# 15.3: Untitled Page 235

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## Chapter 9

*k * IV

Elementary chemical kinetic schema IV:

NO

NO

N O

(5)

2

3

2

5

Since neither NO nor NO appear in the global stoichiometric schema given by 3

Eq. 1, one can assume that these two compounds are *reactive intermediates* or *Bodenstein products* and their rates of production can be set equal to zero *as an* *approximation*

9‐15. In Problem 9‐14 we considered the decomposition of N O to produce 2

5

NO and O by means of the kinetic schemata illustrated by Eqs. 2 through 5 in 2

2

Problem 9‐14. The reactive intermediates were identified as NO and NO . The 3

rate equations for these two reactive intermediates and for N O are given by 2

5

*R*

*k c*

*k c*

*c*

*k c*

*c*

*k c*

*c*

(1)

NO3

I N2O5

II NO3 NO2

III NO NO3

IV NO2 NO3

*R*

*k c*

*c*

*k c*

*c*

(2)

NO

II NO3 NO2

III NO NO3

*R*

*k c*

*k c*

*c*

(3)

N2O5

I N2O5

IV NO2 NO3

If the condition of local reaction equilibrium is imposed according to Local reaction equilibrium:

*R*

0 *, *

*R*

0

(4)

NO3

NO

one can obtain a simple representation for *R*

in terms of only *c*

. In this

N2O5

N2O5

problem you are asked to replace the *assumptions* given by Eqs. 4 with *restrictions* indicating that *R*

and *R*

are *small enough* so that Eqs. 4 become acceptable

NO3

NO

approximations.

*Section* 9.4

9‐16. If species *C * in Eq. 9‐88 is a *reactive intermediate* or *Bodenstein product*, identify the stoichiometric matrix and the Bodenstein matrix associated with Eq. 9‐88. Impose the condition of local reaction equilibrium on the Bodenstein product in order to derive an expression for *R * in terms of *c *, *c * and *c *.

*E*

*A*

*B*

*D*

Appendix A

**A1 Atomic Mass of Common Elements Referred to Carbon‐12 **

**(Details are available at http://www.nist.gov/physlab/data/comp.cfm)**

Element

Symbol

Atomic mass, g/mol

Aluminum

Al

26.9815

Antimony

Sb

121.75

Argon

Ar

39.948

Arsenic

As

74.9216

Barium

Ba

137.34

Beryllium

Be

9.0122

Bismuth

Bi

208.980

Boron

B

10.811

Bromine

Br

79.904

Cadmium

Cd

112.40

Calcium

Ca

40.08

Carbon

C

12.01

Cerium

Ce

140.12

Cesium

Cs

132.905

Chlorine

Cl

35.453

Chromium

Cr

51.996

Cobalt

Co

58.9332

Copper

Cu

63.546

Fluorine

F

18.9984

Gallium

Ga

69.72

Germanium

Ge

72.59

Gold

Au

196.967

Hafnium

Hf

178.49

Helium

He

4.0026

Hydrogen

H

1.00797

Indium

In

114.82

Iodine

I

126.904

Iridium

Ir

192.2

Iron

Fe

55.847

Krypton

Kr

83.80

439

440

*Appendices *

Element

Symbol

Atomic mass, g/mol

Lead

Pb

207.19

Lithium

Li

6.939

Magnesium

Mg

24.312

Manganese

Mn

54.938

Mercury

Hg

200.59

Molybdenum

Mo

95.94

Neon

Ne

20.183

Nickel

Ni

58.71

Niobium

Nb

92.906

Nitrogen

N

14.0067

Oxygen

O

15.9994

Palladium

Pd

106.4

Phosphorus

P

30.9738

Platinum

Pt

195.09

Plutonium

Pu

242

Potassium

K

39.102

Radium

Ra

226

Radon

Rn

222

Rhodium

Rh

102.905

Rubidium

Rb

85.47

Selenium

Se

78.96

Silicon

Si

28.086

Silver

Ag

107.868

Sodium

Na

22.9898

Strontium

Sr

87.62

Sulfur

S

32.064

Tantalum

Ta

180.948

Tellurium

Te

127.60

Thallium

Tl

204.37

Thorium

Th

232.038

Tin

Sn

118.69

Titanium

Ti

47.90

Tungsten

W

183.85

Uranium

U

238.03

Vanadium

V

50.942

Xenon

Xe

131.30

441

Element

Symbol

Atomic mass, g/mol

Yttrium

Y

88.905

Zinc

Zn

65.37

Zirconium

Zr

91.22

442

*Appendices *

**A2 Physical Properties of Various Chemical Compounds **

Name

Formula

Molecular

Liquid @ *T*(K)

*T* melting *T* boiling

Mass, g/mol

density (g/L)

K

K

Argon

Ar

39.948

83.8

87.3

Acetaldehyde

C2H4O

44.054

778

293

150.2

293.6

Acetic acid

C2H4O2

60.052

1049

293

289.8

391.1

Acetone

C3H6O

58.08

790

293

178.2

329.4

Acetylene

C2H2

26.038

192.4

189.2

Acrylic acid

C3H4O2

72.064

1051

293

285

414

Ammonia

NH3

17.031

195.4

239.7

Aniline

C6H7N

93.129

1022

293

267

457.5

Benzaldehide

C7H6O

106.124

1045

293

216

452

Benzene

C6H6

78.114

885

289

278.7

353.3

Benzoic acid

C7H6O2

122.124

1075

403

395.6

523

Bromine

Br2

159.808

3119

293

266

331.9

1,2‐Butadiene

C4H6

54.092

137

284

1,3‐Butadiene

C4H6

54.092

164.3

268.7

n‐Butane

C4H10

58.124

134.8

272.7

i‐Butane

C4H10

58.124

113.6

261.3

n‐Butanol

C4H10O

74.123

810

293

183.9

390.9

1‐Butene

C4H8

56.108

87.8

266.9

i‐Butene

C4H8

56.108

132.8

266.3

Carbon

tetrachloride

CCl4

153.823

1584

298

250

349.7

Carbon dioxide

CO2

44.01

216.6

194.7

Carbon monoxide

CO

28.01

68.1

81.7

Chlorine

Cl2

70.906

172.2

238.7

Chlorobenzene

C6H5Cl

112.559

1106

293.

227.6

404.9

Chloroform

CHCl3

119.378

1498

293

209.6

334.3

Cyclobutane

C4H8

56.108

182.4

285.7

Cyclohexane

C6H12

84.162

779

293

279.7

353.9

Cyclohexanol

C6H12O

100.161

942

303

298

434.3

Cyclopentane

C5H10

70.135

745

293

179.3

322.4

443

Name

Formula

Molecular

Liquid @ *T*(K)

*T* melting *T* boiling

Mass, g/mol

density (g/L)

K

K

Cyclopentene

C5H8

68.119

772

293

138.1

317.4

Ethane

C2H6

30.07

89.9

184.5

Ethanol

C2H6O

46.069

789

293

159.1

351.5

Ethyl amine

C2H7N

45.085

683

293

192

289.7

Ethyl acetate

C4H8O2

88.107

901

293

89.6

350.3

Ethylbenzene

C8H10

106.168

867

293

178.2

409.3

Ethylendiamine

C2H8N2

60.099

896

293

284

390.4

Ethyl ether

C4H10O

74.123

713

293

156.9

307.7

Ethyl propionate

C5H10O2 102.134

895

293

199.3

372

Ethylene

C2H4

28.054

104

169.4

Ethylene Glycol

C2H6O2

62.069

1,114

293

260.2

470.4

Ethylene oxide

C2H4O

44.054

161

283.5

Fluorine

F2

37.997

53.5

85

Formaldehide

CH2O

30.026

156

254

Formic acid

CH2O2

46.025

1,226

288

281.5

373.8

Glycerol

C3H8O3

92.095

1,261

293

291

563

n‐Heptane

C7H16

100.205

684

293

182.6

371.6

1‐Heptanol

C7H16O

116.204

822

293

239.2

449.5

1‐Heptene

C7H14

98.189

697

293

154.3

366.8

n‐Hexane

C6H14

86.178

659

293

177.8

341.9

1‐Hexanol

C6H14O

102.177

819

293

229.2

430.2

Hydrogen

H2

2.016

14.0

20.4

Hydrogen bromide HBr

80.912

187.1

206.1

Hydrogen chloride

HCl

36.461

159.0

188.1

Hydrogen cyanide

CHN

27.026

688

293

259.9

298.9

Hydrogen sulfide

H2S

34.08

187.6

212.8

Iodine

I2

253.808

3,740

453

386.8

457.5

Isopropyl alcohol

C3H8O

60.096

786

293

184.7

355.4

Maleic anhydride

C4H2O3

98.058

1,310

333

326

472.8

Methane

CH4

16.043

90.7

111.7

Mercury

Hg

200.59

13,546

293

234.3

630.1

Methanol

CH4O

32.042

791

293

175.5

337.8

Methyl acetate

C3H6O2

74.08

934

293

175

330.1

444

*Appendices *

Name

Formula

Molecular

Liquid @ *T*(K)

*T* melting *T* boiling

Mass, g/mol

density (g/L)

K

K

Methyl acrylate

C4H7O2

86.091

956

293

196.7

353.5

Methyl amine

CH5N

31.058

179.7

266.8

Methyl benzoate

C8H8O2

136.151

1,086

293

260.8

472.2

Methyl ethyl ketone C4H8O

72.107

805

293

186.5

352.8

Naphtalene

C10H8

128.174

971

363

353.5

491.1

Nitric oxide

NO

30.006

109.5

121.4

Nitrogen

N2

28.013

63.3

77.4

Nitrogen dioxide

NO

30.01

112.2

122.2

Nitrogen tetroxide

N2O4

46.006

261.9

294.3

Nitrous oxide

N2O

44.013

182.3

184.7

Oxygen

O2

31.999

54.4

90.2

n‐Pentane

C5H12

72.151

626

293

143.4

309.2

1‐Pentanol

C5H12O

88.15

815

293

195

411

1‐Pentyne

C5H8

68.119

690

293

167.5

313.3

1‐Pentene

C5H10

70.135

640

293

107.9

303.1

Phenol

C6H6O

94.113

1059

314

313

455

Propane

C3H8

44.097

85.5

231.1

1‐Propanol

C3H8O

60.096

804

293

146.9

370.4

Propionic acid

C3H6O2

74.08

993

293

252.5

414

Propylene

C3H6

42.081

87.9

225.4

Propylene oxide

C3H6O

58.08

829

293

161

307.5

Styrene

C8H8

104.152

906

293

242.5

418.3

Succinic acid

C4H6O4

118.09

456

508

Sulfur dioxide

SO2

64.063

197.7

263

Sulfur trioxide

SO3

80.058

1780

318

290

318

Toluene

C7H8

92.141

867

293

178

383.8

Trimethyl amine

C3H9N

59.112

156

276.1

Vinyl chloride

C2H3Cl

62.499

119.4

259.8

Water

H2O

18.015

998

293

273.2

373.2

o‐Xylene

C8H10

106.168

880

293

248

417.6

m‐Xylene

C8H10

106.168

864

293

225.3

412.3

p‐Xylene

C8H10

106.168

861

293

286.4

411.5

445

**A3 Constants for Antoine’s Equation **

o

log *p*

*A * *B / *( *T*) *, p * is in mm Hg and *T * is in *C *

*vap*

*vap*

Compound

Formula

*A*

*B*

Acetaldehyde

CH3CHO

7.0565

1070.6

236.0

Acetic acid

CH3COOH

7.29963

1479.02

216.81

Acetone

CH3COCH3

7.23157

1277.03

237.23

Acetylene

C2H2

7.0949

709.1

253.2

Acrylic acid

C2H3COOH

7.1927

1441.5

192.66

Ammonia

NH3

7.36050

926.132

240.17

Aniline

C6H7N

7.2418

1675.3

200.01

Benzaldehide

C7H6O

7.1007

1628.00

207.04

Benzene

C6H6

6.90565

1211.03

220.790

Benzoic acid

C7H6O2

7.45397

1820

147.96

1,2‐Butadiene

C4H6

7.1619

1121.0

251.00

1,3‐Butadiene

C4H6

6.85941

935.531

239.554

*n*‐Butane

C4H10

6.83029

945.90

240.00

*n*‐Butanol

C3H7CH2OH

7.4768

1632.39

178.83

*i*‐Butane

C4H10

6.74808

882.80

240.00

*n*‐Butene

C4H8

6.84290

926.10

240.00

*i*‐Butene

C4H8

6.84134

923.200

240.00

Carbon

tetrachloride

CCl4

6.8941

1219.58

227.17

Chlorobenzene

C6H5Cl

6.9781

1431.05

217.56

Chloroform

CHCl3

6.93707

1171.2

236.01

Cyclobutane

C4H8

6.92804

1024.54

241.38

Cyclohexane

C6H12

6.84498

1203.526

222.863

Cyclopentane

C5H10

6.88678

1124.16

231.37

Cyclopentene

C5H8

6.92704

1121.81

233.46

Ethane

C2H6

6.80266

656.40

256.00

Ethanol

CH3CH2OH

8.16290

1623.22

228.98

Ethyl amine

C2H7N

7.38618

1137.3

235.86

Ethyl acetate

C4H8O2

7.01455

1211.9

216.01

Ethene (Ethylene)

C2H4

6.74756

585.00

255.00

446

*Appendices *

Compound

Formula

*A*

*B*

Ethylbenzene

C6H5C2H5

6.95719

1424.55

213.206

Ethylenediamine

C2H8N2

7.12599

1350.0

201.03

Ethyl ether

C4H10O

6.98467

1090.64

231.21

Ethyl propionate

C5H10O2

7.01907

1274.7

209.0

Ethylene glycol

C2H6O2

8.7945

2615.4

244.91

Formaldehyde

HCHO

7.1561

957.24

243.0

Formic acid

HCOOH

7.37790

1563.28

247.06

Glycerol

C3H8O3

7.48689

1948.7

132.96

*n‐* Heptane

C7H16

6.90240

1268.115

216.900

1‐Heptanol

C7H16O

6.64766

1140.64

126.56

1‐Heptene

C7H14

6.90068

1257.5

219.19

*n‐* Hexane

C6H14

6.87776

1171.530

224.366

Hydrogen bromide HBr

6.28370

539.62

225.30

Hydrogen chloride

HCl

7.167160

744.49

258.704

Hydrogen cyanide

HCN

7.17185

1123.0

236.01

Hydrogen sulfide

H2S

6.99392

768.13

247.093

Iodine (c)

I2

9.8109

2901.0

256.00

Isopropyl alcohol

C3H8O

8.11822

1580.92

219.62

Maleic anhydride

C4H2O3

7.06801

1635.4

191.01

Methane

CH4

6.61184

389.93

266.00

Methanol

CH3OH

8.07246

1574.99

238.86

Methyl acetate

C3H6O2

7.00495

1130.0

217.01

Methyl acrylate

C4H7O2

6.99596

1211.0

214.01

Methyl amine

CH5N

7.49688

1079.15

240.24

Methyl benzoate

C8H8O2

7.04738

1629.4

192.01

Methyl ethyl ketone C4H8O

7.20868

1368.21

236.51

Napthalene

C10H8

6.84577

1606.5

187.227

Nitric oxide

NO

8.74300

682.94

268.27

Nitrogen tetroxide

N2O4

7.38499

1185.72

234.18

Nitrous oxide

N2O

7.00394

654.26

247.16

*n‐* Pentane

C5H12

6.85221

1064.63

232.000

1‐Pentanol

C5H12O

7.17758

1314.56

168.16

1‐Pentyne

C5H8

6.96734

1092.52

227.19

1‐Pentene

C5H10

6.84648

1044.9

233.53

447

Compound

Formula

*A*

*B*

Phenol

C6H5OH

7.13457

1516.07

174.569

Phosphorus

trichloride

PCl3

6.8267

1196

227.0

Phosphine

PH3

6.71559

645.512

256.066

Propane

C3H8

6.82973

813.20

248.00

1‐Propanol

CH3CH2CH2OH

6.79498

969.27

150.42

Propionic acid

C3H6O2

7.57456

1617.06

205.68

Propene

(Propylene)

C3H6

6.81960

785.00

247.00

Propylene oxide

C3H6O

6.65456

915.31

208.29

*n*‐Propionic

Acid

CH3CH2COOH

7.54760

1617.06

205.67

Styrene

C8H8

6.95709

1445.58

209.44

Sulfur dioxide

SO2

7.28228

999.900

237.190

Sulfur trioxide

SO3

9.05085

1735.31

236.50

Toluene

C6H5CH3

6.95464

1344.800

219.482

Trimethyl amine

C3H9N

6.97038

968.7

234.01

Vynil chloride

C2H3Cl

6.48709

783.4

230.01

Water

H2O

7.94915

1657.46

227.03

*o*‐Xylene

C6H5(CH3)2

6.99891

1474.679

213.686

*p*‐Xylene

C6H5(CH3)2

6.99052

1453.430

215.307

448

*Appendices *

Appendix B

Iteration Methods

**B1. Bisection method **

Given some function of *x * such as *H*( *x*) , we are interested in the solution of the equation

*H*( *x*) 0 *, *

*x * *x *

(B‐1)

Here we have used *x * to represent the solution. For simple functions such as

*H*( *x*) *x * *b * we obtain a single solution given by *x * *b *, while for a more complex function such as

2

*H*( *x*) *x * *b * we obtain more that one solution as

indicated by *x * *b *. In many cases there is no *explicit* solution for Eq. B‐1. For example, if *H*( *x*) is given by

*H*( *x*) *a * sin ( *x* 2 ) *b * cos(2 *x*

)

(B‐2)

we need to use iterative methods to determine the solution *x * *x *.

The simplest iterative method is the bisection method (Corliss, 1977) that is illustrated in Figure B‐1. This method begins by locating *x * and *x * such that o

1

*H*( *x *) and *H*( *x *) have different signs. In Figure B‐1 we see that *x * and *x * have o

1

o

1

been chosen so that there is a change of sign for *H*( *x*) , i.e., *H*( *x *) 0 *, *

*H*( *x *) 0

(B‐3)

o

1

Thus if *H*( *x*) is a continuous function we know that a solution *H*( *x *) 0 exists somewhere between *x * and *x *. We attempt to locate that solution by means of o

1

a guess (i.e., the bisection) indicated by

*x * *x*

o

1

*x*

2

(B‐4)

2

As illustrated in Figure B‐1, this guess is closer to the solution, *x * *x *, than either *x * or *x *, and if we repeat this procedure we will eventually find a value of *x* o

1

*Material Balances *

449

*Figure B‐1*. Illustration of the bisection method

that produces a value of *H*( *x*) that is arbitrarily close to zero. In terms of the particular graph illustrated in Figure B‐1, it is clear that *x * will be located 3

between *x * and *x *; however, this need not be the case. For example, in 1

2

Figure B‐2 we have represented a slightly different function for which *x * will be 3

located between *x * and *x *. The location of the next guess is based on the idea o

2

that the function *H*( *x*) must change sign. In order to determine the location of the next guess we examine the products *H*( *x *) *H*( *x*

) and *H*( *x *) *H*( *x*

) in

*n*

*n*1

*n*

*n*2

order to make the following decisions:

*n*

*x*

*n*

*x * 1

if *H*( *x *) *H*(

*n*

*n*

*x * 1)

0 *, *

then

*n*

*x *1

2

(B‐5)

*n*

*x*

*n*

*x *2

if *H*( *x *) *H*(

*n*

*n*

*x *2)

0 *, *

then *n*

*x *1

2

Since these two choices are mutually exclusive there is no confusion about the next choice of the dependent variable. The use of Eqs. B‐5 is crucial when the details of *H*( *x*) are not clear, and a program is written to solve the implicit equation.

450

*Appendices *

*Figure B‐2*. Alternate choice for the second bi‐section **B2. False position method **

The false position method is also known as the *method of interpolation* (Wylie, 1951) and it represents a minor variation of the bisection method. Instead of bisecting the distance between *x * and *x * in Figure B‐1 in order to locate the o

1

point *x *, we use the straight line indicated in Figure B‐3. Sometimes this line is 2

called the secant line. The definition of the tangent of the angle provides 0 *H*( *x *)

*H*( *x *) *H*( *x *)

1

o

1

tan

(B‐6)

*x *

2

1

*x*

o

*x*

1

*x*

and we can solve for *x * to obtain

2

*x * *x H*( *x *)

o

1

1

*x*

2

1

*x*

(B‐7)

*H*(

o

*x *)

*H*( 1

*x *)

This replaces Eq. B‐4 in the bisection method and it can be generalized to obtain

*x * *x*

*H*( *x*

)

*n*

*n*1

*n * 1

*n*

*x *2

*n*

*x *1

(B‐8)

*H*( *x *) *H*(

*n*

*n*

*x *1)

*Material Balances *

451

Application of successive iterations will lead to a value of *x * that approaches *x* shown in Figure B‐3.

*Figure B‐3*. False position construction

**B3. Newton’s method **

Newton’s method (Ypma, 1995), which is also known as the Newton‐Raphson method, is named for Sir Isaac Newton and is perhaps the best known method for finding roots of real valued functions. The method is similar to the false position method in that a straight line is used to locate the next estimate of the root of an equation; however, in this case it is a tangent line and not a secant line.

This is illustrated in Figure B‐4a where we have chosen *x * as our first estimate of o

the solution to Eq. B‐1 and we have constructed a tangent line to *H*( *x*) at *x *. The o

slope of this tangent line is given by

*dH*

*H*( *x *) 0

o

(B‐9)

*dx*

*x x*

o

*x*

1

*x*

o

and we can solve this equation to produce our next estimate of the root. This new estimate is given by

452

*Appendices *

*H*( *x *)

o

*x*

*x *

(B‐10)

1

o

( *dH dx*) *x* o *x*

and we use this result to determine *H*( *x *) as indicated in Figure B‐4a.

1

*Figure B‐4a*. First estimate using Newton’s method

Given *H*( *x *) and *x * we can construct a second estimate as indicated in 1

1

Figure B‐4b, and this process can be continued to find the solution given by *x *.

The general iterative procedure is indicated by

*H*( *x *)

*n*

*x*

*x *

*, *

*n * 0 *, * 1 *, * 2 *,..., *

(B‐11)

*n*1

*n*

( *dH dx*) *x* *nx*

Newton’s method is certainly an attractive technique for finding solutions to implicit equations; however, it does require that one know both the function and its derivative. For complex functions, calculating the derivative at each step in the iteration may require more effort than that associated with the bisection method or the false position method. In addition, if the derivative of the function is zero in the region of interest, Newton’s method will fail.

*Material Balances *

453

*Figure B‐4b*. Second estimate using Newton’s method

**B4. Picard’s method **

Picard’s method for solving Eq. B‐1 begins by defining a new function according to Definition:

*f *( *x*) *x * *H*( *x*)

(B‐12)

Given any value of the dependent variable, *x *, we *define* a new value, *x*

, by

*n*

*n*1

Definition:

*x*

*f *( *x *) *, *

*n*

0 *, * 1 *, * 2 *, * 3 *,... *

(B‐13)

*n*

1

*n*

This represents Picard’s method or the *method of direct substitution* or the *method of* *successive substitution*. If this procedure converges, we have

*f *( *x *) *x * *H*( *x *) *x* (B‐14)

In Eq. B‐13 we note that the function *f *( *x *) , maps the point *x * to the new point *n*

*n*

*x*

. If the function *f x * maps the point *x * to itself, i.e., *f *( *x*) *x *, then *x * is *n*

( )

1

called the *fixed point* of *f *( *x*) . In Figure B‐5 we again consider the function represented in Figures B‐1, B‐3 and B‐4, and we illustrate the functions *f *( *x*) , *y*( *x*)

and *H*( *x*) . The graphical representation of the fixed point, *x *, is the intersection

454

*Appendices *

*Figure B‐5*. Picard’s method

of the function of *f *( *x*) with the line *y * *x *. Note that not all functions have fixed points. For example if *f *( *x*) is parallel to the line *y * *x * there can be no intersection and no fixed point. Given our first estimate, *x *, we use Eq. B‐13 to o

compute *x * according to

1

*x*

*f *( *x *)

(B‐15)

1

o

Clearly *x * is further from the solution, *x *, than *x * and we can see from the 1

o

graphical representation in Figure B‐5 that Picard’s method *diverges* for this case.

If *x * were chosen to be less than the solution, *x *, we would also find that the o

iterative procedure diverges. If the slope of *f *( *x*) were less than the slope f o

*y*( *x*) , we would find that Picard’s method converges. This suggests that the method is useful for “weak” fu c

n tions of *x *, i.e., *df dx * 1 and this is confirmed in Sec. B6.

*Material Balances *

455

**B **

**5. Wegstein’s method **

In Figure B‐6 we have

strated

illu

the same function, *f *( *x*) , that appears in Figure B‐5. For some point *x * in the neighborhood of *x * we can approximate

1

o

*Figure B‐6*. Wegstein’s method

the derivative of *f *( *x*) according to

*df*

*f *( *x *) *f *( *x *)

1

o

slope *S*

*dx*

(B‐16)

1

*x*

o

*x*

and we can use this result to obtain an approximation

for the function *f *( *x *) .

1

*f *( *x *)

*f *( *x *) *S x * *x*

(B‐17)

1

o

1 o

At this point we recall Eq. B‐14 in the form

456

*Appendices *

*f *( *x *) *x *

(B‐18)

nd

a

note that if *x * is

in the neighborhood of *x * we obtain the approximation 1

*f *( *x *) *x *

(B‐19)

1

1

e

W use this result in Eq. B‐17 to produce an equation

*x*

*f *( *x *) *S x * *x *

(B‐20)

1

o

1 o

in which *S * is an *adjustable parameter* that is used to determine the next step in the iterative procedure. It is traditional, but not necessary, to define a new *adjustable* *arameter*

*p*

according to

*S*

*q *

*S *

(B‐21)

1

se

U of this representation in Eq. B‐20 leads to

*x*

(1 *q*) *f *( *x *) *q x *

(B‐22)

1

o

o

nd

a

we can generalize this result to Wegstein’s method given by

*x*

(1 *q*) *f *( *x *)

*q x , *

*n * 0 *, * 1 *, * 2 *, * 3 *, * e

*... * tc

*n*

1

*n*

*n*

(B‐23)

When the adjustable parameter is equal to zero, *q * 0 , we obtain Picard’s method described in Sec. B4. When the adjustable parameter greater than zero and less than one, 0 *q * 1 , we obtain a *damped* successive substitution process that improves stability for nonlinear systems. When the adjustable parameter is egative,

n

*q * 0 , we obtain an *accelerated* successive substitution that may lead to an unstable procedure.

**B6. Stability of iteration methods **

In this section we

r

conside the *linear* stability characteristics of Newton’s method, Picard’s method, and Wegstein’s method that have been used to solve e

th implicit equation given by

*H*( *x*) 0 *, *

*x * *x *

(B‐24)

The constraint associated with the linear analysis will be listed below and it must be kept in mind when interpreting results such as those presented in Chapter 7.

We begin by

g

recallin the three iterative methods as

457

*H*( *x *)

Newton’s method:

*n*

*x*

*x *

*, *

*n * 0 *, * 1 *, * 2 *,..., * (B‐25)

*n*1

*n*

( *dH dx*) *x* *nx*

Picard’s method:

*x*

*f *( *x ) , *

*n*

0 *, * 1 *, * 2 *,.... *

(B‐26)

*n*

1

*n*

Wegstein’s method:

*x*

(1 *q*) *f *( *x *)

*q x , *

*n*

0 *, * 1 *, * 2 *, ... *

(B‐27)

*n*

1

*n*

*n*

in which the auxiliary function, *f *( *x*) , is defined by Definition:

*f *( *x*) *x * *H*( *x*)

(B‐28)

The general form of these three iterative methods is given by *x*

(

*G x *) *, *

*n*

0 *, * 1 *, * 2 *,..... *

*n*

(B‐29)

1

*n*

and for each of the three methods on seeks

to find the fixed point *x * of *G(x) *

such that

*x*

(

*G x *)

(B‐30)

Our stability analysis of Eqs. B‐25 through B‐27 is based on linearizing (

*G x*)

bout

a

the fixed point *x *

. We let *x * and *x*

be small perturbations from the

*n*

*n*1

fixed point as indicated by

*x*

*x * *x , *

*x *

*x *

*n*

*n*

*n*

*x*

1

*n*

1

(B‐31)

This allows us to express Eq. B‐29 as

*x *

*x*

(

*G x * *x *) *, *

*n * 0 *, * 1 *, * 2 *,..... *

(B‐32)

*n*1

*n*

and a Taylor series expansion (See Problems 5‐30 and 5‐31 in Chapter 5) leads to

2

*dG*

*d G*

1

2

(

*G x * *x*

)

(

*G x *)

*x*

*x*

*..... *

(B‐33)

*n*

*n*

2

*n *

2

*dx*

*x*

*dx * *x*

On the basis of Eq. B‐30 this infinite series simplifies to

2

*dG*

*d G*

1

2

(

*G x * *x *) *x *

*x*

*x*

*..... *

(B‐34)

*n*

*n*

2

*n *

2

*dx*

*x*

*dx * *x*

nd

a

we can use Eq. B‐32 to represent the left hand side in a simpler form to obtain

458

*Appendices *

2

*dG*

*d G*

1

2

*x * *x*

*x * *x*

*x*

*..... *

(B‐35)

*n*1

*n*

2

*n *

2

*dx*

*x*

*dx * *x*

At this point we impose a constraint on the higher order terms expressed as

2

*dG*

*d G *

Constraint:

1

2

*x*

*x *

*..... *

(B‐36)

*n*

2

*n *

2

*dx*

*x*

*dx * *x*

so that Eq. B‐35 takes the form

*dG *

*x*

*x*

*, *

*n *

0 *, * 1 *, * 2 *, * 3 *, * 4 *,..... *

(B‐37)

*n*1

*n * *dx * *x*

If we write a few of these equations explicitly as

*dG *

*x*

*x*

(B‐38a)

1

o

*dx * *x*

*dG *

*x*

*x*

(B‐38b)

2

1

*dx * *x*

*dG *

*x*

*x*

(B‐38c)

3

2

*dx * *x*

……………………

*dG *

*x*

*x*

(B‐38d)

*n*

*n*1

*dx * *x*

it becomes clear that they can be used to provide a general representation given by

*n*

*x*

*x*

*dG dx * *, *

*n * 0 *, * 1 *, * 2 *,..... *

*n*

o

*x *

(B‐39)

At this point we see that *x*

0 when

*n * provided

that

*n*

*dG dx* 1

(B‐39)

*x*

When *x*

0 as *n * the system *converges* and one says that the fixed point *n*

*x * is *attracting*. The three special cases represented by Eq. B‐39

can be expressed

as

459

*dG dx*

1 *, *

the fixed point *x * is *attracting*

*x*

I.

(B‐41)

and the iteration *converges*

*dG dx*

1 *, *

the fixed point *x * is *repelling*

*x*

II.

(B‐42)

and the iteration *diverges*

*dG dx*

1 *, *

the fixed point *x * is neither

III.

*x*

(B‐43)

*attracting* nor *repelling*

It is extremely important to note that the stability analysis leading to these three results is based on the *linear approximation* associated with Eq. B‐36. In this development, the word *attracting* is used for a system that converges since *x* *n*

*moves toward* *x * as *n * increases, while the word *repelling* is used for a system that

diverges since *x * *moves away from* *x * as *n * increases. The case in which the fixed *n*

point is neither attracting nor repelling can lead to chaos (Gleick, 1988; Peitgen *et* *al*., 1992).

At this point we are ready to return to Eqs. B‐25, B‐26 and B‐27 in order to determine the linear stability characteristics of Newton’s method, Picard’s method, and

Wegstein’s method.

Newton’s method

In this case we have

*H*( *x*)

(

*G x*) *x *

(B‐44)

*dH dx*

and the derivative that is required to determine the stability is given by 2

*H*( *x*)

*d H*

*dG dx*

(B‐45)

*dH dx*2

2

*dx*

Evaluation of this derivative at the fixed point where *H*( *x *) 0 leads to

*dG dx* 0

(B‐46)

*x*

460

*Appendices *

This indicates that Newton’s method will converge provided that *dH dx* 0

and provided that the initial guess, *x *, is close

gh

enou

to *x * so that Eq. B‐36 is

o

atisfied.

s

If Eq. B‐36 is not satisfied, the linear

stability analysis leading to

Eqs. B‐41 through B‐43 is not valid.

Picard’s method

In this case Eqs. B‐26 and B‐29 provide (

*G x*) *f *( *x*) and

*dG dx* *df dx*

(B‐47)

*x*

*x*

and from Eq. B‐41 we conclude that Picard’s method is stable when

*df dx* 1

(B‐48)

*x*

In Example 7.7 of Chapter 7 we used the fixed point iteration (see Eq. 17) that can be expressed as

*x*

*f *( *x *)

1

C 1

*x*

*, *

C

0 3

*. * 0 *, *

*n*

0 *, * 1 *, * 2 *, * 3 *,..... * (B‐49)

*n*

1

*n*

*n *

This leads to the condition

*df dx* 1 C 1

(B‐50)

that produces the *stable* iteration illustrated in Table 7.7a. In Example 7.8 of Chapter 7 we find another example of Picard’s method (see Eq. 40) that we repeat here as

15 1678

*. *

*x*

*x*

*f *( *x *)

*x*

37 *. * 6190

*n * *, *

*n*

0 *, * 1 *, * 2 *, * 3 *,... * (B‐51)

*n*

1

*n*

*n*

0 *. * 8571 *x*

*n *

The solution is

given by *x * 0 *. * 6108 and this leads to

*df dx*

*. *

(B‐52)

*x*

213 1733

dicating

in

that Picard’s method is unstable for this particular fixed point iteration. This result is consistent with the entries in Table 7.8a.

egstein’s

W

method

In this case Eqs. B

‐27 and B‐29 provide

(

*G x*) (1 *q*) *f *( *x*) *q x* (B‐53)

461

which leads to

(

*G x*) (1 *q*) *f *( *x*) *q x* (B‐54)

From this we have

*dG*

*df*

(1 *q*)

*q *

(B‐55)

*dx*

*dx*

and the stability condition given by Eq. B‐41 indicates that Wegstein’s method will converge provided that

*df*

(1 *q*)

*q * 1

(B‐56)

*dx*

Here one can see that the adjustable parameter *q * can often be chosen so that this inequality is satisfied and Wegstein’s method will converge as illustrated in Examples 7.7 and 7.8 of Chapter 7.

462

*Appendices *

Appendix C

Matrices

**Matrix Methods and Partitioning**

In order to support the results obtained for the *atomic matrix* studied in Chapter 6 and for the *mechanistic matrix* studied in Chapter 9, we need to consider that matter of partitioning matrices. All the information necessary for our studies of stoichiometry is contained in Eq. 6‐22; however, that information can be presented in *different forms* depending on how the atomic matrix and the column matrix of net rates of production are partitioned. In our studies of reaction kinetics, all the information that we need is contained in the mechanistic matrix; however, that information can also be presented in *different forms* depending on presence or absence of Bodenstein products. In this appendix we review the methods required to develop the desired *different forms*.

Matrix addition

We begin our study of partitioning with the process of *addition* (or subtraction) as illustrated by the following matrix equation

*a*

*a*

*a*

*a *

*b*

*b*

*b*

*b *

*c*

*c*

*c*

*c *

11

12

13

14

11

12

13

14

11

12

13

14

*a*

*a*

*a*

*a*

*b*

*b*

*b*

*b*

*c*

*c*

*c*

*c*

21

22

23

24

21

22

23

24

21

22

23

24

(C‐1)

*a*

*a*

*a*

*a *

*b*

*b*

*b*

*b *

*c*

*c*

*c*

*c *

31

32

33

34

31

32

33

34

31

32

33

34

*a*

*a*

*a*

*a*

*b*

*b*

*b*

*b*

*c*

*c*

*c*

*c*

41

42

43

44

41

42

43

44

41

42

43

44

This can be expressed in more compact nomenclature according to A B

C

(C‐2)

The fundamental meaning of Eqs. C‐1 and C‐2 is given by the following sixteen (16) equations:

*Material Balances *

463

*a*

*b*

*c*

*a*

*b*

*c*

11

11

11

21

21

21

*a*

*b*

*c*

*a*

*b*

*c*

12

12

12

22

22

22

*a*

*b*

*c*

*a*

*b*

*c*

13

13

13

23

23

23

*a*

*b*

*c*

*a*

*b*

*c*

14

14

14

24

24

24

(C‐3)

*a*

*b*

*c*

*a*

*b*

*c*

31

31

31

41

41

41

*a*

*b*

*c*

*a*

*b*

*c*

32

32

32

42

42

42

*a*

*b*

*c*

*a*

*b*

*c*

33

33

33

43

43

43

*a*

*b*

*c*

*a*

*b*

*c*

34

34

34

44

44

44

These equations represent a *complete partitioning* of the matrix equation given by Eq. C‐1, and we can also represent this complete partitioning in the form (C‐4)

Here we have shaded the particular partition that represents the first of Eqs. C‐3.

The complete partitioning illustrated by Eq. C‐4 is not particularly useful; however, there are other possibilities that we will find to be very useful and one example is the *row/column partition* given by

(C‐5)

Each partitioned matrix can be expressed in the form

(C‐6)

and the partitioned matrix equation is given by

A

A

B

B

C

C

11

12

11

12

11

12

(C‐7)

A

A

B

B

C

C

21

22

21 22

21

22

464

*Appendices *

We usually think of the elements of a matrix as numbers such as *a *, *a *, etc.; 11

12

however, the elements of a matrix can also be matrices as indicated in Eq. C‐7.

The usual rules for matrix addition lead to

A

B

C

(C‐8a)

11

11

11

A

B

C

(C‐8b)

12

12

12

A

B

C

(C‐8c)

21

21

21

A

B

C

(C‐8d)

22

22

22

and the details associated with Eq. C‐8a are given by

*a*

*a *

*b*

*b *

*c*

*c *

11

12

11

12

11

12

(C‐9)

*a*

*a*

*b*

*b*

*c*

*c*

21

22

21

22

21

22

A little thought will indicate that this matrix equation represents the first four equations given in Eqs. C‐3. Other partitions of Eq. C‐1 are obviously available and will be encountered in the following paragraphs.

Matrix multiplication

Multiplication of matrices can also be represented in terms of *submatrices*, provided that one is careful to follow the rules of matrix multiplication. As an example, we consider the following matrix equation

*a*

*a*

*a*

*a * *b*

*b *

*c*

*c *

11

12

13

14

11

12

11

12

*a*

*a*

*a*

*a*

*b*

*b*

*c*

*c*

21

22

23

24 21

22

21

22

(C‐10)

*a*

*a*

*a*

*a * *b*

*b *

*c*

*c *

31

32

33

34

31

32

31

32

*a*

*a*

*a*

*a*

*b*

*b*

*c*

*c*

41

42

43

44 41

42

41

42

which conforms to the rule that the number of columns in the first matrix is equal to the number of rows in the second matrix. Equation C‐10 represents the eight (8) individual equations given by

*a b*

*a b*

*a b*

*a b*

*c *

(C‐11a)

11 11

12 21

13 31

14 41

11

*a b*

*a b*

*a b*

*a b*

*c *

(C‐11b)

11 12

12 22

13 32

14 42

12

*a b*

*a b*

*a b*

*a b*

*c *

(C‐11c)

21 11

22 21

23 31

24 4

21

*a b*

*a b*

*a b*

*a b*

*c *

(C‐11d)

21 12

22 22

23 32

24 42

22

*a b*

*a b*

*a b*

*a b*

*c *

(C‐11e)

31 11

32 21

33 31

34 41

31

*Material Balances *

465

*a b*

*a b*

*a b*

*a b*

*c *

(C‐11f)

31 12

32 22

33 32

34 42

32

*a b*

*a b*

*a b*

*a b*

*c *

(C‐11g)

41 11

42 21

43 31

44 41

41

*a b*

*a b*

*a b*

*a b*

*c *

(C‐11h)

41 12

42 22

43 32

44 42

42

which can also be expressed in compact form according to

AB

C

(C‐12)

Here the matrices A, B, and C are defined explicitly by

*a*

*a*

*a*

*a *

*b*

*b *

*c*

*c *

11

12

13

14

11

12

11

12

*a*

*a*

*a*

*a*

*b*

*b*

*c*

*c*

21

22

23

24

21

22

21

22

A

B

C

(C‐13)

*a*

*a*

*a*

*a *

*b*

*b *

*c*

*c *

31

32

33

34

31

32

31

32

*a*

*a*

*a*

*a*

*b*

*b*

*c*

*c*

41

42

43

44

41

42

41

42

In Eqs. C‐1 through C‐9 we have illustrated that the process of addition and subtraction can be carried out in terms of partitioned matrices. Matrix multiplication can also be carried out in terms of partitioned matrices; however, in order to conform to the rules of matrix multiplication, we must partition the matrices properly. For example, a proper *row partition* of Eq. C‐10 can be expressed as

(C‐14)

In terms of the submatrices defined by

*a*

*a*

*a*

*a *

*a*

*a*

*a*

*a *

11

12

13

14

31

32

33

34

A

*, *

A

11

21

*a*

*a*

*a*

*a*

*a*

*a*

*a*

*a*

21

22

23

24

41

42

43

44

(C‐15)

*c*

*c *

*c*

*c *

11

12

31

32

C

C

11

21

*c*

*c*

*c*

*c*

21

22

41

42

we can represent Eq. C‐14 in the form

A

A B

C

11

11

11

B

(C‐16)

A

A

B

C

21

21

21

466

*Appendices *

Often it is useful to work with the separate matrix equations that we have created by the partition, and these are given by

A

B

C

(C‐17)

11

11

A

B

C

(C‐18)

21

21

The details of the first of these can be expressed as

*b*

*b *

11

12

*a*

*a*

*a*

*a * *b*

*b*

*c*

*c *

11

12

13

14 21

22

11

12

A

B

*, *

C

(C‐19a)

11

11

*a*

*a*

*a*

*a*

*b*

*b *

*c*

*c*

21

22

23

24

31

32

21

22

*b*

*b*

41

42

Multiplication can be carried out to obtain

*a b * *a b * *a b * *a b*

*a b * *a b * *a b * *a b *

11 11

12 21

13 31

14 41

11 12

12 22

13 32

14 42

*a b * *a b * *a b * *a b*

*a b * *a b * *a b * *a b*

21 11

22 21

23 31

24 41

21 12

22 22

23 32

24 42

(C‐19b)

*c*

*c *

11

12

*c*

*c*

21

22

and equating the four elements of each matrix leads to

*a b*

*a b*

*a b*

*a b*

*c*

11 11

12 21

13 31

14 41

11

*a b*

*a b*

*a b*

*a b*

*c*

11 12

12 22

13 32

14 42

12

(C‐19c)

*a b*

*a b*

*a b*

*a b*

*c*

21 11

22 21

23 31

24 41

21

*a b*

*a b*

*a b*

*a b*

*c*

21 12

22 22

23 32

24 42

22

Here we see that these four individual equations (associated with the partitioned matrix equation) are those given originally by Eqs. C‐11a through C‐11d. A little thought will indicate that the matrix equation represented by Eq. C‐18 contains the four individual equations represented by Eqs. C‐11e through C‐11h. All of the information available in Eq. C‐10 is given explicitly in Eqs. C‐11 and partitioning of the original matrix equation does nothing more than arrange the information in a different form.

*Material Balances *

467

If we wish to obtain a *column partition* of the matrix A in Eq. C‐10, we must also create a row partition of matrix B in order to conform to the rules of matrix multiplication. This *column/row partition* takes the form (C‐20)

and the submatrices are identified explicitly according to

*a*

*a *

*a*

*a *

11

12

13