Problem 52 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A sample of \(\frac{233}{92} \mathrm{U} \quad\left(T_{2}=1.59 \times 10^{5} \mathrm{yr}\right)\) contains \(5.50 \times 10^{18}\) nuclei. \((a)\) What is the decay constant? \((b)\) Approximately how many disintegrations will occur per minute?

Step-by-Step Solution

Answer

The decay constant is approximately \(8.287 \times 10^{-12} \text{ min}^{-1}\). There are about \(4.558 \times 10^{7}\) disintegrations per minute.

Step 1: Calculate the Decay Constant

The decay constant \( \lambda \) is related to the half-life \( T_{1/2} \) by the formula:\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]Given, the half-life \( T_{1/2} = 1.59 \times 10^{5} \text{ years} \). First, convert the half-life from years to minutes:\[ 1 \text{ year} = 525600 \text{ minutes} \] So,\[ T_{1/2} = 1.59 \times 10^{5} \text{ years} \times 525600 \text{ minutes/year} = 8.35844 \times 10^{10} \text{ minutes} \]Now, calculate the decay constant:\[ \lambda = \frac{\ln(2)}{8.35844 \times 10^{10}} \approx 8.287 \times 10^{-12} \text{ min}^{-1} \]

Step 2: Calculate the Activity

The activity \( A \) of a radioactive sample, which is the number of disintegrations per time, is given by:\[ A = \lambda N \]where \( N = 5.50 \times 10^{18} \) is the number of nuclei.Substitute the known values:\[ A = 8.287 \times 10^{-12} \text{ min}^{-1} \times 5.50 \times 10^{18} = 4.558 \times 10^{7} \text{ disintegrations per minute} \]

Key Concepts

Decay Constant

The decay constant, denoted by \( \lambda \), is a fundamental property of a radioactive substance. It provides the probability per unit time that a single nucleus will decay.
The decay constant is inversely related to the half-life of the substance, which is the time required for half of the radioactive nuclei to disintegrate. The formula to calculate the decay constant is:

• \( \lambda = \frac{\ln(2)}{T_{1/2}} \)

Here, \( \ln(2) \) is the natural logarithm of 2, which is approximately 0.693. The half-life \( T_{1/2} \) is a measurable characteristic of the radioactive element.
In practical terms, understanding the decay constant helps us compute how quickly or slowly a sample will decay, thus influencing areas such as nuclear medicine and radiometric dating. For example, a large decay constant indicates rapid decay, while a small one suggests slower decay over a longer period.

Half-Life Calculations

Half-life calculations form a critical part of understanding radioactive decay. The half-life of a radioactive substance is the time it takes for half of the radioactive nuclei in a sample to decay. This property is essential as it helps to predict how the concentration of a radioisotope decreases over time.
To perform half-life calculations, consider the equation:

• \( T_{1/2} = \frac{\ln(2)}{\lambda} \)

Here, \( T_{1/2} \) can be used to determine how long a process takes, be it in radioactive waste management, carbon dating, or cancer treatment using radiation. Given a sample of Uranium-233, its half-life \( 1.59 \times 10^{5} \) years could be converted into smaller time units such as minutes, using conversions like 1 year equalling 525,600 minutes. This kind of conversion helps when you need to measure rates or predict decay within a specific timeframe.
Understanding half-life assists both in practical applications and theoretical calculations, enabling scientists to design processes or predict changes appropriately.

Nuclear Disintegration

Nuclear disintegration, commonly referred to as decay, is the process by which an unstable atomic nucleus loses energy by emitting radiation. This process leads to the transformation of the original nucleus into one or more different elements or isotopes.
In the context of radioactive decay, the activity (or rate of disintegration) is expressed as the number of disintegrations that occur per unit of time. This is often measured in units like disintegrations per minute. The formula used to calculate this activity is:

• \( A = \lambda N \)

where \( A \) is the activity, \( \lambda \) is the decay constant, and \( N \) is the number of radioactive nuclei present.
Essentially, the activity provides insight into how active a radioactive sample is at a given time and how quickly it will reach stability. For students and professionals, understanding nuclear disintegration is key for fields that rely on radiation, such as nuclear medicine, energy, and archaeology.