The radioactive decay formula


define the terms activity and decay constant and recall and solve problems using A =
λN.
solve problems using the relation λ = 0.693/t½.
The meaning of the decay constant .
definition of the decay constant : the probability or chance that an individual nucleus will
decay per second.
Units:  is measured in s-1 (or h-1, year-1, etc).
If you have a sample of N undecayed nuclei, what will its activity A be? In other words, how
many of the N will decay in a second?
A=
As time passes, N will get smaller, so A represents a decrease in N. To make the formula reflect
this, it needs a minus sign.
A=
The more undecayed nuclei you have, and the greater the probability that an individual one will
decay, the greater the activity of the sample.
In calculus notation, this is
dN/dT = -N
put this equation into words.
This is the underlying relationship in any process that follows exponential decay.
More generally: if the rate of change is proportional to what is left to change, then exponential
decay follows.
Conversely if data gives an exponential decay graph, then you know something about the
underlying process.
The link with half-life.
If a nuclide has a large value of , will it have a long or short T1/2? (High probability of decay =>
it will decay quite quickly => a short half life, and vice versa.)
What type of proportionality does this suggest?
Thus  ~ 1/T1/2. In fact, T1/2 = ln 2/ = 0.693/
This is a very important and useful formula. Depending which of  or T1/2 is easiest to measure
experimentally, the other can be determined. In particular it allows very long half lives
(sometimes millions of years) to be determined.
How could we measure the 4.5 billion year half-life of uranium-238? ( Think paper 5)
Smoothed out radioactive decay
Here is the first step in making a model – idealisation and simplification of a messy process
Smoothed out radioactive decay
Actual, random decay
N
N
t
t
time t
probability p of decay in short time t is proportional to t:
p =  t
average number of decays in time t is pN
t short so that N much less than N
change in N = N = –number of decays
N = –N
t
N = –pN
N = –N t
Simplified, smooth decay
rate of change
= slope
dN
= dt
time t
Consider only the smooth form of the average behaviour.
In an interval dt as small as you please:
probability of decay p =  dt
number of decays in time dt is pN
change in N = dN = –number of decays
dN = –pN
dN = –N dt
dN = –N
dt
Half-life and time constant
Here is the relationship between these two, explained
Radioactive decay times
dN/dt = – N
N/N0 = e–t
N0
N0 /2
N0 /e
0
t=0
t = t1/2 t = time constant 1/
Time constant 1/
at time t = 1/
N/N0 = 1/e = 0.37 approx.
t = 1/is the time constant of the decay
Half-life t1/2
at time t1/2 number N becomes N0/2
N/N0 =
1
2
= exp(– t1/2)
In 12 = –t1/2
t1/2 = ln 2 = 0.693


In 2 = loge 2
Half-life is about 70% of time constant 1/. Both indicate the
decay time
Now complete questions on p 345 and 347 Adams and Allday