Radioactive decay

1. Random nature of radioactive decay:

• The emission of radiation from a nucleus is both spontaneous and random. A nucleus could remain unchanged for millions of years before suddenly it decays by emitting a radioactive particle.
• Nuclei decay independently of each other, and their behavior is not affected by the proximity of other nuclei or external factors such as temperature and pressure.
• The decay constant (λ) is a fundamental parameter in radioactive decay. It’s a measure of the probability of a nucleus decaying per unit time.
• The decay constant is defined as the number of nuclei decaying per unit time, divided by the total number of nuclei present.
• This definition of λ leads to the equation:
• [math] λ = \text{fractional change in the number of nuclei} \, , \frac{∆N}{N} , \, \text{per unit time} \, ∆t [/math]
• or
• [math] λ = – \frac{∆N ⁄ N}{∆t} \\
  λ=- \frac{∆N}{ N∆t} [/math]
• The significance of the minus sign is that the number of radioactive nuclei in a sample of material decreases with time.
  The unit of [math] λ  \, \text{is} \, s^{-1}[/math].
  In the dice decay’, λ is [math] \frac{1}{6} [/math]  per throw – every time the dice are thrown, (on average) [math] \frac{1}{6} [/math] of them turn up a 6.
  The equation above may be written in this form:
• [math] \frac{∆N}{∆t} = -λN [/math]
• In words, we can say that:
• number of nuclei decaying per second = decay constant * number of nuclei
• This leads to the definition of the activity of a radioactive source, which is the number of emissions per second.
• [math]A=λN [/math]
• Where A is the activity of the source. The unit of activity is the becquerel (Bq), which is a rate of decay of one disintegration per second.

2. Decay constant:

• Number of nuclei decaying per second is equal to the product of decay constant and number of nuclei
• [math] \frac{∆N}{∆t} = -λN [/math]
• A differential equation is solved by separating the variables and integrating both sides. So doing this gives
• [math] \frac{dN}{dt} = -\lambda N \\ \int_{N_0}^{N} \frac{dN}{N} = \int_0^t -\lambda \, dt [/math]
• Note that the limits of the integration are from  to N for the nuclei and from 0 to t for the time.
• [math] [\ln N ]_{N_0}^{N} = [-\lambda t]_0^t \\
  \ln \left( \frac{N}{N_0} \right) = -\lambda t \\
  \frac{N}{N_0} = e^{-\lambda t} \\
  N = N_0 e^{-\lambda t} [/math]
• Where N is the number of nuclei in the radioactive sample at time t, and [math]N_0 [/math] is the number of nuclei at time t=0, which is the time that we start to observe the sample of nuclei.

Example:

(1) A sample of radioactive material contains [math]100 * 10^{12} [/math] The nuclei have a decay constant of [math] 0.01 s^{-1} [/math]. Predict the number of nuclei remaining after 10s.
Given data:
Radioactive material [math] = N_0 = 100 * 10^{12} [/math]
Decay constant [math] = λ = 0.01s^{-1} [/math]
time [math]= t = 10s [/math]

Find data:
Number of nuclei = N =?

Formula:

[math]N = N_0 e^{-λt} [/math]

Solution:

[math]N = N_0 e^{- λt} \\
N = (100 * 10^{12}) e^{- (0.01)(10)} \\
N = (10^{14})e^{- 0.1} \\
N = 9 * 10^{13} [/math]

So, there will be about [math] 9 * 10^{13} [/math] nuclei after 10s.

(2) Draw up a table of the number of nuclei at intervals of 20s up to a time of 160 s.

Solution:
Table 1

Figure 1  The graph shows the exponential decay of a sample of radioactive nuclei.

The numbers in table 1 have been used to plot a graph of the number of nuclei against time; this is shown in figure 1. This graph shows an exponential decay, and it has the following important qualities.

3. Half-life

• Half-life [math] (T_{1⁄2}) [/math] is the time required for half of the initial number of nuclei to decay or transform into a more stable form. It’s a fundamental concept in radioactive decay.
• [math] N = N_0 e^{-λt} [/math]
• After one half-life [math] T_{1⁄2} [/math], there will be [math] \frac{N_0}{2} [/math] nuclei left, so
• [math] \frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}\\
  \frac{1}{2} = e^{-\lambda T_{1/2}} \\
  \ln \frac{1}{2} = -\lambda T_{1/2} \\
  \ln 2 = \lambda T_{1/2} \\
  T_{1/2} = \frac{\ln 2}{\lambda} \\
  T_{1/2} = \frac{0.69}{\lambda} [/math]
• Here are some key points:- Half-life is constant for a given type of nucleus.
  – It’s a measure of the decay rate, but not the total amount of decay.
  – After one half-life, half of the initial nuclei remain.
  – After two half-lives, a quarter of the initial nuclei remain.
  – After three half-lives, an eighth of the initial nuclei remain, and so on.
• Half-life ranges from extremely short (fractions of a second) to extremely long (billions of years).
• Some examples:
  – Carbon-14: 5,730 years
  – Uranium-238: 4.5 billion years
  – Radon-222: 3.8 days
  – Francium-223: 22 minutes

4. Modelling with constant decay probability:

• Modelling with constant decay probability is a fundamental concept in radioactive decay. It assumes that the probability of decay per unit time is constant, leading to an exponential decay law.
• Here are the key aspects:
  – Constant decay probability (λ)

   

  [math] N(t) = N_0 e^{-λt} [/math]

  – Exponential decay law:
  – Half-life : time for half of the initial nuclei to decay
  – Decay constant (λ): related to half-life by

  [math] T_{1⁄2} = \frac{ln⁡2}{λ} \\
  λ = \frac{ln⁡2}{T_{1⁄2}} [/math]

• This model is widely used due to its simplicity and accuracy in many cases. However, it’s important to note that:
  – Real-world decay processes can deviate from this model
  – More complex models may be needed for certain situations (e.g., chain decay, branching ratios)
• Constant decay probability modelling has far-reaching applications in:
  – Nuclear physics
  – Radiocarbon dating
  – Radiation safety
  – Medical applications (radiation therapy)
  – Industrial applications (food irradiation)

5. Molar mass or the Avogadro constant:

• Molar mass and the Avogadro constant are important concepts in chemistry and physics, and they can be related to radioactive decay and nuclear reactions.
• Molar mass is the mass of one mole of a substance, and it’s used to convert between the amount of a substance (in moles) and its mass (in grams or other units).
• The Avogadro constant (NA) is the number of particles (atoms or molecules) in one mole of a substance. It’s used to convert between the amount of a substance (in moles) and the number of particles.
• In the context of radioactive decay, molar mass and the Avogadro constant can be used to:
  – Calculate the activity of a sample (in decays per second)
  – Determine the number of nuclei present in a sample
  – Calculate the mass of a sample required for a certain activity or number of nuclei

Examples:
(1) How many nuclei are present in a 10gram sample of carbon-14?
Given data:
Mass of carbon = m = 10g
Molar mass = 14 g/mol
Avogadro number = NA =[math]6.022 * 10^{23 nuclei /mole}[/math]

Find data:
Number of moles = n =?
Number of nuclei =?

Formula:

[math] \text{number of moles} = \frac{\text{mass}}{\text{molar mass}} \\
\text{nuclei} = \text{moles} * NA [/math]

Solution:

[math] \text{number of moles} =\frac{10}{14} \\
\text{number of moles} = 0.714 moles [/math]

By using the Avogadro constant (NA) to calculate the number of nuclei

[math] \text{nuclei} = moles * NA \\
\text{nuclei} = 0.714 * 6.022 * 10^{23} \\
\text{nuclei} = 4.30 * 10^{23} nuclei [/math]

So, approximately [math]4.30 * 10^{23} nuclei [/math]carbon-14 nuclei are present in a 10g sample.

(2) What is the activity of a 5milligram sample of uranium-238?
Given data:
Mass of uranium-238 [math] = m = 5mg = 5 * 10^{-3}g = 0.005g [/math]
Molar mass [math]= 238 g/mol [/math]
Half-life of uranium-238 [math] = T_{1⁄2} = 4.51 * 10^9 year * 3.15 * 10^7 second/year = 14.2 * 10^{16} s [/math]

Find data:
Number of nuclei = ?
Decay constant = λ= ?
Decay activity = A = ?

Formula:

[math] \text{number of moles} = \frac{\text{mass}}{\text{molar mass}} \\
\text{nuclei} = \text{moles} * NA \\
\lambda = \frac{\ln 2}{T_{1/2}} \\
A = \lambda * \text{nuclei} [/math]

Solution:

[math] \text{Number of moles} = \frac{\text{mass}}{\text{molar mass}} \\
\text{Number of moles} = \frac{0.005}{238} \\
\text{Number of moles} = 2.1 * 10^{-5} \, \text{mole} \\
\text{Nuclei} = \text{moles} * N_A \\
\text{Nuclei} = (2.1 * 10^{-5}) * (6.022 * 10^{23}) \\
\text{Nuclei} = 1.26 * 10^{19} \, \text{nuclei} \\
\lambda = \frac{\ln 2}{T_{1/2}} \\
\lambda = \frac{0.69}{14.2 * 10^{16}} \\
\lambda = 1.55 * 10^{-10} \, \text{dps/nucleus} \\
A = \lambda * \text{nuclei} \\
A = (1.55 *10^{-10}) * (1.26* 10^{19}) \\
A = 1.95 * 10^{9} \, \text{dps} [/math]

So, the activity of a 5mg sample of uranium-238 is approximately [math] 1.95 * 10^{9} \, \text{decays per second} [/math] .

6. Applications:

• The concepts of half-life, decay constant, and exponential decay have numerous applications in various fields:
  1. Radioactive Waste Storage: Understanding the half-life of radioactive isotopes is crucial for storing waste safely. It helps determine the optimal storage time and conditions to minimize radiation exposure.
  2. Radioactive Dating: Half-life is used to date rocks, fossils, and artifacts. By measuring the decay of isotopes like carbon-14, uranium-238, and potassium-40, scientists can determine the age of samples.
  3. Nuclear Medicine: Radioisotopes with specific half-lives are used for diagnosis and treatment. For example, technetium-99m (half-life: 6 hours) is used for imaging, while iodine-131 (half-life: 8 days) is used for thyroid treatment.
  4. Food Irradiation: Gamma radiation is used to sterilize food, extending shelf life. Half-life is essential for calculating the required radiation dose.
  5. Radiation Safety: Understanding half-life helps protect people from radiation exposure. It informs safety protocols for handling radioactive materials and managing radiation emergencies.
  6. Environmental Monitoring: Half-life is used to study the movement and distribution of radioactive isotopes in the environment, helping monitor pollution and assess ecological risks.
  7. Geological Research: Half-life helps scientists study Earth’s history, including geological events and climate change.
  8. Industrial Applications: Radioisotopes are used for various industrial purposes, such as sterilization, food irradiation, and materials modification.
• These applications rely on the principles of radioactive decay, showcasing the significance of half-life and exponential decay in real-world contexts.