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In the intriguing world of radioactive decay, understanding the concept of half-life is crucial. The half-life of a substance is the period it takes for half of its radioactive nuclei to disintegrate. This time frame remains constant for each type of radioactive nuclide. For example, if you have a sample with a half-life of 30 years, like the nuclide in our problem, it will consistently take 30 years for half of the nuclides in the sample to decay.
Simply put, half-life helps us determine how long it will take for a specific amount of a substance to decay to half its original quantity. It doesn't matter how much of the substance you start with; this time frame remains consistent, making it a reliable measure in predicting the decay process of radioisotopes.

A radioactive nuclide, often referred to as a radionuclide, is an atom with an unstable nucleus that loses energy through radiation. When discussing radioactive nuclides like the one in our exercise, these specific radioactive isotopes undergo decay, resulting in the emission of particles or electromagnetic radiation.
Key characteristics to consider when studying radioactive nuclides include:

• They are unstable and tend to release energy to achieve a more stable form.
• Each has a unique half-life, which helps determine the rate at which it decays.
• They play significant roles in various fields, ranging from medicine to energy production.

Understanding radioactive nuclides is essential for comprehending processes involving nuclear reactions and their practical applications.

The term 'fraction remaining' in the context of radioactive decay refers to the amount of the original radioactive sample that still has not decayed. Calculating this fraction is essential to understanding how much of the original radioactive nuclide remains after a certain period.
To find the fraction remaining after a given time, you can use the formula based on the number of elapsed half-lives: \[\text{Fraction remaining} = \left( \frac{1}{2} \right)^n\]where \(n\) is the number of half-lives that have passed.
For example, after 2 half-lives, as seen with 60 years in the exercise, the fraction remaining is \(\left( \frac{1}{2} \right)^2 = \frac{1}{4}\). This means only a quarter of the original sample remains undecayed.

The undecayed sample refers to the portion of the original radioactive material that remains unchanged after a certain time period. This part of the sample has not yet undergone any radioactive decay process and still retains its original properties.
In radioactive decay experiments such as our exercise, the undecayed sample is calculated to observe how much of the radioactive nuclide is left after a series of half-lives.

• The larger the number of half-lives completed, the smaller the undecayed fraction will be.
• For our nuclide with a 30-year half-life, only \(\frac{1}{4}\) of the sample remains undecayed after 60 years, and only \(\frac{1}{8}\) after 90 years.

Understanding the concept of an undecayed sample is crucial for calculating how long a substance will remain active or potent in its original form as it gradually decays over time.