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6.01: Chemical Reactions

  • Page ID
    18258
  • 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

    \[\sum_{A=1}^{A=N} r_{A}=0 \label{6.2}\]

    This leads to the total mass balance given by

    \[\frac{d}{d t} \int_{V} \rho d V+\int_{A} \rho \mathbf{v} \cdot \mathbf{n} d A=0 \label{6.3}\]

    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 II

    \[\frac{d}{d t} \int_{V} c_{A} d V+\int_{A} c_{A} \mathbf{v}_{A} \cdot \mathbf{n} d A=\int_{V} R_{A} d V \label{6.4}\]

    with \(A=1,2, \ldots, N\)

    Here \(R_A\)  is the net molar rate of production of species \(A\) per unit volume owing to chemical reactions. This is related to \(r_A\) by

    \[R_{A}=r_{A} / M W_{A} \label{6.5}\]

    and Equation \ref{6.2} provides a constraint on the net molar rates of production given by

    Axiom II

    \[\sum_{A=1}^{A=N} M W_{A} R_{A}=0\label{6.6}\]

    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 and R 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{6.4} is preferred over Equation 6.1 for two reasons. To begin with, chemical reaction rates are traditionally expressed in terms of molar concentrations, c , c , etc., and one AB 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

    \[R_{A}=-\frac{k c_{A}^{2}}{1+k^{\prime} c_{A}} \label{6.7}\]

    , irreversible decomposition

    where \(k\)  and \(k^{\prime}\) are coefficients to be determined by experiment. In this case the negative sign indicates that species A is being consumed by the chemical reaction. One can use Equation \ref{6.7} along with Equation \ref{6.4} to predict the behavior of a system, i.e., to design a system. Chemical reaction rate equations such as Equation \ref{6.7} are considered in Chapter 9.

    In the absence of specific chemical kinetic information about R , one can only use Equation \ref{6.4} 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{6.4} is preferred over Equation \ref{6.1} is that the net molar rates of production of the various species are easily related to the atomic structure of the molecules involved in the chemical reactions. These relations can be constructed in terms of stoichiometric coefficients and as an example we consider the special case illustrated in Figure 6.1. Here we have suggested that ethane

    clipboard_e445c63f18b1057b0cc8960f62056286c.png

    Figure 6.1. Combustion reaction

    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

    \[\ce{ 1/2 C2H6 + 7/4 O2 -> 3/2 H2O + CO2} \label{6.8}\]

    and we call this a stoichiometric schema. In general, this stoichiometric schema has no connection with the actual kinetics of the reaction, thus Equation 6.8 does not mean that 1 2 a molecule of C H collides with 7 4 of a molecule of O to create 3 2

    2

    6

    2

    of a molecule of H O and one molecule of CO . The actual molecular processes 2

    2

    involved in the oxidation of ethane are far more complicated than is suggested by Equation \ref{6.8}, and an introduction to these processes is given in Chapter 9. The coefficients in Equation \ref{6.8} are often deduced by counting atoms, and this process is based on the idea that