Problem 36 - Physics

Question

To make the dial of a watch glow in the dark, \(1.000 \times 10^{-9} \mathrm{~kg}\) of radium \(\frac{226}{88} \mathrm{Ra}\) is used. The half-life of this isotope has a value of \(1.60 \times 10^{3}\) yr. How many kilograms of radium disappear while the watch is in use for fifty years?

Step-by-Step Solution

Answer

Approximately \(2.11 \times 10^{-11} \mathrm{~kg}\) of radium disappears in 50 years.

Step 1: Understanding Half-life

The half-life of a substance is the time it takes for half of the substance to decay. For this problem, the half-life of radium-226 is given as \(1.60 \times 10^3\) years.

Step 2: Calculate Number of Half-lives

We need to determine how many half-lives pass during the time the watch is in use. The watch is used for 50 years. Thus, the number of half-lives is calculated as follows: \[ n = \frac{50}{1.60 \times 10^3} \] This results in \(n = 0.03125\) half-lives.

Step 3: Calculate Remaining Mass After Decay

Using the formula for exponential decay, the remaining mass \(m_t\) after \(n\) half-lives is given by: \[ m_t = m_0 \times (\frac{1}{2})^n \] where \(m_0 = 1.000 \times 10^{-9} \mathrm{~kg}\). Plugging in, \[ m_t = 1.000 \times 10^{-9} \times (\frac{1}{2})^{0.03125} \].

Step 4: Calculate the Amount of Radium That Disappeared

We calculate the remaining mass and subtract it from the initial mass to find the mass of radium that has decayed: \[ m_t = 9.789 \times 10^{-10} \mathrm{~kg} \] \[ \text{Disappeared mass} = 1.000 \times 10^{-9} - 9.789 \times 10^{-10} \].

Step 5: Final Value

From the calculations, the amount of radium that disappeared due to decay in the 50 years watch's usage is \[2.11 \times 10^{-11} \mathrm{~kg}\].

Key Concepts

Understanding Half-Life

Half-life is a concept used to describe the time it takes for half of a radioactive substance to decay. This period is unique to each radioactive isotope.
For instance, radium-226 (used in glow-in-the-dark watch dials) has a half-life of 1600 years. This means that if you start with a certain amount of radium-226, it will take 1600 years for half of that substance to decay or reduce in mass. It's a recurring process. After another 1600 years, half of the remaining amount would have decayed, leaving a quarter of the initial mass still radioactive.

Understanding half-life is crucial for predicting how long a radioactive substance will remain active and for determining how much of the material will remain after a given time.

The Nature of Radium-226

Radium-226 is a radioactive isotope of the element radium. It was historically used for its glow-in-the-dark properties, which are a result of its radioactive decay. Radium-226 is known for its emissions of alpha particles during its decay process.
This isotope's long half-life makes it suitable for devices requiring steady, long-lasting activity, like watch dials. However, its use has decreased significantly due to safety concerns since radioactive decay releases radiation that can be harmful to human health over time.

• Radium-226 decays at a predictable rate, making it a good candidate for time-based applications.
• Despite its practical applications, the radiation risk has led to the development of safer, non-radioactive alternatives.

Exponential Decay of Radioactive Material

Exponential decay describes how the quantity of a substance diminishes over time. For radioactive materials, this decay process occurs as unstable atoms release energy and particles, transforming into more stable forms.
The process follows an exponential pattern, meaning the rate at which the substance decays is proportional to its current amount.
For the radium-226 in a watch, the formula used to calculate the remaining mass after a certain period is: \[ m_t = m_0 \times \left(\frac{1}{2}\right)^n \] where - \( m_0 \) is the initial mass, - \( m_t \) is the mass after time \( t \), - \( n \) is the number of half-lives elapsed.
This formula helps estimate how much radium is left after a specific timeframe.

Basics of Nuclear Physics

Nuclear physics is the field of physics that studies atomic nuclei and their interactions. It encompasses various phenomena, including radioactive decay, fission, and fusion.
In the context of radium-226, understanding nuclear physics helps explain why certain isotopes are radioactive and how their decay can be calculated using principles like half-life and exponential decay.

• Radioactive decay, as studied in nuclear physics, fundamentally involves changes in the nucleus of an atom.


• The study of isotopes like radium-226 provides insight into both natural processes and applications in technology and medicine.

Nuclear physics also contributes to technology development: from medical imaging techniques to power generation and understanding elemental formation in the universe.