Radioactive Decay
Activity:
Units:
the number of atoms that decay per unit time: (disintegrations per
second).
Becquerel (Bq) = 1 dps
Curie (Ci) [old unit] = 3.7 x 1010 Bq exactly (originally defined as the
activity of 1.0 g of radium
Exponential Decay:
Activity (A) of a radioactive nuclide decreases exponentially with time.
dN = - λN dt
Let N = # atoms present
The constant of proportionality, λ, has units of sec-1.
A=
− dN
=
dt
λN
Each radioactive nuclide has a unique decay constant λ.
dN
= −λ dt
N
dN
= λdt
N
∫
∫
ln N = -λt + c
When t = 0, N0 atoms are present - implies that ln N0 = c
ln N = - λt + ln N0
N
ln N = -λt
0
N
= e-λt
N0
or N = N0 e-λt
or A = A0 e-λt
1
Half-Life (t1/2 or T)
1
When N = N o
2
- ln 2 = - λt½
λ=
0.693
t½
1
N0
2
= e −λt1 / 2
N0
1 −λt1 / 2
=e
2
0.693 = λt½
t1 / 2 =
0.693
λ
Image removed due to copyright restrictions.
Fig. 4.1 in Turner J. E. Atoms, Radiation, and Radiation
Protection, 2nd ed. New York, NY: Wiley-Interscience, 1995.
2
Specific Activity
Specific Activity (SA) defined as activity per unit mass.
Ci
Bq
Units: g or g
A = λN
N = # of atoms
N
=
g
atoms
mole = atoms
grams
g
M
mole
6.02 x10 23
6.02 x10 23
λ
SA =
M
A λN
SA = g = g
Example: Specific activity of radium
M = 226
SA =
g
mole
t½ = 1600 y
.693
λ = t½
⎛
23 atoms ⎞
⎜ 6.02 x10
⎟
mole ⎠ ⎛⎜ .693 ⎞⎟⎛ 1 y ⎞⎛ 1d ⎞⎛ 1h ⎞⎛ 1m ⎞
⎝
⎜ 1600 y ⎟⎜ 365d ⎟⎜ 24h ⎟⎜ 60 min ⎟⎜ 60 sec ⎟
g
⎠⎝
⎠⎝
⎠
⎠⎝
⎝
⎠⎝
226
mole
3.66 x1010 atoms
SA =
g ⋅ sec
=
3.66 x1010 Bq
g
1 Ci = 3.66 x 1010 dps
1 Ci orig. defined as activity associated with 1 g of Radium.
Ci is now defined as 3.7 x 1010 dps exactly.
3
Count rates - vs half-life
Example:
Compound A: t½ = 45 min
Compound B: t½ = 45 years
Given 1010 atoms of each - find the activity (A)
A = λN
AA =
λ=
0.693
t½
[λ = 2.5 x 10-4 sec-1]
0.693
10
⎛ 60 sec ⎞ 10 atoms
(45 min )⎜
⎟
⎝ min ⎠
AA = 2.56 x 106 Bq
AB =
0.693
1010
(45 y )(365)(24)(60)(60)
[λ = 4.8 x 10-10 sec-1]
AB = 4.8 Bq
239
Pu t½ = 24,065 y
235
U t½ = 7.038 x 108 y
4
Serial Radioactive Decay
N1 → N2
N10 = # parent atoms present at t = 0.
N20 = # daughter atoms present at t = 0.
General Case
dN 2
= λ1 N1 − λ2 N 2
dt
⇒ ⇒ A2 = A10
λ2
λ2 − λ1
(e −λ1t − e −λ2t ) + A20 e −λ2t
Secular equilibrium (T1 >> T2)
Simplifying assumptions:
A20 = 0
T1 is large, ∴ λ1 is small; λ2 - λ1 = λ2 e
- λ1t
≅1
A2 = A10 (1 − e − λ2t )
General Case simplifies to
e − λ2 t ≈ 0
after ~ seven half-lives (of N2 daughter),
A2 = A10
A2 = A1
Activities
A1
Secular
Equilibrium
A2
T1 >> T2
0
~7T2
t
Activity A2 of relatively short-lived radionuclide daughter (T2 << T1) as a
function of time t with initial condition A20 = 0. Activity of daughter builds
up to that of the parent in above seven half-lives (~7T2). Thereafter, daughter
decays at the same rate it is produced (A2 = A1), and secular equilibrium
is said to exist.
Figure by MIT OCW.
5
Radon Decay
Mass
Number
226
Radium
1620y
α
222
Radon
3.82d
Short-lived
radon
daughters
α
218
Polonium
β
3.05m
(Radium A) 0.019%
Lead
26.8m
(Radium B)
β
Radon
0.035s
α
Bismuth
19.7m
(Radium C)
α
210
β
α
α
214
Astatine
2s
β
α
0.021%
Thallium
3.1m
Polonium
0.000164s
(Radium C)
β
Lead
22y
α
206
β
0.000002%
Mercury
8m
β
Bismuth
5d
α
β
0.00013%
Thallium
4m
β
Polonium
138d
α
Lead
• Radon itself, due to its fairly short half-life (222Rn) is not a major concern.
• Radon is also an inert gas and is typically exhaled after breathing it in
(although some will dissolve in the blood).
• The concern is over the daughter products of radon that are particulate
(attached to aerosol particles), α-emitting, and decay within hours to 210Pb
(T1/2 = 22 years).
6
Transient equilibrium (T1 ≥ T2)
General Case
dN 2
= λ1 N1 − λ2 N 2
dt
⇒ ⇒ A2 = A10
Simplifying assumptions:
λ2
λ2 − λ1
(e −λ1t − e −λ2t ) + A20 e −λ2t
A20 = 0
after ~ 10t½s
A2 = A10
λ2
λ2 − λ2
−λ t
by definition: A10 e 1 = A1
λ
A
2
2
or A = λ − λ - at equilibrium A1 and A2 present in a
1
2
1
constant ratio
T1 >
~ T2
Activities
A2 = A1
λ2
e -λ1t
λ 2 − λ1
−λ t
e − λ 2 t << e 1
A1+A2
A10
A1
Transient
Equilibrium
A2
0
A1+A2
A2
A1
t
Activities as functions of time when T1 is somewhat larger than
T2 (T1 >
~ T2) and N20 = 0. Transient equilibrium is eventually reached,
in which all activities decay with the half-life T1 of the parent.
Figure by MIT OCW.
7
No equilibrium (T1 < T2)
[no simplifying assumptions possible]
dN 2
= λ1 N1 − λ2 N 2
dt
⇒ ⇒ A2 = A10
λ2
λ2 − λ1
(e −λ1t − e −λ2t ) + A20 e −λ2t
A10
Activities
T2 > T1
A1
A1+A2
A2
0
t
Activities as functions of time when T2 > T1 and N20 = 0.
No equilibrium conditions occur. Eventually, only the daughter
activity remains.
Figure by MIT OCW.
8
The 99mTc Generator: Transient equilibrium in action
99
•
Mo is adsorbed on an alumina column as ammonium molybdate
(NH4MoO4)
•
99
•
99
•
99m
Mo (T = 67 hrs) decays (by β -decay) to
MoO4 ion becomes the
99m
Tc (T = 6 hrs)
99m
TcO4 (pertechnetate) ion (chemically different)
TcO4 has a much lower binding affinity for the alumina and can be
selectively eluted by passing physiological saline through the column.
Image removed due to copyright restrictions.
9
EDTA
ethylenediaminetetraacetate
O
O-
O
O-
N
O-
N
O
O
-
O
DTPA
10
Chelator Kits
Image removed due to copyright restrictions.
11