All?gre Claude J. Isotope Geology

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(5) Agesofthreepopulationsofzircons

Little B eltMountains 1935^2 000 Ma and 2630^265 0 Ma

Fin land 2700^2800Ma and 2000 Ma

Maryland 600 60Ma

It canbeseen thattwo ofthe zircon populations are double.

(6) A straightline.

Chapter 4

(1) C / N ¼14.4 million!PlantshaveC/Nratios of100^200. Draw yourown conclusions!

(2) About1350years! Itis thereforenottheshroud inwhichChristwaswrapped.

(3)



cosm



cosm





measured



measured





normal



cosm





normal

(4) 8.9 10

years.

(5) a ¼1.62 0. 1mm Ma

1

Chapter 5

(1) (i) Basalt >1.6 mg, granite > 3mg.

(ii) Yes: basalt (2 Ga) ¼0.7072; basalt (0.5 Ga) ¼0.700 98; granite (2 G a) ¼0.7829;

granite(0.5Ga) ¼0.7 197 .

(iii) 3ø for Sr, 2% for the Rb /Sr ratio.

(iv) Five 200-g pieces arebetter than one1-kg piece.The result is cross-referenced and

thereis a hope ofconstructing an isochron.

(2) 110 16 ka.

(3) 2 77 5Ma.

(4) (i) (a) 1.12 ionsper minute.

(b) Theuncertainty is 11540 years, or 20%, comparedwith 0.7% uncertaintyon

the present-day measurement.

(d) Countingfor 8 days,300 mgofcarbonwould havetobe extracted.

(ii) The error by the

230

Th^

234

U method is 420 years for an age of 55 000 years, which

corresponds to 0.76%. It is therefore an excellent method for this half-life, far

superior to

C in theory. Geochemical conditions mu st hold for it to apply, that is:

closedsystem andsu⁄cient abundancelevels.

(5) (i) The

Rb/

Srageofthegneissesis1.05Ga.

(ii) Theageofthe granite determinedbythe concordiadiagram is 2.11Gawith an inter-

cept less than about1Ga (see ¢gure).The apparent ages andthegeometric relations

establishedbygeologyaretherefore contradictory. Agranitethatcross-cuts agneiss

cannot be older than the gneiss! Eitherof two hypotheses may hold.The ¢rst (A) is

that both gneiss and granite are about 1 Ga old (given by the concordia intercept

and the R b ^Sr isochron).The second (B) is toaccept thatthe gneiss is 2 Ga old and

499 Solutions to problems

thegranitealittleyounger,butthatthewholewassubjectedtoalargetectonic crisis1

Ga ago, making the zircons discordant and partially re-homogenizing the

Rb^

Sr system.

How do we choose between A and B? Hypothesis B is the more likely for two

reasons. First, it is the c oncordia diagram that gives two ages of 2 Ga and ab out

1Ga. Now, the1Gacorresponds tothe

Rb^

Srageofth e gneisswhich has apoor

alignment and ab ove all a (

Sr/

Sr)

initial

ratio of about 0.713, which is very clear

and indicates isotope re-homogenization. If we take the average of the

Sr/

and

Rb/

Sr ratios, we get an average p oint. Taking an initial ratio of 0.705, we

¢nd an age of 1.95 Ga. Everything seems to be coherent therefore. However, of

course,aseriesof U^Pbmeasu rementsonthezircon ofthegranitewouldbeneeded

to con¢r m this.

(6) Initialage: 2.7Ga, Grenvilleorogeny1.1Ga.Thezircon agesof2.8Gashowthatsomeof

the zi rcon is inherited.

(7) 235 5Ma.

(8) (i) All the time intervals are mathematically possible. Allowing for di¡usion

T < 30 Ma.

(ii) Rockofinterestwould contain zircon, apatite, andsphene.

0.1 0.2 0.3 0.4 0.5

Rb/

Sr/

0.6 0.7 0.8 0.9

0.705

0.710

0.715

0.720

0.725

0.718

0.35

~ 2 Ga

0.719

0.45

0.722

0.6

0.723

0.9

0.730

0.724

0.8

0.713

0.1

(207/235)*

5 10

2.5 Ga

2 Ga

1 Ga

0.1

0.2

0.3

0.4

0.5

0.6

(206

238)

500 Solutions to problems

(9) The rhyolite is the resultofpartial m elting (whether followed bydi¡erenti ation or not)

ofthe ancientbasaltcrust.Thisburied hydratedcrustisheated and meltsalittle togive

rise tothe rhyolite.

(10) See Figure5.11for how to constructthe diagram.

Chapter 6

(1) (i) The initial two p roportions in the twovolcanogenic sediments are 0.576 and 0.423,

respectively, giving:

Sr=

SrÞ

J:Sedian

¼ 0:709 46 and ð

Rb=

SrÞ

Sedian

¼ 0:1239:

(ii) Initial ratios: (

Sr/

Sr)

granite

¼0. 718 65, (

Rb/

Sr)

granite

¼3.885.

Final ratio:(

Sr/

Sr)

granite

¼0.7738.

(2) (i) Mass of

Ar assumed tobe contained in the core approx. 2 0 10

g, mass of

in the lower mantle 60  10

g.This putative evaluation does not alter the general

idea behind the

Ar balance, a missing part of which must be in the mantle.

(ii) Concentration ofnon-radiogenicl ead inthe core C

¼ 1:81 ppm.

(iii) For T ¼3 Ga, (

206

Pb/

204

Pb)

lower mantle

¼17.4 0, (

207

Pb/

204

Pb)

lower mantle

¼14.74.

Thevalues for the closedsystem are17.35 and14.53, respectively.Thepointisslightly

tothe rightofthe geochronat 4.5Ga.

(iv) TakingT ¼4 Ga, we have (

206

Pb/

204

Pb)

lower mantle

¼17.9 0 an d (

207

Pb/

204

Pb)

lower

mantle

¼16.10.

(v) This phenomenon places the rep resentative points i n the J domain but is insu⁄-

cientto give leadvalues with an isotope signature like the islan d of St. Helena. For

th is,50% ofthelead in the mantle wou ld havehad topass into the coreafter 4.4 Ga,

which seems a lot.The idea of explaining OIBs with high  values by this mechan-

ismdoesnotseemtobe corrob oratedbythe data.

(3) (i) For the mantle, the re injection of continental cr ust playsvirtually no part.This is

becaus e ofthe mass di¡e rence.Only the initial di¡erentiation is seen.

U/Pb

Whole

rock

e A

Granites 0.2

–3

–4

–5

Basalts 0.1

Ultramafic 0.05

4.55 Ga 1 Ga

0.1

100

0.01 Ga

0.0001 Ga

100

0.001

4.55

Ga 1 Ga

Basalts

Ultramafic

Granites

Sphenes, Apotites

Zircons

0.1 Ga

100

0.01 Ga

0.0001 Ga

100

0.001

501 Solutions to problems

Forthe continental crust, itdoes playa role, butits evolution does notcorrespond

to observation. The concavity observed with tim e is the opposite of what is

observed, whi ch seems on the contrary to involve increasing recycling of conti-

nental cru st.

(ii) Neither the Nd nor S r curves of mantle evolution show the evolution really

observed, which seeaprogressiveonsetofprimitivemantle evolution.

(iii) Overall, withthe s implifyingassumptions madehere, this model does notaccount

for the observations very well (see¢gure).

(4) (i) The upper mantle has a mass of 1  10

g with 5 ppb uranium, corresponding to

2 10

moles.

(ii) Production of

Hein1Ga is 2.12 times thatofuranium in moles, therefore 4.45 10

moles. The quantity of

He outgassed is about 10

moles yr

1

.Therefore, the resi-

dence time

He ¼440 Ma is equal to half the residence time of the lithosphere,

whi ch means that when the oc eani c lithosphere forms at the mid-ocean ridges,

there is enrichmentin

Hetowardsthe melting zone.

(5) (i) Values of W

are calculated i n both hypotheses 

¼ 0:5110 and



¼ 0:5120, which are the extremes using the balance equation .Nextthe

model ages are cal culated using the two extre me data by the age formula

deduced from continental crust data. This gives a pair of values (W

, T

)

which are (0.2465; 2.84 Ga) and (0.4608; 1.08 Ga).Values of W

are calculated

with the  balance equation under both extreme c onditions 

Sm=Nd

¼ 0:227 and



Sm=Nd

¼ 0:28.

ThenT

is calculatedusing th e formula for the depleted mantle.We get a pairof

values (W

) of (0.256; 2.6 Ga) and (0.4882; 0.964Ga).We then takea (W

,T)

plan e and draw the two straight lines corresponding to the pair of values (W, T)

(I)

Time (Ga)

Mantle

Bulk Earth

Continental

crust

Continental

crust

without

filtering

1 2 3 4

-10

-15

-20

Time (Ga) Time (Ga)

Sr/

Residual

mantle

Bulk Earth

Continental

crust

Continental

crust

without

exchange

1 2 3 4

0.725

0.720

0.715

0.710

0.705

Nd model age

Sediments

2 134 0

Concordia

Armstrong’s model of continent growth.

502 Solutions to problems

taken two by two (see ¢gure). Their point of intersection gives W

¼0.335,

¼2. 05 10

years.

Returning then to the balance equations, we get: 

¼ 0:511 567 and



Sm=Nd

¼ 0:241.

(ii) Using W

is calculated as 0.22. From the balance equation for  values we

deduc e 

¼0.409. From the balance equation for the  values, we deduce



¼ 0:713 28.We check by calculating theT

mo del age on the depleted mantle

values T

¼2.1 Ga; with continental crust values T

¼1.90 Ga, for an average of

2 Ga.Theoverall pictureisfairlycoherent.

Chapter 7

(1) The ¢rst job is to esti mate the fractionation factors atthe missing temperatures þ10 8C

and 30 8C.

(i) We plot the two curves 

, d

against temperature and interpolate and extrapo-

late linearly, which is warranted because they seem to vary linearly. The complete

table isgivenbelow.



þ20 1.085 1.00 98

þ10 1 .10 1 .0106

01.11231.0117

^20 1.1492 1.0141

^30 1.175 1.0155

Time (Ga)

1 2 3

0.5

0.4

0.3

0.2

503 Solutions to problems

(ii) The results areasgivenbelow.

vapor

rain

vap

rain

1 ^1 3 3 ^33 ^1 6.3 ^6.5

2 ^208 ^96 ^24.13 ^12.4

3 ^307 ^158 ^33.57 ^1 9.47

4 ^424 ^249 ^43.95 ^28.38

(iii) See ¢gure.

(2) The equationis:

d ¼ d

þ 10

ð  1Þln f:

For th e ¢ rst phase, f varies from 1 to 0.7. The results are given in the table and ¢gure

below.

As canbeseen, greatprecision is required tobringoutthesevariations.

–100

–200

–300

–100

–200

–300

–40 –30 –20

0.9 0.8 0.7 0.6

Rain

Vapor

0.5 0.4 0.3 0.2

–10

P = 8

δD



1 ¼0.0002



1 ¼0.00025



1 ¼0.00035

f ¼0.9 f ¼0. 7 f ¼59, f ¼0.84 f ¼0.49, f ¼0. 7 f ¼0.388,f ¼0. 79

Melt 5.52 5.571 5.614 5.66 5.74

Olivine 5 .32 5 .37 5 .41 4 5 .46 5 .54

Pyroxene 5.3 1 5.36 5.44

Plagioclase 5.14

504 Solutions to problems

(3) W ith d

¼70% and Craig’s straightline ofprecipitation, we deduce d

¼10%.W e cal-

culateth e partition coe⁄cientat 550 8C.



A B

¼A 10

2

þB

AB 

mineral^ water

Quartz 4.10 3.7 2.35

Magnetite 1.47 3.7 5.8

Muscovite 1 .9 3.10 0.41

Feldspar 3.13 3.41 1.20

Calcite 2.78 2.89 1.21

Plagioclase 2.15 2.0 1.1 49

Thebalance equationiswritten:

 d



þ d

 d



þ d

 d



 x

feld

 d

ðÞx

þ d

cal

 d

ðÞx

þ d

plag

 d



¼ Q:

Fromthisweget:

silicate

 d

¼ 0:99 ð2Þ;

butwe alsoknowth e initi al balance:

initial silicate

 y

þ d

initial water

¼ d

initial

where y

and y

are calculatedal lowing for thefactthatbymass:

O ¼ 15% and silicate ¼ 85%:

In the silicates, oxygen ¼54%; in the water, oxygen ¼88.8%.Therefore the initialover-

allvalue ofthesystem, ifassumed closed, is:

initial

¼ 7  0:77 ð10Þ0:23 ¼ 5:39  2:3 ¼ 3:09:

0.9

5.7

5.6

5.5

5.4

5.3

5.2

5.1

0.8

Magma

Olivine

Pyroxene

Plagioclase

0.7 0.6

0.5 0.4 0.3 0.2

505 Solutions to problems

This is pres erved, sowehave:

silicate

þ d

¼ 3:09 ð2Þ:

We th en de du c e d

byeliminating d

silicate

between (1) and (2):

¼ 1:05

All thatisleftis to calculatethe minerald values:

quartz

¼þ3:4

magnetite

¼4:7

muscovite

¼ 0:68

feldspar

¼þ2:29

calcite

¼þ2:28

plagioclase

¼þ2:39:

(4) (i) The quantityofcarbonburnt is 5.9 10

g, which corresponds to410 ppm of CO

(ii) Now, the content is 330 ppm.The remainder has been dissolved in the ocean after

homogenization.

(iii) The d

calcite

hasgonefrom þ3to0.

(iv) If d canbe measuredwithaprecision of 0.1 d or 0.05 d,this criterion canbeused as a

pollution control.

(5) D

gm

¼ d

 d

maj

¼ d

 xd

2 þ 1  xðÞd

2

x ¼

2

þ SO

2

x ¼

KTðÞ



þ KTðÞðÞ

gm

¼ 7:5x  4:5:

Thereforeifx < 0.6 

g^m

itis negative, otherwiseitispositive.

(6) (i) d ¼0.344and1.386,respectively.

(ii) d ¼^1.05.

(iii) 0 .008 1 d m

1

and 0.011 08 d m

1

for the two snowball scenarios and for the

scorched Earth scenario 0.0148d m

1

(iv) No.

Chapter 8

(1) W e ¢nd:

R ¼

ﬃﬃﬃﬃﬃ

506 Solutions to problems

A decay reaction, for example, decay of organic matter, obeys a second-order kinetic

equation:

x ¼

kt þ 1

(2) Th efrequency100 kawillbe continuedunchange d.The40 kaand20kafrequencieswill

bephas e-shiftedand dampedand thereforethe100kafrequency willpredominate.

(3) (i) The upper mantle 1  10

g with 5 ppm of U, corresponding to 2  10

moles.

Productionof

He in1Ga is 2.2 tim es thatofuranium in moles, therefore4.4 10

yr.

Thequantityof

Hed egassedis about10

molesperyear.

(ii) Therefore the residence time of

He of 440 Ma is twice th e residence time in the

lithosphere.

(iii) This means that when the oceanic lithosphere forms at the ocean ridges, there is

enrichmentof

Hetowardsthe melting zones.

(4) Mixing times in the upper mantle

Mixingtime ( Ma)

NorthAtlantic 350

South Atlantic 138

South-westIndian Ocean 533

NorthPaci¢c 42

CentralIndianOcean 77

Yes,thereis a relationwiththe expansion rate(see¢gure).

100

200 300

N. Pacific

Central

Indian

S. Atlantic

N. Atlantic

S. W.

Indian

400

Time

500

507 Solutions to problems

INDEX OF NAMES

Ahrens,L. H.,70

Albare

de, F., 247,438

Aldrich, L.T.,9,35,282

Alle

gre,C. J.,35,53,73,74,75,85,89,

90,95 ,96,98,199 , 201, 202, 239,

242, 24 4,248,260,271,287, 289,

301,303,305,306,309,310,311,

313,315,31 7,323,336,337,338,

449,467,468,471

Alvarez, L.,282

Arndt,N.T.,185

Arnold,J. R.,

1,126

Aston,F.,1

Atkinson, J.,139

Bainbridge, K.T., 2, 9

Bard, E.,121

Barrell, J.,189,190

Becquerel, H.,18,25

Beltrani, E.,438

BenOthman, D., 239, 271 ,272

Bennett,V. C., 272, 273

Berger, G.W.,66,67,77,181,182

Berner,R., 424,438,451,457

Bigeleisen, J. ,365,367,382

Birck,J. L.,35,53

Blichert-Toft,J.,84,103,247, 248

Boato,G.,361,362

Bohr, N.,1

Boltwood, B. B.,188

Bottinga,Y.,386

Bourdon,B.,349

Bowen,N., 226,401

Boyet, M.,352

Bradley, P.,41 3 ,416

Bre

vard, O .

Broecker,W.,125,143,144,

147

Burbidge,J.,139

Burbidge,M.,139

Carlson,R.,352

Castaing, R.,99

Chaussidon, M.,54

Chupka,W.,2,8

Clarke,W., 282

C l ay ton , D. D. , 138, 206

Clayton,R.,371 ,372,385,

386,432

Compston,W., 2,79,100,275

Condomines,M.,89, 90,323,326

Cornog, R.,282

Craig,H.,282,285,359 ,361,362,

363,394,397,399,400

Curie,M.,19,36

Curie,P.,18,19,25,188,194

Damon, P., 288,313

Dansgaard,W.,381,395,396,

409,41 9

Dars, R.,96

Darwin, C.,25

Davis,G.

DeNiro,M.,431,432

DePaolo, D., 234,236,237, 259,

272, 273, 274

Doe,D.,301

Duplessy,J. C.,408

Dupre

,B.,185,186,302,305,306

Edmond, J.,425

Eiler,J.,389

Einstein,A.,1

Elliot,T.,327

Emiliani,C.,406,407,408,41 5

Epstein, S.,362,370,382,385,389,

394,397,406,409,41 1

Er icson, J.,407

Farquhar, J.,295,429,430

Farquhar, R.

Fermi, E.,1

Fourcade,S.,379

Fou r ier, J., 25,456

Fowler, W ., 139

Galer, S.,446

Gamow, G.,139

Gangarz,A. J.,301

Ganoun, H.

Gast, P.,67 ,313

,318

Gaudette, H., 275

Geiss,J., 298

Gerling,E.,195,196,198

Ghosh, P.,389,392

Goldstein, S.,426

pel,C.,200,201,309

Gourlan,A.,426

Grand, S.,339

Gru

menfelder, M.,73,98,99

Haberman, R.,438,458

Halliday, A.,53 ,54,93, 202

Hamelin, B.,303,305

Hamilton, P. J.,83 ,185

Hanson, G.,67 ,303,304

Harper, C. L.,53,34 8

Harrison,T. M.,66,350,429

Hart,S. R.,67,68,181,182,234,

248,305,311,313

Hauri, E.,311

Hawkesworth,C.,276

Hayden, R. J.,65

Hedge, C.,313

Heinrich, H.,

419

Hemming, S.,426

Herzog, R., 9

Hirt,B.,35 ,84

Hofmann, A.,307,313,315,330,

332,334,335

Hohenberg, C., 92,93,205

Holland, R.,430

Holmes, A.,189,190,195,197,296

Honda, M.,126

Houtermans, F., 139, 195,

198, 296

Hoyle, F.,139

Hubble,E.,195

Hurley, P., 253, 265,26 6,313

Hutton,J.,193

Inghram, M.G.

Ito , E.,403