Nick Connor

The radioactive decay law states that the probability per unit time that a nucleus will decay is a constant, independent of time. This constant is called the decay constant and is denoted by λ, “lambda”. This constant probability may vary greatly between different types of nuclei, leading to the many different observed decay rates. The radioactive decay of certain number of atoms (mass) is exponential in time.

Radioactive decay law: N = N.e^-λt

The rate of nuclear decay is also measured in terms of half-lives. The half-life is the amount of time it takes for a given isotope to lose half of its radioactivity. If a radioisotope has a half-life of 14 days, half of its atoms will have decayed within 14 days. In 14 more days, half of that remaining half will decay, and so on. Half lives range from millionths of a second for highly radioactive fission products to billions of years for long-lived materials (such as naturally occurring uranium). Notice that short half lives go with large decay constants. Radioactive material with a short half life is much more radioactive (at the time of production) but will obviously lose its radioactivity rapidly. No matter how long or short the half life is, after seven half lives have passed, there is less than 1 percent of the initial activity remaining.

The radioactive decay law can be derived also for activity calculations or mass of radioactive material calculations:

(Number of nuclei) N = N.e^-λt     (Activity) A = A.e^-λt      (Mass) m = m.e^-λt

, where N (number of particles) is the total number of particles in the sample, A (total activity) is the number of decays per unit time of a radioactive sample, m is the mass of remaining radioactive material.

Decay Constant and Half-Life

In calculations of radioactivity one of two parameters (decay constant or half-life), which characterize the rate of decay, must be known. There is a relation between the half-life (t_1/2) and the decay constant λ. The relationship can be derived from decay law by setting N = ½ N_o. This gives:

where ln 2 (the natural log of 2) equals 0.693. If the decay constant (λ) is given, it is easy to calculate the half-life, and vice-versa.

Decay Constant and Radioactivity

The relationship between half-life and the amount of a radionuclide required to give an activity of one curie is shown in the figure. This amount of material can be calculated using λ, which is the decay constant of certain nuclide:

The following figure illustrates the amount of material necessary for 1 curie of radioactivity. It is obvious, that the longer the half-life, the greater the quantity of radionuclide needed to produce the same activity. Of course, the longer lived substance will remain radioactive for a much longer time. As can be seen, the amount of material necessary for 1 curie of radioactivity can vary from an amount too small to be seen (0.00088 gram of cobalt-60), through 1 gram of radium-226, to almost three tons of uranium-238.

Example – Calculation of Radioactivity

A sample of material contains 1 mikrogram of iodine-131. Note that, iodine-131 plays a major role as a radioactive isotope present in nuclear fission products, and it a major contributor to the health hazards when released into the atmosphere during an accident. Iodine-131 has a half-life of 8.02 days.

Calculate:

1. The number of iodine-131 atoms initially present.
2. The activity of the iodine-131 in curies.
3. The number of iodine-131 atoms that will remain in 50 days.
4. The time it will take for the activity to reach 0.1 mCi.

Solution:

1. The number of atoms of iodine-131 can be determined using isotopic mass as below.

N_I-131 = m_I-131 . N_A / M_I-131

N_I-131 = (1 μg) x (6.02×10²³ nuclei/mol) / (130.91 g/mol)

N_I-131 = 4.6 x 10¹⁵ nuclei

2. The activity of the iodine-131 in curies can be determined using its decay constant:

The iodine-131 has half-live of 8.02 days (692928 sec) and therefore its decay constant is:

Using this value for the decay constant we can determine the activity of the sample:

3) and 4) The number of iodine-131 atoms that will remain in 50 days (N_50d) and the time it will take for the activity to reach 0.1 mCi can be calculated using the decay law:

As can be seen, after 50 days the number of iodine-131 atoms and thus the activity will be about 75 times lower. After 82 days the activity will be approximately 1200 times lower. Therefore, the time of ten half-lives (factor 2¹⁰ = 1024) is widely used to define residual activity.

 

See also:

Decay Law

Curie to Becquerel – Conversion

The original unit for measuring the amount of radioactivity was the curie (symbol Ci), which is a non-SI unit of radioactivity defined in 1910. A curie was originally named in honour of Pierre Curie, but was considered at least by some to be in honour of Marie Curie as well. A curie was originally defined as equivalent to the number of disintegrations that one gram of radium-226 will undergo in one second. Currently, a curie is defined as 1Ci = 3.7 x 10¹⁰ disintegrations per second. Therefore:

1Ci = 3.7 x 10¹⁰ Bq = 37 GBq

The SI unit for measuring the amount of radioactivity is the becquerel (symbol Bq). The becquerel is named in honour of Henri Becquerel, a French physicist who discovered radioactivity in 1896. One becquerel (1Bq) is equal to 1 disintegration per second.

Curie to Becquerel – Problem with Solution

A sample of material contains 1 mikrogram of iodine-131. Note that, iodine-131 plays a major role as a radioactive isotope present in nuclear fission products, and it a major contributor to the health hazards when released into the atmosphere during an accident. Iodine-131 has a half-life of 8.02 days.

Calculate:

1. The number of iodine-131 atoms initially present.
2. The activity of the iodine-131 in curies.
3. The number of iodine-131 atoms that will remain in 50 days.
4. The time it will take for the activity to reach 0.1 mCi.

Solution:

1. The number of atoms of iodine-131 can be determined using isotopic mass as below.

N_I-131 = m_I-131 . N_A / M_I-131

N_I-131 = (1 μg) x (6.02×10²³ nuclei/mol) / (130.91 g/mol)

N_I-131 = 4.6 x 10¹⁵ nuclei

2. The activity of the iodine-131 in curies can be determined using its decay constant:

The iodine-131 has half-live of 8.02 days (692928 sec) and therefore its decay constant is:

Using this value for the decay constant we can determine the activity of the sample:

3) and 4) The number of iodine-131 atoms that will remain in 50 days (N_50d) and the time it will take for the activity to reach 0.1 mCi can be calculated using the decay law:

As can be seen, after 50 days the number of iodine-131 atoms and thus the activity will be about 75 times lower. After 82 days the activity will be approximately 1200 times lower. Therefore, the time of ten half-lives (factor 2¹⁰ = 1024) is widely used to define residual activity.

See also:

Units of Radioactivity

Historically, the original unit for measuring the amount of radioactivity was the curie (symbol Ci), which is a non-SI unit of radioactivity defined in 1910.  The SI unit for measuring the amount of radioactivity is the becquerel (symbol Bq). The becquerel is named in honour of Henri Becquerel.  Rutherford (symbol Rd) is also a non-SI unit defined as the activity of a quantity of radioactive material in which one million nuclei decay per second. This unit was introduced in 1946, but after the becquerel was introduced in 1975, the rutherford became obsolete.

Calculation of Radioactivity

A sample of material contains 1 mikrogram of iodine-131. Note that, iodine-131 plays a major role as a radioactive isotope present in nuclear fission products, and it a major contributor to the health hazards when released into the atmosphere during an accident. Iodine-131 has a half-life of 8.02 days.

Calculate:

1. The number of iodine-131 atoms initially present.
2. The activity of the iodine-131 in curies.
3. The number of iodine-131 atoms that will remain in 50 days.
4. The time it will take for the activity to reach 0.1 mCi.

Solution:

1. The number of atoms of iodine-131 can be determined using isotopic mass as below.

N_I-131 = m_I-131 . N_A / M_I-131

N_I-131 = (1 μg) x (6.02×10²³ nuclei/mol) / (130.91 g/mol)

N_I-131 = 4.6 x 10¹⁵ nuclei

2. The activity of the iodine-131 in curies can be determined using its decay constant:

The iodine-131 has half-live of 8.02 days (692928 sec) and therefore its decay constant is:

Using this value for the decay constant we can determine the activity of the sample:

3) and 4) The number of iodine-131 atoms that will remain in 50 days (N_50d) and the time it will take for the activity to reach 0.1 mCi can be calculated using the decay law:

As can be seen, after 50 days the number of iodine-131 atoms and thus the activity will be about 75 times lower. After 82 days the activity will be approximately 1200 times lower. Therefore, the time of ten half-lives (factor 2¹⁰ = 1024) is widely used to define residual activity.

See also:

Radiation Protection

Stable Nuclei – Unstable Nuclei

A nuclear stability is determined by the competition between two fundamental interactions. Atomic nuclei consist of protons and neutrons, which attract each other through the nuclear force, while protons repel each other via the electromagnetic force due to their positive charge. These two forces compete, leading to various stability of nuclei. There are only certain combinations of neutrons and protons, which forms stable nuclei. Neutrons stabilize the nucleus, because they attract each other and protons , which helps offset the electrical repulsion between protons. As a result, as the number of protons increases, an increasing ratio of neutrons to protons is needed to form a stable nucleus. If there are too many (neutrons also obey the Pauli exclusion principle) or too few neutrons for a given number of protons, the resulting nucleus is not stable and it undergoes radioactive decay. Unstable isotopes decay through various radioactive decay pathways, most commonly alpha decay, beta decay, or electron capture. Many other rare types of decay, such as spontaneous fission or neutron emission are known.

The Pauli exclusion principle also influences the critical energy of fissile and fissionable nuclei. For example, actinides with odd neutron number are usually fissile (fissionable with slow neutrons) while actinides with even neutron number are usually not fissile (but are fissionable with fast neutrons). Heavy nuclei with an even number of protons and an even number of neutrons are (due to Pauli exclusion principle) very stable thanks to the occurrence of ‘paired spin’. On the other hand, nuclei with an odd number of protons and neutrons are mostly unstable.

Magic Numbers of Protons and Neutrons

A magic number is a number of nucleons in a nucleus, which corresponds to complete shells within the atomic nucleus. Atomic nuclei consisting of such a magic number of nucleons have a higher average binding energy per nucleon than one would expect based upon predictions such as the mass formula of von Weizsaecker (also called the semi-empirical mass formula – SEMF) and are hence more stable against nuclear decay. Magic numbers are predicted by the nuclear shell model and are proved by observations that have shown that there are sudden discontinuities in the proton and neutron separation energies at specific values of Z and N. These correspond to the closing of shells (or sub-shells). Nuclei with closed shells are more tightly bound than the next higher number. The closing of shells occurs at Z or N = 2, 8, 20, 28, (40), 50, 82, 126. It is found that nuclei with even numbers of protons and neutrons are more stable than those with odd numbers. Nuclei which have both neutron number and proton number equal to one of the magic numbers can be called “doubly magic“, and are found to be particularly stable.There are further special propertis of nuclei, which have a magic number of nucleons:

1. Higher abundance in nature. For example, helium-4 is among the most abundant (and stable) nuclei in the universe.
2. The stable elements at the end of the decay series all have a “magic number” of neutrons or protons. The nuclei He-4, O-16, and Pb-208 (82 protons and 126 neutrons) that contain magic numbers of both neutrons and protons are particularly stable. The relative stability of these nuclei is reminiscent of that of inert gas atoms (closed electron shells).
3. Nuclei with N = magic number have much lower neutron absorption cross-sections than surrounding isotopes.
4. These nuclei appear to be perfectly spherical in shape; they have zero quadrupole electric moments.
5. Magic number nuclei have higher first excitation energy.

Unstable Nuclei – Decay Modes

Nuclear decay (Radioactive decay) occurs when an unstable atom loses energy by emitting ionizing radiation. Radioactive decay is a random process at the level of single atoms, in that, according to quantum theory, it is impossible to predict when a particular atom will decay. During radioactive decay an unstable nucleus spontaneosly and randomly decomposes to form a different nucleus (or a different energy state – gamma decay), giving off radiation in the form of atomic partices or high energy rays. This decay occurs at a constant, predictable rate that is referred to as half-life. A stable nucleus will not undergo this kind of decay and is thus non-radioactive. There are many modes of radioactive decay:

• Alpha radioactivity. Alpha decay is the emission of alpha particles (helium nuclei). Alpha particles consist of two protons and two neutrons bound together into a particle identical to a helium nucleus. Because of its very large mass (more than 7000 times the mass of the beta particle) and its charge, it heavy ionizes material and has a very short range.
• Beta radioactivity. Beta decay is the emission of beta particles. Beta particles are high-energy, high-speed electrons or positrons emitted by certain types of radioactive nuclei such as potassium-40. The beta particles have greater range of penetration than alpha particles, but still much less than gamma rays.The beta particles emitted are a form of ionizing radiation also known as beta rays. The production of beta particles is termed beta decay.
• Gamma radioactivity. Gamma radioactivity consist of gamma rays. Gamma rays are electromagnetic radiation (high energy photons) of an very high frequency and of a high energy. They are produced by the decay of nuclei as they transition from a high energy state to a lower state known as gamma decay. Most of nuclear reactions are accompanied by gamma emission.
• Neutron emission. Neutron emission is a type of radioactive decay of nuclei containing excess neutrons (especially fission products), in which a neutron is simply ejected from the nucleus. This type of radiation plays key role in nuclear reactor control, because these neutrons are delayed  neutrons.

The rate of nuclear decay is also measured in terms of half-lives. The half-life is the amount of time it takes for a given isotope to lose half of its radioactivity. Half lives range from millionths of a second for highly radioactive fission products to billions of years for long-lived materials (such as naturally occurring uranium). Notice that short half lives go with large decay constants. Radioactive material with a short half life is much more radioactive (at the time of production) but will obviously lose its radioactivity rapidly. No matter how long or short the half life is, after seven half lives have passed, there is less than 1 percent of the initial activity remaining.

See also:

Atomic Nucleus

The original unit for measuring the amount of radioactivity was the curie (symbol Ci), which is a non-SI unit of radioactivity defined in 1910. Rutherford (symbol Rd) is also a non-SI unit defined as the activity of a quantity of radioactive material in which one million nuclei decay per second. This unit was introduced in 1946, but after the becquerel was introduced in 1975, the rutherford became obsolete.

Currently, one rutherford is equivalent to one megabecquerel:

1Rd = 1 x 10⁶ Bq = 1 MBq

One rutherford is quite large amount of activity. The typical human body contains roughly 0.0037 Rd (14 mg) of naturally occurring potassium-40. As well, a human body containing 16 kg of carbon would also have about 0.0037 Rd of carbon-14 (24 nanograms).

Rutherford – Examples

The relationship between half-life and the amount of a radionuclide required to give an activity of one rutherford is shown in the figure. This amount of material can be calculated using λ, which is the decay constant of certain nuclide:

The following figure illustrates the amount of material necessary for 37 kilorutherford of radioactivity. It is obvious, that the longer the half-life, the greater the quantity of radionuclide needed to produce the same activity. Of course, the longer lived substance will remain radioactive for a much longer time. As can be seen, the amount of material necessary for 37 kRd of radioactivity can vary from an amount too small to be seen (0.00088 gram of cobalt-60), through 1 gram of radium-226, to almost three tons of uranium-238.

Example – Calculation of Radioactivity

A sample of material contains 1 mikrogram of iodine-131. Note that, iodine-131 plays a major role as a radioactive isotope present in nuclear fission products, and it a major contributor to the health hazards when released into the atmosphere during an accident. Iodine-131 has a half-life of 8.02 days.

Calculate:

1. The number of iodine-131 atoms initially present.
2. The activity of the iodine-131 in curies.
3. The number of iodine-131 atoms that will remain in 50 days.
4. The time it will take for the activity to reach 0.1 mCi.

Solution:

1. The number of atoms of iodine-131 can be determined using isotopic mass as below.

N_I-131 = m_I-131 . N_A / M_I-131

N_I-131 = (1 μg) x (6.02×10²³ nuclei/mol) / (130.91 g/mol)

N_I-131 = 4.6 x 10¹⁵ nuclei

2. The activity of the iodine-131 in curies can be determined using its decay constant:

The iodine-131 has half-live of 8.02 days (692928 sec) and therefore its decay constant is:

Using this value for the decay constant we can determine the activity of the sample:

3) and 4) The number of iodine-131 atoms that will remain in 50 days (N_50d) and the time it will take for the activity to reach 0.1 mCi can be calculated using the decay law:

As can be seen, after 50 days the number of iodine-131 atoms and thus the activity will be about 75 times lower. After 82 days the activity will be approximately 1200 times lower. Therefore, the time of ten half-lives (factor 2¹⁰ = 1024) is widely used to define residual activity.

See also:

Units of Radioactivity

The SI unit for measuring the amount of radioactivity is the becquerel (symbol Bq). The becquerel is named in honour of Henri Becquerel, a French physicist who discovered radioactivity in 1896. One becquerel (1Bq) is equal to 1 disintegration per second.

An older unit of radioactivity is the curie. The curie was originally defined as equivalent to the number of disintegrations that one gram of radium-226 will undergo in one second. Currently, a curie is defined using becquerels as 1Ci = 3.7 x 10¹⁰ disintegrations per second. Therefore

1Ci = 3.7 x 10¹⁰ Bq = 37 GBq

One becquerel is very small amount of activity. The typical human body contains roughly 3.7 kBq (14 mg) of naturally occurring potassium-40. As well, a human body containing 16 kg of carbon would also have about 3.7 kBq of carbon-14 (24 nanograms). Activities measured in a nuclear power plant (except irradiated fuel) often have usually higher activity than becquerel, and the following multiples are often used:

1 kBq (kilobecquerel) = 1E3 Bq

1 MBq (megabecquerel) = 1E6 Bq

1 GBq (gigabecquerel) = 1E9 Bq

1 TBq (terabecquerel) = 1E12 Bq

Becquerel – Examples

The relationship between half-life and the amount of a radionuclide required to give an activity of 37 GBq (1 Ci) is shown in the figure. This amount of material can be calculated using λ, which is the decay constant of certain nuclide:

The following figure illustrates the amount of material necessary for 37 GBq of radioactivity. It is obvious, that the longer the half-life, the greater the quantity of radionuclide needed to produce the same activity. Of course, the longer lived substance will remain radioactive for a much longer time. As can be seen, the amount of material necessary for 37 GBq of radioactivity can vary from an amount too small to be seen (0.00088 gram of cobalt-60), through 1 gram of radium-226, to almost three tons of uranium-238.

 

Example – Calculation of Radioactivity

A sample of material contains 1 mikrogram of iodine-131. Note that, iodine-131 plays a major role as a radioactive isotope present in nuclear fission products, and it a major contributor to the health hazards when released into the atmosphere during an accident. Iodine-131 has a half-life of 8.02 days.

Calculate:

1. The number of iodine-131 atoms initially present.
2. The activity of the iodine-131 in curies.
3. The number of iodine-131 atoms that will remain in 50 days.
4. The time it will take for the activity to reach 0.1 mCi.

Solution:

1. The number of atoms of iodine-131 can be determined using isotopic mass as below.

N_I-131 = m_I-131 . N_A / M_I-131

N_I-131 = (1 μg) x (6.02×10²³ nuclei/mol) / (130.91 g/mol)

N_I-131 = 4.6 x 10¹⁵ nuclei

2. The activity of the iodine-131 in curies can be determined using its decay constant:

The iodine-131 has half-live of 8.02 days (692928 sec) and therefore its decay constant is:

Using this value for the decay constant we can determine the activity of the sample:

3) and 4) The number of iodine-131 atoms that will remain in 50 days (N_50d) and the time it will take for the activity to reach 0.1 mCi can be calculated using the decay law:

As can be seen, after 50 days the number of iodine-131 atoms and thus the activity will be about 75 times lower. After 82 days the activity will be approximately 1200 times lower. Therefore, the time of ten half-lives (factor 2¹⁰ = 1024) is widely used to define residual activity.

See also:

Units of Radioactivity

The original unit for measuring the amount of radioactivity was the curie (symbol Ci), which is a non-SI unit of radioactivity defined in 1910. A curie was originally named in honour of Pierre Curie, but was considered at least by some to be in honour of Marie Curie as well. A curie was originally defined as equivalent to the number of disintegrations that one gram of radium-226 will undergo in one second. Currently, a curie is defined as 1Ci = 3.7 x 10¹⁰ disintegrations per second. Therefore:

1Ci = 3.7 x 10¹⁰ Bq = 37 GBq

One curie is a large amount of activity. The typical human body contains roughly 0.1 μCi (14 mg) of naturally occurring potassium-40. As well, a human body containing 16 kg of carbon would also have about 0.1 μCi of carbon-14 (24 nanograms). Activities measured in a nuclear power plant (except irradiated fuel) often have usually lower activity than curie, and the following multiples are often used:

1 mCi (milicurie) = 1E-3 Ci

1 µCi (microcurie) = 1E-6 Ci

While its continued use is discouraged by many institutions, the curie is still widely used throughout the government, industry and medicine in the world.

Curie – Examples

The relationship between half-life and the amount of a radionuclide required to give an activity of one curie is shown in the figure. This amount of material can be calculated using λ, which is the decay constant of certain nuclide:

The following figure illustrates the amount of material necessary for 1 curie of radioactivity. It is obvious, that the longer the half-life, the greater the quantity of radionuclide needed to produce the same activity. Of course, the longer lived substance will remain radioactive for a much longer time. As can be seen, the amount of material necessary for 1 curie of radioactivity can vary from an amount too small to be seen (0.00088 gram of cobalt-60), through 1 gram of radium-226, to almost three tons of uranium-238.

Example – Calculation of Radioactivity

A sample of material contains 1 mikrogram of iodine-131. Note that, iodine-131 plays a major role as a radioactive isotope present in nuclear fission products, and it a major contributor to the health hazards when released into the atmosphere during an accident. Iodine-131 has a half-life of 8.02 days.

Calculate:

1. The number of iodine-131 atoms initially present.
2. The activity of the iodine-131 in curies.
3. The number of iodine-131 atoms that will remain in 50 days.
4. The time it will take for the activity to reach 0.1 mCi.

Solution:

1. The number of atoms of iodine-131 can be determined using isotopic mass as below.

N_I-131 = m_I-131 . N_A / M_I-131

N_I-131 = (1 μg) x (6.02×10²³ nuclei/mol) / (130.91 g/mol)

N_I-131 = 4.6 x 10¹⁵ nuclei

2. The activity of the iodine-131 in curies can be determined using its decay constant:

The iodine-131 has half-live of 8.02 days (692928 sec) and therefore its decay constant is:

Using this value for the decay constant we can determine the activity of the sample:

3) and 4) The number of iodine-131 atoms that will remain in 50 days (N_50d) and the time it will take for the activity to reach 0.1 mCi can be calculated using the decay law:

As can be seen, after 50 days the number of iodine-131 atoms and thus the activity will be about 75 times lower. After 82 days the activity will be approximately 1200 times lower. Therefore, the time of ten half-lives (factor 2¹⁰ = 1024) is widely used to define residual activity.

See also:

Units of Radioactivity

Units of Radioactivity

As was written, radioactivity (activity of certain radionuclides) is the process by which an unstable nucleus spontaneosly and randomly disintegrates to form a different nucleus (or a different energy state – gamma decay), giving off radiation in the form of atomic partices or high energy rays. This decay occurs at a constant, predictable rate that is referred to as decay constant. As a result, a measure of radioactivity is based on counting of disintegrations per second. The activity depends only on the number of decays per second, not on the type of decay, the energy of the decay products, or the biological effects of the radiation. It can be used to characterize the rate of emission of ionizing radiation.

Units of radioactivity (the curie and the becquerel) can be also used to characterize an overall quantity of controlled or accidental releases of radioactive atoms. Because the probability of decay is a fixed physical quantity, for a known number of atoms of a particular radionuclide, a predictable number will decay in a given time. For example, according to the Nuclear Safety Commission (NSC) of Japan, the total amounts of activity released during Fukushima Daiichi accident between 11 March and 5 April were about 130 PBq (petabecquerels, 3.5 megacuries) for iodine-131 and 11 PBq for caesium-137.

According to the UNSCEAR, from 1998 to 2002, the global annual average releases of tritium to the atmosphere and to the aqueous environment from nuclear facilities were estimated to be 11.7 PBq and 16.0 PBq, respectively. Note that, tritium emits low-energy beta particles with a short range in body tissues and, therefore, poses a risk to health as a result of internal exposure only following ingestion in drinking water or food, or inhalation or absorption through the skin. According to the ICRP, a biological half-time of tritium is 10 days for HTO and 40 days for OBT (organically bound tritium) formed from HTO in the body of adults.

See also: SOURCES, EFFECTS AND RISKS OF IONIZING RADIATION, UNSCEAR 2016, ISBN: 978-92-1-142316-7.

Historically, the original unit for measuring the amount of radioactivity was the curie (symbol Ci), which is a non-SI unit of radioactivity defined in 1910.  The SI unit for measuring the amount of radioactivity is the becquerel (symbol Bq). The becquerel is named in honour of Henri Becquerel.  Rutherford (symbol Rd) is also a non-SI unit defined as the activity of a quantity of radioactive material in which one million nuclei decay per second. This unit was introduced in 1946, but after the becquerel was introduced in 1975, the rutherford became obsolete.

Example – Calculation of Radioactivity

A sample of material contains 1 mikrogram of iodine-131. Note that, iodine-131 plays a major role as a radioactive isotope present in nuclear fission products, and it a major contributor to the health hazards when released into the atmosphere during an accident. Iodine-131 has a half-life of 8.02 days.

Calculate:

1. The number of iodine-131 atoms initially present.
2. The activity of the iodine-131 in curies.
3. The number of iodine-131 atoms that will remain in 50 days.
4. The time it will take for the activity to reach 0.1 mCi.

Solution:

1. The number of atoms of iodine-131 can be determined using isotopic mass as below.

N_I-131 = m_I-131 . N_A / M_I-131

N_I-131 = (1 μg) x (6.02×10²³ nuclei/mol) / (130.91 g/mol)

N_I-131 = 4.6 x 10¹⁵ nuclei

2. The activity of the iodine-131 in curies can be determined using its decay constant:

The iodine-131 has half-live of 8.02 days (692928 sec) and therefore its decay constant is:

Using this value for the decay constant we can determine the activity of the sample:

3) and 4) The number of iodine-131 atoms that will remain in 50 days (N_50d) and the time it will take for the activity to reach 0.1 mCi can be calculated using the decay law:

As can be seen, after 50 days the number of iodine-131 atoms and thus the activity will be about 75 times lower. After 82 days the activity will be approximately 1200 times lower. Therefore, the time of ten half-lives (factor 2¹⁰ = 1024) is widely used to define residual activity.

See also:

Radiation Protection

In analyzing nuclear decay reactions, we apply the many conservation laws. Nuclear decay reactions are subject to classical conservation laws for charge, momentum, angular momentum, and energy(including rest energies).  Additional conservation laws, not anticipated by classical physics, are:

• Law of Conservation of Lepton Number
• Law of Conservation of Baryon Number
• Law of Conservation of Electric Charge

Certain of these laws are obeyed under all circumstances, others are not. We have accepted conservation of energy and momentum. In all the examples given we assume that the number of protons and the number of neutrons is separately conserved. We shall find circumstances and conditions in which  this rule is not true. Where we are considering non-relativistic nuclear reactions, it is essentially true. However, where we are considering relativistic nuclear energies or those involving the weak interactions, we shall find that these principles must be extended.

Some conservation principles have arisen from theoretical considerations, others are just empirical relationships. Notwithstanding, any reaction not expressly forbidden by the conservation laws will generally occur, if perhaps at a slow rate. This expectation is based on quantum mechanics. Unless the barrier between the initial and final states is infinitely high, there is always a non-zero probability that a system will make the transition between them.

For purposes of analyzing non-relativistic reactions, it is sufficient to note four of the fundamental laws governing these reactions.

1. Conservation of nucleons. The total number of nucleons before and after a reaction are the same.
2. Conservation of charge. The sum of the charges on all the particles before and after a reaction are the same
3. Conservation of momentum. The total momentum of the interacting particles before and after a reaction are the same.
4. Conservation of energy. Energy, including rest mass energy, is conserved in nuclear reactions.

Reference: Lamarsh, John R. Introduction to Nuclear engineering 2nd Edition

See also:

Radioactivity

Nuclear decay (Radioactive decay) occurs when an unstable atom loses energy by emitting ionizing radiation. Radioactive decay is a random process at the level of single atoms, in that, according to quantum theory, it is impossible to predict when a particular atom will decay. During radioactive decay an unstable nucleus spontaneosly and randomly decomposes to form a different nucleus (or a different energy state – gamma decay), giving off radiation in the form of atomic partices or high energy rays. This decay occurs at a constant, predictable rate that is referred to as half-life. A stable nucleus will not undergo this kind of decay and is thus non-radioactive. There are many modes of radioactive decay:

• 

  Alpha radioactivity. Alpha decay is the emission of alpha particles (helium nuclei). Alpha particles consist of two protons and two neutrons bound together into a particle identical to a helium nucleus. Because of its very large mass (more than 7000 times the mass of the beta particle) and its charge, it heavy ionizes material and has a very short range.

• Beta radioactivity. Beta decay is the emission of beta particles. Beta particles are high-energy, high-speed electrons or positrons emitted by certain types of radioactive nuclei such as potassium-40. The beta particles have greater range of penetration than alpha particles, but still much less than gamma rays.The beta particles emitted are a form of ionizing radiation also known as beta rays. The production of beta particles is termed beta decay.
• Gamma radioactivity. Gamma radioactivity consist of gamma rays. Gamma rays are electromagnetic radiation (high energy photons) of an very high frequency and of a high energy. They are produced by the decay of nuclei as they transition from a high energy state to a lower state known as gamma decay. Most of nuclear reactions are accompanied by gamma emission.
• Neutron emission. Neutron emission is a type of radioactive decay of nuclei containing excess neutrons (especially fission products), in which a neutron is simply ejected from the nucleus. This type of radiation plays key role in nuclear reactor control, because these neutrons are delayed  neutrons.

 

See also:

Radioactivity