nuclear stability and radioactive decay

7/26/2019 Nuclear Stability and Radioactive Decay

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Nuclear stability and radioactive decay

Introduction:

For a nucleus to be stable, the number of neutrons should in most cases be little higher than the number of

protons. For example, oxygen has three stable isotopes ]10,9,8[,, 18

8

17

8

16

8 =NOOO and five known

unstable (i.e. radioactive) isotopes ]12,11,7,6,5[,,, 20

8

19

8

15

8

14

8

13

8 =NOandOOOO .

In the case of the isotopes ]7,6,5[,, 15

8

14

8

13

8 =NOOO there are not enough neutrons for stability while

the isotopes ]12,11[, 20

8

19

8 =NOO have too many neutrons.

Different radioactive decay processes:

1. Beta Decay:

Nuclei such as O15

8 which are lacking in neutrons, undergo +-decay. In this process one of the protons in

the nucleus is transformed into a neutron and a positron and a neutrino are emitted. his transformation iswritten as

+ +

NO 15

7

15

8

where !"signifies the emitted positron which in this context is called a !#ray and $ denotes the neutrino.

%y contrast, nuclei like O19

8 which are excessively rich in neutron, decay by decay emitting a

negative electron and an antineutrino&

+

FO 19

9

19

8

It should be noted that in both !"#decay and decay the atomic mass number remains the same.

2. Electron capture:

' nucleus lacking in neutrons can also increase its neutron number by electron capture. In this process, anatomic electron interacts with one of the protons in the nucleus and a neutron is formed of the union. hisleaves a vacancy in the electron cloud which is latter filled by another electron. sually the electron that iscaptured by the nucleus is the innermost or #electron and so this mode of decay is also called K- capture.

' classic example of #capture is the transformation of vanadium#*+ into titanium *+ &

TieV 49220

1

49

23 +

. !lp"a Decay:

'nother way by which some unstable nuclei undergo radioactive decay is by the emission of an alpha

particle - He4

2. For example

HeThU 4

2

234

90

238

92

+

/ecay by #-e$ission is comparatively rare in nuclides lighter than lead, but it is common for the heaviernuclei.

%. &a$$a 'adiation:

he nucleus formed as the result of !#decay (" or #), electron capture or 0#decay is often left in an excitedstate following the transformation. he excited nucleus then decays by the emission of one or more (-rays(photons).

'n example of gamma ray production follows&First cobalt#12 decays to excited nickel#12 by beta decay&

hen the nickel#12 drops down to the ground state by emitting a gamma ray&

3amma rays of 4.45 6e7 and 4.88 6e7 are produced.

1
http://en.wikipedia.org/wiki/Cobalthttp://en.wikipedia.org/wiki/Nickelhttp://en.wikipedia.org/wiki/Beta_rayshttp://en.wikipedia.org/wiki/Cobalthttp://en.wikipedia.org/wiki/Nickelhttp://en.wikipedia.org/wiki/Beta_rays


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)a*s of radioactive decay:

here is only one law which governs all decay processes. his law states that the probability per unit timethat a nucleus will decay is a constant, independent of time. his constant is called the 9decay constant: andis denoted by ;.


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1

}1

]/{[

1

0

0

0

0000

==

+=

===

dte

dtete

tedtetdNtN

t

tt

tt

Now, we know

t

teAeAtA

== 00)(

In can be seen from this e=uation that in one mean (average) life, the activity falls to (4Be) of its initial value.

Bindin, ener,y:

he force of electrostatic repulsion between like charges which varies inversely us the s=uare of theirseparation, would be expected to be so large that nuclei could not be formed. he fact that they do exist isevidence that there is an even larger force of attraction. his nuclear force acts only when the nucleons arevery close to each other and binds them into a compact stable structure. o disrupt a nucleus and separate itinto its component nucleons, energy must be supplied from the outside.

' given nucleus is lighter than the sum of its separate nucleons, the difference being the binding mass#energy. otal mass of separate particles 6ass of the atom

>N mn" C mp#6

@mn" 4mp6

>@-4.22D11"4-4.225D@#8.2452@

>2.22D4@ amu>2.22D4@x+84 6e7>5.E1 6e7

Note that we have given the mass in amu -'tomic 6ass nit. 'tomic mass units are so defined that the

mass of a C126 atom, the most abundant isotope of carbon, is exactly 4@ amu. ne amu is defined as12

1

-mass of C126 atom. he value of 4 amu is

4 amu > 4.112E*x42#@5

kg whose e=uivalent energy

> 4.11x42#@5x(8.2x42D)@> 4.*+x42#42G> MeVx

x931

1061

10491

19

10

=

.

.

.

-6ass of C126 atom>4@ g. herefore, 4 amu>12

1-mass of C126 atom>4 g.

Now, one gram atom of carbon contains 1.2@x42@8atoms.

herefore, 4 amu >23100226

1

x.>4.112E*x42#@5kg

otal binding energy % and the binding energy per nucleon %B' for UandFe 238

92

56

26 can be shown to be&

3


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For Fe56

26 , % > *[email protected] 6e7, %B' > *[email protected] > D.5+4 6e7

For U238

92 , % > 4D2@ 6e7, %B' > 4D2@B@8D > 5.E54 6e7

In Figure below, we show the binding energy per nucleon as a function of mass number.

Fig. & %inding energy per nucleon as a function of mass number.

he peak at '>* corresponds to the exceptionally stable He4

2 nucleus which is alpha particle. he

binding energy per nucleon is a maximum for nuclei of mass number ' >E1.

his figure suggests that we can liberate energy from the nucleus in two different ways. If we split a

heavy nucleus into two lighter nuclei, energy is released because the binding energy per nucleon is

greater for the two lighter fragments than it is for the original nucleus. his process is known asnuclear fission. For example, if the uranium nucleus is broken into two smaller nuclei, the binding

energy difference per nucleon as about 2.D 6e7. he total energy given off is therefore

-2.D 6e7Bnucleon-@8E nucleons>4DD 6e7

'lternatively when we combine two light nuclei into a heavier nucleus, again, energy is released

when the binding energy per nucleon is greater in the final nucleus than it is in the two original

nuclei. he process is known as nuclear fusion. For instance, if two deuterium- H21 nuclei

combine to form a He4

2 helium nucleus, over @8 6e7 is released. In fact, nuclear fusion is the

main energy source of the sun and other stars.

Nuclear ission

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Hhen a nucleus fissions, it splits into several smaller fragments. hese fragments, or fission products, are

about e=ual to half the original mass. wo or three neutrons are also emitted.

Fig& Nuclear Fission

he sum of the masses of these fragments is less than the original mass. his missing mass (about 2.4percent of the original mass) has been converted into energy according to Ainsteins e=uation.

Fission can occur when a nucleus of a heavy atom captures a neutron, or it can happen spontaneously.

' typical fission reaction might be

nCsRbnU 2861415556

9337

23592 +++

f course, many different fission reactions are possible, with many different final products. he number of

neutrons produced in the fission process can likewise vary, but the average is about @.E. he two neutrons

emitted in the fission process shown in the reaction above areprompt neutrons# they are emitted essentially

at the instant of fission. 'bout 4J of the neutrons in the fission process are delayed neutrons emitted

following the decay of the heavy fragments. Delayed neutronsplay significant role in the mechanical control

of thermal reactors.

Ener,y released fro$ eac" fission:

41E 6e75 6e71 6e75 6e71 6e7+ 6e7

2 /e0

K kinetic energy of fission productsK gamma raysK kinetic energy of the neutronsK energy from fission productsK gamma rays from fission productsK anti#neutrinos from fission products

6ost of the energy is released as kinetic energy of the fission fragments. hese relatively heavy fragments

do not travel very far through the reactor fuel element before they dissipate most of their kinetic energy

5

?ketch of induced nuclear fission, a neutron (n) strikes auranium nucleus which splits into similar products (F. L.), andreleases more neutrons to continue the process, and energyin the form of gamma and other radiation.
http://en.wikipedia.org/wiki/Image:Nuclear_fission.gif


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in collisions with the atoms of the fuel element. he energy can be extracted as heat and used to boil

waterM the resulting steam can be used in a conventional way to drive a turbine to generate electricity.

Nuclear "ain 'eactions

' chain reaction refers to a process in which neutrons released in fission produce an additional fission in at

least one further nucleus. his nucleus in turn produces neutrons, and the process repeats. he process

may be controlled (nuclear power) or uncontrolled (nuclear weapons).

If each neutron releases two more neutrons, then the number of fissions doubles each generation. In that

case, in 42 generations there are 4,2@* fissions and in D2 generations about 1 x 42 @8(a mole) fissions.

@8E" n fission " @ or 8 n " @22 6e7

ontrolled Nuclear ission

o maintain a sustained controlled reaction, for every @ or 8 neutrons released, only one must be allowed to

strike another uranium nucleus. If this ratio is less than one then the reaction will die outM if it is greater thanone it will grow uncontrolled (an atomic explosion). Nuclear reactions are controlled by a neutron#absorbing

material, such as cadmium.

Nuclear uels:

@ & (@8E" @8D)@

h@ &@8@h " @

6O & Lu@" @

Natural uranium contains three isotopes& #@8* (2.221J), #@8E (2.5J), and #@8D (++.8J). he speed

re=uired for a fission event vs. non#fission capture event is different for different isotopes.

Fissile 6aterial & @8E, @88, @8+Lu, @*4LuFertile 6aterial & @8@h, @8D, @*2Lu,

Fertile 6aterial " Neutron Fissile 6aterial

6
http://en.wikipedia.org/wiki/Uranium-235http://en.wikipedia.org/wiki/Uranium-238http://en.wikipedia.org/wiki/Uranium-238http://en.wikipedia.org/wiki/Uranium-235http://en.wikipedia.org/wiki/Uranium-238


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deuterium-tritium!u"ion re#$tion i" $on"idered t%e

mo"t &romi"ing !or &rodu$ing !u"ion &o'er.

It may be mentioned that fissile materials like @8E, @88, @8+Lu will fission with very low energy neutrons, while@8D re=uires fast neutrons of 4 to @ 6e7 of energy to fission.

Natural fission reactor:

his natural fission reactoris believed to have operated in 'frica two billion years ago for a period of perhaps

several hundred thousand years. his reactor used naturally occurring uranium as a fuel and naturally

occurring water as a moderator. It would not be possible to build such a reactor today, because the capture

of neutrons by the protons in water results in too few neutrons remaining to sustain a chain reaction in

uranium with only 2.5 percent of @8E. Powever, two billion years ago, naturally occurring uranium contained

a much larger fraction of @8E than does present#day uranium. %oth @8E and @8D are radioactive, but the half#

life of @8E is only about on#sixth as great as the half#life of @8D. If we go back in time about @x42+ y, which is

half of one half#life of @8D, there was about *2 percent more @8D than there is today, but there was @8>D

times as much @8E. Naturally uranium was then about 8 percent @8E, and at such enrichments, ordinary

water can serve as an effective moderator. ' deposit of such uranium, in a large enough mass and with

ground water present to act as moderator, could have gone 9critical: and began to react. he reaction could

have been controlled by the boiling of the water# when enough heat had been generated to evaporate some

of the water, the reaction would slow down and perhaps stop, because of the lack of a moderator. Hhen the

uranium had cooled sufficiently to allow more li=uid water to collect, the reactor would have started up again.his cycle could in principle have continued indefinitely, until enough @8E were used up or until geological

changes resulted in the removal of the water.

he discovery of this reactor followed the observation of a French researcher that the uranium that was

being mined from that region in 'frica contained too little @8E. he discrepancy was a very small one#the

samples contained 2.5454 percent, compared to the usual 2.5@2@ percent# but it was enough to stimulate the

curiosity of the French workers. hey guessed that the only mechanism that could result in the consumption

of @8E was the nuclear fission process, and this guess was tested by searching in the ore for stable isotopes

that result from the radioactive decay of fission products. Hhen such isotopes were found in abundances

very different from what would be expected from 9natural: mineral deposits, the existence of the natural

reactor was confirmed.

Nuclear usion

Fig& Nuclear Fusion

Nuclear energy can also be released by fusion of two light

elements (elements with low atomic numbers). he power

that fuels the sun and the stars is nuclear fusion. In a

hydrogen bomb, two isotopes of hydrogen, deuterium and

tritium are fused to form a nucleus of helium and a neutron. his

fusion releases 45.1 6e7 of energy. nlike nuclear fission, there

is no limit on the amount of the fusion that can occur.

I$portant fusion reactions

7
http://en.wikipedia.org/wiki/Deuteriumhttp://en.wikipedia.org/wiki/Image:FusionintheSun.pnghttp://en.wikipedia.org/wiki/Image:FusionintheSun.pnghttp://en.wikipedia.org/wiki/Image:FusionintheSun.pnghttp://en.wikipedia.org/wiki/Image:FusionintheSun.pnghttp://en.wikipedia.org/wiki/Image:FusionintheSun.pnghttp://en.wikipedia.org/wiki/Image:FusionintheSun.pnghttp://en.wikipedia.org/wiki/Image:FusionintheSun.pnghttp://en.wikipedia.org/wiki/Image:FusionintheSun.pnghttp://en.wikipedia.org/wiki/Image:D-t-fusion.pnghttp://en.wikipedia.org/wiki/Deuterium


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!strop"ysical reaction c"ains

he most important fusion process in nature is that which powers the stars. he net result is the fusion of

four protons into one alpha particle, with the release of two electrons, two neutrinos, and energy, but several

individual reactions are involved, depending on the mass of the star. For stars of the siQe of the sun or

smaller, the proton#proton chain dominates. In heavier stars, the RN cycle is more important.

roton-proton cycle

?ince the sun is composed of ordinary hydrogen, rather than deuterium, it is first necessary to convert the

hydrogen to deuterium. his is done according to the reaction

+++ +eHHH 211

1

1

1

his process involves converting a proton to a neutron and is analogous to the beta#decay processes

discussed earlier. nce we have obtained @P (deuterium), the next reaction that can occur is

++ HeHH 311

1

2

1

followed by

HHeHeHe 1

1

4

2

3

1

3

1 2++

Note that the first two reactions must occur twice in order to produce the two 8Pe we need for the third

reaction. He can write the net process as

he net result is the fusion of four protonsinto one alpha particle, with the release of twoelectrons, two

neutrinos, and energy, but several individual reactions are involved, depending on the mass of the star. ?incethe two positrons disappear in this process, the only masses remaining are four hydrogen atoms and the one

helium atom, and so

MeVuMeVuuxcmmQ fi 7.26)/5.931()002603.4007825.14()( 2

===

Aach fusion reaction liberates about @1.5 6e7 of energy.


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In heavier stars, the RN cycle is more important. ' more likely se=uence of reactions in the carbon cycle isshown below&

HeCHN

eNO

OHN

NHC

eCN

NHC

412115

1515

15114

14113

1313

13112

++

++

++

++

++

++

+

+

Notice that the 4@R plays the role of catalystM we neither produce nor consume any 4@R in these reactions, but

the presence of the carbon permits this se=uence of reactions to take place at a much greater rate than the

previously discussed proton#proton cycle. he net process is still described by * 4P *Pe, and of course the

S value is the same. ?ince the coulomb repulsion between P and R is larger than the Roulomb repulsion

between two P nuclei, more thermal energy and a correspondingly higher temperature are needed for the

carbon cycle. he carbon cycle probably becomes important at a temperature of about @2 O 421, while the

?unTs interior temperature is only 4E O 421.

Hhen all of the hydrogen has been converted to helium, the ?un will contract and its temperature will

increase until helium burning occurs, by processes such as

8*Pe4@R

wo Pe nuclei have a larger mutual Roulomb repulsion than two P nuclei, so helium fusion needs more

thermal energy than hydrogen fusion.

Hhen the helium is used up, a still higher temperature will allow carbon fusion to make even heavier

elements, for example, @*6g. ?uch processes will continue until E1Fe is reachedM beyond this point no further

energy is gained by fusion.

3"e ydro,en Bo$b: 3"e Basics

' fission bomb, called the primary, produces a flood of radiation including a large number of neutrons. his

radiation impinges on the thermonuclear portion of the bomb, known as the secondary. he secondary

consists largely of lithium deuteride. he neutrons react with the lithium in this chemical compound,

producing tritium and helium.

his reaction produces the tritium on the spot, so there is no need to include tritium in the bomb itself. In the

extreme heat which exists in the bomb, the tritium fuses with the deuterium in the lithium deuteride.

he most interesting fusion reactions are the following&

(4) / " *Pe (8.E 6e7) " n (4*.4 6e7)

(@) / " / (4.24 6e7) " p (8.2@ 6e7) (E2J)

(8) 8Pe (2.D@ 6e7) " n (@.*E 6e7) (E2J)

(*) / " 8Pe *Pe (8.1 6e7) " p (4*.5 6e7)

(E) " *Pe " @ n " 44.8 6e7

(1) 8Pe " 8Pe *Pe " @ p " 4@.+ 6e7

(5) 8Pe " *Pe " p " n " [email protected] 6e7 (E4J)

(D) *Pe (*.D 6e7) " / (+.E 6e7) (*8J)

(+) *Pe (2.E 6e7) " n (4.+ 6e7) " p (44.+ 6e7) (1J)(42) / " 1


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(4@) 8Pe " 1 (*.28"45.1"8.@5)B@ > [email protected] 6e7 and the energy in charged particlesas Ech> (*.28"8.E"2.D@)B@ > *.@ 6e7.

'nother uni=ue aspect of the /#/ reaction is that there is only one reactant, which must be taken intoaccount when calculating the reaction rate.

Hith this choice, we tabulate parameters for four of the most important reactions.

fuel Z Efus4/e05 Ec"4/e05 neutronicity

/# 4 45.1 8.E 2.D2

/#/ 4 [email protected] *.@ 2.11

/#8Pe @ 4D.8 4D.8 K2.2E

p#44% E D.5 D.5 K2.224

he last column is the neutronicityof the reaction, the fraction of the fusion energy released as neutrons.

his is an important indicator of the magnitude of the problems associated with neutrons like radiation

damage, biological shielding, remote handling, and safety. For the first two reactions it is calculated as (Efus#

Ech)BEfus. For the last two reactions, where this calculation would give Qero, the values =uoted are rough

estimates based on side reactions that produce neutrons in a plasma in thermal e=uilibrium.

10
http://en.wikipedia.org/wiki/Protonhttp://en.wikipedia.org/wiki/Deuteriumhttp://en.wikipedia.org/wiki/Tritiumhttp://en.wikipedia.org/wiki/Aneutronic_fusionhttp://en.wikipedia.org/wiki/Protonhttp://en.wikipedia.org/wiki/Deuteriumhttp://en.wikipedia.org/wiki/Tritiumhttp://en.wikipedia.org/wiki/Aneutronic_fusion


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Nuclear 'eactors:

' nuclear reactoris a system in which a controlled nuclear chain reaction is used to liberate energy. In a

nuclear power plant, this energy is used to generate steam, which operates a turbine and turns an electricalgenerator.

uel:In a typical reactor, the uranium fuel is in the form of uranium oxide pellets, which are inserted end to

end into long hollow metal tubes. n an average, each fission of a U235 nucleus produces about @.E free

neutrons, so *2J of the neutrons are needed to sustain a chain reaction. ' U235 nucleus is much more

likely to absorb a low#energy neutron -less than 4 e7 than one of the high energy neutrons -4 6e7 or so

that are liberated during fission. he isotope U238 can also absorb neutrons, leading to 239U , but not with

high enough probability for it to sustain a chain reaction by itself. hus uranium that is used in thermal

reactors is often 9enric"ed: by increasing the proportion of U235 above the natural value of 2.5J, typically

to 8J or so, by isotope#separation processing.

11


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/oderator: In a thermal nuclear reactor, the higher#energy neutrons are slowed down by collisions with

nuclei of a light material -like water, deuterium, graphite surrounding the fuel, called the $oderator. hese

slowed down neutrons called thermal neutrons are used to initiate further fissions in a thermal reactor.

ontrol rod: he rate of the fission chain reaction is controlled by inserting or withdrawing control rods

made of elements -such as boron or cadmium whose nuclei absorb neutrons without undergoing any

additional reaction.

oolant: Peat produced in the core of the reactor is taken out by pumping the coolant -water, / @, R@, etc.

through the core and transferred to a "eat e6c"an,er to produce steam to run the generator. %efore

transferring the coolant to the core it is cooled in the condenser.

7afety devices: ?afety is very important in a nuclear reactor. It is shut down automatically in case of

malfunctioning. he radioactive material is never allowed to diffuse outside.

12


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78)0ED E9!/)E7

Lroblem No. 4& ?how that after 42 half#lives a radioactive material is reduced to 4B4222 part approximately.

?olution& From radioactive decay law, one can show that a radioactive material, after n half#lives, will decay

tonNN )2/1(0= .

1000/1024/)2/1( 0010

0 NNNN ==

Lroblem No. @& he half#life of radium is 4122 years. 'fter how much time (4B41) thpart of radium will remain

un#disintegrated in the sampleU

?olution& 3iven 16/0NN= , therefore,nNN )2/1()2/1()16/1(/ 40 ===

4= n

herefore, time of disintegration > No. of half#lives x half#lives > * x 4122 > 1*22 years.

Lroblem No. 8& Find the half#lives of a radioactive material if its activity drops to (4B41) thof its initial value in

82 years.

?olution& He know thatnNN )2/1(0= .

4

)2

1()

2

1(

16

1 4

0

=

===

nor

N

N n

herefore, Palf#life > otal time of disintegration B No. of half#lives > 82 years B * > 5.E years.

Lroblem No. *& Hhat percentage of initial amount of a radioactive material decays during the time, e=ual to

mean lifetime of this materialU

?olution& Number of radioactive nuclei left after one mean lifetime

>

/1,/000 ====

eNeNeNN t

63100)7.2

11(100)

11(100

0

0 ===

= xxe

xN

NNdecayofPercentae

Lroblem No. E& here is a stream of neutrons of kinetic energy of 2.2@E e7. If the half#life of neutron is 522

seconds, what fraction of neutrons will decay, before they travel a distance 42 mU

?olution& He know that smxx

xxx

m

!x"#m#!" /1019.21067.1

106.1025.2.2

2

1. 327

192 ====

ime needed to travel a distance of 42 m > 42 m B (@.4+ x42 8) > *.E5x42#8seconds.

13


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6

0

)700/1057.4(

0

1053.61

99999347.03

==

===

xN

Ndecayedfraction

eeN

N xt

Lroblem No. 1& ' sample of uranium is a mixture of three isotopes UandUU 238

92

235

92

234

92 , , present in the

ratio 2.221J, 2.54J and ++.@D*J respectively. he half#lives of these isotopes are @.Ex42 Eyears, 5.4x42D

years and *.Ex42+years respectively. Ralculate the contribution to activity (J) of each isotope in sample.

?olution& No. of U234

92 nuclei in the mixture >0

24N >234

1002.6006.0 23xx(aking total mass of 422gm)

No. of U235


25N >235

1002.671.0 23xx

No. of U238


28N >238

1002.6284.99 23xx

he relative contribution of the isotopes in the activity (J) would be

45.46*13.2*41.51926.0*0425.0*02.110926.0*100425.0*1002.1

105.4238

.01002.6284.99*

101.7235

693.01002.671.0*

105.2234

693.01002.6006.0**

101010

9

23

8

23

5

23

28

0

2825

0

2524

0

24

xxxor

xx

xxx

xx

xxx

xx

xxxNNN

Lroblem No. 5& he half#lives of U235

92 and U238

92 are 5.4x42Dyears and *.Ex42+years respectively. oday

the isotopic abundance of U235

92 and U238

92 are respectively 2.5@J and ++.@DJ. 'ssuming that initially

these isotopes were in e=ual abundance and no isotopic separation has occurred, calculate the age of these

elements on the earth.

?olution&

t

t

eNN

eNN

28

25

0

2828

0

2525

=

=

In the beginning,0

28

0

25 NN = .

yearsxxt

year$erxxT

year$erxxTNo%

tor

tor

e

et

t

910

10928

2/128

10825

2/125

2825

2528

1099.510]54.176.9/[927.4

1054.1105.4/693.0/693.0

1076.9101.7/693.0/693.0

]/[927.4

927.488.137ln][

88.13772.0

28.9925

28

==

===

===

=

==

==

14


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Lroblem sheet on radioactivity

Lroblem No. 4& ' radioactive source has a half#life of 4 min. 't time t>2, it is placed near a detector, and the

counting rate (the number of decay particles detected per unit time) is observed to be @222 countsBsec. Find

the counting rate at times t>4 min, @ min, 8 min, and 42 min.

Lroblem No. @& If the detection efficiency in Lroblem No. 4 is @2 J, (a) how many radioactive nuclei are there

at time t>2U +b) Pow many nuclei decay in the first minuteU

Lroblem No. 8& Hhat percentage of initial amount of a radioactive element decays during the time, e=ual to

mean lifetime of this elementU

Lroblem No. *& ?how that after 42 half life, the material is reduced to -4B4222 part approximately.

Lroblem No. E& he half life of radium is 4122 years. 'fter how much time th.- 161 part of radium will

remain undisintegrated in a sampleU

Lroblem No.1& Find the number of alpha#decays that occur in a one gram sample of @8@h in one year if the

disintegration constant ; of @8@h is 4.EDx42#4Dsec#4.

Lroblem No. 5& Ralculate the activity of one gram of @8@h using the data of Lroblem No. 1.

Lroblem No. D& It is found that *1.8 mgm of naturally occurring potassium show a beta# activity of 4.E disBsec.

he isotope responsible for this activity is *2 which makes up 2.24@ J of the natural mixture. Ralculate the

half#life of *2.

Lroblem No. +& Ra226

88 decays to radon gas. Ralculate (a) the decay constant ; and (b) the initial activity of

one micro#gram of Ra226

88 . 3iven half#life of Ra226

88 > 41@2 years.

Lroblem No. 42& 6ass spectrometric analysis of potassium and argon atoms in a moon rock sample shows

that the ratio of the numbers of (stable) *2'r atoms present to the number of (radioactive) *2 atoms is 42.8.

'ssume that all the argon atoms were produced by the decay of potassium atoms, with a half#life of 4.@Ex42+

years. Pow old is the rockU

Lroblem No. 44& he radioactive isotope E5Ro decays by electron capture with a half#life of @5@ days. (a) Find

the decay constant and the lifetime. (b) If you have a radiation source containing E5Ro, with activity @.22 VRi,

how many radioactive nuclei does it containU (c) Hhat will be the activity of your source after one yearU

Lroblem No. 4@& %efore 4+22 the activity per mass of atmospheric carbon due to the presence of 4*R

averaged about 2.@EE %= per gram of carbon. (a) Hhat fraction of carbon atoms were 4*RU (b) In analyQing

an archeological specimen containing E22 mg of carbon, you observe 45* decays in one hour. Hhat is the

age of the specimen, assuming that its activity per mass of carbon when it died was that average value of the

airU

Lroblem No. 48& ' bone containing @22 gm of carbon has a !#decay rate of *22 decaysBmin. Pow old is the

boneU (In radioactive carbon datin, 4*R decays by emitting !# and its half#life is E582 years. he

radioactive 4*R is produced in the upper atmosphere in nuclear reactions caused by cosmic rays. ?ince living

organisms continually exchange R@ with the atmosphere, the ratio of4*R to 4@R in a living organism is the

same as the e=uilibrium ratio in the atmosphere, which is about 4.8x42 #4@. 'fter an organism dies, it no longer

absorbs 4*R from the atmosphere, so the ratio of 4*R to 4@R continually decreases due to radioactive decay of4*R. here are about 4E.2 decays per minute per gram of carbon in a living organism. sing this result and

the measured number of decays per minute per gram in a nonliving sample of bone, wood or other obWect

containing carbon, we can determine the age of the sample.)

Lroblem No. 4*& 'n old wooden piece has @E.1J of radioactive carbon as compared to ordinary wood. Find

its age, if its half life is E512 years.

Lroblem No. 4E& he activity of a radioactive substance is decreased to D5.EJ in a course of E years. Hhat

is its half lifeU Ralculate the time in which the activity will fall by D5.EJ.

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Lroblem No. 41& he ratio of #@8E to #@8D in natural uranium deposits today is 2.225@. Hhat was this ratio

two billion years ago. he half#lives of the two isotopes are 2.52*x42+ y and *.*5x42+ y, respectively.

LX%


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Lroblem No.4*. In a thermonuclear reaction 4x42#8kg hydrogen is converted into 2.++8x42 #8kg Pe. Raculate

the energy released. If the efficiency of the generator is EJ, calculate the electrical energy produced in

HP.

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nuclear stability and radioactive decay

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