Calculation of Decay Activity of One Gram of Natural Uranium
The following problem demonstrates the calculation of the decay rate in
atoms per second (becquerels), given the mass of the
sample and the mass number and halflife of the
nuclide.
Photo: Blocks of uranium metal produced for the Manhattan project in the 1940s. The smallest cubes each have a mass of about 40 grams. Courtesy of the U. S. Department of Energy.
Problem
How many atoms decay each second in a 1.000 gram sample of natural uranium?
Solution
Short answer: 12,000 atoms per second (12 kBq) if you consider
uranium238 alone; 25,000 atoms per second (25 kBq) if you consider all
the uranium isotopes present in natural uranium; or 50,000 atoms per
second (50 kBq) if you also consider the decay products that accumulate
over the course of a few months or longer following extraction from the
uranium ore.
Detailed Calculation
Natural uranium consists of the following isotopes:
Uranium isotope

Percentage

Halflife

uranium238

99.275%

4.47 billion years

uranium235

0.720%

704 million years

uranium234

0.00548%

245 thousand years

Note: The numbers don't add up to exactly 100 percent due to rounding.
Let's consider the uranium238 by itself first, since it makes up more than 99% of natural uranium.
Decay of uranium238 alone
The 1.000 gram of uranium contains 0.993 gram of uranium238.
Uranium238 has a
halflife
of 4.47 billion years. Let's convert that to seconds:
(4.47 x 10
^{9} years) x (365.25 days/year) x
(24 hours/day) x (60 minutes/hour) x (60 seconds/minute) = 1.411 x 10
^{17} seconds
The get the
mean
lifetime of a radioactive isotope, designated τ (Greek letter tau), divide the
halflife by the natural logarithm of 2:
mean lifetime τ = halflife
/
ln(2) = (1.411 x 10
^{17} seconds) / 0.6931 = 2.035 x 10
^{17} seconds
The
decay
rate, designated λ
(Greek letter lambda), is the fraction of the total mass that decays in
one unit of time. It is equal to the inverse of the mean lifetime:
decay rate λ = 1/τ = 1 / (2.035 x 10
^{17} seconds) = 4.914 x 10
^{18} per second
This is the fraction of the uranium238 that decays in one second, so a
mass of 0.993 gram of uranium decays at a rate of 0.993 x 4.914 x 10
^{18}
= 4.88 x 10
^{18} gram per second. To convert this decay rate from grams per second to
atoms per second, you use the atomic mass of uranium238, 238 grams per mole, and the
Avogadro
constant,
6.022 x 10
^{23}
atoms per mole:
(4.88 x 10
^{18} gram / second) x (1.0
mole / 238 grams) x (6.022 x 10
^{23} atoms
/ mole) = 1.23 x 10
^{4}
atoms/second
This is 12,300 atoms/second or 12,300
becquerels
(12.3 kBq)
Decay rate of uranium235 alone
The 1.000 gram of uranium contains 0.0072 gram of uranium235.
Uranium235 has a halflife of 704 million years. Let's convert that to seconds:
(704 x 10
^{6} years) x (365.25 days/year) x
(24 hours/day) x (60 minutes/hour) x (60 seconds/minute) = 2.22 x 10
^{16} seconds
mean lifetime τ = halflife
/
ln(2) = (2.22 x 10
^{16} seconds) / 0.6931 = 3.205 x 10
^{16} seconds
decay rate λ = 1/τ = 1 / (3.205 x 10
^{16} seconds) = 3.12 x 10
^{17} per second
The decay rate of 0.0072 gram of uranium235 is:
(0.0072 gram) x (3.12 x 10
^{17} per second) x (1.0 mole / 235 grams) x (6.022 x 10
^{23} atoms
/ mole) = 570
atoms/second
This is 570 atoms/second or 570 becquerels.
Decay rate of uranium234 alone
The 1.000 gram of uranium contains 5.48 x 10
^{5} gram of uranium234.
Uranium234 has a halflife of 245 thousand years. Let's convert that to seconds:
(245 x 10
^{3} years) x (365.25 days/year) x
(24 hours/day) x (60 minutes/hour) x (60 seconds/minute) = 7.73 x 10
^{12} seconds
mean lifetime τ = halflife
/
ln(2) = (7.73 x 10
^{12} seconds) / 0.6931 = 1.115 x 10
^{13} seconds
decay rate λ = 1/τ = 1 / (1.115 x 10
^{13} seconds) = 8.97 x 10
^{14} per second
The decay rate of 5.48 x 10
^{5} gram of uranium235 is:
(5.48 x 10
^{5} gram) x ( 8.97 x 10
^{14} per second) x (1.0 mole / 234 grams) x (6.022 x 10
^{23} atoms
/ mole) = 12,600
atoms/second
This is 12,600 atoms/second or 12,600 becquerels (12.6 kBq)
Decay rate of natural uranium per gram
The total decay activity for the three uranium isotopes (U238, U235, and U234) in one gram of natural uranium is:
12.3 kBq + 0.570 kBq + 12.6 kBq = 25.5 kBq
Note: Your answer might be slightly different due to rounding errors.
Relationship between uranium238 and uranium234 decay rates
You might be surprised to find that nearly half of the decay activity
comes from uranium234, which makes up only 1/200 of 1% of the total
amount of uranium. The reason is that it has a much shorter halflife,
just 245,000 years compared to 4.5 billion years for uranium238, so it
decays much faster.
You might also notice the similar decay rates of uranium238 and
uranium234, both around 12 kBq per gram of natural uranium. This is not
a
coincidence. Uranium234 exists in nature only because it is a product
in the
decay chain of uranium238. A gram of uranium238 decays at a rate of 12
kBq, producing thorium234 at a rate of 12 kBq, which decays into
protactinium234, which decays into uranium234, and so on. Each nuclide
in the decay chain accumulates to an equilibrium concentration such
that it is produced at, and decays away at, the same rate as the
ultimate ancestor nuclide, uranium238. (The results obtained in the
calculations above are slightly different due to rounding errors.)
Uranium235, on the other hand, is a primordial nuclide like
uranium238. It exists in nature because is was created when the Earth
was formed and has a long enough halflife that it still exists today,
4.5 billion years later. There is no particular relationship between the
decay rates of uranium235 and uranium238. Because uranium235
has a shorter half life than
uranium238, its relative concentration in natural uranium decreases
over geological time spans.
Decay of uranium decaychain products
The answer obtained earlier for the activity of natural uranium, about
25 kBq per gram, is correct for natural uranium freshly
extracted from uranium ore. However, that's not the end of the story. As
uranium decays, its decay products begin to accumulate and they themselves
begin to decay, thereby increasing the total activity of the material
significantly, even after just a few weeks.
Let's take a look at the primary decay products and halflives of the first few nuclides in the
uranium238 decay chain:
Nuclide

Halflife

Decay product

uranium238

4.47 billion years

thorium234

thorium234

24.1 days

protactinium234

protactinium234

1.16 minutes

uranium234

uranium234

245,000 years 
thorium230

thorium230

75,400 years

radium226

radium226

1,600 years

radon222

... and so on ...

In undisturbed natural uranium ore, before the uranium is extracted, all
of the decaychain products of uranium238 have accumulated to their
equilibrium concentrations over geological time periods, such that they
are all are being produced, and are decaying away, at the same rate, as
determined by the amount and halflife of the ultimate ancestor
nuclide, uranium238. The
equilibrium quantity of each nuclide is inversely proportional to its
halflife.
Uranium238 DecayChain Nuclides in Uranium Ore (Containing 1 g Uranium)
Nuclide

Halflife

Quantity present

Decay rate

uranium238

4.47 billion years

1/238 mole

12.3 kBq

thorium234

24.1 days

1/238 mole x (1.48 x 10^{11})

12.3 kBq 
protactinium234

1.16 minutes

1/238 mole x (4.93 x 10^{16}) 
12.3 kBq 
uranium234

245,000 years 
1/238 mole x (5.48 x 10^{5}) 
12.3 kBq 
thorium230

75,400 years

1/238 mole x (1.69 x 10^{5}) 
12.3 kBq 
radium226

1,600 years

1/238 mole x (6.53 x 10^{7}) 
12.3 kBq 
... and so on ...

In the
Quantity present column above, the number in parentheses
is the halflife of the nuclide divided by the halflife of the ultimate
ancestor nuclide, uranium238. For example, the amount of thorium230
present under equilibrium conditions is 1.0/238 mole (the amount of
the ancestor uranium238 present) multiplied by the ratio of the two
halflives (75,400 / 4,470,000,000 = .0000169).
When uranium is extracted chemically from the ore, all the uranium isotopes are retained, and all the decay
products are left behind in the waste product, the
mill tailings. The following table shows what the extracted uranium contains (not including the uranium235, which we will consider later).
Uranium238 DecayChain Nuclides in 1 g Freshly Extracted Uranium
Nuclide

Halflife

Quantity present

Decay rate

uranium238

4.47 billion years

1/238 mole

12.3 kBq

thorium234

24.1 days

none

none 
protactinium234

1.16 minutes

none

none 
uranium234

245,000 years 
1/238 mole x (5.48 x 10^{5}) 
12.3 kBq 
thorium230

75,400 years

none

none 
radium226

1,600 years

none

none 
... and so on ...

However, thorium234 immediately begins to accumulate as the decay
product of uranium238 and itself begins to decay. Because it is
being produced at a constant rate (12,300 atoms per second), and its
own decay rate is proportional to the amount present, it accumulates to
onehalf
its equilibrium amount in one halflife, to threefourths of that
amount in two halflives, to seveneights of that
amount in three halflives, and so on.
Therefore, after one halflife (24 days), we have
the following amounts of uranium238 decaychain products.
Uranium238 DecayChain Nuclides 24 Days After Extraction of 1 g Uranium
Nuclide

Halflife

Quantity present

Decay rate

uranium238

4.47 billion years

1/238 mole

12.3 kBq

thorium234

24.1 days

1/238 mole x (1.48 x 10^{11}) x 0.5

6.1 kBq 
protactinium234

1.16 minutes

1/238 mole x (4.93 x 10^{16}) x 0.5

6.1 kBq 
uranium234

245,000 years 
1/238 mole x (5.48 x 10^{5}) 
12.3 kBq 
thorium230

75,400 years

negligible

none 
radium226

1,600 years

none 
none 
... and so on ...

Protactinium234 has a halflife of one minute, so it decays just about
as fast as it is produced. So it also reaches onehalf of its
equilibrium amount in 24 days, because it is being produced at onehalf
its equilibrium rate. Thorium230 is also accumulating at 12,300 atoms
per second as a decay product of uranium234. However, because
thorium230 has such a long halflife on a human time scale (75,400
years), it can be considered, for practical purposes, a stable element
that accumulates without decay. As a result, none of the decaychain
nuclides below thorium230, such as radium and radon, will be produced
in the next several tens of thousands of years.
By the time a few months have passed, thorium234 and protactinium234
accumulate to their equilibrium amounts and decay at the same
equilibrium rate of 12.3 kBq.
Uranium238 DecayChain Nuclides Beyond a Few Months After Extraction of 1 g Uranium
Nuclide

Halflife

Quantity present

Decay rate

uranium238

4.47 billion years

1/238 mole

12.3 kBq

thorium234

24.1 days

1/238 mole x (1.48 x 10^{11})

12.3 kBq 
protactinium234

1.16 minutes

1/238 mole x (4.93 x 10^{16})

12.3 kBq 
uranium234

245,000 years 
1/238 mole x (5.48 x 10^{5}) 
12.3 kBq 
thorium230

75,400 years

negligible

none 
radium226

1,600 years

none 
none 
... and so on ...

Therefore, total activity of uranium238 and uranium234, as long as it has been sitting around
for a few months or more after extraction from the ore, has an activity
of 4 x 12.3 kBq = 49.2 kBq.
The diagram on the right shows the decay activity of one gram of natural
uranium on the logarithmic time scale, ranging from 3 days to 30 billion
years. When freshly extracted from the ore, the uranium has an decay
rate of 25 kBq, arising equally from the uranium238 and uranium234. Within a
few months, the decay products thorium234 and protactinium234 accumulate to their
equilibrium concentrations, doubling the activity to 50 kBq.
The activity remains constant for tens of thousands of years, until the
thorium230 has a chance to accumulate to significant levels. After
that, all
of the decaychain nuclides accumulate to their equilibrium levels, in
what is called secular equilibrium, or equilibrium over the ages. This
is what you see in old uranium ore rocks. Eventually, after tens of
billions of years, the uranium238 will run out and all activity will
die
away.
Diagram source: WISE Uranium Project, Uranium Radiation Properties

Meanwhile, the small amount of uranium235 in natural uranium also
produces a nuclide that contributes to the total activity. A similar
consideration of the uranium235 decaychain produces the following
results.
Uranium235 DecayChain Nuclides Beyond a Few Days After Extraction of 1 g Uranium
Nuclide

Halflife

Quantity present

Decay rate

uranium235

704 million years

0.0072 / 235 mole

570 Bq

thorium231

25.52 hours

0.0072 / 235 mole x (4.1 x 10^{12})

570 Bq 
protactinium231

32,800 years

negligible

none 
actinium227

21.8 years

none 
none

thorium227

18.7 days

none

none 
radium223

11.4 days

none 
none 
... and so on ...

Thus, the activity of one gram of natural uranium due to the presence of
uranium235 that has been sitting around for a few days or more is 2 x
570 Bq = 1,140 Bq = 1.14 kBq.
The final answer
The total activity of one gram of natural uranium that has been sitting
around for a few months or longer is the activity of uranium238 and its
decay products thorium234, protactinium234, and uranium234 (49.2
kBq); plus the activity of the 0.72% of uranium235 present and its
decay product thorium231 (1.1 kBq). Thus, the total activity is
50.3 kBq per gram of natural uranium.
Why is the decay of uranium isotopes important?
Shortly after the discovery of radioactivity by
Henri Becquerel in 1896,
Marie Curie
found that only uranium and thorium were radioactive among the known
elements. She also found that the form of uranium, whether pure metal,
uranium salt, or uranium oxide, made no difference in the strength of
the radioactivity. Only the amount of uranium present was important,
with one exception  pitchblende ore was four times more radioactive
than expected for its uranium content.
Curie correctly hypothesized that the extra radioactivity was caused by
the presence of undiscovered radioactive elements, present in small
quantities but with greater activity that uranium. By separating
pitchblende ore into is constituent parts, she discovered two new
elements, polonium and radium, which are decaychain
products of uranium238.
In assessing the environmental dangers of uranium mining, one must
consider not only the uranium extracted from the ore, but also the
leftover waste rock left behind after extraction, the mill tailings,
which are more radioactive than the uranium itself. For more information
on this subject, see Peter Diehl's
Uranium Radiation Properties web page and its toplevel index page, the World Information Service on Energy
Uranium Project.
Among the isotopes of natural uranium, only uranium235 is
fissile, that is, can be used in nuclear reactors or to make nuclear weapons. For these purposes, the uranium must be
enriched from the natural 0.72% concentration to at least 3% for nuclear fuel or at least 90% for nuclear weapons.
Because all isotopes of uranium are chemically identical, the only way
to separate out the uranium235 from uranium238 is to take advantage
of the small difference in mass of the two nuclides. In the modern
gas centrifuge
method, a gaseous compound of uranium is spun in a rapidly rotating
cylinder, causing the heavier uranium238 molecules to accumulate at the
outside, forcing the lighter uranium235 molecules to the inside.
The uranium from which most of the uranium235 isotope has been removed is called
depleted uranium,
which has several practical uses but can be dangerous because it is
radioactive, and the radioactivity grows over time. It is also toxic in
the same way as lead or cadmium.