# 4.2: Untitled Page 57

- Page ID
- 18189

## Chapter 4

**v ****n**. This follows from the fact that **v****n** is negative over surfaces where mass is *entering* the control volume and **v ****n** is positive over surfaces where mass is *leaving* the control volume. In addition, one should remember that the control volume, *V *, in Eq. 4‐2 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 Eq. 4‐1 to the form given by

(4‐3)

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

(4‐4)

(4‐5)

(4‐6)

Here it is important to understand that

represents both the *creation* of species

*A*

*r*

*A* (when

is positive) and the *consumption* of species *A* (when is negative).

*A*

*r*

*A*

*r*

In terms of these primitive quantities, we can make use of an arbitrary fixed control volume to express Eq. 4‐3 as

98

Axiom I:

(4‐7)

Within the volume, *V *, the *total mass produced by chemical reactions* must be zero.

This is our second axiom that we express in words as

Axiom II:

(4‐8)

and in terms of the definition given by Eq. 4‐6 this word statement takes the form (4‐9)

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 Eq. 4‐9 as

(4‐10)

Since the volume *V * is arbitrary, the integrand must be zero and we extract the preferred form of the second axiom given by

Axiom II:

(4‐11)

Here it should be clear that *A*

*r * represents both the *creation* of species *A* (when *A* *r *

is positive) and the *consumption* of species *A* (when is negative). In Eqs. 4‐7

*A*

*r*

and 4‐11 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 *A* *r *,

and

. When dealing with systems containing *N* different molecular species, *B*

*r*

*C*

*r*

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. 4‐7 and 4‐11. We