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10.5: Appendix E - Conservation of Charge

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    51847
  • In Chapter 6 we represented the conservation of atomic species by

    Axiom II:

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

    and in matrix form we expressed this result as

    Axiom II:

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

    If some of the species undergoing reaction are charged species (ions), we need to impose conservation of charge1 in addition to conservation of atomic species. This is done in terms of the additional axiomatic statement given by

    Axiom III:

    \[\sum_{A = 1}^{A = N}N_{\mathbf{e}A} R_{A} =0 \label{3}\]

    in which \(N_{\mathbf{e}A}\) represents the electronic charge associated with molecular species \(A\). In terms of matrix representation, Axiom III can be added to Equation \ref{2} to obtain a combined representation for conservation of atomic species and conservation of charge. This combined representation is given by

    \[\begin{bmatrix} {N_{11} } & {N_{12} } & {N_{13} } & {......} & {N_{1, N-1,} } & {N_{1N} } \\ {N_{21} } & {N_{22} } & . & {......} & {N_{2, N-1} } & {N_{2N} } \\ {N_{31} } & {N_{32} } & {} & {......} & . & . \\ {.} & . & {} & {......} & . & . \\ {.} & . & {} & {......} & . & . \\ {N_{T1} } & {.} & {} & {......} & {.} & {.} \\ {N_{\mathbf{e}1} } & {N_{\mathbf{e}2} } & {} & {......} & {N_{\mathbf{e} N-1} } & {N_{\mathbf{e}N} } \end{bmatrix} \begin{bmatrix} {R_{1} } \\ {R_{2} } \\ {R_{3} } \\ {.} \\ {.} \\ {R_{N-1} } \\ {R_{N} } \end{bmatrix} = \begin{bmatrix} {0} \\ {0} \\ {0} \\ {.} \\ {.} \\ {0} \end{bmatrix} \label{4}\]

    Here the elements in the last row of the \((T+1) \times N\) matrix take on the values associated with the charge on species \(1, 2, …N\) such as

    \[\begin{align} & N_{\mathbf{e}1} = 0, && \text{ non-ionic species} \nonumber\\ & N_{\mathbf{e}2} = -2, && \text{ ionic species such as } \ce{SO4-} \label{5} \\ & N_{\mathbf{e}3} = +1, && \text{ ionic species such as } \ce{Na+} \nonumber \end{align}\]

    As an example of competing reactions in a redox system2 we consider a mixture consisting of \( \ce{ClO2-}\), \(\ce{H3O+}\), \(\ce{Cl2}\), \(\ce{H2O}\), \(\ce{ClO3-}\), and \(\ce{ClO2}\). The visual representation for the atomic/electronic matrix is given by

    \[\text{ Molecular Species and Charge} \to \ce{ClO2-} \quad \ce{H3O+} \quad \ce{Cl2} \quad \ce{H2O} \quad \ce{ClO3-} \quad \ce{ClO2} \\ \begin{matrix} {chlorine} \\ { oxygen} \\ {hydrogen} \\ {charge} \end{matrix} \begin{bmatrix} { 1} & { 0} & {2} & {0} & { 1} & {1 } \\ { 2} & { 1} & {0} & {1} & { 3} & {2 } \\ { 0} & { 3} & {0} & {2} & { 0} & {0 } \\ {-1} & { +1} & {0} & {0} & { -1} & {0 } \end{bmatrix} \label{6}\]

    and use of this result with Equation \ref{4} leads to Axiom II & III:

    \[ \begin{bmatrix} { 1} & { 0} & {2} & {0} & { 1} & {1 } \\ { 2} & { 1} & {0} & {1} & { 3} & {2 } \\ { 0} & { 3} & {0} & {2} & { 0} & {0 } \\ {-1} & { +1} & {0} & {0} & { -1} & {0 } \end{bmatrix} \begin{bmatrix} R_{\ce{ClO2-}} \\ R_{\ce{H3O+}} \\ R_{\ce{Cl2}} \\ R_{\ce{H2O}} \\ R_{\ce{ClO3-}} \\ R_{\ce{ClO2}} \end{bmatrix} = \begin{bmatrix} {0} \\ {0} \\ {0} \\ {0} \end{bmatrix} \label{7}\]

    At this point we follow the developments given in Chapters 6 through 9 and search for the optimal form of the atomic/electronic matrix. We begin with

    \[ \mathbf{A_e} = \begin{bmatrix} { 1} & { 0} & {2} & {0} & { 1} & {1 } \\ { 2} & { 1} & {0} & {1} & { 3} & {2 } \\ { 0} & { 3} & {0} & {2} & { 0} & {0 } \\ {-1} & { +1} & {0} & {0} & { -1} & {0 } \end{bmatrix} \label{8}\]

    and apply a series of elementary row operations to find the row reduced echelon form given by

    \[ \mathbf{A^*_e} =\begin{bmatrix} { 1} & { 0} & {0} & {0} & { 5/3} & {4/3 } \\ { 0} & { 1} & {0} & {0} & { 2/3} & {4/3 } \\ { 0} & { 0} & {1} & {0} & { -1/3} & {-1/6 } \\ {0} & { 0} & {0} & {1} & { -1} & {-2 } \end{bmatrix} \label{9}\]

    Use of this result in Equation \ref{7} leads to

    \[ \begin{bmatrix} { 1} & { 0} & {0} & {0} & { 5/3} & {4/3 } \\ { 0} & { 1} & {0} & {0} & { 2/3} & {4/3 } \\ { 0} & { 0} & {1} & {0} & { -1/3} & {-1/6 } \\ {0} & { 0} & {0} & {1} & { -1} & {-2 } \end{bmatrix} \begin{bmatrix} R_{\ce{ClO2-}} \\ R_{\ce{H3O+}} \\ R_{\ce{Cl2}} \\ R_{\ce{H2O}} \\ R_{\ce{ClO3-}} \\ R_{\ce{ClO2}} \end{bmatrix} = \begin{bmatrix} {0} \\ {0} \\ {0} \\ {0} \end{bmatrix} \label{10}\]

    We now follow the type of analysis given in Sec. 6.4 and apply the obvious partition \(column/row\) to obtain

    \[ \underbrace{\begin{bmatrix} { 1} & { 0} & {0} & {0} \\ { 0} & { 1} & {0} & {0} \\ { 0} & { 0} & {1} & {0} \\ {0} & { 0} & {0} & {1} \end{bmatrix}}_{\text{non-pivot submatrix}} \begin{bmatrix} R_{\ce{ClO2-}} \\ R_{\ce{H3O+}} \\ R_{\ce{Cl2}} \\ R_{\ce{H2O}} \end{bmatrix} + \underbrace{\begin{bmatrix} { 5/3} & {4/3 } \\ { 2/3} & {4/3 } \\ { -1/3} & {-1/6 } \\ { -1} & {-2 } \end{bmatrix}}_{\text{pivot submatrix}} \begin{bmatrix} R_{\ce{ClO3-}} \\ R_{\ce{ClO2}} \end{bmatrix} = \begin{bmatrix} {0} \\ {0} \\ {0} \\ {0} \end{bmatrix} \label{11}\]

    Making use of the property of the identity matrix leads to

    \[\begin{bmatrix} { 1} & { 0} & {0} & {0} \\ { 0} & { 1} & {0} & {0} \\ { 0} & { 0} & {1} & {0} \\ {0} & { 0} & {0} & {1} \end{bmatrix} \begin{bmatrix} R_{\ce{ClO2-}} \\ R_{\ce{H3O+}} \\ R_{\ce{Cl2}} \\ R_{\ce{H2O}} \end{bmatrix} = \begin{bmatrix} R_{\ce{ClO2-}} \\ R_{\ce{H3O+}} \\ R_{\ce{Cl2}} \\ R_{\ce{H2O}} \end{bmatrix} \label{12}\]

    and substituting this result in Equation \ref{11} provides the desired result

    \[\underbrace{\begin{bmatrix} R_{\ce{ClO2-}} \\ R_{\ce{H3O+}} \\ R_{\ce{Cl2}} \\ R_{\ce{H2O}} \end{bmatrix}}_{\text{non-pivot species}} = \underbrace{ \begin{bmatrix} { -5/3} & {-4/3 } \\ { -2/3} & {-4/3 } \\ { 1/3} & {1/6 } \\ { 1} & {2 } \end{bmatrix}}_{ \text{pivot matrix}} \underbrace{ \begin{bmatrix} R_{\ce{ClO3-}} \\ R_{\ce{ClO2}} \end{bmatrix}}_{\text{pivot species}} \label{13}\]

    As discussed in Chapter 6, the choice of pivot and non-pivot species is not completely arbitrary. Thus one must arrange the atomic/electronic matrix in row reduced echelon form as illustrated in Equation \ref{9} in order to make use of the pivot theorem indicated by Equation \ref{13}.

    Use of Equation \ref{1} with Equation \ref{3} is a straightforward matter leading to Equation \ref{4}. Within the framework of Chapter 6, one can apply Equation \ref{4} in a routine manner in order to solve problems in which charged species are present. In Chapters 6 and 7 we dealt with problems in which net rates of production had to be measured experimentally, and Equation \ref{13} is an example of this type of situation. There we see that the net rates of production of the pivot species (\(\ce{CLO3-}\) and \(\ce{CLO2}\)) must be determined experimentally so that the pivot theorem can be used to determine the net rates of production of the non-pivot species (\(\ce{ClO2-}\), \(\ce{H3O+}\), \(\ce{Cl2}\) and \(\ce{H2O}\)).

    Mechanistic Matrix

    In our studies of reaction kinetics in Chapter 9 we made use of chemical reaction rate expressions so that all the net rates of production could be calculated in terms of a series of reference reaction rates. These reaction rates were developed on the basis of mass action kinetics and thus contained rate coefficients and the concentrations of the chemical species involved in the reaction. That development made use of elementary stoichiometry which we express as

    Elementary stoichiometry:

    \[\sum_{A = 1}^{A = N}N_{JA} R^k_{A} =0, \quad J = 1,2,...,T, \quad k = I,II,...,K \label{14}\]

    This result insures that atomic species are conserved in each elementary kinetic step, and Equation \ref{1} is satisfied by imposition of the condition (see Eqs. \((9.1.49)\) and \((9.1.50)-(9.1.53)\))

    \[\sum_{k = 1}^{k = K}R^k_{A} = R_{A} \quad k = I,II,...,K\label{15}\]

    When confronted with charged species (ions) in a study of reaction kinetics, one makes use of elementary conservation of charge as indicated by

    Elementary conservation of charge:

    \[\sum_{A = 1}^{A = N}N_{\mathbf{e}A} R^k_{A} =0 \quad k = I,II,...,K\label{16}\]

    Thus charge is conserved in each elementary step of a chemical kinetic schema, and total conservation of charge indicated by Equation \ref{3} is automatically achieved in the construction of a mechanistic matrix.


    1. Feynman, R.P., Leighton, R.B. and Sands, M. 1963, The Feynman Lectures on Physics, Vol. I, page 4-7, AddisonWesley Pub. Co., New York. ↩
    2. Porter, S.K. 1985, How should equation balancing be taught?, J. Chem. Education 62, 507-508 ↩
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