Question

Find the half-life of a radioactive material if after 1 year \(99.57 \%\) of the initial amount remains.

Short answer

Solving this gives a half-life \(T_{1/2} \approx 161 \) years

Step-by-step solution

Identify the knowns and unknowns

In this problem, the portion decayed is 0.43% or \(0.0043\) after 1 year. The objective is to find the half-life, represented by \(T_{1/2}\), of the radioactive material.

Apply the exponential decay formula

The formula for exponential decay is \(N = N_0e^{-\lambda t}\), where \(N\) is the final amount of the substance, \(N_0\) is the initial amount, \(\lambda\) is the decay constant, and \(t\) is the time. Since we know that the the decayed portion is \(0.0043 = N_0 - N\), we can rewrite this equation using these variables and solve for \(\lambda\). Which gives us the equation \(0.0043 = N_0 - N_0e^{-\lambda * 1}\)

Solve for the decay constant

Divide both sides of the equation by \(N_0\) and add \(e^{-\lambda}\) to both sides to get \(e^{-\lambda} = 0.9957\). Then, take the natural log of both sides to get \(-\lambda = ln(0.9957)\), which gives you \(\lambda \approx 0.0043\)

Calculate the half-life

The relationship between the decay constant and the half-life is given by \(T_{1/2} = \frac{ln(2)}{\lambda}\). Substituting the earlier found value for \(\lambda\) we get \(T_{1/2} = \frac{ln(2)}{0.0043}\)

Key concepts

Exponential Decay Formula

The phenomenon of radioactive substances reducing in quantity over time is modeled by the exponential decay formula. This mathematical representation is fundamental in understanding how rapidly a radioactive element loses its potency. The general formula used to describe this process is \[N = N_0e^{-\text{λ}t}\],where

• \(N\) refers to the quantity of the substance remaining after time \(t\),
• \(N_0\) is the initial quantity present,
• \(e\) is the base of the natural logarithms,
• \(λ\) is the decay constant which indicates the rate of decay,
• \(t\) is the time elapsed.

In the context of the exercise, using the exponential decay formula, students can determine the decay constant by observing the remaining quantity of substance after a certain time frame. Simplifying the formula with given values allows us to see how much of the material has decayed and thus infer the behavior of radioactive decay over time.

Decay Constant

The decay constant, symbolized as \(λ\), is an essential part of understanding the kinetics of radioactive decay. It is unique to each radioactive substance and quantifies the rate at which the substance undergoes decay. The value of \(λ\) tells us the proportion of a substance that decays per unit time. Thus, a higher decay constant means a quicker reduction in the amount of the radioactive substance.

To find the half-life of a radioactive material, understanding the decay constant is crucial. As demonstrated in the exercise, we extract the decay constant from the exponential decay formula by rearranging the terms and isolating \(λ\). Here's a breakdown of the process:

• We first determine how much of the substance has decayed over a known period,
• We then reformulate the exponential decay formula to solve for \(λ\),
• Using logarithmic functions allows us to extract \(λ\) from an exponent,
• A calculated decay constant can then be utilized to find the half-life of the radioactive material.

In short, the decay constant represents the intrinsic rate of radioactive decay, serving as a bridge between the remaining amount of substance after a period and the half-life, which is a concept frequently used to describe the longevity of the substance's activity.

Radioactive Decay Mathematics

Radioactive decay is intrinsically probabilistic and is described using mathematics originating from statistics and differential equations. The mathematics underlying radioactive decay provide a detailed framework for predicting the behavior of unstable isotopes. This process is expressed using the exponential decay formula, which incorporates the decay constant to capture the stochastic nature of nuclear decay processes.By applying calculus and logarithmic transformations, as in the steps outlined in the solution, students can determine various useful pieces of information, such as the rate of decay (decay constant) and the half-life. The half-life, \(T_{1/2}\), represents the time taken for half of the radioactive material to decay. The mathematical relationship between half-life and the decay constant is described as \(T_{1/2} = \frac{\ln(2)}{λ}\).This equation allows students to connect the theoretical concepts of exponential decay and decay constants with a tangible measure of time, showcasing the practical application of mathematics in understanding natural radioactive processes.