Question

A freshly isolated sample of \(90 \mathrm{Y}\) was found to have an activity of \(9.8 \times 10^{5}\) disintegrations per minute at 1: 00 P.M. on December \(3,2000 .\) At 2: 15 P.M. on December \(17,2000,\) its activity was redetermined and found to be \(2.6 \times 10^{4}\) disintegrations per minute. Calculate the half-life of \(90 \mathrm{Y}\).

Short answer

The half-life of \(90 \mathrm{Y}\) is around \(64\) minutes.

Step-by-step solution

Understand Definitions & Constants

Here we are dealing with the concept of half-life of a radioactive element, which is the time required for the activity of a given quantity of a radioactive substance to decay to half of its initial value. We need to find this half-life for \(90 \mathrm{Y}\). Take note of the decay constant \(\lambda\), which is related to half-life \(T_{1/2}\) by the formula \(T_{1/2} = \frac{0.693}{\lambda}\).

Define the knowns

We know the initial activity \(N_0 = 9.8 \times 10^{5}\) decays per minute, the final activity \(N = 2.6 \times 10^{4}\) decays per minute, and the time interval which is 14 days and 15 mins equal to \(20085\) minutes. And we are to solve for \(T_{1/2}\), the half-life.

Apply the Radioactive Decay Law

Since the decay laws states \(N = N_0 e^{-\lambda t}\), after rearranging it to solve for decay constant we have \(\lambda = - \frac{1}{t} \ln \frac{N}{N_0}\), substituting the known values and calculating will yield \(\lambda\).

Find the half-life

Having found \(\lambda\), we can substitute it in the relation \(T_{1/2} = \frac{0.693}{\lambda}\) to find \(T_{1/2}\), the half-life of the \(90 \mathrm{Y}\).

Key concepts

Radioactive Decay

Radioactive decay is a natural process in which unstable atomic nuclei lose energy by emitting radiation. This emission occurs when a nucleus transforms into a smaller nucleus with less energy, often converting into a more stable element or isotope.
This change happens over time, leading to a decrease in the activity of the radioactive substance. Radioactive decay is a statistical process. It happens randomly for each atom, but collectively, it can be predicted accurately over time. This prediction is possible through an essential property known as the decay constant, which we'll discuss next.
For example, in our exercise, a sample of the radioisotope 90Y (Yttrium-90) undergoes radioactive decay, reducing its activity from 9.8 x 10^5 disintegrations per minute to 2.6 x 10^4 disintegrations per minute over a specified period. This decrease in activity is a direct result of the decay process.

Decay Constant

The decay constant, often denoted by the Greek letter \( \, \lambda \), is a probability measure that quantifies the likelihood of a decay event per unit time. This constant is crucial because it directly influences the rate of radioactive decay and helps in determining the half-life of a radioisotope.A large decay constant indicates rapid decay, while a small decay constant suggests slow decay. Mathematically, the relationship between the decay constant and activity \( N \) over time \( t \) is given by the radioactive decay law: \( N = N_0 e^{-\lambda t} \).
In our exercise, to find the decay constant for 90Y, we rearrange this decay law to \( \lambda = - \frac{1}{t} \ln \frac{N}{N_0} \), where \( N_0 \) is the initial activity and \( N \) is the activity after time \( t \). Once we have \( \lambda \), we can further use it to calculate the half-life of 90Y, showcasing its critical role in understanding radioactive decay.

Radioisotope 90Y

Radioisotope 90Y, or Yttrium-90, is a radioactive isotope used in various medical and industrial applications. It is particularly noted for its use in radiation therapy to treat certain cancers, taking advantage of its decay properties to target and destroy cancer cells. 90Y emits beta particles during its decay process, which makes it useful in targeted radiation treatments, minimizing damage to surrounding healthy tissue.
One key characteristic of 90Y is its half-life, which determines how long it remains active as a radioisotope. Understanding the half-life, which we've calculated in our exercise, helps in safely managing its storage and usage in treatments. By analyzing its decay over a given timeframe, like in the exercise, scientists can determine its half-life and ensure it is used effectively both in medicine and industry. This ability to measure and predict the behavior of radioisotopes is vital for their safe and beneficial application in real-world scenarios.