Question

If a friend creates a nucleus whose half-life is 4 hours and gives it to you at noon, what is the probability that it will not have decayed by noon the following day?

Short answer

Answer: The probability of the nucleus not having decayed after 24 hours is approximately 3.63%.

Step-by-step solution

Recall the exponential decay formula

The exponential decay formula is given by N(t) = N₀ e^(-λt), where N(t) is the number of nuclei remaining at time t, N₀ is the initial number of nuclei, λ is the decay constant, and t is the time elapsed. Since half-life is given, we can first find the decay constant λ using the relation T = ln(2) / λ, where T is the half-life.

Find the decay constant λ

We are given that the half-life T is 4 hours. Use the formula T = ln(2) / λ to find λ. Rearrange the formula to solve for λ: λ = ln(2) / T Plugging in the values: λ = ln(2) / 4 ≈ 0.1733/hour

Use the exponential decay formula to find the remaining percentage of the nucleus after 24 hours

We can use the exponential decay formula N(t) = N₀ e^(-λt) to find the percentage of the nucleus remaining after 24 hours. For this, we can consider that the initial number of nuclei N₀ is 1. The variable t indicates the elapsed time, in this case, 24 hours. N(24) = 1 * e^(-0.1733 * 24) N(24) ≈ 0.0363

Interpret the remaining percentage as the probability

The remaining percentage of the nucleus (0.0363 or 3.63%) can be interpreted as the probability that the nucleus will not have decayed after 24 hours. So, the probability of the nucleus not having decayed by noon the following day is approximately 3.63%.

Key concepts

Understanding Nuclear Physics

Nuclear physics is a branch of physics that deals with the constituents and interactions of atomic nuclei. The most prominent aspect of nuclear physics is how it explains the principles behind the energy that powers the sun, nuclear reactors, and medical imaging devices like PET scans. Notably, nuclear physics describes the behavior of nuclei under various conditions, including the natural phenomenon of radioactive decay.

Radioactive decay refers to the process by which an unstable atomic nucleus loses energy by radiating particles or electromagnetic waves. This process can change one type of element into another and is a spontaneous event that occurs at the level of the atomic nuclei. Understanding this process requires a grasp of other related concepts such as half-life, the exponential decay formula, and the decay constant, each playing a key role in comprehending how elements transform over time.

Delving into Half-Life Calculation

Half-life is a term used in nuclear physics to describe the time it takes for half of the radioactive nuclei in a given sample to decay. It is a constant amount of time specific to each radioactive substance and is crucial for gauging the longevity and decay rate of radioactive materials. Calculation of half-life is fundamental in various fields, including archaeology for carbon dating, medicine for drug clearance from the body, and environmental science for understanding pollutant degradation.

To calculate half-life, one has to understand and employ the relationship between half-life and decay constant. As demonstrated in the textbook exercise, the concept of half-life can also be applied to determine the probability of a nucleus decaying over a period, providing insightful predictions about how matter evolves over time.

Exploring the Exponential Decay Formula

The exponential decay formula is a mathematical representation of the decay process of unstable nuclei. It takes the form of N(t) = N₀ e^(-λt), where N(t) denotes the number of undecayed nuclei at a certain time, N₀ represents the initial quantity, λ is the decay constant, and t symbolizes time elapsed.

The formula's exponential nature reflects how a substance's quantity decreases at a rate proportional to its current amount. To put this in perspective, if a substance has a large number of unstable nuclei, more of them will decay in a given timeframe compared to a substance with fewer unstable nuclei. This formula helps scientists and engineers calculate the remaining amount of a substance over time, which is pivotal for safety, dosage, and material integrity considerations.

Grasping the Decay Constant

The decay constant λ in the exponential decay formula is a pivotal factor that characterizes the rate at which a radioactive substance decays. It is distinct for each radioactive isotope, symbolizing the probability of decay of a nucleus over a specific time interval. The decay constant is directly related to half-life and is found using the relationship λ = ln(2) / T, where ln(2) is the natural logarithm of 2 and T is the half-life.

Understanding the decay constant is crucial for all kinds of practical applications, from nuclear energy management to setting safety standards in hospitals where radioactive substances are used for diagnoses or treatments. The decay constant not only informs us about the intrinsic stability of a nucleus but also provides insights into the potential risks and necessary precautions needed when handling radioactive materials.