_{A}\), be independent of the atomic masses, \(AW_{J}\). We prefer Equation \ref{20} as Axiom II since it can be used to prove Equation \ref{29} without imposing any constraints; however, one must accept the idea that different people state the laws of physics in different ways.

    Local and Global Forms of Axiom II

    Up to this point we have discussed the local form of Axiom II, i.e., the form that applies at a point in space. However, when Axiom II is used to analyze the reactors shown in Figures \(6.1.1\) and \(6.1.2\), we will make use of a an integrated form of Equation \ref{20} that applies to the control volume illustrated in Figure \(\PageIndex{1}\).

    6.3.PNG
    Figure \(\PageIndex{1}\): Local and global rates of production

    Here we have illustrated the local rate of production for species \(A\), designated by \(R_{A}\), and the global rate of production for species \(A\), designated by \(\mathscr{R}_{ A}\). The latter is defined explicitly by

    \[\mathscr{R}_{ A} =\int_{\mathscr{V}}R_{A} dV =\left\{\begin{array}{c} {net \text{ macroscopic molar rate}} \\ \text{ of production of species }A \\ \text{ owing to chemical reactions} \end{array}\right\} \label{30}\]

    and we will often use an abbreviated description given by

    \[\mathscr{R}_{ A} =\left\{\begin{array}{c} \text{ global rate of} \\ \text{ production of} \\ \text{ species }A \end{array}\right\} \label{31}\]

    When dealing with a problem that involves the global rate of production, we need to form the volume integral of Equation \ref{20} to obtain

    \[\int_{\mathscr{V}}\sum_{A = 1}^{A = N}N_{JA} R_{A} dV=0 , \quad J = 1,2,...,T \label{32}\]

    The integral can be taken inside the summation operation, and we can make use of the fact that the elements of \(N_{JA}\) are independent of space to obtain

    \[\sum_{A = 1}^{A = N}N_{JA} \int_{\mathscr{V}}R_{A} dV =0 , \quad J = 1,2,...,T \label{33}\]

    Use of the definition of the global rate of production for species \(A\) given by Equation \ref{30} leads to the following global form of Axiom II:

    Axiom II (global form): \[\sum_{A = 1}^{A = N}N_{JA} \mathscr{R}_{ A} =0 , \quad J = 1,2,...,T \label{34}\]

    Here one must remember that \(\mathscr{R}_{ A}\) has units of moles per unit time while \(R_{A}\) has units of moles per unit time per unit volume, thus the physical interpretation of these two quantities is different as illustrated in Figure \(\PageIndex{1}\). In our study of complex systems described in Chapter 7, we will routinely encounter global rates of production and Axiom II (global form) will play a key role in the analysis of those systems.

    Solutions of Axiom II

    In the previous paragraphs we have shown that Equation \ref{20} and Equation \ref{22} represent the fundamental concept that atomic species are conserved during chemical reactions. In addition, we made use of the concept that atomic species are conserved by counting atoms or balancing chemical equations (see Eqs. \((6.1.7)\), \((6.1.10)\), and \((6.1.11)\)). The fact that the process of counting atoms is not unique for the partial oxidation of ethane is a matter of considerable interest that will be explored carefully in this chapter.

    In order to develop a better understanding of Axiom II, we carry out the matrix multiplication indicated by Equation \ref{22} for a system containing three (3) atomic species and six (6) molecular species. This leads to the following set of three (3) equations containing six (6) net rates of production:

    Atomic Species 1: \[N_{11} R_{1} + N_{12} R_{2} + N_{13} R_{3} + N_{14} R_{4} + N_{15} R_{5} + N_{16} R_{6} =0 \label{35a}\]

    Atomic Species 2: \[N_{21} R_{1} + N_{22} R_{2} + N_{23} R_{3} + N_{24} R_{4} + N_{25} R_{5} + N_{26} R_{6} =0 \label{35b}\]

    Atomic Species 3: \[N_{31} R_{1} + N_{32} R_{2} + N_{33} R_{3} + N_{34} R_{4} + N_{35} R_{5} + N_{36} R_{6} =0 \label{35c}\]

    This \(3 \times 6\) system of equations always has the trivial solution \(R_{A} =0\), for \(A=1, 2, ..., 6\), and the necessary and sufficient condition for a non-trivial solution to exist is that the rank of the atomic matrix be less than the number of molecular species. For this special case of three atomic species and six molecular species, we express this condition as5

    Non-trivial solution: \[r = rank [N_{JA} ]<6 \label{36}\]

    By rank we mean explicitly the row rank which represents the number of linearly independent equations contained in Eqs. \ref{35a} - \ref{35c}. It is possible that all three of Eqs. \ref{35a} - \ref{35c} are independent and the rank associated with the atomic matrix is three, i.e., \(r = rank = 3 \). On the other hand, it is possible that one of the three equations is a linear combination of the other two equations and the rank is two, i.e., \(r = rank = 2\). The general condition concerning the rank of the atomic matrix in Equation \ref{22} is given by

    Non-trivial solution (General): \[r = rank [N_{JA} ]<N \label{37}\]

    When the rank is equal to \(N-1\), we have a special case of Equation \ref{22} that leads to a single independent stoichiometric reaction. In that special case, the \(N-1\) net rates of production can be specified in terms of \(R_N\) and Equation \ref{22} can be expressed as

    \[R_{A} = \nu_{AN} R_{N} \label{38a}\]

    \[R_{B} =\nu_{BN} R_{N} \label{38b}\]

    \[R_{C} =\nu_{CN} R_{N} \label{38c}\]

    \[. \nonumber\]

    \[. \nonumber\]

    \[R_{N-1} =\nu_{N-1N} R_{N} \label{38d}\]

    Here \(\nu_{AN}\), \(\nu_{BN}\), etc., are often referred to as stoichiometric coefficients; however, the authors prefer to identify these quantities as elements of the pivot matrix as indicated in Example \(\PageIndex{1}\).

    Stoichiometric equations

    It is crucial to understand that the equations given by Eqs. \ref{38a} - \ref{38d} are based on the concept that atoms are conserved in the absence of nuclear reactions. The bookkeeping associated with the conservation of atoms is known as stoichiometry, thus it is appropriate to refer to the equations given by Eqs. \ref{38a} - \ref{38d} as stoichiometric equations. In addition, it is appropriate to identify Eqs. \ref{38a} - \ref{38d} as a case in which there is a single independent rate of production, and that this single independent rate of production is identified as \(R_{N}\). In order to be clear about stoichiometry and chemical kinetics, we place Equation \((6.1.6)\) side-by-side with Equation \ref{38a} to obtain

    \[R_{A} =-\frac{k c_{A}^{2} }{1+k^{\prime}c_{A} } , \quad \text{ chemical kinetics} \label{39a}\]

    \[R_{A} =\nu_{AN} R_{N} , \quad \text{ stoichiometry} \label{39b}\]

    Here we note that the symbol, \(R_{A}\), on the left hand side of Equation \ref{39a} has exactly the same physical significance as \(R_{A}\) on the left hand side of Equation \ref{39b}. In both cases, \(R_{A}\) is defined by Equation \ref{17}. However, the description of the right hand side of these two representations of \(R_{A}\) is quite different. The right hand side of Equation \ref{39a} is a chemical kinetic relation while the right hand side of Equation \ref{39b} is a stoichiometric relation. The chemical kinetic representation depends on the complex processes that occur at the quantum mechanical level where molecules are dissociated, active complexes are formed, and various molecular fragments coalesce to form products. The stoichiometric representation is based solely on the concept that atoms are conserved. The chemical kinetic representation may depend on temperature, pressure, and the presence of catalysts, while the stoichiometric representation remains invariant depending only on the conservation of atoms. In this chapter, and throughout most of the text, we will deal only with stoichiometric equations such as those given by Axiom II. In Chapter 8 we will introduce some examples of chemical kinetic equations, and in that treatment we will be very careful to identify stoichiometric constraints that are associated with elementary stoichiometry.

    Example \(\PageIndex{1}\): Complete combustion of ethane

    In this example we consider the complete combustion of ethane, thus the molecular species under consideration are identified as (see Figure \(6.1.1\))

    \[\ce{C2} \ce{H6} , \quad \ce{O2} , \quad \ce{H2O} , \quad \ce{CO2} \nonumber\]

    One form of the atomic matrix for this group of molecular species can be visualized as

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

    and for this particular arrangement the atomic matrix is given explicitly by

    \[ \left[N_{JA} \right]=\begin{bmatrix} {2} & {0} & {0} & {1} \\ {6} & {0} & {2} & {0} \\ {0} & {2} & {1} & {2} \end{bmatrix} \label{2a}\tag{2}\]

    A simple calculation (see Section 6.5, Problem 5) shows that the rank of the matrix is three

    \[ r = rank [N_{JA} ]=3 \label{3a}\tag{3}\]

    thus we have three equations and four unknowns. The three homogeneous equations corresponding to Equation \ref{35a} - \ref{35c} are given by

    \[2R_{ \ce{C2} \ce{H6} } + 0 + 0 + R_{\ce{CO2}} =0 \label{4aa}\tag{4a}\]

    \[6R_{ \ce{C2} \ce{H6} } + 0 + 2R_{ \ce{H2O}} + 0 =0 \label{4ba}\tag{4b}\]

    \[0 + 2R_{ \ce{O2} } + R_{ \ce{H2O}} + 2R_{\ce{CO2}} =0 \label{4ca}\tag{4c}\]

    while the analogous matrix equation corresponding to Equation \ref{22} takes the form

    \[ \begin{bmatrix} {2} & { 0} & { 0} & {1 } \\ {6} & { 0} & { 2} & {0 } \\ {0} & { 2} & { 1} & {2 } \end{bmatrix} \begin{bmatrix} R_{ \ce{C2} \ce{H6} } \\ R_{\ce{O2}} \\ R_{\ce{H2O}} \\ R_{\ce{CO2}} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \label{5a}\tag{5}\]

    It is possible to use intuition and the picture given by Eq. \((6.1.7)\) to express the net rates of production in the form

    \[R_{ \ce{C2} \ce{H6} } =-\frac{1}{2} R_{\ce{CO2}} \label{6aa}\tag{6a}\]

    \[R_{ \ce{O2} } =-{\frac{7}{4}} R_{\ce{CO2}} \label{6ba}\tag{6b}\]

    \[R_{ \ce{H2O}} =+{\frac{3}{2}} R_{\ce{CO2}} \label{6ca}\tag{6c}\]

    however, the use of Eqs. \ref{4aa} - \ref{4ca} to produce this result is more reliable. Finally, we note that these results for the net rates of production can be expressed in the form of the pivot theorem that is described in Sec. 6.4. In terms of the pivot theorem that can be extracted from Equation \ref{5a} we have

    \[ \underbrace{\begin{bmatrix} R_{\ce{C2} \ce{H6} } \\ R_{\ce{O2}} \\ R_{\ce{H2O}} \end{bmatrix}}_{\text{column matrix of non-pivot species}} = \underbrace{\begin{bmatrix} -\frac{1}{2} \\ -\frac{7}{4} \\ +\frac{3}{2} \end{bmatrix}}_{\text{pivot matrix}} \underbrace{\left[R_{\ce{ CO2} }\right]}_{\text{column matrix of pivot species}} \label{7a}\tag{7}\]

    This indicates that all the rates of production are specified if we can determined the rate of production for carbon dioxide, \(R_{\ce{ CO2} }\). Indeed, all the rates of production can be determined if we know any one of the four rates; however, we have chosen carbon dioxide as the pivot species in order to arrange Eqs. \ref{4aa} - \ref{4ca} and \ref{5a} in the forms given by Eqs. \ref{6aa} - \ref{6ca} and \ref{7a}.

    In the previous example we illustrated how Axiom II can be used to analyze the stoichiometry for the complete combustion of ethane. The process of complete combustion was described earlier by the single stoichiometric schema given by Equation \((6.1.7)\) and the coefficients that appeared in that schema are evident in Eqs. \ref{6aa} - \ref{6ca} and \ref{7a} of Example \(\PageIndex{1}\).

    Elementary row operations and column/row interchange operations

    In working with sets of equations such as those represented by Equation \ref{22}, we will make use of elementary row operations and column/row operations in order to arrange the equations in a convenient form. Elementary row operations were described earlier in Sec. 4.9.1 and we list them here as they apply to the atomic matrix:

    I. Any row in the atomic matrix can be modified by multiplying or dividing by a non-zero scalar without affecting the system of equations.

    II. Any row in the atomic matrix can be added or subtracted from another row without affecting the system of equations.

    III. Any two rows in the atomic matrix can be interchanged without affecting the system of equations.

    The column/row interchange operation that we will use in the treatment of Equation \ref{22} is described as follows:

    IV. Any two columns in the atomic matrix can be interchanged without affecting the system of equations provided that the corresponding rows of the column matrix of net rates of production are also interchanged.

    In terms of Equation \ref{20} this latter operation can be described mathematically as

    \[N_{J B} R_{B} \rightleftarrows N_{J D} R_{D}, \quad B, D=1,2, \ldots, N, \quad J=1,2, \ldots, T \label{40}\]

    We can use these operations to develop row equivalent matrices, row reduced matrices, row echelon matrices, and row reduced echelon matrices. In order to illustrate these concepts, we consider the following example of Axiom II:

    Axiom II: \[\begin{bmatrix} 2 & 2 & 0 & 2 & 0 & 4 \\ 6 & 4 & 2 & 4 & 2 & 6 \\ 2 & 2 & 0 & 1 & 1 & 3 \\ 1 & 0 & 1 & 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} R_{1} \\ R_{2} \\ R_{3} \\ R_{4} \\ R_{5} \\ R_{6} \end{bmatrix}=0 \label{41}\]

    Directing our attention to the atomic matrix, we subtract three times the first row from the second row to obtain a row equivalent matrix given by

    \(R2-3R1\): \[\begin{bmatrix} 2 & 2 & 0 & 2 & 0 & 4 \\ 0 & -2 & 2 & -2 & 2 & -6 \\ 2 & 2 & 0 & 1 & 1 & 3 \\ 1 & 0 & 1 & 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} R_{1} \\ R_{2} \\ R_{3} \\ R_{4} \\ R_{5} \\ R_{6} \end{bmatrix}=0 \label{42}\]

    Dividing the first row by two will create a coefficient of one in the first row of the first column. This operation leads to

    \(R1/2\): \[\begin{bmatrix} 1 & 1 & 0 & 1 & 0 & 2 \\ 0 & -2 & 2 & -2 & 2 & -6 \\ 2 & 2 & 0 & 1 & 1 & 3 \\ 1 & 0 & 1 & 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} R_{1} \\ R_{2} \\ R_{3} \\ R_{4} \\ R_{5} \\ R_{6} \end{bmatrix}=0 \label{43}\]

    Multiplication of the second row by \(-1/2\) provides

    \(R2(-1/2)\): \[\begin{bmatrix} 1 & 1 & 0 & 1 & 0 & 2 \\ 0 & 1 & -1 & 1 & -1 & 3 \\ 2 & 2 & 0 & 1 & 1 & 3 \\ 1 & 0 & 1 & 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} R_{1} \\ R_{2} \\ R_{3} \\ R_{4} \\ R_{5} \\ R_{6} \end{bmatrix}=0 \label{44}\]

    Using several elementary row operations, we construct a row echelon form of the atomic matrix that given by

    \[\begin{bmatrix} 1 & 1 & 0 & 1 & 0 & 2 \\ 0 & 1 & -1 & 1 & -1 & 3 \\ 0 & 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} R_{1} \\ R_{2} \\ R_{3} \\ R_{4} \\ R_{5} \\ R_{6} \end{bmatrix}=0 \label{45}\]

    The row of zeros indicates that one of the four equations represented by Equation \ref{41} is not independent, i.e., it is a linear combination of two or more of the other equations. This means that the rank of the atomic matrix represented in Equation \ref{41} is three, \(r=rank[N_{JA}]=3\).

    We can make further progress toward the row reduced echelon form by subtracting row two from row one to obtain

    \(R1-R2\): \[\begin{bmatrix} 1 & 0 & 1 & 0 & 1 & -1 \\ 0 & 1 & -1 & 1 & -1 & 3 \\ 0 & 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} R_{1} \\ R_{2} \\ R_{3} \\ R_{4} \\ R_{5} \\ R_{6} \end{bmatrix}=0 \label{46}\]

    In this form, the first term in each row is one and all of the entries in the column below the first term in each row are zero. At this point we have not yet achieved the row reduced echelon form; however, use of the following column/row interchange

    \[N_{J 3} R_{3} \rightleftarrows N_{J 4} R_{4}, \quad J=1,2, 3 \label{47}\]

    provides a step in that direction given by

    \[\begin{bmatrix} 1 & 0 & 0 & 1 & 1 & -1 \\ 0 & 1 & 1 & -1 & -1 & 3 \\ 0 & 0 & 1 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} R_{1} \\ R_{2} \\ R_{3} \\ R_{4} \\ R_{5} \\ R_{6} \end{bmatrix}=0 \label{48}\]

    We now subtract row three from row two in order to obtain the row reduced echelon form given by

    \(R2-R3\): \[\begin{bmatrix} 1 & 0 & 0 & 1 & 1 & -1 \\ 0 & 1 & 0 & -1 & 0 & 2 \\ 0 & 0 & 1 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} R_{1} \\ R_{2} \\ R_{3} \\ R_{4} \\ R_{5} \\ R_{6} \end{bmatrix}=0 \label{49}\]

    The last row of zeros produces the null equation that we express as

    \[0 \times R_{1}+0 \times R_{2}+0 \times R_{4}+0 \times R_{3}+0 \times R_{5}+0 \times R_{6}=0 \label{50}\]

    thus we can discard that row to obtain

    Axiom II: \[\begin{bmatrix} 1 & 0 & 0 & \vdots & 1 & 1 & -1 \\ 0 & 1 & 0 & \vdots & -1 & 0 & 2 \\ 0 & 0 & 1 & \vdots & 0 & -1 & 1 \end{bmatrix} \begin{bmatrix} R_{1} \\ R_{2} \\ R_{3} \\ R_{4} \\ R_{5} \\ R_{6} \end{bmatrix}=0 \label{51}\]

    This form has the attractive feature that the submatrix located to the left of the dashed line is a unit matrix, and this is a useful result for solving sets of equations. Finally, it is crucial to understand that any atomic matrix can always be expressed in row reduced echelon form, and uniqueness proofs are given in Sec. 3.8 of Noble6 and Sec. 1.5 of Kolman7.

    Example \(\PageIndex{2}\): Experimental determination of the rate of production

    Here we consider the experimental determination of a global rate of production for the steady-state, catalytic dehydrogenation of ethane as illustrated in Figure \(\PageIndex{2}\). We assume (see Sec. 6.1.1) that the reaction produces ethylene and hydrogen, thus only \(\ce{C2} \ce{H6}\), \(\ce{C2} \ce{H4}\) and \(\ce{H2}\) are present in the system. We are given that the feed is pure ethane and the feed flow rate is 100 kmol/min.

    image
    Figure \(\PageIndex{2}\): Experimental reactor

    The product Stream #2 is subject to a measurement indicating that the molar flow rate of hydrogen in that stream is 30 kmol/min, and we wish to use this information to determine the global rate of production for ethane. For steady-state conditions, the axiom given by Equation \((6.1.3)\) takes the form

    Axiom I: \[\int_{\mathscr{A}}c_{A } \mathbf{v}_{A} \cdot \mathbf{n} dA=\int_{\mathscr{V}}R_{A} dV , \quad A \Rightarrow \ce{C2} \ce{H6} , \quad \ce{C2} \ce{H4} , \quad \ce{H2} \label{2b}\tag{1}\]

    Application of this result to the control volume illustrated in Figure \(\PageIndex{2}\) provides the following three equations:

    Ethane: \[- (y_{\ce{C2H6}} )_{1} \dot{M}_{1} + (y_{\ce{C2H6}} )_{2} \dot{M}_{2} =\mathscr{R}_{ \ce{C2} \ce{H6} } \label{3b}\tag{2}\]

    Ethylene: \[- (y_{\ce{C2} \ce{H4} } )_{1} \dot{M}_{1} + (y_{\ce{C2} \ce{H4} } )_{2} \dot{M}_{2} =\mathscr{R}_{ \ce{C2} \ce{H4} } \label{4b}\tag{3}\]

    Hydrogen: \[- (y_{\ce{H2}} )_{1} \dot{M}_{1} + (y_{\ce{H2}} )_{2} \dot{M}_{2} =\mathscr{R}_{ \ce{H2} } \label{5b}\tag{4}\]

    Here we have used \(\mathscr{R}_{ A}\) to represent the global (net) rate of production for species \(A\) that is defined by (see Equation \ref{30})

    \[ \mathscr{R}_{ A} =\int_{\mathscr{V}}R_{A} dV , \quad A \Rightarrow \ce{C2} \ce{H6} , \quad \ce{C2} \ce{H4} , \quad \ce{H2} \label{6b}\tag{5}\]

    The units of the global rate of production, \(\mathscr{R}_{ A}\), are moles/time while the units of the rate of production, \(R_{A}\), are moles/(time \(\times\) volume), and one must be careful to note this difference.

    At the entrance and exit of the control volume, we have two constraints on the mole fractions given by

    Stream #1: \[(y_{\ce{C2H6}} )_{1} + (y_{\ce{C2} \ce{H4} } )_{1} + (y_{\ce{H2}} )_{1} =1 \label{7b}\tag{6}\]

    Stream #2: \[(y_{\ce{C2H6}} )_{2} + (y_{\ce{C2} \ce{H4} } )_{2} + (y_{\ce{H2}} )_{2} =1 \label{8b}\tag{7}\]

    For this particular process, the global form of Axiom II is given by

    Axiom II \[\sum_{A = 1}^{A = N}N_{JA} \mathscr{R}_{ A} =0 , \quad J \Rightarrow \ce{ C } , \quad \ce{ H } \label{9b}\tag{8}\]

    The visual representation of the atomic matrix is given by

    \[\text{ Molecular Species} \to \ce{C2H6} \quad \ce{C2H4} \quad \ce{H2} \\ \begin{matrix} {carbon} \\ { hydrogen} \end{matrix} \begin{bmatrix} {2} & { 0} & { 0} \\ {6} & { 4} & { 2} \end{bmatrix} \label{10b}\tag{9}\]

    and we express the explicit form of the matrix as

    \[\mathbf{A} = \begin{bmatrix} {2} & {2} & {0} \\ {6} & {4} & {2} \end{bmatrix}, \quad \text{ or } \quad \left[N_{JA} \right]=\begin{bmatrix} {2} & {2} & {0} \\ {6} & {4} & {2} \end{bmatrix} \label{11b}\tag{10}\]

    Use of this result for the atomic matrix with Equation \ref{9b} leads to

    \[\begin{bmatrix} {2} & {2} & {0} \\ {6} & {4} & {2} \end{bmatrix} \begin{bmatrix} \mathscr{R}_{ \ce{C2} \ce{H6} } \\ \mathscr{R}_{ \ce{H2} } \\ \mathscr{R}_{ \ce{C2} \ce{H4} } \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \label{12b}\tag{11}\]

    At this point we can follow the development in Sec. 6.2.5 to obtain

    \[\begin{bmatrix} {1} & {0} & {1} \\ {0} & {1} & {-1} \end{bmatrix} \begin{bmatrix} \mathscr{R}_{ \ce{C2} \ce{H6} } \\ \mathscr{R}_{ \ce{H2} } \\ \mathscr{R}_{ \ce{C2} \ce{H4} } \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \label{13b}\tag{12}\]

    in which \(\ce{C2} \ce{H4}\) has been chosen to be the pivot species (see Sec. 6.4). Carrying out the matrix multiplication leads to

    \[\mathscr{R}_{ \ce{C2} \ce{H6} } =-\mathscr{R}_{ \ce{C2} \ce{H4} } \label{14ab}\tag{13a}\]

    \[\mathscr{R}_{ \ce{H2} } =\mathscr{R}_{ \ce{C2} \ce{H4} } \label{14bb}\tag{13b}\]

    in which \(R_{ \ce{C2} \ce{H4} }\) is to be determined experimentally. A degree of freedom analysis will show that a unique solution is available and we can summarize the various equations as

    Ethane mole balance: \[- { 100 \ kmol/min} + (y_{\ce{C2H6}} )_{2} \dot{M}_{2} =\mathscr{R}_{ \ce{C2} \ce{H6} } \label{14b}\tag{14}\]

    Ethylene mole balance: \[(y_{\ce{C2} \ce{H4} } )_{2} \dot{M}_{2} =\mathscr{R}_{ \ce{C2} \ce{H4} } \label{15b}\tag{15}\]

    Hydrogen mole balance: \[{ 30 \ kmol/min}=\mathscr{R}_{ \ce{H2} } \label{16b}\tag{16}\]

    Stream #1: \[(y_{\ce{C2H6}} )_{1} = 1 , \quad (y_{\ce{C2} \ce{H4} } )_{2} = 0 , \quad (y_{\ce{H2}} )_{2} = 0 \label{17b}\tag{17}\]

    Stream #2: \[(y_{\ce{C2H6}} )_{2} + (y_{\ce{C2} \ce{H4} } )_{2} + (y_{\ce{H2}} )_{2} =1 \label{18b}\tag{18}\]

    Axiom II constraint: \[\mathscr{R}_{ \ce{C2} \ce{H6} } =-\mathscr{R}_{ \ce{C2} \ce{H4} } \label{19b}\tag{19}\]

    Axiom II constraint: \[\mathscr{R}_{ \ce{H2} } =\mathscr{R}_{ \ce{C2} \ce{H4} } \label{20b}\tag{20}\]

    The solution to Eqs. \ref{14b} through \ref{20b} is given by

    \[ \dot{M}_{2} ={ 130 \ kmol/min} \label{21b}\tag{21}\]

    \[ (y_{\ce{C2} \ce{H6} } )_{2} =\frac{7}{13} , \quad (y_{\ce{C2} \ce{H4} } )_{2} =\frac{3}{13} , \quad (y_{\ce{H2} } )_{2} = \frac{3}{13} \label{22b}\tag{22}\]

    \[ \mathscr{R}_{ \ce{C2} \ce{H4} } ={ 30 \ kmol/min} \label{23b}\tag{23}\]

    Here we see how the experimental system illustrated in Figure \(\PageIndex{2}\) can be used to determine the global rate of production for ethylene, \(R_{ \ce{C2} \ce{H4} }\).

    In this example we have made use of the global form of Axiom II given by Equation \ref{34} as opposed to the local form given by Equation \ref{20}. In addition, we can integrate the local form given by Equation \((6.1.5)\) to obtain

    \[ \sum_{A = 1}^{A = N}MW_{A} \mathscr{R}_{ A} =0 \label{24b}\tag{24}\]

    This form of Axiom II reminds us that, in general, moles are not conserved and they are certainly not conserved in this specific example.

    In the previous example, we illustrated how a rate of production could be determined experimentally for the case of a single independent stoichiometric reaction. When this condition exists in an \(N\)-component system, we can express \(N-1\) rates of production in terms of a single rate of production, \(R_{N}\). For the complete combustion of ethane described in Example \(\PageIndex{1}\), there are four molecular species, and the rates of production for \(\ce{C2} \ce{H6}\), \(\ce{O2}\), \(\ce{H2O}\) can be related to the rate of production for \(\ce{CO2}\). For the rate of production of ethylene described in Example \(\PageIndex{2}\), we have another example of a single independent reaction. In more complex systems, the stoichiometry is represented by multiple independent stoichiometric reactions, and we consider such a case in the following example.

    Example \(\PageIndex{3}\): Partial oxidation of carbon

    Carbon and oxygen can react to form carbon monoxide and carbon dioxide, thus the reaction involves four molecular species and two atomic species as indicated by

    Molecular Species: \[\ce{ C } , \quad \ce{O2} , \quad {\ce{ CO}} , \quad \ce{CO2} \label{1c}\tag{1}\]

    Atomic Species: \[\ce{ C } \text{ and } \ce{ O} \label{2c}\tag{2}\]

    A visual representation of the atomic matrix for this system is given by

    \[\text{ Molecular Species} \to \ce{C} \quad \ce{O2} \quad \ce{CO} \quad \ce{CO2} \\ \begin{matrix} {carbon} \\{oxygen} \end{matrix} \begin{bmatrix} { 1} & { 0} & {1} & { 1 } \\ { 0} & { 2} & {1} & { 2 } \end{bmatrix} \label{3c}\tag{3}\]

    and this can be used with Equation \ref{22} to obtain

    Axiom II: \[\begin{bmatrix} {1} & {0} & {1} & {1} \\ {0} & {2} & {1} & {2} \end{bmatrix} \begin{bmatrix} {R}_{ \ce{C} } \\ {R}_{ \ce{O2} } \\ R_{ \ce{CO} } \\ R_{ \ce{CO2} } \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \label{4c}\tag{4}\]

    A simple calculation shows that the rank of the atomic matrix is two

    \[ r=rank\left[N_{JA} \right]=2 \label{5c}\tag{5}\]

    thus we have two equations and four unknowns. Here we note that the atomic matrix can be expressed in row reduced echelon form (see Equation \ref{51}) leading to

    Axiom II: \[\begin{bmatrix} {1} & {0} & {1} & {1} \\ {0} & {1} & {1/2} & {1} \end{bmatrix} \begin{bmatrix} {R}_{ \ce{C} } \\ {R}_{ \ce{O2} } \\ R_{ \ce{CO} } \\ R_{ \ce{CO2} } \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \label{6c}\tag{6}\]

    and the homogeneous system of equations corresponding to this form is given by

    \[R_{\ce{ C }} + 0 +R_{\ce{ CO}} + R_{\ce{ CO2} } =0 \label{7ac}\tag{7a}\]

    \[0 + R_{\ce{ O2} } + \frac{1}{2} R_{\ce{ CO}} + R_{\ce{ CO2} } =0 \label{7bc}\tag{7b}\]

    Given two equations and four rates of production, it is clear that we must determine two rates of production in order to determine all the rates of production. We will associate these two rates with two pivot species, and if we choose the pivot species to be carbon monoxide and carbon dioxide the rates of production for carbon and oxygen are given by

    \[R_{\ce{ C }} =-R_{\ce{ CO}} - R_{\ce{ CO2} } \label{8ac}\tag{8a}\]

    \[R_{\ce{O2} } =-\frac{1}{2} R_{\ce{ CO}} - R_{\ce{ CO2} } \label{8bc}\tag{8b}\]

    Here we have followed the same style used in Example \(\PageIndex{2}\) and placed the pivot species on the right hand side of Eqs. \ref{8ac} - \ref{8bc}. In matrix notation this result can be expressed as (see Sec. 6.4)

    \[ \underbrace{\begin{bmatrix} R_{\ce{C} } \\ R_{\ce{O2}} \end{bmatrix}}_{\text{column matrix of non-pivot species}} = \underbrace{\begin{bmatrix} -1 & -1 \\ -1/2 & -1 \end{bmatrix}}_{\text{pivot matrix}} \underbrace{\begin{bmatrix} R_{\ce{CO}} \\ R_{\ce{ CO2} } \end{bmatrix}}_{\text{column matrix of pivot species}} \label{9c}\tag{9}\]

    in which the \(2 \times 2\) matrix is referred to as the pivot matrix since it is the matrix that maps the net rates of production of the pivot species onto the net rates of production of the non-pivot species. Other possibilities can be constructed by using different pivot species and the development of these has been left as an exercise for the student.

    The partial oxidation of carbon is an especially simple example of multiple independent stoichiometric equations, i.e., \(rank \left[N_{JA} \right] < N-1\). The partial oxidation of ethane, illustrated in Eqs. \((6.1.9)\) through \((6.1.11)\), provides a more challenging problem.

    Example \(\PageIndex{4}\): Partial oxidation of ethane

    As an example only, we imagine that the process illustrated in Figure \(6.1.2\) is carried out so that ethane is partially oxidized to produce ethylene oxide, carbon dioxide, carbon monoxide and water. Thus the molecular species involved in the process are assumed to be

    Molecular species: \[\ce{C2} \ce{H6} { , \quad }\ce{O2} , \quad {\ce{ CO}} , \quad \ce{C2} \ce{H4O} , \quad \ce{H2O} , \quad \ce{CO2} \label{1d}\tag{1}\]

    and the rates of production for these species are constrained by

    Axiom II \[\sum_{A = 1}^{A = 6}N_{JA} R_{A} =0 , \quad J \Rightarrow \ce{ C } , \quad \ce{ H } , \quad \ce{ O} \label{2d}\tag{2}\]

    A visual representation of the atomic matrix is given by

    \[\text{ Molecular Species} \to \ce{C2H6} \quad \ce{O2} \quad \ce{H2O} \quad \ce{CO} \quad \ce{CO2} \quad \ce{C2H4O} \\ \begin{matrix} {carbon} \\ { hydrogen} \\ {oxygen} \end{matrix} \begin{bmatrix} { 2} & { 0} & {0} & {1} & { 1} & {2 } \\ { 6} & { 0} & {2} & {0} & { 0} & {4 } \\ { 0} & { 2} & {1} & {1} & { 2} & {1 } \end{bmatrix} \label{3d}\tag{3}\]

    while the matrix representation of Equation \ref{2d} takes the form

    \[ \begin{bmatrix} {2} & {0} & {0} & {1} & {1} & {2} \\ {6} & {0} & {2} & {0} & {0} & {4} \\ {0} & {2} & {1} & {1} & {2} & {1} \end{bmatrix} \begin{bmatrix} R_{\ce{C2} \ce{H6} } \\ R_{\ce{ O2} } \\ R_{\ce{H2O}} \\ R_{\ce{ CO}} \\ R_{\ce{CO2} } \\ R_{\ce{C2} \ce{H4O} } \end{bmatrix}=\begin{bmatrix} {0} \\ {0} \\ {0} \end{bmatrix} \label{4d}\tag{4}\]

    By a series of elementary row operations we can transform the atomic matrix to the row reduced echelon form so that Equation \ref{4d} can be expressed as (see Sec. 6.5.2)

    Axiom II: \[ \begin{bmatrix} {1} & {0} & {0} & {1/2} & {1/2} & {1} \\ {0} & {1} & {0} & {5/4} & {7/4} & {1} \\ {0} & {0} & {1} & {-3/2} & {-3/2} & {-1} \end{bmatrix}\begin{bmatrix} R_{\ce{C2} \ce{H6} } \\ R_{\ce{ O2} } \\ R_{\ce{H2O} } \\ R_{\ce{ CO}} \\ R_{\ce{CO2}} \\ R_{\ce{C2} \ce{H4O} } \end{bmatrix}=\begin{bmatrix} {0} \\ {0} \\ {0} \end{bmatrix} \label{5d}\tag{5}\]

    Here we see that the rank of the atomic matrix is three, \(r=3\), thus the rank is less than \(N\) which is equal to six. Since the rank \(< N\) a non-trivial solution exists consisting of three independent equations that can be expressed as

    \[R_{\ce{C2} \ce{H6} } =-\frac{1}{2} R_{\ce{ CO}} - \frac{1}{2} R_{\ce{CO2} } - R_{\ce{C2} \ce{H4O}} \label{6ad}\tag{6a}\]

    \[R_{\ce{O2}} =-\frac{5}{4} R_{\ce{ CO}} - \frac{7}{4} R_{\ce{CO2} } - R_{\ce{C2} \ce{H4O}} \label{6bd}\tag{6b}\]

    \[R_{\ce{H2O}} =\frac{3}{2} R_{\ce{ CO}} + \frac{3}{2} R_{\ce{CO2} } + R_{\ce{C2} \ce{H4O}} \label{6cd}\tag{6c}\]

    Here we have chosen \(\ce{ CO}\), \(\ce{CO2}\), and \(\ce{C2} \ce{H4O}\) as the pivot species with the idea that the rates of production for these species will be determined experimentally. Given the rates of production for the pivot species, Eqs. \ref{6ad} - \ref{6cd} can be used to determine the rates of production for the non-pivot species, \(\ce{C2} \ce{H6}\), \(\ce{O2}\), and \(\ce{H2O}\).

    In this section we have illustrated how the Axiom II, given by Eqs. \ref{20} or by Equation \ref{22}, is used to constrain the net rates of production. When we have a single independent stoichiometric reaction, such as the complete combustion of ethane, one need only measure a single net rate of production in order to determine all the net rates of production. This case is illustrated in Eq. \((6.1.7)\) and discussed in detail in Example \(\PageIndex{2}\). When we have multiple independent stoichiometric reactions, such as the partial oxidation of carbon (Example \(\PageIndex{3}\)) or the partial oxidation of ethane (Example \(\PageIndex{4}\)), we need to determine more than one net rate of production in order to determine all the net rates of production.

    Matrix partitioning

    Axiom II provides an example of the multiplication of a \(T \times N\) matrix with a \(1 \times N\) column matrix. 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

    \[\begin{bmatrix} {a_{11} } & {a_{12} } & {a_{13} } & {a_{14} } & {a_{15}} \\ {a_{21} } & {a_{22} } & {a_{23} } & {a_{24} } & {a_{25}} \\ {a_{31} } & {a_{32} } & {a_{33} } & {a_{34} } & {a_{35}} \end{bmatrix} \begin{bmatrix} {b_{1} } \\ {b_{2} } \\ {b_{3} } \\ {b_{4} } \\ {b_{5} } \end{bmatrix}=\begin{bmatrix} {c_{1} } \\ {c_{2} } \\ {c_{3} } \end{bmatrix} \label{52}\]

    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{52} represents the three individual equations given by

    \[{a_{11}b_{1} } + {a_{12}b_{2} } + {a_{13}b_{3} } + {a_{14}b_{4} } + {a_{15}b_{5} } = {c_{1} }\label{53a}\]

    \[{a_{21}b_{1} } + {a_{22}b_{2} } + {a_{23}b_{3} } + {a_{24}b_{4} } + {a_{25}b_{5} } = {c_{2} } \label{53b}\]

    \[{a_{31}b_{1} } + {a_{32}b_{2} } + {a_{33}b_{3} } + {a_{34}b_{4} } + {a_{35}b_{5} } = {c_{3} }\label{53c}\]

    which can also be expressed in compact form according to

    \[\mathbf{AB}=\mathbf{C}\label{54}\]

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

    \[\mathbf{A} = \begin{bmatrix} {a_{11} } & {a_{12} } & {a_{13} } & {a_{14} } & {a_{15}} \\ {a_{21} } & {a_{22} } & {a_{23} } & {a_{24} } & {a_{25}} \\ {a_{31} } & {a_{32} } & {a_{33} } & {a_{34} } & {a_{35}} \end{bmatrix} \quad \mathbf{B} = \begin{bmatrix} {b_{1} } \\ {b_{2} } \\ {b_{3} } \\ {b_{4} } \\ {b_{5} } \end{bmatrix} \quad \mathbf{C} =\begin{bmatrix} {c_{1} } \\ {c_{2} } \\ {c_{3} } \end{bmatrix} \label{55}\]

    In addition to the matrix multiplication that we have used up to this point, matrix multiplication can also be carried out in terms of partitioned matrices.

    If we wish to obtain a column partition of the matrix \(\mathbf{A}\) in Equation \ref{52}, we must also create a row partition of matrix \(\mathbf{B}\) in order to conform to the rules of matrix multiplication that are discussed in detail in Appendix C1. This column /row partition takes the form

    \[\begin{bmatrix} {a_{11} } & {a_{12} } & {a_{13} } & \vdots & {a_{14} } & {a_{15}} \\ {a_{21} } & {a_{22} } & {a_{23} } & \vdots & {a_{24} } & {a_{25}} \\ {a_{31} } & {a_{32} } & {a_{33} } & \vdots & {a_{34} } & {a_{35}} \end{bmatrix} \begin{bmatrix} {b_{1} } \\ {b_{2} } \\ {b_{3} } \\ \hdashline {b_{4} } \\ {b_{5} } \end{bmatrix}=\begin{bmatrix} {c_{1} } \\ {c_{2} } \\ {c_{3} } \end{bmatrix} \label{56}\]

    and the submatrices are identified explicitly according to

    \[\mathbf{A}_{11} = \begin{bmatrix} {a_{11} } & {a_{12} } & {a_{13} } \\ {a_{21} } & {a_{22} } & {a_{23} } \\ {a_{31} } & {a_{32} } & {a_{33} } \end{bmatrix} \quad \mathbf{B}_1 = \begin{bmatrix} {b_{1} } \\ {b_{2} } \\ {b_{3} } \end{bmatrix} \quad \mathbf{A}_{12} =\begin{bmatrix} {a_{14} } & {a_{15}} \\ {a_{24} } & {a_{25}} \\ {a_{34} } & {a_{35}} \end{bmatrix} \quad \mathbf{B}_2 = \begin{bmatrix} {b_{4} } \\ {b_{5} } \end{bmatrix} \label{57}\]

    Use of these representations in Equation \ref{56} leads to

    \[\begin{bmatrix} \mathbf{A}_{11} & \mathbf{A}_{12} \end{bmatrix}\begin{bmatrix} \mathbf{B}_1 \\ \mathbf{B}_2 \end{bmatrix} = \mathbf{C} \label{58}\]

    and matrix multiplication in terms of the submatrices provides

    \[\mathbf{A}_{11}\mathbf{B}_{1} + \mathbf{A}_{12}\mathbf{B}_{2} = \mathbf{C} \label{59}\]

    To verify that this result is identical to Eqs. \ref{53a} - \ref{53c}, we use Eqs. \ref{57} and the third of Eqs. \ref{55} to obtain

    \[\begin{bmatrix} {a_{11} } & {a_{12} } & {a_{13} } \\ {a_{21} } & {a_{22} } & {a_{23} } \\ {a_{31} } & {a_{32} } & {a_{33} } \end{bmatrix} \begin{bmatrix} {b_{1} } \\ {b_{2} } \\ {b_{3} } \end{bmatrix} + \begin{bmatrix} {a_{14} } & {a_{15}} \\ {a_{24} } & {a_{25}} \\ {a_{34} } & {a_{35}} \end{bmatrix} \begin{bmatrix} {b_{4} } \\ {b_{5} } \end{bmatrix} =\begin{bmatrix} {c_{1} } \\ {c_{2} } \\ {c_{3} } \end{bmatrix} \label{60}\]

    Carrying out the matrix multiplication on the left hand side of this result leads to

    \[\begin{bmatrix} {a_{11}b_{1} } & {a_{12}b_{2} } & {a_{13}b_{3} } \\ {a_{21} b_{1}} & {a_{22} b_{2}} & {a_{23}b_{3} } \\ {a_{31}b_{1} } & {a_{32}b_{2} } & {a_{33}b_{3} } \end{bmatrix} + \begin{bmatrix} {a_{14}b_{4} } & {a_{15}b_{5}} \\ {a_{24}b_{4} } & {a_{25}b_{5}} \\ {a_{34}b_{4} } & {a_{35}b_{5}} \end{bmatrix} = \begin{bmatrix} {c_{1} } \\ {c_{2} } \\ {c_{3} } \end{bmatrix} \label{61}\]

    At this point we add the two matrices on the left hand side of this result following the rules for matrix addition given in Sec. 2.6 in order to obtain

    \[\begin{bmatrix} {a_{11}b_{1} } & {a_{12}b_{2} } & {a_{13}b_{3} } & {a_{14}b_{4} } & {a_{15}b_{5}} \\ {a_{21} b_{1}} & {a_{22} b_{2}} & {a_{23}b_{3} } & {a_{24}b_{4} } & {a_{25}b_{5}} \\ {a_{31}b_{1} } & {a_{32}b_{2} } & {a_{33}b_{3} } & {a_{34}b_{4} } & {a_{35}b_{5}} \end{bmatrix} = \begin{bmatrix} {c_{1} } \\ {c_{2} } \\ {c_{3} } \end{bmatrix} \label{62}\]

    This is an alternate representation of Equation \ref{52} that immediately leads to Eqs. \ref{53a} - \ref{53c}. A more detailed discussion of matrix multiplication and partitioning is given in Appendix C1; however, the results in this section are sufficient for our treatment of Axiom II. In the following paragraphs we learn that partitioning of the atomic matrix leads to especially useful forms of Axiom II.

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