Problem 34 - College Algebra with Corequisite Support

Question

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of \(\begin{array}{ll}1.15 \% & \text { per day. }\end{array}\) The half-life of Radium-226 is 1590 years. What is the annual decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.

Step-by-Step Solution

Answer

The annual decay rate is approximately 0.000434 or 0.0434%.

Step 1: Understand the Problem

We are dealing with two isotopes: Iodine-125 and Radium-226. However, the question only pertains to the annual decay rate of Radium-226. We are not concerned with Iodine-125 for this problem.

Step 2: Determine Decay Constant

The half-life of Radium-226 is 1590 years. The decay constant \( k \) can be found using the formula for half-life: \( t_{1/2} = \frac{ln(2)}{k} \). Therefore, \( k = \frac{ln(2)}{1590} \).

Step 3: Convert Decay Constant to Annual Decay Rate

The decay constant \( k \) gives us a good indication of how fast the substance decays per year. The annual decay rate can be found by using the formula: \( r = 1 - e^{-k} \). First, compute \( k \) then use it to find \( e^{-k} \) and the rate \( r \).

Step 4: Calculate Numerical Values

Using \( k = \frac{ln(2)}{1590} \), calculate \( k \) approximately as \( 0.000435 \). Then calculate the annual decay rate \( r = 1 - e^{-0.000435} \), which approximately equals \( 0.000434 \).

Step 5: Convert to Percentage Form

Convert the decay rate to a percentage by multiplying by 100, yielding an annual decay rate of 0.0434%.

Key Concepts

Decay Constant

The decay constant, often represented as \( k \), is a crucial element in understanding radioactive decay. It provides a measure of the probability of a decay event per unit time. This value allows us to predict how quickly a radioactive substance will decay over time.
To determine \( k \), you need the half-life of the substance, which is the time it takes for half of the radioactive atoms to decay. The decay constant is calculated using the formula:

• \( t_{1/2} = \frac{\ln(2)}{k} \)

Subsequently, \( k = \frac{\ln(2)}{t_{1/2}} \). For Radium-226, with a half-life of 1590 years, the decay constant is approximately \( 0.000435 \).
The decay constant is integral to calculate how many atoms will decay over a given period.

Half-Life

Half-life is the time required for half of the radioactive nuclei in a sample to undergo decay. It remains constant, regardless of the initial amount of substance available and is a fundamental feature of radioactive isotopes like Radium-226 and Iodine-125.
For instance, Radium-226 has a half-life of 1590 years. This means that after 1590 years, only half of the original Radium-226 atoms remain; the rest have transformed into other elements through radioactive decay.
Understanding half-life is crucial because it helps predict how long a radioactive isotope will remain active. In medicine, like with Iodine-125, it helps calibrate doses to minimize harm while maintaining effectiveness.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value, common in radioactive decay scenarios. It's governed by the relationship where the decay rate changes over time, following an exponential function format:
\[ N(t) = N_0 e^{-kt} \]
where:

• \( N(t) \) is the remaining quantity at time \( t \)
• \( N_0 \) is the initial quantity
• \( k \) is the decay constant
• \( t \) is time

For Radium-226, despite the slow decay over 1590 years, this formula accurately portrays a gradual decrease in material, characterizing the slow decay of long-lived isotopes. Iodine-125 also follows exponential decay, which is why it’s used in precise time-dosed applications, like treating tumors.

Iodine-125

Iodine-125 is a radioactive isotope commonly used in medical treatments, particularly in brachytherapy for cancer, due to its effective radiation properties and relatively short half-life.
This isotope has a half-life of approximately 60 days, which makes it suitable for targeted applications where the radioactive effects diminish relatively quickly, minimizing long-term risks to the patient.
Iodine-125's decay characteristics make it possible to use in smaller doses while still maintaining effectiveness. During the decay process, it emits gamma radiation, which facilitates imaging and treatment in medical applications, helping to treat localized cancerous cells.

Radium-226

Radium-226 is a naturally occurring radioactive isotope of radium, renowned for its historical use in luminescent paints. Today, it is of interest due to its radiogenic properties and long half-life of 1590 years.
The longevity of Radium-226 poses both scientific interest and safety challenges. It decays slowly into radon gas, which is highly radioactive and represents a health hazard if not properly managed.
Radium-226's slow rate of decay corresponds with its decay constant; although minimal annually, it's consistent, necessitating careful handling over long durations. Its properties are useful in studies of radioactive decay processes and can be employed in certain types of radiotherapy in very specific circumstances.