Problem 29 - Physics

Question

The isotope \({ }_{88}^{224}\) Ra of radium has a decay constant of \(2.19 \times 10^{-6} \mathrm{~s}^{-1}\). What is the halflife (in days) of this isotope?

Step-by-Step Solution

Answer

The half-life of the isotope is approximately 3.66 days.

Step 1: Understanding decay constant and half-life relationship

The decay constant \( \lambda \) is related to the half-life \( t_{1/2} \) of a radioactive substance by the formula \( t_{1/2} = \frac{\ln(2)}{\lambda} \). This equation derives from the exponential decay law and the definition of half-life, which is the time taken for the substance to reduce to half its initial amount.

Step 2: Calculate the half-life in seconds

Substitute the given decay constant \( \lambda = 2.19 \times 10^{-6} \mathrm{~s}^{-1} \) into the formula. \[t_{1/2} = \frac{\ln(2)}{2.19 \times 10^{-6}}\]Calculate \( \ln(2) \approx 0.693 \) and solve:\[t_{1/2} = \frac{0.693}{2.19 \times 10^{-6}} \approx 3.163 \times 10^5 \text{ seconds}\]

Step 3: Convert seconds to days

Now convert the half-life from seconds to days. There are 86400 seconds in one day (60 seconds per minute, 60 minutes per hour, 24 hours per day).\[t_{1/2} = \frac{3.163 \times 10^5}{86400} \approx 3.66 \text{ days}\]

Step 4: Final Calculation and Solution

The half-life of the isotope \({ }_{88}^{224}\) Ra is calculated to be approximately 3.66 days.

Key Concepts

Half-life Calculation

A core concept in radioactive decay is calculating the half-life of an isotope, which is crucial for understanding the time it takes for half of a radioactive substance to transform. The half-life is symbolized as \( t_{1/2} \) and can be determined using the formula:

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

This formula links the half-life to the decay constant, \( \lambda \). \( \ln(2) \) is the natural logarithm of 2, approximately 0.693, which reflects the property of the exponential function describing the decay.
To calculate the half-life in seconds, you plug the decay constant given into the formula. For example, with a decay constant of \( 2.19 \times 10^{-6} \) s\(^{-1}\), inputting this value provides the half-life in seconds.
Finally, converting seconds to more manageable units like days helps illustrate how quickly or slowly a substance decays, particularly if it involves human timescales.

Decay Constant

The decay constant, represented as \( \lambda \), is a vital parameter in the mathematics of radioactive decay. It reveals the probability per unit time that a nucleus will decay. A higher decay constant means the isotope decays more quickly. The decay constant is measured in units of inverse time, such as s\(^{-1}\).
Here's how it's used:

• Linked to half-life by the formula \( t_{1/2} = \frac{\ln(2)}{\lambda} \)
• Provides insight into the stability of the isotope, with larger constants indicating rapid decay.

In practical applications, it helps predict how long it takes for a certain amount of radioactive material to lose its radioactivity. Understanding \( \lambda \) is essential, especially when dealing with nuclear processes or any field where radioactive materials are utilized.

Exponential Decay

Exponential decay is a mathematical concept used to describe processes where quantities decrease at a rate proportional to their current value. In the context of radioactivity, it is used to illustrate how radioactive substances reduce over time. This process can be expressed with the equation:

• \( N(t) = N_0 e^{-\lambda t} \)

Here:

• \( N(t) \) is the quantity of substance that still remains after time \( t \)
• \( N_0 \) is the initial quantity
• \( e \) is the base of the natural logarithm
• \( \lambda \) is the decay constant

This equation shows how the number of undecayed nuclei decreases over time. The concept of half-life fits into this model as it represents the time needed for \( N(t) \) to reduce to \( N_0/2 \). Understanding exponential decay is crucial in fields such as nuclear physics and environmental science, where it predicts the behavior of unstable atoms and informs measures for safety and usage.