A maior rede de estudos do Brasil

Grátis
534 pág.

## Pré-visualização | Página 9 de 50

exp(–1.283 � 104� 2000)¼ 5.492 � 10�13 moles.
After 2000 years there will be 5.492 � 10�13 moles of 14C and so 7.688 � 10�12 g of 14C.
After 106 years there will remain 1.271 � 10�68 moles of 14C and so 1.7827 � 10�67 g, which
is next to nothing.
In fact there will be no atoms left because 1:271�10
�68 moles
6:02�1023 � 2 � 10�44 atoms!
The number of disintegrations per unit time dN/dt is equal to lN.
The number of atoms is calculated bymultiplying the number ofmoles by Avogadro’s number
6.023 � 1023. This gives, for 2000 years, 5.4921 � 10�13� 6.023 � 1023� 1.283 � 10�4¼ 4.24 � 107
disintegrations per year. If 1 year� 3 � 107 s, that corresponds to 1.4 disintegrations per
second (dps), which is measurable.
10 This value is not quite exact (see Chapter 4) but was the one used when the method was ﬁrst introduced.
For 106 years, 1.27 � 10�68� 6.023 � 1023� 1.283 � 10�4¼ 9.7 � 10�49 disintegrations per
year. This figure shows one would have to wait for an unimaginable length of time to observe
a single disintegration! (1048 years for a possible disintegration, which is absurd.)
This calculation shows that the geochronometer has its limits in practice! Even if the 14C
content was initially 1 g (which is a substantial amount) no decay could possibly have been
detected after 106 years!
This means that if radioactivity is to be used for dating purposes, the half-life of the chosen
form of radioactivity must be appropriate for the time to be measured.
Exercise
Wewish tomeasure the age of the Earth with 14C, the mean life of which is 5700 years. Can it
be done? Why?
No. The surviving quantity of 14C would be too small. Calculate that quantity.
Four types of radioactivity are known. Their laws of decay all obey the
Curie^Rutherford^Soddy formula.
The nucleus emits an electron spontaneously. AsEnrico Fermi suggested in1934, the neu-
tron disintegrates spontaneously into aproton and an electron.To satisfy the lawof conser-
vation of energy andmass, it is assumed that the nucleus emits an antineutrino along with
the electron.Thedecayequation iswritten:
n! pþ �� þ ��
neutron! protonþ electronþ antineutrino
Too¡setthe (þ ) charge created inthenucleus, theatomcapturesan electronandso‘‘moves
forwards’’ in theperiodic table:
A
ZA! AZþ1Bþ e� þ ��:
In the (Z,N) diagram, the transformation corresponds to adiagonal shiftup and to the left.
Forexample,87Rbdecays into 87Srby thismechanism(seeFigure1.6).
Wewrite, then:
87Rb! 87Srþ �� þ ��:
This long-lived radioactivity is very important in geochemistry. Its decay constant is l
¼ 1.42 � 10�11yr�1. Itshalf-life isT1
2
¼ 49 � 109years.
The nucleus emits a positron (anti-electron) at the same time as a neutrino. A proton
disintegrates into a neutron. A similar but di¡erent process is electron capture by a
proton.
pþ e� ! nþ �
protonþ electron! neutronþ neutrino
Theatomemits aperipheral electronto ensurethenuclide remainsneutral.
A
ZA! AZ�1Bþ eþ þ � �þ radioactivity
or
A
ZAþ e� ! AZ�1Bþ � electron capture:
This is represented in the (Z, N) diagram by a diagonal shift down and to the right.
Notice that neither of these forms of radioactivity involves a change in mass number. It
Neutron number (N)
Sm α
Nd
Pr
o
to
n
n
u
m
b
er
(
Z)
N increases by 1
Z decreases by 1
N decreases by 2
N decreases by 1
Z decreases by 1
Z decreases by 2
or electron
capture
144 147 148 149 150 152 154
142
Sr
Rb
84 86
85
87 88
87
Al
Mg 24
26
25
27
26
143 144 145 146 148 150
Va
lle
y o
f s
tab
ilit
y
Figure 1.6 The various types of radioactivity in the neutron–proton diagram. Notice that all forms of
disintegration shift the decay products towards the valley of stability. Radioactivity seems to restore the
nuclear equilibrium of nuclides lying outside the valley of stability and so in disequilibrium.
is said to be isobaric radioactivity. For example, potassium-40 (40K) decays into argon-
40 (40Ar):
40Kþ e� ! 40Arþ �:
This is avery important form of radioactivity for geologists and geochemists. Its radio-
active constant is l40K¼ 0.581 �10�10 yr�1 and its half-lifeT12¼ 1.19 � 1010 years.11We shall
be lookingat it again.
The radioactive nucleus expels a helium nucleus 42He (in the form of He
þ ions) and heat is
servationofmassandcharge, thedecayequation canbewritten:
A
ZA! A�4Z�2Bþ42 He:
Inthe(N,Z)diagram,thepath isthediagonalofslope1shiftingdowntotheleft.Forexam-
ple, samarium-147 (147Sm)decays intoneodymium-143 (143Nd)by thedecayscheme:
147Sm! 143Ndþ 42He
withl¼ 6.54 � 10�12 yr�1andT1
2
¼ 1.059 � 1011years.
This formofdecayhasplayed an importanthistorical role in the developmentof isotope
geologyandweshallbeusing itonmanyoccasions.
Spontaneous ¢ssion
Fission is a chain reaction caused by neutrons when they have su⁄cient energy. The
elementary reaction splits a uranium nucleus into two unequal parts ^ for example a
krypton nucleus and a xenon nucleus, a bromine nucleus and an iodine nucleus ^ and
many neutrons.These neutrons in turn strike other uranium atoms and cause new ¢ssion
reactions, and neutron reactions on the nuclei formed by ¢ssion.This is ‘‘statistical break-
up’’of uranium atoms into two parts of unequal masses. The nucleus that splits does not
always produce the same nuclei but awhole series ofpairs. Figure1.7 shows the abundance
ofthevarious isotopesproducedbyspontaneous¢ssionof 238U.
EXAMPLE
The Oklo natural reactor
The isotope 238U undergoes spontaneous fissionwhile 235U is subject to fission induced by the
impact of neutrons. Both these forms of fission occur naturally.
11 This is for disintegration of 40K into 40Ar. 40K also disintegrates giving 40Ca, as shall be seen later.
Spontaneous fission of 238U has an extremely low decay constant l¼ 8.62 � 10�17 yr�1.
Induced fission of 235U is a reaction produced in the laboratory or in nuclear reactors by
bombarding uranium with neutrons.
In 1973, a natural nuclear reactor some 2 billion years old was discovered in the Oklo
uranium mine in Gabon. This uranium deposit worked like an atomic pile, that is, with
induced fission of 235U. Apart from a negative anomaly in the abundance of 235U, the
whole series of fission-induced products corresponding to this isotope was detected. This
fission of 235U, which was believed to be confined to laboratories or industrial nuclear
reactors, therefore occurred naturally, probably triggered by � disintegration of 235U,
which was much more abundant at the time. Nature had discovered nuclear chain
reactions and atomic piles some 2 billion years before we did! Oklo is a unique example
to date.
Exercise
Given that the 238U/235U ratio nowadays is 137.8, what was the activity level of 235U per gram
of ore 2 billion years ago for a uranium ore that today contains 30% uranium?
The decay constants are l238¼ 0.155 125 � 10�9 yr�1 and l235¼ 0.9885 � 10�9 yr�1.
The activity of 235U was 1247 disintegrations per second per gram (dsg). Today the activity of
235U is 172 dsg.
Light nuclei
Fissile nuclei
Neutrons
Neutrons generated
by fission process
a
10
1.0
0.1
0.01
60 80 100 120
Mass number
Y
ie
ld
%
140 160
0.001
b
Figure 1.7 Spontaneous