Question

Iodine-129 is a product of nuclear fission, whether from an atomic bomb or a nuclear power plant. It is a \(\beta^{-}\) emitter with a half-life of \(1.7 \times 10^{7}\) years. How many disintegrations per second would occur in a sample containing \(1.00 \mathrm{mg}^{129} \mathrm{I} ?\)

Short answer

The disintegrations per second for this 1 mg sample of Iodine-129 is calculated by first establishing necessary constants, then converting the mass of Iodine-129 to moles. From there, use Avogadro's number to calculate the number of atoms. The decay constant can then be calculated from the half-life given, which is used in the decay law formula along with the number of atoms to determine the amount of disintegrations per second.

Step-by-step solution

Establish Necessary Constants

Before starting the calculation, some constants must be established. The atomic mass of Iodine-129 is \(129 \, g/mol\). Avogadro's number is \(6.022 \times 10^{23} \, atoms/mol\). It is also important to note that 1 year has approximately \(3.1536 \times 10^{7}\) seconds.

Convert Mass to Moles

First, convert the mass of the Iodine-129 sample from mg to g, knowing that 1 g has a thousand mg. Then, convert the mass in grams to moles using the atomic mass of Iodine-129: \[ moles = \frac{mass}{atomic \, mass} = \frac{0.001 \, g}{129 \, g/mol} \] You'll obtain an amount in moles of Iodine-129.

Find the Number of Atoms

Next, find the number of atoms by multiplying the amount in moles by Avogadro's number: \[ atoms = moles \times Avogadro's \, number\] You'll obtain the total number of Iodine-129 atoms in the 1 mg sample.

Calculate the Decay Constant

Since the half-life has been provided in years, it must be converted to seconds using the constant provided in step 1. Then, recall the formula for the decay constant, \(λ\), given a half-life, \(T\): \[ λ = \frac{ln2}{T} \] Where \(T\) is the half-life of the substance in seconds. This will give you the decay constant in \(s^{-1}\).

Find the Rate of Decay

Finally, use the decay law formula to find the rate of decay, or the disintegrations per second: \[ rate = λ \times atoms \] This calculation will yield the disintegrations per second in the given sample of Iodine-129.

Key concepts

Half-life

Half-life is a term that stands front and center in nuclear physics, referring to the period it takes for half the atoms in a radioactive substance to decay. That is, after one half-life, a 1 kg sample of a radioactive element would diminish to 0.5 kg. This concept is crucial for understanding how quickly unstable atoms transition to more stable forms. It's also a key factor in dating archaeological finds, assessing environmental contamination, and managing nuclear waste.

For our Iodine-129 example, with a half-life of 17 million years, this isn't just a piece of trivia; it's a numerical glimpse into its long-term persistence in the environment and its potential impact on living organisms over extended periods.

Avogadro's Number

Avogadro's number, a staple in chemistry and physics discussions, is a constant that connects the macroscopic and microscopic worlds. By definition, it's the number of atoms or molecules in one mole of a substance. This whopping figure, approximately 6.022 x 10²³, enables us to count individual atoms by weighing them en masse — think of it like knowing the count of rice grains in a sack by its total weight.

When dealing with decay calculations, Avogadro's number lets us jump from the mass of the substance directly to the number of atoms present. This paves the way for a deeper understanding of how much material remains after a certain period and the activity of a radioactive sample, as seen in the case of our 1 mg of Iodine-129.

Radioactive Decay

Radioactive decay occurs spontaneously in unstable atomic nuclei, leading to the emission of particles or radiation and the transformation into a different element or isotope. It's the engine behind phenomena like carbon dating and nuclear medicine.

Radiation comes in various forms, such as alpha particles, beta particles, and gamma rays, each carrying different levels of energy and penetration capabilities. Our iodine sample emits beta particles (β^-), indicating that a neutron in its nucleus has transformed into a proton, electron, and an electron antineutrino; the electron is the beta particle that zips away at high speed.

Decay Constant

The decay constant (represented by the Greek letter lambda, \(λ\)) is a probability factor that quantifies how likely it is for a single atom of a radioactive substance to decay per unit of time. Inversely related to the half-life, it underlines the relationship between the time aspect and the transformation process.

Calculating the decay constant, which involves natural logarithms and time conversions, is necessary to determine the disintegration rate of our Iodine-129 sample. It's a measure that helps to bridge the abstract world of probability with the tangible effects of nuclear decay.

Nuclear Fission Products

When a heavy nucleus splits into smaller parts — a process known as nuclear fission — it releases a significant amount of energy and a variety of smaller atoms, which we call fission products. These byproducts are often radioactive themselves, inheriting the unstable mantle from their precursor and perpetuating the chain of radioactive decay.

Iodine-129 is one such fission product. Its very existence in the environment or waste from a nuclear reactor is a strong indicator of past fission events. Understanding these products allows us to trace the history of nuclear material and manage its life cycle with an eye on safety and environmental impact.