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6.1: Chemical Reactions

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    44458
  • In the presence of chemical reactions, the total mass balance is obtained directly from Equation \((6.1)\) by summing that result over all \(N\) species and imposing the second axiom

    Axiom II: \[\sum_{A = 1}^{A = N}r_{A} =0 \label{2}\]

    This leads to the total mass balance given by

    \[\frac{d}{dt} \int_{\mathscr{V}}\rho dV + \int_{\mathscr{A}}\rho \mathbf{v}\cdot \mathbf{n} dA=0 \label{3}\]

    For problems involving a gas phase and the use of an equation of state (like the ideal gas law), the molar form of Equation \((6.1)\) is more convenient and can be written as

    Axiom I: \[\frac{d}{dt} \int_{\mathscr{V}}c_{A } dV + \int_{\mathscr{A}}c_{A } \mathbf{v}_{A} \cdot \mathbf{n} dA=\int_{\mathscr{V}}R_{A} dV , \quad A = 1, 2,...,N \label{4}\]

    where \(R_{A}\) is the net molar rate of production of species \(A\) per unit volume owing to chemical reactions. This is related to \(r_{A}\) by

    \[R_{A} ={r_{A} / MW_{A} } \label{5}\]

    and Equation \ref{2} provides a constraint on the net molar rates of production given by

    Axiom II: \[\sum_{A = 1}^{A = N}MW_{A} R_{A} =0 \label{6}\]

    Here we note that \(MW_A\) represents the molecular mass of species \(A\) and that we have chosen a nomenclature based on the traditional phrase, molecular weight. It is important to remember that \(r_{A}\) and \(R_{A}\) represent both the creation of species \(A\) (when \(r_{A}\) and \(R_{A}\) are positive) and the consumption of species \(A\) (when \(r_{A}\) and \(R_{A}\) are negative). For systems involving chemical reactions, Equation \ref{4} is preferred over Equation \((6.1)\) for two reasons. To begin with, chemical kinetic rate expressions are traditionally given in terms of molar concentrations, \(c_{A}\), \(c_{B}\), etc., and one needs to determine how \(R_{A}\) is related to these molar concentrations. For example, if species \(A\) is undergoing an irreversible decomposition, the net molar rate of production might be expressed as

    \[R_{A} =-\frac{k c_{A}^{2} }{1+k^{\prime}c_{A} } , \quad \text{irreversible decomposition} \label{7}\]

    where \(k\) and \(k^{\prime}\) are coefficients to be determined by experiment or by quantum mechanical calculations. In this case the negative sign indicates that species \(A\) is being consumed by the chemical reaction. One can use Equation \ref{7} along with Equation \ref{4} to predict the behavior of a system, i.e., to design a system. Chemical reaction rate equations such as Equation \ref{7} are considered in Chapter 9.

    In the absence of specific chemical kinetic information about \(R_{A}\), one can only use Equation \ref{4} to analyze a system in terms of the flow rates, global rates of production, and composition of the streams entering and leaving the system. The second reason that Equation \ref{4} is preferred over Equation \((6.1)\) is that the net molar rates of production of the various species are related to the atomic structure of the molecules involved in the reactions. These relations can be constructed in terms of stoichiometric coefficients, and as an example we consider the special case illustrated in Figure \(\PageIndex{1}\).

    image
    Figure \(\PageIndex{1}\): Combustion reaction

    Here we have suggested that ethane reacts with oxygen to form carbon dioxide and water, a process that is often referred to as complete combustion. The stoichiometry of this process can be visualized as

    \[{\frac{1}{2}} \ce{C2H6} + {\frac{ 7}{ 4}} \ce{O2} \to {\frac{3}{2}} \ce{H2O} + \ce{CO2} \label{8}\]

    and we call this a stoichiometric schema 2. In general, this stoichiometric schema has no connection with the actual kinetics of the reaction, thus Equation \ref{8} does not mean that \({1 / 2}\) a molecule of \(\ce{C2H6}\) collides with \({7 / 4}\) of a molecule of \(\ce{O2}\) to create \({3 / 2}\) of a molecule of \(\ce{H2O}\) and one molecule of \(\ce{CO2}\). The actual molecular processes involved in the oxidation of ethane are far more complicated than is suggested by Equation \ref{8}, and an introduction to these processes is given in Sec. 8.6. The coefficients in Equation \ref{8} are often deduced by counting atoms, and this process is based on the idea that

    Axiom II: \[\left\{\begin{array}{c} \text{atomic species are} \\ \text{neither created nor} \\ \text{destroyed by } \\ \text{chemical reactions} \end{array}\right\} \label{9}\]

    If the process illustrated in Figure \(\PageIndex{1}\) is carried out with a stoichiometric mixture of ethane and oxygen, one might find that the product stream contains mostly \(\ce{CO2}\) and \(\ce{H2O}\), but one might also find small amounts of \(\ce{ CO}\), \( \ce{CH3} \ce{OH}\), \(\ce{C2H4}\), etc. and it is not always obvious that these small amounts can be ignored. In fact, we believe that these small amounts should be a matter of constant concern.

    When a system of the type represented by Figure \(\PageIndex{1}\) is encountered, it is appropriate to immediately consider the alternative illustrated in Figure \(\PageIndex{2}\).

    image
    Figure \(\PageIndex{2}\): Incomplete combustion reaction

    It is always possible that the “other molecular species” suggested in Figure \(\PageIndex{2}\) may be present in small enough amounts so that Equation \ref{8} is a satisfactory approximation; however, what is meant by small enough may be difficult to determine since the “other molecular species” may consist of biocides or carcinogens or other species that could be damaging to the environment even in small amounts. Under certain circumstances, small amounts may produce major consequences, and we want students to react to any proposed stoichiometric schema in the manner indicated by Figure \(\PageIndex{2}\). This idea will be especially important in terms of our studies of reaction kinetics in Chapter 9 where small amounts of reactive intermediates or Bodenstein products actually control the macroscopic process suggested in Figure \(\PageIndex{2}\).

    One can indeed postulate that ethane and oxygen will react to produce carbon dioxide and water, but the postulate needs to be verified by experiment. For example, the process illustrated in Figure \(\PageIndex{2}\) might be carried out in such a manner that ethane is partially oxidized to produce ethylene oxide, carbon dioxide, carbon monoxide and water. Under these circumstances, the stoichiometry of the reaction might be represented by the following undetermined schema:

    \[? \ce{C2H6} + { ?} \ce{O2} \to { ?} \ce{CO} + { ?} \ce{C2H4O} + { ?} \ce{H2O} + \ce{CO2} \label{10}\]

    In this case the stoichiometric coefficients could be found by counting atoms to obtain

    \[2\ce{C2H6} + 4\ce{O2} \to \ce{CO} + \ce{C2H4O} + 4 \ce{H2O} + \ce{CO2} \label{11}\]

    and one could also count atoms to develop a different schema given by

    \[{ 2} \ce{C2H6} + {\frac{19}{4}} \ce{O2} \to 2 \ce{CO} + {\frac{1}{2}} \ce{C2H4O} + 5 \ce{H2O} + \ce{CO2} \label{12}\]

    Here it should be clear that we need more information to treat the case of partial oxidation of ethane, and to organize this additional information efficiently we need a precise mathematical representation of the concept that atomic species are neither created nor destroyed by chemical reactions. It is important to understand that Eqs. \ref{11} and \ref{12} are pictures of the concept that atoms are conserved, and what we need are equations describing the concept that atoms are conserved. In this text we use arrows to represent pictures and equal signs to represent equations.

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