Axiom I: $\frac{d}{dt} \int_{\mathscr{V}}\rho_{A } dV + \int_{\mathscr{A}}\rho_{A } \mathbf{v}_{A} \cdot \mathbf{n} dA=\int_{\mathscr{V}}r_{A} dV , \quad A=1, 2,...N \label{1}$
has been used to solve problems dealing with liquid and solid systems when it was convenient to work directly with the species mass density $$\rho_{A}$$ or with the mass fraction $$\omega_{A}$$. Those problems did not involve chemical reactions, thus the net mass rate of production of species $$A$$ owing to chemical reactions was zero, i.e., $$r_{A} =0$$. When it was convenient to work with the species molar concentration $$c_{A}$$ or the mole fraction $$x_{A}$$, we used the molar form of Equation \ref{1} with $$R_{A} =0$$.
In this chapter we begin our study of material balances with chemical reactions, and we continue this study throughout the remainder of the text. Aris1 pointed out that stoichiometry is essentially the bookkeeping of atomic species, and we use this bookkeeping to develop constraints on the net molar rates of production, $$R_A,R_B,R_C,...,R_N$$. These stoichiometric constraints reduce the degrees of freedom and they represent a key aspect of macroscopic balance analysis for systems with chemical reaction. The net molar rates of production can be determined experimentally without any detailed knowledge of the chemical kinetics, and we show how this is done in Example $$6.2.2$$. Knowledge of the net global rates of production allows us to specify the flow sheets referred to in Sec. 1.3; however, they do not allow us to specify the size of the units illustrated in Figure $$1.3.1$$. To determine the size of a chemical reactor, we must know how the net rates of production depend on the concentration of the participating chemical species and this matter is explored in Chapter 9.