# 9.16: Untitled Page 232

- Page ID
- 18365

## Chapter 9

present, and they usually are, it is appropriate to partition the mechanistic matrix into a *stoichiometric matrix* and a *Bodenstein matrix* as indicated by Eqs. 9‐110 through 9‐112. The general partitioning of Eq. 9‐127 can be expressed as

R

S

R

M r

r

(9‐129)

M

R

B

B

and this leads to forms analogous to Eqs. 9‐111 and 9‐112. We list the first of these results as

Stoichiometric matrix:

R

S r

(9‐130)

in which S is the stoichiometric matrix composed of *stoichiometric coefficients*.

The second of Eqs. 9‐129 is given by

Bodenstein matrix:

R

B r

(9‐131)

B

in which B represents the Bodenstein matrix. In general, the Bodenstein products are subject to the approximation of local reaction equilibrium that is expressed as

Local reaction equilibrium:

R

0

(9‐132)

B

and this allows one to extract additional constraints on the elementary chemical reaction rates. The result given by Eq. 9‐130 represents a key aspect of reactor design that can be expressed in more detailed form by

*B * K

*R*

*S*

*r , *

*A * 1 *, * 2 *, ..... N *

(9‐133)

*A*

*AB B*

*B * I

Here *S*

represents the stoichiometric coefficients, *r * represents the elementary *AB*

*B*

chemical reaction rates, and K represents the number of elementary reactions as indicated by Eq. 9‐31.

**9.5 Problems **

*Section* 9.1

9‐1. Apply a column/row partition to show how Eq. 9‐6 is obtained from Eq. 9‐5.

9‐2. Illustrate how a row/row partition leads from Eq. 9‐22 to Eq. 9‐23.

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9‐3. Use Eqs. 9‐34 through 9‐42 to obtain a local chemical reaction rate equation for species *D*.

9‐4. Develop the local chemical reaction rate equation for species *C* on the basis of Eqs. 9‐34 through 9‐42.

9‐5. Re‐write Eqs. 9‐40 and 9‐42 using the stoichiometric coefficients,

,

,

*A* I

*D* II

etc., in place of , , , etc. Show that this change in the nomenclature leads to the *form* encountered in Eq. 6‐38.

9‐6. Develop the representation for *R * given in Eqs. 9‐43.

*B*

9‐7. It is difficult to find a real system containing three species for which the reactions can be described by

*k*

1

*k* 2

*A*

*B * *C *

(1)

however, this represents a useful model for the exploration of the condition of local reaction equilibrium. The stoichiometric constraint for this series of first order reactions is given by

*R*

*R * *R*

0

(2)

*A*

*B*

*C*

since the three molecular species must all contain the same atomic species. For a constant volume batch reactor, and the initial conditions given by IC.

o

*c*

*c , *

*c*

0 *, *

*c*

0 *, *

*t * 0

(3)

*A*

*A*

*B*

*C*

one can determine the concentrations and reaction rates of all three species as a function of time. If one thinks of species *B* as a *reactive intermediate*, the condition of local equilibrium takes the form

Local reaction equilibrium:

*R*

0

(4)

*B*

In reality, the reaction rate for species *B* cannot be exactly zero; however, we can have a situation in which

*R*

*R *

(5)

*B*

*A*

Often this condition is associated with a *very large* value of *k *, and in this 2

problem you are asked to develop and use the exact solution for this process to determine how large is *very large*. You can also use the exact solution to see why

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