Secular Equilibrium and Radioactive Decay Chains
In a radioactive decay chain
a long-lived ancestor element decays into another element, which itself
decays into another element, and so on, resulting in a chain of
transformations ending with the formation of a stable element.
In an undisturbed sample of the ancestor element, after enough time has
passed, each intermediate decay-chain product accumulates to an
equilibrium concentration, such that the element is created at, and
decays away at, the same rate. This rate is the same rate for all elements
in the chain. The equilibrium amount of each element is proportional to
. This condition is called secular equilibrium, or equilibrium over the ages.
For example, in a sample of uranium ore rock containing 1.00 gram of uranium, the uranium-238 decays at a rate of 12,300 atoms per second
into throium-234. The thorium-234 in the rock also decays at a rate of 12,300 atoms
per second into protactinium-234. The protactinium-234 in the rock also decays at a
rate of 12,300 atoms per second into uranium-234, and so on down the
chain, which ends with lead-206, a stable element that accumulates at a rate of 12,300 atoms per second,
without any further decay.
238U -> 234Th -> 234Pa -> 234U -> 230Th -> 226Ra -> 222Rn -> 218Po -> 214Pb -> 214Bi -> 214Po -> 210Pb -> 210Bi -> 201Po -> 206Pb
What causes this exquisite match-up in decay rates for all the elements
in the decay chain? The answer lies in the behavior of all radioactive
materials -- the decay rate is proportional to the amount of material
present. So when a long-lived ancestor element decays at a constant
rate, each descendent eventually accumulates to the level at which it
decays at the same rate as it is produced. This concept is easier to understand
when you consider the analogy described in the next section.
|First, a brief note on terminology. So far, I've used the term element to describe a distinctive radioactive material. However, this term does not distinguish between different isotopes of the same element, for example, uranium-238 and uranuim-234. The word isotope
can be used to describe different radioactive forms of the same
element, but not different elements, such as uranium-238 and its decay
product thorium-234. Therefore, I'll use the less common term nuclide
to describe an "atomic species," that is, an atom with a nucleus having
a given number of protons and a given number of neutrons.
The Dripping Bucket Analogy
Imagine that you have the job of maintaining your city's water
supply tower, a huge tank carrying enough water to supply the whole
city. On your first day of work, you notice that water is leaking from
the tank and dripping on the ground -- drip, drip, drip, drip -- at a
steady 4 drops per second.
You hate to see water being wasted and making a messy puddle, so you
find a bucket to catch the water. Unfortunately, the bucket has a tiny
hole on the bottom. You wonder how well this bucket will hold
up. As an experiment, you fill it with water and check underneath.
When the bucket is full, it leaks and drips at a rate of 10 drops per
second. When the bucket is half full, it drips at a rate of 5 drops
per second. When the bucket is nearly empty, it hardly drips at all.
After further testing, you discover that the bucket always leaks at a
to the amount of water in the bucket.
Since you can't find another container, you put the empty bucket on a
ladder and start catching the drips from the water tower. What will
happen over time? Take a moment to think about this, and then read on.
Water enters the bucket at a rate of 4 drops per second. A first,
nothing leaks from the bucket, so the water accumulates at a rate of 4
drops per second. However, as the bucket starts filling up, it starts
leaking, very slowly at first: drip, ... , drip, ...
Remember that the bucket's drip rate is proportional to its fullness.
When the bucket is 10% full, it's losing water at a rate of 1
drop per second out of the bottom, while water enters
at a rate of 4 drops
per second. So it still has a net accumulation rate of 3 drops per
second. So it's still filling up, but more slowly.
When the bucket it 20% full, it's losing water at a rate of 2
drop per second out of the bottom, so it still has a net accumulation
rate of 2 drops per second. It's still filling up, but even more slowly.
When the bucket it 30% full, it's losing water at a rate of 3
drop per second out of the bottom, so it still has a net accumulation
rate of 1 drop per second. It's still filling up, but even more
Let's say that you observe the following behavior.
Water accumulates in the bucket more and more
slowly as the water level approaches 40% full. If you wait long enough,
the bucket reaches 40% full and leaks at a rate of 4 drops per second
while fresh water enters at 4 drops per second. This is the stable,
The amount of time it takes for the bucket to fill to one-half the equilibrium level is the half-life
of the bucket, which is one hour for this particular bucket, with this
particular leaking hole. This becomes apparent when you remove the
the dripping water supply and allow it to drain out through the hole. It
starts dripping at 4
drops per second and gradually slows down as the bucket empties. As
each hour goes by, the bucket contains half as much water as it did an
An Interesting Note
If you superimpose the picture above (buckets leaking out water) with
the previous picture (buckets being being filled from above while also
leaking), you'll notice that all five buckets look the same: each has 4
drops per second entering, a total of 4 drops per second leaking out, and a total water
level at 40% full. More about this later.
|For the Mathematically Inclined (skip this box otherwise)|
Because the amount of water in the tower is very large, the few drips
leaking out if it won't affect the total stored amount significantly.
Therefore, we can consider the drip rate into the bucket to be a
constant, 4 drops per second. When the bucket is catching this water, the amount of water in the bucket as
function of time is:
x(t) = x(final) + [x(initial) - x(final)] e -λt
where x(final) is the final equilibrium amount in the bucket, x(initial)
is the starting amount in the bucket, and λ (Greek letter lambda) is the decay constant for
the bucket, expressed as the number of
drops per unit of time leaking from the bucket divided by the number of drops
in the bucket. (For the derivation of this equation, see the Wikipedia
article on radioactive decay rates.)
For this particular bucket, the half-life is 1.0 hour. The mean lifetime τ (Greek letter tau) (1/e life) is the half-life divided by the natural logarithm of 2:
τ = half-life/[ln(2)] = 1.0/.693 = 1.44 hours
The mean lifetime τ is the amount of time it takes for the water to fall to 1/e of its previous level, the "1/e"-life of the bucket.
The decay rate λ is the inverse of the mean lifetime τ:
λ = 1/τ = 1/1.44 = 0.693 per hour
(For a description of the decay constant and half-life, see the Wikipedia article on exponential decay.)
If you start with an empty bucket, the amount grows exponentially and approaches the equilibrium level of 40% full:
x(t) = x(final) - x(final) e -λt
So the bucket starts out empty at t=0, reaches 20% full in one
half-life, 30% full in two half-lifes, 35% full in three half-lifes, and
so on, approaching 40% full as the elapsed time gets very large.
If you start with a 100% full bucket instead, it starts by leaking at 10 drops
per second. With 4 drops entering per second, the net loss is 6 drops per second, and the amount in the
bucket falls exponentially to the equilibrium level of 40% full:
x(t) = x(final) + [2.5*(x(final)) - x(final)] e -kt
In this case, the bucket starts out 100% full at t=0, falls to 70% full
in one half-life, 55% full in two half-lifes, and so on, again
approaching 40% full as the elapsed time gets very large.
Dripping Bucket = Radioactive Decay Product
Now what has all this dripping have to do with radioactive decay?
water in the huge water tank is like uranium-238, a long-lived nuclide
with a tiny decay constant. The water in the bucket is like thorium-234,
the immediate decay product of uranium-238. If you extract the
from its ore, you separate it from its decay
products. After extraction, the
uranium-238 immediately begins decaying into thorium-234. The throrium-234 accumulates, quickly at first, then not so
quickly, and then slowly approaches its equilibrium amount.
The half-life of thorium-234 is 24 days, so after that many days has
passed, the thorium-234 accumulates to one-half of its equilibrium
amount. After 48 days, it accumulates to three-fourths of its
equilibrium amount, and after 72 days, to seven-eights the final amount,
and so on. By the time a few months have passed, thorium-234 reaches
its equilibrium amount and decays at the same rate that it is produced
by its uranium-238 source.
How much is the equilibrium amount of thorium-234 relative to the amount
of the parent nuclide, uranium-238? It is simply the ratio of the
half-lifes of the two nuclides. The half-life of uranum-238 is 4.47
billion years, and that of thorium-234 is 24.1 days, After
you convert the time units to a common base (left as an exercise for
the reader), you find that the half-life of thorium-234 is only 1.48 x 10-11
that of uranium-238. Therefore, under equilibrium, the number of thorium-234
atoms is 1.48 x 10-11
times the number of uranium-238 atoms.
|For the Mathematically Inclined|
Problem: A sample of uranium contains 1.00 gram of uranium-238.
The uranium was extracted from its ore more than a year ago, so it is
in equilibrium with thorium-234. How many grams of thorium-234 are
present in the sample?
Solution: Under equilibrium conditions, the number of uranium-238 atoms multiplied by 1.48 x 10-11 (the ratio of the two nuclide half-lifes) gives you the number of thorium-234 atoms.
(1.00 gram U-238) x (1 mole U-238 / 238 grams U-238) (6.02 x 1023 atoms U-238 / mole U-238) x
(1.48 x 10-11 atoms Th-234 / atoms U-238) x
(1 mole Th-234 / 6.02 x 1023 atoms Th-234) x (234 grams Th-234 / mode Th-234)
= (1.00) x (1.48 x 10-11) x (234/238) gram Th-234
= 1.46 x 10-11 gram Th-234
The Next Containers
Let's get back to the dripping bucket story. Now that the bucket is 40%
full and dripping just as fast as the water tank, you find another
container, a tin cup, to catch the drips from the bucket. This works
fine for awhile, but soon you find that it too has a
small hole in the bottom. As the cup fills up, it starts leaking faster
and faster. When it's almost full, it's dripping at the same rate as
the water tank and the bucket, 4 drops per second.
You need a big container to catch all these drips. So you go next door
and find a 55-gallon barrel. You place it under the tin cup. That should
take care of the problem!
The next day, you come back and find that the barrel is about one-third full, and
unfortunately, it must have a tiny hole in the bottom, because it's
leaking a couple of drops per second. The next day, the barrel is about half full and dripping
at about 3 drops per second. After several days, the barrel is about 3/4 full
and dripping at 4 drops per second.
You decide to take care of this once and for all. You put a railroad
tank car beneath the barrel! This works fine for quite a long time. But
after a few months, you notice a slow drip from the bottom of the tank car.
By the time a year has passed, the tank car is about one-third full and
dripping at about 2 drops per second.
So you get another container, and so on. The containers are different
shapes and sizes. Sometimes it's just a thimble-sized container with a
big hole, which fills quickly. Sometimes it's like the railroad tank
car, very large
and with a tiny hole, taking years to fill up. But in every case, after
you wait long enough, water enters and leaves each container at 4 drops
Eventually, you find a container that doesn't leak at all, ending the
long chain of containers. As time passes, all of
the containers fill up to their respective equilibrium levels, and all have water
entering at a rate of 4 drops per second, and all of them leak at 4
drops per second, except for the very last container at the bottom,
which accumulates water at 4 drops per second endlessly.
Uranium-238 Decay Chain
This string of containers is like the uranium-238 decay chain. One gram of uranium-238
decays at a rate of 12,300 atoms per second
. In a uranium ore rock
containing one gram of uranium-238, which has been left undisturbed for
millions of years, all of the nuclides in the decay chain have accumulated to
their secular equilibrium levels, such that each nuclide is produced at a rate of
12,300 atoms per second and decays at 12,300 atoms per second. At the
end of the chain is lead-206, a stable isotope that accumulates
continuously at 12,300 atoms per second.
The following table shows the amount of each decay-chain product
existing in an old uranium ore rock, relative to the amount of
uranium-238 present in the rock. For each decay-chain product, the "Mass
present" is calculated as the half-life of the nuclide divided by
the half-life of the ancestor uranium-238, multiplied
the ratio of the atomic masses of the nuclide and uranium-238.
Uranium-238 Decay Chain in an Undisturbed Uranium Ore Rock
(relative to ancestor U-238)
||4.47 billion years
||(1.48 x 10-11) x (234/238)
||(4.93 x 10-16) x (234/238)
|(5.49 x 10-5 ) x (234/238)
|(1.69 x 10-5 ) x (230/238)
|(3.58 x 10-7 ) x (226/238)
|(2.34 x 10-12) x (222/238)
||(1.32 x 10-15) x (218/238)
||(1.14 x 10-14) x (214/238)
||(8.47 x 10-15) x (214/238)
||(1.16 x 10-21) x (214/238)
|(4.99 x 10-9 ) x (210/238)
|(3.07 x 10-12) x (210/238)
||(8.46 x 10-11) x (210/238)
||Depends on age of rock
What Happens When You Empty the Containers (Extract the Uranium)?
When uranium is extracted chemically from its ore, it is separated from
its decay products. How much of each decay product is produced by pure
uranium after it's extracted? For the time being, let's ignore the small amount of uranium-234
and uranium-235 in the rock, and assume that all the uranium freshly
extracted from its ore is pure uranium-238. In that case, uranium
extraction is like emptying all the containers below the
tower, and starting with a fresh set of empty containers.
At first, water drips into the bucket at the top, and nothing else
happens. As the bucket starts filling up, it starts dripping into the
tin cup. After the cup starts filling up, it starts dripping into the
barrel, and so on. The containers fill up in sequence, starting with the
top containers, and gradually trickling down to the lower containers. A
large, slow container such as a railway tank car creates a bottleneck
in the chain,
causing all the containers below it to remain empty for a long time.
In the case of uranium-238, the initial bucket in the chain is
thorium-234, which has a half-life of 24 days. So 24 days after pure
uranium is extracted from its ore, the amount of thorium-234 reaches
one-half its equilibrium level, and is dripping into the next container,
the "tin cup," protactinium-234, at half the equilibrium rate. Protactinium-234 has a half-life of
just 1 minute, so it empties almost immediately into the "barrel,"
uranium-234. Uranium-234 has a half-life of 245,000 years, so on a human
time scale, it can be considered a stable end product. True secular
equilibrium for the entire chain takes at least a million years. For a
detailed accounting of the decay rates of natural uranium, see Calculation of Decay Activity of One Gram of Natural Uranium
What Happens When You Move the Containers Aside (Examine the Mining Waste)?
You might be surprised to learn that in a uranium ore rock, the ancestor
nuclide, uranium-238, contributes only a small fraction of the total
radioactivity of the rock. Most of the radioactivity comes from the
decay-chain products, not from the uranium-238 itself. If you look at
the whole decay chain under secular equilibrium conditions, each drip
from each container
in the chain represents a radioactive decay, each releasing an amount
of energy similar to each of the other drips in the chain. Since
there are 14 dripping containers
(including the municipal water tank), the decay of the ancestor
uranium-238 nuclide contributes only one-fourteenth of the total
radioactivity of the rock.
Separating uranium-238 from its ore is like taking a set of dripping
containers under secular equilibrium conditions, moving them aside, and
putting a new string of empty containers beneath the water tank, as
shown in the picture on the left.
The water tower and its empty containers represent the
extracted urnaium-238, while the string of full containers on the right
represents the discarded mining waste. The moved-over set of full
containers continue to drip, while the municipal water tower starts
dripping into the first container, the bucket. Initially, on the left there are only 4
drops per second entering the empty bucket on the top, but on the right,
all the containers are dripping at 4 drops per second. The mining waste
is 13 times more radioactive than the uranium-238 itself.
What happens over time? On the left, as the bucket on the
left starts filling up, it stars dripping into the tin cup, and as the
tin cup starts filling up, it starts dripping into the next container,
and so on. On the right, as the top bucket's water level drops, the drip
rate into the tin cup starts slowing down, causing the water level to
drop in the tin cup, and the drip rate from the tin cup starts slowing
down, and so on.
On the left, the bucket at the top eventually fills up to its
equilibrium level, and feeds the tin cup at its equilibrium rate. On the
right, the bucket at the top eventually empties completely, and stops
feeding the tin cup entirely.
As it turns out, all the actions on the left and right are exactly
complementary. As each container on the left fills up, the water level
in the corresponding container on the right drops by the same amount. If
you combine the water from the left bucket and the right bucket at any
given time, the total amount of water will be the equilibrium amount for
the bucket. This is true for the tin cup, the barrel, and and all the
other containers in the chain. Eventually, all the containers on the
left fill up and all the containers on the right become empty, starting
from the top and trickling down (or emptying down) the chain of
The diagram on the right shows the decay activity of a one-gram uranium
ore rock containing 0.1% uranium on a logarithmic time scale, ranging
from 3 days to 30 billion
years. The colored bars represent the decay activity of the uranium-238
and each of its decay-chain products in the rock. Because the rock is
in secular equilibrium, all of the decay products have the same decay
rate of 12 atoms per second. Eventually, after many billions of years,
the uranium-238 will run out and the decay will diminish to nothing.
You might wonder why the activity appears to remain constant for such a
long time, and then falls off rapidly. It's because of the logarithmic
time scale. During the first 50 million years, the activity falls less
than 1%, so that part of the graph looks constant. A linear
time scale would show the classic exponential decay curve.
Source of diagram: WISE Uranium Project, Uranium Radiation Properties
Compare the activity of uranium ore with the activity of one gram of
natural uranium extracted from its ore and one gram of
uranium mining waste (mill tailings), below.
Source of decay activity diagrams: WISE Uranium Project, Uranium Radiation Properties
The natural uranium extracted from its ore initially consists of
uranium-238 (yellow) and a tiny amount of its third-level decay product,
uranium-234 (blue), both decaying at a rate of 25 kBq. Uranium-238, within a few months, produces its
equilibrium amount of thorium-234 and protactinium-234; it fills up its
bucket and tin cup to their equilibrium levels. The uranium-234 starts
producing thorium-230, but the latter has a half-life of 75,000 years,
so it takes many thousands of years to accumulate to significant levels.
Eventually, after about half a million years, the uranium reaches
secular equilibrium, just like an old uranium ore rock. After many billions of years, the supply of
uranium-238 runs out, and the activity declines to nothing.
In the mining waste discarded after uranium extraction, the decay-chain
products are present and decaying. The products at the top of the chain,
thorium-234 and protactinium-234 (yellow and red), run out after a few months; the
bucket and tin cup are completely emptied. The remaining decay-chain
elements continue to be supplied for a long time by the thorium-234,
which has half-life of 75,000 years. Eventually, after about half a
million years, the thorium runs out, and the decay rate diminishes to
nothing. (In the Uranium Mill Tailings Activity diagram above, there is
still a small amount of residual activity after half a million years
because of the 10% of uranium-238 that was left behind during the
Like the empty and full strings of containers below the water tank, the
decay activity of the uranium and its extracted mining waste are
complementary. As the activity of the extracted uranium increases, the
activity of the mining waste decreases by a like amount.
This seemingly peculiar behavior of a radioactive elements and their
decay products led to the discovery of the nature of radioactivity
itself. When Ernest Rutherford
and Frederick Soddy
examined the behavior of radium and its decay products (which have
laboratory-observable half-lives), they deduced from this behavior the
random nature of radioactive decay. For the whole story of this
discovery, see Soddy's book The Interpretation of Radium and the Structure of the Atom
Radioactive Decay Problem Examples