# 4.1: Axioms for the Mass of Multicomponent Systems

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In Chapter 3, we studied the concept of conservation of mass for single-component systems, both for a *body* and for a *control volume*. The *words* associated with the control volume representation were

\[\left\{\begin{array}{l} \text{ time rate of change } \\ \text{ of mass in a } \\ \text{ control volume} \end{array}\right\}=\left\{\begin{array}{l} \text{ rate at which} \\ \text{ mass }enters\text{ the} \\ \text{ control volume} \end{array}\right\}-\left\{\begin{array}{l} \text{ rate at which} \\ \text{ mass }leaves\text{ the} \\ \text{ control volume} \end{array}\right\} \label{1}\]

and for a *fixed control volume*, the *mathematical representation* was given by

\[\frac{d}{dt} \int_{\mathscr{V}}\rho dV + \int_{\mathscr{A}}\rho \mathbf{v} \cdot \mathbf{n} dA =0 \label{2}\]

One must keep in mind that the use of vectors allows us to represent both the mass *entering* the control volume and mass *leaving* the control volume in terms of \(\rho \mathbf{v} \cdot \mathbf{n}\). This follows from the fact that \(\mathbf{v} \cdot \mathbf{n}\) is negative over surfaces where mass is *entering* the control volume and \(\mathbf{v} \cdot \mathbf{n}\) is positive over surfaces where mass is *leaving* the control volume. In addition, one should remember that the control volume, \(\mathscr{V}\), in Equation \ref{2} is arbitrary and this allows us to choose the control volume to suit our needs.

Now we are ready to consider \(N\)-component systems in which chemical reactions can take place, and in this case we need to make use of the *two axioms* for the mass of multicomponent systems. The first axiom deals with the mass of species \(A\), and when this species can undergo chemical reactions we need to extend Equation \ref{1} to the form given by

\[\left\{\begin{array}{l} \text{ time rate of change} \\ \text{ of mass of species }A \\ \text{ in a control volume} \end{array}\right\}=\left\{\begin{array}{l} \text{ rate at which} \\ \text{ mass of species }A \\ enters\text{ the} \\ \text{ control volume} \end{array}\right\} - \left\{\begin{array}{l} \text{ rate at which} \\ \text{ mass of species }A \\ leaves\text{ the} \\ \text{ control volume} \end{array}\right\} + \left\{\begin{array}{l} \text{ net rate of production} \\ \text{ of the mass of species }A \\ \text{ owing to} \\ chemical \ reactions \end{array}\right\} \label{3}\]

In order to develop a precise mathematical representation of this axiom, we require the following quantities:

\[\rho_{A} =\left\{\begin{array}{l} \text{ mass density} \\ \text{ of species }A\end{array}\right\} \label{4}\]

\[\mathbf{v}_{A} =\left\{\begin{array}{l} \text{ velocity of} \\ \text{ species }A \end{array}\right\} \label{5}\]

\[r_{A} =\left\{\begin{array}{l} \text{ net mass rate of production} \\ \text{ per unit volume of species }A \\ \text{ owing to }chemical \ reactions \end{array}\right\} \label{6}\]

Here it is important to understand that \(r_{A}\) represents both the *creation* of species \(A\) (when \(r_{A}\) is positive) and the *consumption* of species \(A\) (when \(r_{A}\) is negative). In terms of these primitive quantities, we can make use of an arbitrary fixed control volume to express Equation \ref{3} as

Axiom I: \[\frac{d}{dt} \int_{\mathscr{V}}\rho_{A} dV + \int_{\mathscr{A}}\rho_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA =\int_{\mathscr{V}}r_{A} dV , \quad A=1, 2, ...., N \label{7}\]

In the volume \(\mathscr{V}\) the *total mass produced by chemical reactions* must be zero. This is our second axiom that we express in words as

\[\left\{\begin{array}{l} \text{ total rate of production} \\ \text{ of mass owing to } \\ \text{ }chemical \ reactions \end{array}\right\}=0 \label{8}\]

and in terms of the definition given by Equation \ref{6} this word statement takes the form

\[\sum_{A=1}^{A=N} \int_{\mathscr{V}}r_{A} dV =0 \label{9}\]

The summation over all \(N\) molecular species can be interchanged with the volume integration in this representation of the second axiom, and this allows us to express Equation \ref{9} as

\[\int_{\mathscr{V}}\sum_{A=1}^{A=N} r_{A} dV =0 \label{10}\]

Since the volume \(\mathscr{V}\) is arbitrary, the integrand must be zero and we extract the preferred form of the second axiom given by

Axiom II: \[\sum_{A=1}^{A=N}r_{A} =0 \label{11}\]

In Eqs. \ref{7} and \ref{11} we have used a *mixed mode nomenclature* to represent the chemical species, i.e., we have used both letters and numbers simultaneously. Traditionally, we use upper case Roman letters to designate various chemical species, thus the rates of production for species \(A\), \(B\) and \(C\) are designated by \(r_{A}\), \(r_{B}\) and \(r_{C}\). When dealing with systems containing \(N\) different molecular species, we allow an indicator, such as \(A\) or \(D\) or \(G\), to take on values from 1 to \(N\) in order to produce compact forms of the two axioms given by Eqs. \ref{7} and \ref{11}. We could avoid this mixed mode nomenclature consisting of letters and numbers by expressing Equation \ref{11} in the form;

Axiom II: \[r_{A} + r_{B} + r_{C} + r_{D} + .... + r_{N} =0 \label{12}\]

however, this approach is rather cumbersome when dealing with \(N\)-component systems.

The concept that mass is neither created nor destroyed by chemical reactions (as indicated by Equation \ref{11}) is based on the work of Lavoisier^{1} who stated:

"We observe in the combustion of bodies generally four recurring phenomena which would appear to be invariable laws of nature; while these phenomena are implied in other memoirs which I have presented, I must recall them here in a few words.”

Lavoisier went on to list four phenomena associated with combustion, the third of which was given by

Third Phenomenon.In all combustion, pure air in which the combustion takes place is destroyed or decomposed and the burning body increases in weight exactly in proportion to the quantity of air destroyed or decomposed.

It is this *Third Phenomenon*, when extended to all reacting systems, that supports Axiom II in the form represented by Equation \ref{9}. The experiments that led to the *Third Phenomenon* were difficult to perform in the 18\(^{th}\) century and those difficulties have been recounted by Toulmin^{2}.

## Molar concentration and molecular mass

When chemical reactions occur, it is generally more convenient to work with the *molar form* of Eqs. \ref{7} and \ref{11}. The appropriate measure of concentration is then the *molar concentration* defined by

\[c_{A} ={\rho_{A} / MW_{A} } =\left\{\begin{array}{l} moles\text{ of species }A \\ \text{ per unit volume} \end{array}\right\} \label{13}\]

while the appropriate net rate of production for species \(A\) is given by^{3}:

\[R_{A} ={r_{A} / MW_{A} } =\left\{\begin{array}{l} \text{ net }molar\text{ rate of production} \\ \text{ per unit volume of species }A \\ \text{ owing to chemical reactions} \end{array}\right\} \label{14}\]

Here \(MW_{A}\) represents the *molecular mass *^{4} of species \(A\) that is given explicitly by

\[MW_{A} =\frac{\text{ kilograms of }A}{\text{ moles of }A} \label{15}\]

The numerical values of the molecular mass are obtained from the atomic masses associated with any particular molecular species, and values for both the atomic mass and the molecular mass are given in Tables A1 and A2 in Appendix A. In those tables we have represented the atomic mass and the molecular mass in terms of grams per mole, thus the definition given by Equation \ref{15} for water leads to

\[MW_{\ce{H2O}} =\frac{0.018015 \ kg}{ mol} =\frac{18.015 \ g}{ mol} \label{16}\]

In terms of \(c_{A}\) and \(R_{A}\), the two axioms given by Eqs. \ref{7} and \ref{11} take the form

Axiom I: \[\frac{d}{dt} \int_{\mathscr{V}}c_{A} dV + \int_{\mathscr{A}}c_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA =\int_{\mathscr{V}}R_{A} dV , \quad A=1, 2, ...., N \label{17}\]

Axiom II: \[\sum_{A = 1}^{A = N}MW_{A} R_{A} =0 \label{18}\]

Here it is important to note that mass is conserved during chemical reactions while moles need not be conserved. For example, the decomposition of calcium carbonate (solid) to calcium oxide (solid) and carbon dioxide (gas) is described by

\[\left(\ce{CaCO}_{ 3} \right)_{solid} \to \left(\ce{CaO}\right)_{solid} + \left(\ce{CO}_{ 2} \right)_{gas} \label{19}\]

thus *one mole* is consumed and *two moles* are produced by this chemical reaction.

One must be very careful to understand that the *net molar rate of production per unit volume of species *\(A\) *owing to chemical reactions*, \(R_{A}\), may be the result of many different chemical reactions. For example, in the chemical production system illustrated in Figure \(\PageIndex{1}\), carbon dioxide may be produced by the

oxidation of carbon monoxide, by the complete combustion of methane, or by other chemical reactions taking place within the control volume illustrated in Figure \(\PageIndex{1}\). The combination of all these individual chemical reactions is represented by \(R_{ \ce{CO2}}\). It is important to note that in Figure \(\PageIndex{1}\) we have suggested the *stoichiometry* of the reactions taking place while the actual *chemical kinetic processes* taking place may be much more complicated. The subject of stoichiometry will be discussed in detail in Chapter 6, and a brief discussion of chemical kinetics is given in Section 8.6.

## Moving control volumes

In Sec. 3.4 we developed the macroscopic mass balance for a single component system in terms of an *arbitrary* moving control volume, \(\mathscr{V}_{a} (t)\). The speed of displacement of the surface of a moving control volume is given by \(\mathbf{w} \cdot \mathbf{n}\), and the flux of species \(A\) that crosses this moving surface is given by the normal component of the *relative velocity* for species \(A\), i.e., \((\mathbf{v}_{A} -\mathbf{w}) \cdot \mathbf{n}\). On the basis of this concept, we can express the first axiom for the mass of species \(A\) in the form

Axiom I: \[\frac{d}{dt} \int_{\mathscr{V}_{a} (t)}\rho_{A} dV + \int_{\mathscr{A}_{a} (t)}\rho_{A} (\mathbf{v}_{A} -\mathbf{w}) \cdot \mathbf{n} dA =\int_{\mathscr{V}_{a} (t)}r_{A} dV , \quad A=1, 2, ...., N \label{20}\]

When it is convenient to work with molar quantities, we divide this result by the molecular mass of species \(A\) in order to obtain the form for an *arbitrary* moving control volume form given by

Axiom I: \[\frac{d}{dt} \int_{\mathscr{V}_{a} (t)}c_{A} dV + \int_{\mathscr{A}_{a} (t)}c_{A} (\mathbf{v}_{A} -\mathbf{w}) \cdot \mathbf{n} dA =\int_{\mathscr{V}_{a} (t)}R_{A} dV , \quad A = 1, 2, ...., N \label{21}\]

The volume associated with a *specific* moving control volume will be designated by \( \mathscr{V}(t)\) while the volume associated with a fixed control volume will be designated by \( \mathscr{V}\). Throughout this chapter, we will restrict our studies to fixed control volumes in order to focus our attention on the new concepts associated with multicomponent systems. However, the world of chemical engineering is filled with moving, dynamic systems and the analysis of those systems will require the use of Eqs. \ref{20} and \ref{21}.