Radioactivity and
radioisotopes
• Half-life
• Exponential law of decay
Half-life
The half-life of a radioactive element means:
a) The time taken for half the radioactive
atoms in the element to disintegrate
Radioactive
atoms
HALF-TIME
Radioactive
atoms
Decayed
atoms
b) The time taken by the radiation from the
element to drop to half its original level
HALF-TIME
What does decay rate depend on?
In other words, there is a 50% chance that any
radioactive atom within the sample will
decay during a half-life time T½.
Consider the b emitter Fe-59. Its half-life is 46
days.
Plot a graph of the fraction of undecayed atoms
vs time (days).
Click here for radioactive decay simulation
Half-life of Fe-59
Fraction of undecayed atoms
1 1/4
1
1
3/4
1/2
1/2
1/4
1/4
1/8
0
0
46
92
Time (days)
138
0
184
What does decay rate depend on?
Can you now answer by considering the graph
you drew? Explain your answer.
The rate of radioactive decay of Fe-59 atoms
depends on the number of atoms itself. In
fact, our graph is not a straight line, which
means that the number of atoms decaying
changes with time, i.e. with the number of
radioactive nuclides left. The number of
radioactive nuclides left after each half-life
drops to ½, not of the original amount, but
of the amount left. This means that not all
Fe-59 has decayed after 2 x 46 days, but
only ¼ of the original amount is left.
Exponential law of decay
Now,
Interpolate
plot the
thegraph
logarithm
of theoflogarithm
the fraction
to base
remaining
10 of
the
afterfraction
120 days.
of Fe-59 remaining against time.
Logarithm of fraction remaining - time
Log of fraction remaining
0
46
92
0
-0.301
-0.602
0.785
-0.903
-1.204
Time (days)
138
184
Exponential law of decay
Consider the table of data from the example
on the previous slides.
Time (day)
Fraction
remaining (F)
log(F)
0
1
0
46
1/2
-0.301
92
1/4
-0.602
138
1/8
-0.903
184
1/16
-1.204
Exponential law of decay
Can you notice any pattern in the log(F)?
Explain your answer.
Each reading of F is divided by 2 (1, ½, ¼, …),
therefore, the value of log(F) must have log2
subtracted from it to get the next reading.
In fact;
log(a/b) = log(a) – log(b)
log(1/2) = log(1) – log(2) = 0 – 0.301 = -0.301
The same applies to the other fractions.
Exponential law of decay
Using the table and similar triangles find the
fraction remaining after 150 days.
138days 150days

 0.903
x
x = -0.982
 x = log(F150)
 F150 = antilog(-0.982) = 10-0.982 = 0.10
Exponential law of decay
From the previous discussion, we can conclude
that the rate of radioactive decay is
proportional to the number of radioactive
atoms:
dN

N
dt
Where N is the number of radioactive atoms still
present at time t.
Exponential law of decay
The previous proportionality gives the following
equation:
dN
 lN
dt
The constant l the decay constant, and it is
measured in s-1.
A solution to the above equation is:
N  N 0e
 lt
N0
N x
2
Exponential law of decay
In the previous formulae, N0 is the number of
radioactive atoms at time t = 0, and x is the
number of half-lives elapsed, which could also
be not integer.