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Radioactivity and radioisotopes
• Half-life • Exponential law of decay
Half-lifeThe half-life of a radioactive element means:
a) The time taken for half the radioactive atoms in the element to disintegrate
b) The time taken by the radiation from the element to drop to half its original level
Radioactive atoms HALF-TIME
Radioactive atoms
Decayed atoms
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 emitter Fe-59. Its half-life is 46 days.
Plot a graph of the fraction of undecayed atoms vs time (days).
Half-life of Fe-59
1
1/2
1/4
1/80
0
1/4
1/2
3/4
1
1 1/4
0 46 92 138 184
Time (days)
Fra
cti
on
of
un
de
ca
ye
d a
tom
sClick here for radioactive decay simulation
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 decayNow, plot the graph of the logarithm to base 10 of
the fraction of Fe-59 remaining against time.
Logarithm of fraction remaining - time
-1.204
-0.903
-0.602
-0.301
0
0 46 92 138 184
Time (days)
Lo
g o
f fr
acti
on
rem
ain
ing
Interpolate the logarithm of the fraction remaining after 120 days.
0.785
Consider the table of data from the example on the previous slides.
Exponential law of decay
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
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.
x = -0.982 x = log(F150)
F150 = antilog(-0.982) = 10-0.982 = 0.10
Exponential law of decay
x
daysdays 150
903.0
138
From the previous discussion, we can conclude that the rate of radioactive decay is proportional to the number of radioactive atoms:
Where N is the number of radioactive atoms still present at time t.
Exponential law of decay
Ndt
dN
The previous proportionality gives the following equation:
The constant the decay constant, and it is measured in s-1.
A solution to the above equation is:
Exponential law of decay
Ndt
dN
teNN 0 x
NN
20
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.
Exponential law of decay
