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9.1: Chemical Kinetics

  • Page ID
    47705
  • In order to predict the concentration changes that occur in reactors, we need to make use of Axiom I (see Eq. \((6.1.3)\)) and Axiom II (see Eq. \((6.2.8)\)) in addition to chemical reaction rate equations that allow us to express the net rates of production, \(R_A\), \(R_B\), etc., in terms of the concentrations, \(c_A\), \(c_B\), etc. The subject of chemical kinetics brings us in contact with the chemical kinetic schemata that are used to illustrate reaction mechanisms. To be useful these schemata must be translated to equations and we will illustrate how this is done in the following paragraphs.

    Hydrogen bromide reaction

    As an example of both stoichiometry and chemical kinetics, we consider the reaction of hydrogen with bromine to produce hydrogen bromide. One could assume 3 that the molecular species involved are \(\ce{H2}\), \(\ce{Br2}\) and HBr, and this idea is illustrated in Figure \(\PageIndex{1}\).

    9.1.PNG
    Figure \(\PageIndex{1}\): Production of hydrogen bromide

    There we have suggested that the reaction does not go to completion since both hydrogen and bromine appear in the product stream. Here it is important to note that the products of a chemical reaction are determined by experiment, and in this case experimental data are available indicating that hydrogen bromide can be produced by reacting hydrogen and bromine. For the process illustrated in Figure \(\PageIndex{1}\), the visual representation of the atomic matrix takes the form

    \[\text{ Molecular species } \to \ce{H} \quad \ce{Br2} \quad \ce{HBr} \\ \begin{matrix} {hydrogen} \\ {bromine} \end{matrix} \begin{bmatrix} { 2} & { 0} & { 1} \\ { 0} & { 2} & { 1} \end{bmatrix} \label{1}\]

    and the elements of this matrix can be expressed explicitly as

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

    The components of \(N_{JA}\) are used with Axiom II

    Axiom II

    \[\sum^{A=N}_{A=1} N_{JA}R_A = 0, \quad J = 1,2,...,T \label{3}\]

    to develop the stoichiometric relations between the three net rates of production represented by \(R_{\ce{H2}}\), \(R_{\ce{Br2}}\), and \(R_{\ce{HBr}}\). For the atomic matrix given by Equation \ref{2} we see that Axiom II provides

    Axiom II:

    \[\begin{bmatrix} { 2} & { 0} & { 1} \\ { 0} & { 2} & { 1} \end{bmatrix} \begin{bmatrix} R_{\ce{H2}} \\ R_{\ce{Br2}} \\ R_{\ce{HBr}} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \label{4}\]

    and the use of the row reduced echelon form of the atomic matrix leads to

    \[\begin{bmatrix} { 1} & { 0} & { 1/2} \\ { 0} & { 1} & { 1/2} \end{bmatrix} \begin{bmatrix} R_{\ce{H2}} \\ R_{\ce{Br2}} \\ R_{\ce{HBr}} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \label{5}\]

    If hydrogen bromide (HBr) is chosen to be the single pivot species (see Sec. 6.4) we can express Axiom II in the form

    Pivot Theorem:

    \[\begin{bmatrix} R_{\ce{H2}} \\ R_{\ce{Br2}} \end{bmatrix} = \underbrace{\begin{bmatrix} -1/2 \\ -1/2 \end{bmatrix}}_{\text{pivot matrix}} \begin{bmatrix} R_{\ce{HBr}} \end{bmatrix} \label{6}\]

    This matrix equation provides the following representations for the net rates of production of hydrogen and bromine

    Local Stoichiometry:

    \[R_{\ce{H2}} = - \frac{1}{2} R_{\ce{HBr}}, \quad R_{\ce{Br2}} = -\frac{1}{2} R_{\ce{HBr}} \label{7}\]

    At this point we wish to apply Axioms I and II to the control volume illustrated in Figure \(\PageIndex{1}\). The macroscopic forms of the axioms are given by Eqs. \((7.1.4)\) and \((7.1.5)\) and repeated here as

    Axiom I:

    \[ \frac{d}{dt} \int_{\mathscr{V}} c_A dV + \int_{\mathscr{A}} c_A \mathbf{v}_A \cdot \mathbf{n} dA = \mathscr{R}_A, \quad A = 1,2,...,N \label{8}\]

    Axiom II

    \[\sum^{A=N}_{A=1} N_{JA} \mathscr{R}_A = 0, \quad J = 1,2,...,T \label{9}\]

    For the particular case under consideration, Equation \ref{8} leads to

    \[-Q \langle c_{A} \rangle_{entrace} + Q \langle c_{A} \rangle_{exit} = \mathscr{R}_A, \quad A \Rightarrow \quad \ce{H2}, \quad \ce{Br2}, \quad \ce{HBr} \label{10}\]

    while Equation \ref{9} takes the special form given by

    Global Stoichiometry:

    \[\mathscr{R}_{\ce{H2}} = - \frac{1}{2} \mathscr{R}_{\ce{HBr}}, \quad \mathscr{R}_{\ce{Br2}} = -\frac{1}{2} \mathscr{R}_{\ce{HBr}} \label{11}\]

    Since there is no hydrogen bromide in the inlet stream illustrated in Figure \(\PageIndex{1}\), the steady-state macroscopic balance provides the following result for hydrogen bromide

    \[Q \langle c_{\ce{HBr}} \rangle_{exit} = \mathscr{R}_{\ce{HBr}} \label{12}\]

    Here we see that measurements of the volumetric flow rate and the exit concentration of HBr provide an experimental determination of \(\mathscr{R}_{\ce{HBr}} \).

    The concepts of local and global stoichiometry are illustrated in Figure \(\PageIndex{2}\) where we suggest that hydrogen, H, and bromine, Br, may participate in the reaction at the local level, but may not be detectable at the macroscopic level.

    9.2.PNG
    Figure \(\PageIndex{2}\): Production of hydrogen bromide: Local and global stoichiometry

    What is not detectable at the macroscopic level is often neglected, and we will do so in this first exploration of the hydrogen bromide reaction. This is indicated by the global stoichiometry shown in Figure \(\PageIndex{2}\). At this point we assume that the reactor is perfectly mixed and this provides the simplification indicated by

    \[\underbrace{\langle c_{A} \rangle}_{\begin{array}{c} \text{volume average} \\ \text{concentration in} \\ \text{the reactor} \end{array}} = \underbrace{\langle c_{A} \rangle_{exit}}_{\begin{array}{c} \text{area average} \\ \text{concentration} \\ \text{in the exit} \end{array}} = \underbrace{c_A}_{\begin{array}{c} \text{concentration} \\ \text{at a point in} \\ \text{the reactor} \end{array}} , \quad A = 1,2,...,N \label{13}\]

    For the specific system illustrated in Figure \(\PageIndex{2}\) the assumption of perfect mixing leads to

    \[R_A = \mathscr{R}_A/\mathscr{V}, \quad A \Rightarrow \quad \ce{H2}, \quad \ce{Br2}, \quad \ce{HBr} \label{14a}\]

    \[\langle c_{A} \rangle = \langle c_{A} \rangle_{exit}, \quad A \Rightarrow \quad \ce{H2}, \quad \ce{Br2}, \quad \ce{HBr} \label{14b}\]

    Given these simplifications we can discuss the process illustrated in Figure \(\PageIndex{1}\) in terms of local conditions for which the chemical kinetics may be illustrated by a schema of the form

    Local chemical kinetic schema:

    \[\ce{H2}+\ce{Br2} \stackrel{k}{\longrightarrow} \ce{2HBr} \label{15}\]

    This schema suggests that that a molecule of hydrogen collides with a molecule of bromine to produce two molecules of hydrogen bromide as illustrated in Figure \(\PageIndex{3}\).

    9.3.PNG
    Figure \(\PageIndex{3}\): Molecular collision leading to a chemical reaction

    The frequency of the collisions that cause the reaction depends on the product of the two concentrations, \(c_{\ce{H2}}\) and \(c_{\ce{Br2}}\), and this leads to the local chemical reaction rate equations given by

    Local chemical reaction rate equations:

    \[R_{\ce{H2}} = -k c_{\ce{H2}}c_{\ce{Br2}} , \quad R_{\ce{Br2}} = -k c_{\ce{H2}} c_{\ce{Br2}} , \label{16}\]

    Chemical kinetic schemata are traditionally represented in local form, as indicated in Equation \ref{15}, even when they are based on macroscopic observations as we have suggested in Figures \(\PageIndex{1}\) and \(\PageIndex{2}\). If we make use of the chemical reaction rate equations given by Eqs. \ref{16} and the stoichiometric equations given by Eqs. \ref{7}, the local chemical reaction rate equation for the production of hydrogen bromide takes the form

    Local chemical reaction rate equation:

    \[R_{\ce{HBr}} = 2k c_{\ce{H2}}c_{\ce{Br2}} \label{17}\]

    This rate equation is based on the concept of mass action kinetics which, in turn, is based on the picture illustrated by Equation \ref{15} or the picture illustrated by Figure \(\PageIndex{3}\). The words associated with Equation \ref{16} and with Equation \ref{17} depend on what aspect of the equations we wish to emphasize. In this text we attempt to use a consistent set of phrases indicated by

    \[\underbrace{\underbrace{R_{\ce{HBr}}}_{\text{net rate of production}} = \underbrace{2 k c_{\ce{H2}} c_{\ce{Br2}}}_{\text{chemical reaction rate}}}_{\text{chemical reaction rate equation}} \label{18}\]

    In general equations are unambiguous while verbal descriptions can sometimes be misleading. When in doubt, study the equations.

    Experimental studies of the reaction of hydrogen and bromine to form hydrogen bromide were carried out by Bodenstein and Lind4 in a well-mixed batch reactor, and those experiments indicate that the net rate of production of hydrogen bromide can be expressed as

    Experimental:

    \[R_{\ce{HBr}} = \frac{kc_{\ce{H2}} \sqrt{c_{\ce{Br2}}}}{1+k^{\prime} \left( c_{\ce{HBr}}/c_{\ce{Br2}}\right)} \label{19}\]

    This experimental result is certainly not consistent with the chemical reaction rate equation given by Equation \ref{17}, thus the picture represented by Equation \ref{15} is not consistent with the kinetics of the real physical process. Clearly we need a new picture of the reaction of hydrogen with bromine to form hydrogen bromide and that new picture is considered in Section 9.3.

    Decomposition of azomethane

    As another example of an apparently simple reaction, we consider the gas-phase decomposition of azomethane [\(\ce{(CH3)2N2}\)] to produce ethane (\(\ce{C2H6}\)) and nitrogen (\(\ce{N2}\)). This reaction is illustrated in Figure \(\PageIndex{4}\) where we have indicated that azomethane appears in both the input and the output streams.

    9.4.PNG
    Figure \(\PageIndex{4}\): Decomposition of azomethane

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

    \[\text{ Molecular species } \to \ce{C2H6} \quad \ce{N2} \quad \ce{(CH3)2N2} \\ \begin{matrix} {carbon} \\ {nitrogen} \\ {hydrogen} \end{matrix} \begin{bmatrix} { 2} & { 0} & { 2} \\ { 0} & { 2} & { 2} \\ 6 & 0 & 6 \end{bmatrix} \label{20}\]

    and use of this representation with Axiom II provides

    Axiom II:

    \[\begin{bmatrix} { 2} & { 0} & { 2} \\ { 0} & { 2} & { 2} \\ 6 & 0 & 6 \end{bmatrix} \begin{bmatrix} R_{\ce{C2H6}} \\ R_{\ce{N2}} \\ R_{\ce{(CH3)2N2}} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \label{21}\]

    This can be expressed in terms of the row reduced echelon form of the atomic matrix to obtain

    \[\begin{bmatrix} { 1} & { 0} & { 1} \\ { 0} & { 1} & { 1} \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} R_{\ce{C2H6}} \\ R_{\ce{N2}} \\ R_{\ce{(CH3)2N2}} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \label{22}\]

    and a row-row partition of this matrix leads to

    \[\begin{bmatrix} { 1} & { 0} & { 1} \\ { 0} & { 1} & { 1} \end{bmatrix} \begin{bmatrix} R_{\ce{C2H6}} \\ R_{\ce{N2}} \\ R_{\ce{(CH3)2N2}} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \label{23}\]

    Use of the pivot theorem (see Sec. 6.4) allows us to express the net rates of production for ethane and nitrogen in terms of azomethane according to

    \[\begin{bmatrix} R_{\ce{C2H6}} \\ R_{\ce{N2}} \end{bmatrix} = \underbrace{\begin{bmatrix} -1 \\ -1 \end{bmatrix}}_{\text{pivot matrix}} \begin{bmatrix} R_{\ce{(CH3)2N2}} \end{bmatrix} \label{24}\]

    and this result leads to the local stoichiometric relations given by

    Local Stoichiometry:

    \[R_{\ce{C2H6}} = - R_{\ce{(CH3)2N2}}, \quad R_{\ce{N2}} = - R_{\ce{(CH3)2N2}} \label{25}\]

    The result for global stoichiometry is based on Equation \ref{9} and it obviously leads to

    Global Stoichiometry:

    \[\mathscr{R}_{\ce{C2H6}} = - \mathscr{R}_{\ce{(CH3)2N2}}, \quad \mathscr{R}_{\ce{N2}} = - \mathscr{R}_{\ce{(CH3)2N2}} \label{26}\]

    The relation between local and global stoichiometry is illustrated in Figure \(\PageIndex{5}\). The fact that these two relations are identical in form is based on the assumption that only azomethane, ethane, and nitrogen are present at both the local level and the macroscopic level. At this point we accept Eqs. \ref{25} and \ref{26} as being valid; however, we note that the principle of stoichiometric skepticism discussed in Sec. 6.1.1 should always be kept in mind. As we did in the case of the hydrogen bromide reaction, we begin with the simplest possible chemical kinetic schema given by

    Local chemical kinetic schema:

    \[\left(\ce{CH3}\right)_{2} \ce{N2} \stackrel{k}{\longrightarrow} \ce{C2H6}+\ce{N2} \label{27}\]

    This schema suggests that a molecule of azomethane spontaneously decomposes into a molecule of ethane and a molecule of nitrogen, and this decomposition is illustrated in Figure \(\PageIndex{6}\).

    9.5.PNG
    Figure \(\PageIndex{5}\): Local and global stoichiometry for the decomposition of azomethane

    On the basis of the chemical kinetic schema indicated by Equation \ref{27} and illustrated in Figure \(\PageIndex{6}\), the local rate equation for the production of ethane takes the form

    Local chemical reaction rate equation:

    \[R_{\ce{C2H6}} = k c_{\ce{(CH3)2N2}} \label{28}\]

    Here the rate constant, \(k\), is a parameter to be determined by experiment and should not be confused with the rate constant that appears in Equation \ref{17} for the production of hydrogen bromide.

    9.6.PNG
    Figure \(\PageIndex{6}\): Spontaneous decomposition to leading to products

    This result is not in agreement with experimental observations5 which indicate that the reaction is first order with respect to azomethane at high concentrations and second order at low concentrations. The experimental observations can be expressed as

    Experimental:

    \[R_{\ce{C2H6}} = \frac{k[c_{\ce{(CH3)2N2}}]^2}{1+k^{\prime} c_{\ce{(CH3)2N2}}} \label{29}\]

    in which \(k\) and \(k^{\prime}\) are not to be confused with the analogous coefficients in Equation \ref{19}.

    The experimental results represented by Equation \ref{19} and Equation \ref{29} indicate that both reaction processes are more complex than suggested by Figure \(\PageIndex{3}\) and Figure \(\PageIndex{6}\). The fundamental difficulty results from the fact that global observations cannot necessarily be used to correctly infer local processes, and we need to explore the local processes more carefully if we are to correctly predict the forms given by Equation \ref{19} and Equation \ref{29}. In order to do so, we need to examine mass action kinetics in more detail and this is done in subsequent paragraphs.

    Local and elementary stoichiometry

    The concept of local stoichiometry was introduced in Chapter 6, identified above by Equation \ref{3} and repeated here as

    Axiom II:

    \[\sum^{A=N}_{A=1} N_{JA}R_A = 0, \quad J = 1,2,...,T \label{30}\]

    If we consider a set of \(K\) elementary reactions involving the species indicated by \(A = 1,2,...,N \), we encounter a set of net rates of production that are designated by \(R^I_A, R^{II}_A, R^{III}_A, …, R^K_A\). Associated with each elementary reaction is a condition of elementary stoichiometry that we express as

    Elementary Stoichiometry:

    \[\sum^{A=N}_{A=1} N_{JA}R^k_A = 0, \quad J = 1,2,...,T, \quad k = I,II,...,K \label{31}\]

    The sum of the \(K\) elementary net rates of production for species \(A\) is the total net rate of production for species \(A\) indicated by

    \[\sum^{k=K}_{k=I} R^k_A = R_A \label{32}\]

    Since \(N_{JA}\) is independent of \(k = I, II,....,K \), we can sum Equation \ref{31} over all \(K\) reactions and interchange the order of summation to recover the local stoichiometric condition given by

    Local Stoichiometry:

    \[\sum^{A=N}_{A=1} N_{JA} \sum^{k=K}_{k=I} R^k_A = \sum^{A=N}_{A=1} N_{JA} R_A = 0, \quad J = 1,2,...,T \label{33}\]

    Clearly when there is a single elementary reaction, the elementary stoichiometry is identical to the local stoichiometry.

    Mass action kinetics and elementary stoichiometry

    In this section we want to summarize the concept of mass action kinetics and indicate how it is connected to elementary stoichiometry. As an example we consider a system in which there are four participating molecular species indicated by \(A\), \(B\), \(C\), and \(D\). The chemical kinetic schema for one possible reaction associated with these molecular species is indicated by6

    Elementary chemical kinetic schema I:

    \[ \alpha A+\beta B \stackrel{k_{I}}{\longrightarrow} \gamma C+\delta D\label{34}\]

    At this point we need to translate this picture to an equation associated with mass action kinetics and then explore what can be extracted from this picture in terms of elementary stoichiometry. According to the rules of mass action kinetics, the chemical kinetic translation of Equation \ref{34} is given by

    Elementary chemical reaction rate equation I:

    \[R^I_A = -k_I c^{\alpha}_A c^{\beta}_B \label{35}\]

    Here we have used the first species in the chemical kinetic schema as the basis for the proposed rate equation, and this represents a reasonable convention but not a necessary one. One should remember that binary collisions dominate chemical reactions and that ternary collisions are rare. This means that we expect the sum of the integers \(\alpha\) and \(\beta\) to be less than or equal to two. Often there is a second reaction involving species \(A\), \(B\), \(C\), and \(D\), and we express the second chemical kinetic schema as

    Elementary chemical kinetic schema II:

    \[\varepsilon B+\eta C \stackrel{k_{II}}{\longrightarrow} \xi D \label{36}\]

    This second chemical kinetic schema leads to a chemical reaction rate equation of the form

    Elementary chemical reaction rate equation II:

    \[R^{II}_B = -k_{II} c^{\varepsilon}_B c^{\eta}_C \label{37}\]

    In general we are interested in the net rate of production which is given by the sum of the elementary rates of production according to

    \[R_{A}=R_{A}^{I}+R_{A}^{II}, \quad R_{B}=R_{B}^{I}+R_{B}^{II}, \quad R_{C}=R_{C}^{I}+R_{C}^{II}, \quad R_{D}=R_{D}^{I}+R_{D}^{II} \label{38}\]

    At this point we need stoichiometric information to develop useful chemical reaction rate equations. Since stoichiometry is associated with the conservation of atomic species, we need to be very careful when using a representation in which there are no identifiable atomic species. The translation associated with kinetic schemata and elementary stoichiometry must be consistent with Axiom II. In terms of stoichiometry, we identify the meaning of Equation \ref{34} as follows:

    \(\alpha\) moles of species \(A\) react with \(\beta\) moles of species \(B\) to form \(\gamma\) moles of species \(C\) and \(\delta\) moles of species \(D\).

    To make things very clear, we consider the highly unlikely prospect that 8 moles of species \(A\) react with 3 moles of species \(B\). This would lead to the condition

    \[\frac{R^I_A}{8} = \frac{R^I_B}{3} \label{39}\]

    and a little thought will indicate that the general stoichiometric translation of Equation \ref{34} is given by

    Elementary stoichiometry I:

    \[\frac{R^I_A}{\alpha} = \frac{R^I_B}{\beta}, \quad \frac{R^I_A}{\alpha} = - \frac{R^I_C}{\gamma} , \quad \frac{R^I_A}{\alpha} = -\frac{R^I_D}{\delta} \label{40}\]

    This result is based on the assumption that species \(A\), \(B\), \(C\), and \(D\) and are all unique species. For example, if species \(C\) is actually identical to species \(A\) the second of Eqs. \ref{40} takes the form

    Unacceptable stoichiometry:

    \[\frac{R^I_A}{\alpha} = -\frac{R^I_A}{\gamma} \label{41}\]

    In this special case, it should be clear that Equation \ref{34} cannot be used as a picture of the stoichiometry. If species \(B\), \(C\), and \(D\) are all unique species, we can follow the same thought process that led to Equation \ref{40} to conclude that the stoichiometric translation of Equation \ref{36} is given by

    Elementary stoichiometry II:

    \[\frac{R^{II}_B}{\varepsilon} = \frac{R^{II}_C}{\eta}, \quad \frac{R^{II}_B}{\varepsilon} = - \frac{R^{II}_D}{\xi} , \quad R^{II}_A = 0 \label{42}\]

    Throughout our study of stoichiometry in Chapter 6 we used representations such as \(\ce{C2H5OH}\) and \(\ce{CH3OC2H3}\) to identify the atomic structure of various molecular species, and with those representations it was easy to keep track of atomic species. The representations given by Equation \ref{34} and Equation \ref{36} are less informative, and we need to proceed with greater care when the atomic structure is not given explicitly.

    With the elementary stoichiometry now available in terms of Eqs. \ref{40} and \ref{42}, we can develop the local chemical reaction rate equations for species \(A\) and \(B\) on the basis of Eqs. \ref{35} and \ref{37}. This leads to

    \[ R_{A}=-k_{I} c_{A}^{\alpha} c_{B}^{\beta}, \quad R_{B}=-(\beta / \alpha) k_{I} c_{A}^{\alpha} c_{B}^{\beta}-k_{I I} c_{B}^{\varepsilon} c_{C}^{\eta} \label{43}\]

    and the rate equations for the other species can be constructed in the same manner.

    Decomposition of azomethane and reactive intermediates

    We are now ready to return to the decomposition of azomethane to produce ethane and nitrogen. The rate equation given by Equation \ref{29} is based on the work of Ramsperger7 and an explanation of that rate equation requires the existence of reactive intermediates 8, 9 or Bodenstein products 10. Most chemical reactions involve reactive intermediate species, and this idea is illustrated in Figure \(\PageIndex{7}\) where we have indicated the existence of an activated form of azomethane identified as \(\ce{(CH3)2N2}*\). This form exists in such small concentrations that it is difficult to detect in the exit stream and thus does not appear in the representation of the global stoichiometry. A key idea here is that the expression for a chemical reaction rate is based on experiments, and when a specific chemical species cannot be detected experimentally it often does not appear in the first effort to construct a chemical reaction rate expression.

    9.7.PNG
    Figure \(\PageIndex{7}\): Local and global stoichiometry for decomposition of azomethane

    For simplicity we represent the molecular species suggested by Figure \(\PageIndex{7}\) as

    \[A = \ce{(CH3)2N2}, \quad B = \ce{C2H6}, \quad C = \ce{N2}, \quad A* = \ce{(CH3)2N2}* \label{44}\]

    in which \(A*\) represents the activated form of azomethane or the so-called reactive intermediate. On the basis of the analysis of Lindemann11 we explore the following set of elementary chemical kinetic schemata:

    Elementary chemical kinetic schema I:

    \[ 2A \stackrel{k_{I}}{\longrightarrow} A + A* \label{45a}\]

    Elementary chemical kinetic schema II:

    \[ A* \stackrel{k_{II}}{\longrightarrow} B+C \label{45b}\]

    Elementary chemical kinetic schema III:

    \[ A* + A \stackrel{k_{III}}{\longrightarrow} 2A \label{45c}\]

    The schema represented by Equation \ref{45a} is illustrated in Figure \(\PageIndex{8}\) where we see that a collision between two molecules of azomethane leads to the creation of the reactive intermediate denoted by \(\ce{(CH3)2N2}*\).

    9.8.PNG
    Figure \(\PageIndex{8}\): Creation of a reactive intermediate for the decomposition of azomethane

    Equation \ref{45a} represents an example of the situation illustrated by Eqs. \ref{40} and \ref{41}, and one must be careful in terms of the stoichiometric interpretation. In this case we draw upon Figure \(\PageIndex{8}\) to conclude that the stoichiometric schema associated with Equation \ref{45a} is the activation of an azomethane molecule that we represent in the form

    Stoichiometric schema:

    \[\ce{(CH3)2N2} \rightarrow \ce{(CH3)2N2}* \quad \text{ or } \quad A \rightarrow A* \label{46}\]

    Given this stoichiometric schema for the first elementary step, we see that Equation \ref{45a} leads to the following four representations:

    Elementary stoichiometric schema I:

    \[A \rightarrow A* \label{47a}\]

    Elementary chemical kinetic schema I:

    \[ 2A \stackrel{k_{I}}{\longrightarrow} A + A* \label{47b}\]

    Elementary stoichiometry I:

    \[R^I_A = - R^I_{A*} \label{47c}\]

    Elementary chemical reaction rate equation I:

    \[R^I_A = - k_I c^2_A \label{47d}\]

    The second elementary step involves the decomposition of the activated molecule to form ethane and nitrogen according to:

    Elementary stoichiometric schema II:

    \[A* \rightarrow B + C \label{48a}\]

    Elementary chemical kinetic schema II:

    \[A * \stackrel{k_{II}}{\longrightarrow} B + C \label{48b}\]

    Elementary stoichiometry II:

    \[R^{II}_{A*} = - R^{II}_B, \quad R^{II}_A* = -R^{II}_C \label{48c}\]

    Elementary chemical reaction rate equation II:

    \[R^{II}_{A*} = -k_{II}c_{A*} \label{48d}\]

    The final elementary step consists of the recombination of an activated molecule with azomethane to form two molecules of azomethane. This final step is described by the following representations:

    Elementary stoichiometric schema III:

    \[A* \rightarrow A \label{49a}\]

    Elementary chemical kinetic schema III:

    \[A*+A \stackrel{k_{III}}{\longrightarrow} 2A \label{49b}\]

    Elementary stoichiometry III:

    \[R^{III}_{A*} = -k_{III}c_Ac_{A*} \label{49c}\]

    Elementary chemical reaction rate equation III:

    \[R^{III}_{A*} = -k_{III} c_A c_{A*} \label{49d}\]

    According to Equation \ref{32} the local net rates of production are given by

    \[R_A = R^I_A + R^{II}_A + R^{III}_A \label{50a}\]

    \[R_{A* }= R^I_{A* } + R^{II}_{A* } + R^{III}_{A* } \label{50b}\]

    \[R_B = R^I_B + R^{II}_B + R^{III}_B \label{50c}\]

    We now have a complete description of the reaction process for the schemata represented by Eqs. \ref{45a} - \ref{45c}, and from these results we would like to extract a representation for \(R_B\) in terms of \(c_A\). The classic simplification of this algebraic problem is to assume that the net rate of production of the reactive intermediate or the Bodenstein product can be approximated by

    Local Reaction Equilibrium:

    \[R_{A*} = 0 \label{51}\]

    This simplification is often referred to as the steady-state assumption or the steady state hypothesis or the pseudo steady state hypothesis. These are appropriate phrases when kinetic mechanisms are being studied by means of a batch reactor; however, the phrase local reaction equilibrium is preferred since it is not process-dependent. Use of Equation \ref{51} with Equation \ref{50b} leads to

    \[ R^I_{A* } + R^{II}_{A* } + R^{III}_{A* } = k_Ic^2_A - k_{II}c_{A*} - k_{III} c_A c_{A*} = 0 \label{52}\]

    and from this we determine the concentration of the reactive intermediate to be

    \[c_{A*} = \frac{k_1c^2_A}{k_{II} + k_{III} c_A} \label{53}\]

    We now make use of Equation \ref{50c} to express the net rate of production of ethane as

    \[R_B = R^I_B + R^{II}_B + R^{III}_B = R_{\ce{C2H6}} \label{54}\]

    and application of Eqs. \ref{48c} and \ref{48d} provides the chemical reaction rate equation given by

    \[R_{\ce{C2H6}} = k_{II} c_{A*} \label{55}\]

    At this point we use Equation \ref{53} in order to express the net rate of production of ethane as

    \[R_{\ce{C2H6}} = \frac{k_Ik_{II}c^2_A}{k_{II}+k_{III}c_A} \label{56}\]

    in which \(c_A\) represents the concentration of azomethane, \(\ce{(CH3)2 N2}\). Here we can see that the two limiting rate expressions for high and low concentrations are given by

    \[R_{\ce{C2H6}} = \frac{k_Ik_{II}c^2_A}{k_{II}+k_{III}c_A} = \begin{cases} (k_Ik_{II}/k_{III})c_A, && \text{ high concentration} \\ k_I c^2_A, && \text{ low concentration} \end{cases} \label{57}\]

    which is consistent with the experimental results of Ramsperger7 illustrated by Equation \ref{29}. We can be more precise about what is meant by high concentration and low concentration by expressing these ideas as

    \[\begin{align} c_A >> k_{II}/k_{III}, && \text{ high concentration} \label{58}\\ c_A << k_{II}/k_{III}, && \text{ low concentration} \nonumber \end{align}\]

    Here we see that the relatively simple process suggested by Equation \ref{27} is governed by the relatively complex rate equation indicated by Equation \ref{56}. The analysis leading to this result is based on three concepts:

    (A) local and elementary stoichiometry,

    (B) mass action kinetics, and

    (C) the approximation of local reaction equilibrium.

    The simplifying assumptions associated with this development are discussed in the following paragraphs.

    Assumptions and Consequences

    A reasonable assumption concerning the continuous stirred tank reactor shown in Figure \(\PageIndex{5}\) is that only azomethane, ethane and nitrogen participate in the reaction. This assumption, in turn, leads to the chemical kinetic schema illustrated both in Equation \ref{27} and in Figure \(\PageIndex{6}\). Experimental measurement of the concentrations in the inlet and outlet streams might confirm the assumption that only \(\ce{(CH3)2N2}\), \(\ce{C2H6}\) and \(\ce{N2}\) are present in the reactor. However, the experimental determination of the reaction rate is not in agreement with Equation \ref{28}. In reality, our analysis is based on the restriction that no significant amount of reactive intermediate enters or exits the reactor, and we state this idea as

    Restriction:

    \[c_{A*} << c_A,c_B,c_C, \begin{cases} \text{ at the entrance and} \\ \text{ exit of the reactor} \end{cases} \label{59}\]

    While the concentration of the reactive intermediate might be small compared to the other species, it is certainly not zero. If it were zero, Equation \ref{55} would indicate that the rate of production of ethane would be zero and that is not in agreement with experimental observation.

    Given that \(c_{A*}\) is not zero, one can wonder about the assumption (see Equation \ref{51}) that \(R_{A*}\) is zero. In reality, \(R_{A*}\) must be small enough so that it can be approximated by zero, and we need to know how small is small enough. To find out, we make use of Equation \ref{50b} to determine that the net rate of production of \(A*\) is given by

    \[R_{A*} = k_Ic^2_A - ( k_{II} + k_{III} c_A ) c_{A*} \label{60}\]

    and we use this result to show that the concentration of the reactive intermediate takes the form

    \[c_{A*} = \frac{k_I c^2_A}{(k_{II}+k_{III}c_A)} - \frac{R_{A*}}{(k_{II}+k_{III}c_A)} \label{61}\]

    Use of this result in Equation \ref{55} leads to the net rate of production of given by

    \[R_{\ce{C2H6}} = \frac{k_Ik_{II}c^2_A}{k_{II}+k_{III}c_A} - \frac{R_{A*}k_{II}}{(k_{II}+k_{III}c_A)} \label{62}\]

    Here we see that if the second term on the right hand side is negligible compared to the first term, we obtain the result given earlier by Equation \ref{56}. This indicates that the assumption given by is a reasonable substitute for the restriction given by \(R_{A*} = 0\)

    Restriction:

    \[R_{A*} << k_I c^2_A \label{63}\]

    When this inequality is imposed on Equation \ref{62} we obtain the result given previously by Equation \ref{56} provided that we are willing to assume that small causes produce small effects 12. Even though Eqs. \ref{51} and \ref{63} lead to the same result, Equation \ref{63} should serve as a reminder that neglecting something that is small always requires the crucial assumption that small causes produce small effects.

    One important part of this analysis is the fact that the assumption concerning \(c_{A*}\) at entrances an exits cannot be extended into the reactor where finite values of the concentration of the reactive intermediate control the rate of reaction. This is clearly indicated by Equation \ref{55}. The situation we have encountered in this study occurs often in the transport and reaction of chemical species and can generalized as:

    Sometimes a small quantity, such as \(R_{A*}\) or \(c_{A*}\), can be ignored and set equal to zero for the purposes of analysis. Sometimes a small quantity cannot be ignored and setting it equal to zero represents a serious mistake.

    Knowing when small causes produce small effects requires experience, intuition, experiment and analysis. These are skills that are acquired steadily over time.

    In this section we have examined the concepts of global, local, and elementary stoichiometry, along with the concept of mass action kinetics. We have made use of pictures to describe both elementary stoichiometry and elementary chemical kinetics, and we have illustrated how these pictures are related to equations. The concept of local reaction equilibrium, also known as the steady-state assumption or the steady state hypothesis or the pseudo steady-state hypothesis, has been applied in order to develop a simplified rate expression for the production of ethane and nitrogen from azomethane. The resulting rate expression compares favorably with experimental observations.

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