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Bradley, R. S. (ed.) Global Changeofthe Past, Boulder, CO: University Corporation for

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Urey, H. C. (1947). Thethermodynamic propertiesofistopicsubstanc es.J. Chem.Soc.Lond., 562^81.

Urey, H. C. , Lowenstam,H., Epstein, S.,and McKinney, C. R.(1951). Measurementof

paleotemperaturesofthe Upper Cretaceous of England, Denmarkandthesouth-eastern

United States. B ull.Geol.Soc.Amer.62,399^426.

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de,F.(1995).Introductionto GeochemicalModeling.Cambridge, UK: Cambridge University

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gre, C. J. andJaupart, C. (1985). Continentaltectonicsand continentalkinetics.EarthPlanet.Sci.

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gre, C. J.and Lewin, E.(1995). Isotopicsystems and stirringoftheEarth’s mantle. EarthPlanet.

Sci. Lett.136,629^46.

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gre, C. J.and Turcotte, D. (1985).Implications ofatwo-componentmarble-cake mantle. Nature

323,123^7.

Alle

gre, C. J., Bre

vard, O., Dupre

,B., and Minster,J. F.(1980). Isotopicand chemical e¡ectsproduced

in a continuouslydi¡erentiatingconvecting Earth mantle. Phil.Trans.Roy.Soc.Lond. A 29 7,

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Alle

gre, C. J., Moreira,M., and Staudacher,T. (1995).

He/

Hedispersion andmantle c onvection.

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Beltrani,E. (1987). MathematicsforDynamic Modeling. NewYork:AcademicPress.

Galer, S. and O’Nions, R. K. (1985). Residencetime ofthorium,uranium andleadinthe mantlewith

implication s for mantle convection.Natu re 316,778^82.

Haberman, R.(1977). Mathematical Models.NewYork: Prentice Hall.

Jacobsen, S. (1988). Isotopic c onstraintson cr ustalgrowthandrecycling. EarthPlanet. Sci. Lett. 90,

315^29.

Jacobsen, S. andWasserburg, G. J. (1981).Transport modelsforcrustand m antle evolution.

Tectonophysics 75,163^79.

Javoy,M.,Pineau,F.,andAlle

gre, C. J.(1982). Carb on geodynamic cycle. Natu re 30 0,171^3.

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489 References

APPENDIX

Table A.1 Symbols and orders of magnitude

Pre¢x Factor

exa (E) 1 0

peta( P) 10

tera(T) 10

giga (G) 10

mega (M) 10

kilo(k) 10

hecto(h) 10

deca (da) 10

deci (d) 10

1

centi (c) 10

2

milli(m) 10

3

micro ()10

6

nano(n) 10

9

pico(p) 10

12

femto (f) 10

15

atto (a) 10

18

Table A.2 Constants

Symbol Value

Speedoflight c 2.997 924 58 10

1

Electron charge e 1.6 02 177 33 10

19

Planckconstant h 6. 620 60755 10

34

Boltzmann constant k 1.380 658 10

23

1

Gravitational constant G 6.6726 10

11

2

Electron rest mass m

0.910 938 97 10

30

Atomic mass unit u 1.66 05402 10

27

Avogadro constant N

6.022136 7 10

mole

1

Idealgas constant R 8.314 510 J mole

1

1.989cal mole

1

Table A.3 Geological data

Mass ofthe Earth 5.9737 10

Vo l u m e 1.0 83 2 0 10

Mean radius assumingspherical Earth 6371000 m

Mean densityof Earth 5515 kg m

3

Mass ofatmosphere 5.1 10

Mass ofoceans 1.37 10

Mass of ice caps 2.9 10

Mass offreshwater (rivers and lakes) 3 10

Mass ofcontinental crust 2.36 10

Mass ofwhole mantle 4 10

Mass ofupper mantle (above670 km) 1 10

Mass ofcore 1.950  10

Mass ofoutercore 1.85 10

Mass of innercore 9.7 10

Areaof E arth 5.100 655 10

Areaofoceans 3.62 10

Areaofcontinents (and continental margins) 2 10

Areaofexpos ed continents 1.48 10

Areaof Atlantic Ocean 9.8 10

Areaof Indian Ocean 7.7 10

Areaof Paci¢c Ocean 1.7 10

Hydrothermal£ux atoceanridges 12.3 10

kg yr

1

Fluxofriversto ocean 4.24 10

1

¼4.24 10

kg yr

1

Fluxofriverse diment load 1.56 10

kg yr

1

Fluxofoceanic crustformed 8.4  10

kg yr

1

Fluxofoceanic lithosphere created 8.4 10

kg yr

1

Length ofoceanridges 50 000 km

Flux from hotspots 2.5 10

kg yr

1

Average sea-£oorspreading rate 3 cmyr

1

Fluxoflithospheric subduction 8.4  10

kg yr

1

Average altitude oflandmasses 875 m

Average depthofoceans 3794 m

Average thicknessofcontinents 35 km

Average thicknessofoceanic cr ust 6.0 km

Averageheat£ow 87 mWm

2

Totalgeothermal£ow 44.3 TW

Average continentalheat£ow 65 mWm

2

Average oceanicheat£ow 101mWm

2

Solar £ux 1373 Wm

2

491 Appendix

Table A.4 Long-lived radioactivity

Percentageofelement Decay constant,l (yr

1

) Meanlife,T (yr) Half-life,T

1=2

(yr) Product

1.167 10

4

total





4.962 10

10



cap0.581 10

10

Total5.5 43 10

10

2. 0 15 10

17. 21 10

1.8 0  10

1.397 10

11.93 10

1.25 10

Rb



0.25 1.4210

11

70.42 10

48.8 10

987

138





0:089



cap





2.25. 10

12

4.44 10

3.0 8 10

138



total



cap4.4 10

12

Total6.65 10

12

2.2 10

1.5 10

1.57 10

1.0 4 10

147

Sm (15 .07) 6.54 10

12

152.88 10

1.0 6 10

11 143

176

Lu 



(2.6) 2 10

11

50  10

3.5 10

10 1 76

187

Re 



(63.93) 1.5 10

11

66.66 10

4.6 10

10 187

190

Pt(0.0127) 1.16 10

12

862 10

6 10

11 186

232

Th(100)(6,4



)4.947510

11

20.21 10

1.4 010  10

10 208

235

U(0.73)(7,5



)9.848510

10

1.015 38 10

0.703 809 9  10

9207

238

U(99.27)(8,6



)1.5512510

10

6.44 10

4.4683 10

9206

Table A.5 Extinct radioactivity

Percentageof

element

Decayconstant,l

(yr

1

)

Meanlife, T

(Ma)

Hal f-life,T

1=2

(Ma)

Pro duct

Al

9.7 10

7

1.03 0.714 58

Be



4.6 10

8

2.16 10

1.5 10

710

Cl



2.25 10

0.44 0.308

Ar (98.1% )

S(1.9%)

Ca

6. 7 10

6

0. 1 5 0.1

Mn 

1.886 10

7

5.3 3.3867

Fe 2



4. 76 1 10

7

2.2 1.456

Nb

2.777 10

8

36 25.67

107

Pd 



1.0 63 10

7

9.4 6.53 8

107

129

I 



4 10

8

25 17. 327

129

146

Sm 6.849  10

9

146 101.19

142

182

Hf2



7.6 92 10

8

13 9.01

182

205

Pb

4.62 10

8

21.6 4 15

205

244

Pu ¢ssion 8.26 4 10

9

121 83.91 Xe

¢ssion

247

Cm2,3

4.4 10

8

22.5 15.6

235

492 Appendix

Table A.6 Half-lives of radium isotopes used in geology

Parent Ra isotope Half-life

238

226

Ra 1622years

235

223

Ra 11.435 days

232

228

Ra 6.7 years

232

224

Ra 3 . 64 day s

Table A.7 Half-lives and decay constants of disintegration

reactions for radioactive chains used in geology

Nuclide Half-life (years) Decayconstantl (yr

1

)

234

U 2.48 10

2.794 10

6

230

Th 7.52 10

9.217 10

6

226

Ra 1.622 10

4.272 10

4

210

Pb 22.26 3.11 10

2

231

Pa 3.248 10

2.1 34 10

5

Table A.8 Mass ratios of selected elements

Chemical massratio Isotope ratio

(Rb/Sr) 0.341 (

Rb/

Sr)

(Sm/Nd) 1.645 (

147

Sm/

144

Nd)

(Lu/Hf) 1.992(

176

Lu/

177

Hf)

(Re/Os) 0.212 (

187

Re/

188

Os)

(Re/Os) 0.0252(

187

Re/

186

Os)

(U/Pb) 70(

238

204

Pb)

This is indicativeonlybecause the ¢gurevarieswiththe leadisotope composition. Care must

betakenwithchemi calvaluesintheliteratureforrockswhen comparedwith,say,carbonaceous

meteorites,which haveprimitiveisotopic compositions.

Table A.9 Values of selected isotope ratios

Bulk silicate earth Initialvalues

143

Nd/

144

Nd 0.512 638 0.505 83

147

Sm/

144

Nd 0.196 6

176

Hf/

177

Hf 0.282 95 0.279 78

176

Lu/

17 7

Hf 0.036

Sr/

Sr 0.70 47 0.698 998

Rb/

Sr 0.031

187

Os/

186

Os 1.06 0.805

187

Os/

188

Os 0.130 0.0987

187

Re/

186

Os 3.3

187

Re/

188

Os 0.412

206

Pb/

204

Pb 18.426 9.3 07

207

Pb/

204

Pb 15.518 10.294

208

Pb/

204

Pb 39.081 29.476

238

204

Pb 9.2

232

Th/

238

U4.25

493 Appendix

Table A.10 Atomic number (Z), chemical symbol, and atomic mass

(A) of the natural elements

Atomic number Element Atomic mass

1Hydrogen(H)1.0079

2Helium(He)4.00260

3Lithium(Li)6.941

4Beryllium(Be)9.01218

5 Boron (B) 10.81

6Carbon(C)12.011

7 Nitrogen (N) 14.0067

8Oxygen(O)15.9994

9 Fluorine (F) 18.998 40

10 Neon (Ne) 20.179

11 Sodium (Na) 22.9898

12 Magnesium (Mg) 24.305

13 Aluminum(Al) 26.981 54

14 Silicon (Si) 28.086

15 Phosphorus (P) 30.97376

16 Sulfur (S) 32.06

17 Chlorine (Cl) 35.453

18 Argon (Ar) 39.948

19 Potassium(K) 39.098

20 Calciu m (Ca) 40.0 8

21 Scandium (Sc) 44.9559

22 Titanium (Ti) 47.90

23 Vanadium (V) 50.9414

24 Chromium(Cr) 51.996

25 Manganese (Mn) 54.9380

26 Iron(Fe) 55.847

27 Cobalt(Co) 58.9332

28 Nickel (Ni) 58.71

29 Copp er (Cu) 63.545

30 Zinc(Zn) 65.38

31 Galliu m (Ga) 69.72

32 Germanium(Ge) 72.59

33 Arsenic (As) 74.9216

34 Selenium (Se) 78.96

35 Bromine(Br) 79.904

36 Krypton (K r) 83.80

37 Rubidium (Rb) 85.468

38 Strontium (Sr) 87.63

39 Yttrium(Y) 88.9059

40 Zirconium(Zr) 91.22

41 Niobium(Nb) 92.9064

42 Molybdenum(Mo) 95.94

43 Technetium (Tc) (97)

44 Ruthenium (Ru) 101.07

45 Rhodiu m (Rh) 102.9055

46 Palladium (Pd) 106.4

47 Silver (Ag) 107 .868

48 Cadmium (Cd) 112.40

49 Indium (In) 114.82

494 Appendix

Table A.10 (cont.)

Atomic number El ement Atomic mass

50 Tin(Sn) 118.69

51 Antimony (Sb) 121.75

52 Tellurium (Te) 127 .60

53 Iodine(I) 126.9045

54 Xenon (Xe) 1 31.30

55 Cesium (Cs) 132.9054

56 Bariu m (Ba) 137.34

57 Lanthanum (La) 138.9055

58 Cerium (Ce) 140.12

59 Praseodymium(Pr) 140.9077

60 Neo dym ium (Nd) 144.24

61 Promethium (Pm) (145)

62 Samarium (Sm) 150.4

63 Europium (Eu) 151.96

64 Gadolinium (Gd) 157.25

65 Terbium (Tb) 158.9524

66 Dysprosium (Dy) 1 62. 50

67 Holmium (Ho) 1 64.9304

68 Erbium(Er) 167.26

69 Thulium ( Tm) 16 8.9342

70 Ytterbium (Yb) 173.04

71 Lutetium (Lu) 1 74.97

72 Hafnium(H f) 178.49

73 Tantalum (Ta) 180.9479

74 Tungsten(W) 183.85

75 Rhenium (Re) 186.2

76 Osmium (Os) 190.2

77 Iridium (Ir) 192.2

78 Platinum (Pt) 195 .09

79 Gold (Au) 196.9665

80 Mercury(Hg) 200.61

81 Thalliu m (Tl) 204.37

82 Lead(Pb) 207.2(variable)

83 Bismuth(Bi) 208.9804

84 Polonium(Po) (209)

85 Astatine (At) (210)

86 Radon(Rn) (222)

87 Francium (Fr) (223)

88 Radium (Ra) 22 6.0254

89 Actinium (Ac) 227.0278

90 Thorium(Th) 232.0381

91 Protactinium (Pa) 231.0359

92 Uranium (U) 238.029

93 Neptunium( Np) 237.0 4 8

94 Pluton ium (Pu) (244)

95 Americium (Am) (243)

96 Curium (Cm) (247)

495 Appendix

FURTHER READING

Albare

de, F., IntroductiontoGeochemical Modeling, Cambridge University Press,1995.A ¢nebook

on modelingalthough a littlehardgoingfor those notkeyedup in mathematics.

Bourdon,B.,Turner,S., Hend erson,G., and Lundstrom,C. C., IntroductiontoU-SeriesGeochemistry,

Reviewsin Mineralogyand Geochemistry vol.52, Washington,DC, GeochemicalSocietyand

MineralogicalSocietyof America,2003. A very good book for the di¡erent applications of

radioactives ser ies.

Bradley, R.,Paleoclimatology,NewYork, Academic Press,1999. An excellentclari¢cation of isotopic

methods andproblemsin climatology.

Broecker,W. S. and Peng,T.-H.,Tracersinthe Sea, NewYork,Eldigio Press of ColumbiaUniversity,

1982.Thereferencebook forstudentsandanyoneinterestedin marin e isotopegeochemistry.

C l ayton , D. D. , Principlesof Stellar Evolution and Nucleosynthesis,UniversityofChicago Press,1968.

Aclearandwell-writtenbookforanyonewantingtogointoth e nitty-grittyofnuclearastrophysics

and nucleosynthesis.

Dalrymple,B.,TheAgeofthe Earth, StanfordUniversity Press,1991. An outstandi ngbookon the age

ofthe Earth and agood introductiontoisotopegeology.

De Paolo, D., Neodymium Isotope Geochemistry, Heidelberg, Springe r-Verlag, 1988. The reference

book for neodymium isotope geochemistry.

Dickin, A., Radiogenic Isotope Geology, 2nd edn, Cambridge University Press, 2005. The book

for advanced students to move on to after this one. Excellent.

Encyclopedia ofEarthSystems Science, London,Academic Press,2005. An exhaustiveintroduction

to studyingexternal geodynamicsystems.

Faure, G., PrinciplesofIsotope Geology, NewYork,JohnWiley,1986.Aprimer for ageneration now.

Muse

um nationald’histoire naturelle,LesMe

orites, Paris,Bordas,1996. Aclearand simple review

ofmeteoritesaccessibletoall. (InFrench.)

Roth,E. and Poty, B., NuclearMethodsof Dating, New York,Springer-Verlag,1989. Acollection of

papersreviewing the main methodsofnucleardating.

Rowlinson, H., UsingGeochemical Data,Harlow, UK , Longman,19 93. Abooktoinitiate studentsin

theuse ofgeochemi cal data.

Tu rcotte,D. andSchubert,G., Geodynamics,CambridgeUniversity Press,2002.A referencebookfor

geophysics.

Valley, J. M.,Taylor, H. P., and O’Neil, J. R.,StableIsotopesin HighTemperatureGeologicalProcesses,

Reviewsin Mineralogy vol.16,Washington, DC,MineralogicalSocietyof America.Agood

round-up ofresearchon high-temperaturestable isotopegeochemistry, foradvanced students.

Walker, J., Num erical Adventureswith GeochemicalCycles,OxfordUniversity Press,1990. An

excellentintroductiontogeochemical cycles us ingsimple computer technology.

SOLUTIONS TO PROBLEMS

Chapter 1

(1) The mass of one atom of

Ois28:455 75  10

27

kg. Since the ato mic mass u nit is

1:660 540 2  10

27

kg,themassof

Oinmassunitis17.1364415.

The mass of each of the two molecules

CDH

and

is17.133367 132 in mass

u nits. The di¡erence is 0.002 77, whi ch in relative mass is about 1.610

4

whi ch corre-

sponds toaresolutionpowerof 6100for the interference ofseparation.

(2) Th ereare31.7 ppmoflithium inthe rock

(3) The measured isotopic ration

¼ R

mes

is equal to the real ratios þ the pollution

(estimatedbytheblank).ThesupposedisotopicratiooftheblankisR

.Therealisotopic

ratio ofthe sampleis R

.So:

mes

¼ R

ð1  xÞþR

ðXÞ;

where (X) is the mass fraction of the blank in the mixture (this is the same formula

as isotope dilution).

If the precision is measured at 1.10

4

, R

mes

^ R

should be greater than 10 times this

value:

mes

 R

1:10

5

whi ch translates to

ðR

 R

ÞX

5

;

sinc e R

^ R

0.006 and

1:610

3

x ’

blank

sample

So if the sample is10

6

gofSr,x < 1.6 10

9

g. If we increase the accuracy by10 times

theblankshouldbe0.1610

9

(4) The radius must be 63 meters if the angle of in cidence is at 908 to the electromagnet’s

inputfaces,31meters ifthe angle is 27%.

(5) Presentday:13.59 dps; 4.5 10

yearsago: 77.15 dps.

(6) Production ofradioactiveheatfora mantleofprimitive composition:19.9 10

W. Fo r a

mantleanalogoustothe upper mantle:2.6 10

Urey ratios: (i) 47%, (ii) 6%.The second ratio makes the production of radioactive

heat virtually negligible. All the internal heat would thereforebe related to the Earth’s

earlyhistory.

Chapter 2

(1) Answers on pp. 288and following.

(2) Apparent ages in Ma

206

Pb/

238

207

Pb/

235

207

Pb/

206

208

Pb/

232

473 510 677 502

472 489 572 469

442 460 548 471

439 45 7 547 492

(3) 5.14 Ma;4.2Ma.Theycorrespondtolavafromdi¡erenteruptions.

(4) In activity:

234

238



234

238



l

þð1 

l

230

238



230

 l

234

238

 1



ð1 

eðl

230

l

234

Þþ½1  e

230

:

(5) (i)

230

231



excess

230

231



initial

excess

ðl

230

l

231

Þt

(ii) About300 000 years.

Chapter 3

(1) 0. 75 Ga.

(2) WithT being thetemperature, wehave:

Ar ¼ð

ARÞ

exp 

þ GT



(3) (i) The ages are: PykeHil l: 2.75 G a;Fred’s Flow: 2.58 Ga;Theo’s Flow: 2.46Ga.

(ii) Thesul¢desnotcontaininguraniumlieontheisochrons andgiveroughlytheiniti al

values oftheleadisotope compositions.

(4) Lu^Hfages: M101 ¼24.7 1.2 Ma, M 214 ¼30.6 2Ma.

Sm^Ndages: M101 ¼23.6 4.3 Ma, M 214 ¼20.0 7Ma.

The Sm^Nd ages seemyounger thanthe Lu^Hfages,butthepyroxenitebeds areprob-

ably24 Mayearsold.

498 Solutions to problems