Problem 16 - Algebra 2

Question

Public works Iodine- 131 is used to find leaks in water pipes. It has a half- life of 8.14 days. Write the exponential decay function for a \(200-\) - Find the amount of iodine- 131 remaining after 72 days.

Step-by-Step Solution

Answer

The remaining amount of Iodine-131 after 72 days is calculated by substituting the given values into the exponential decay function

Step 1: Derive the decay constant (k)

The decay constant (\(k\)) can be derived by using the half-time (\(t_{1⁄2}\)) formula, which is \(k = ln (2) / t_{1⁄2}\). Here, \(t_{1⁄2} = 8.14 \) days.

Step 2: Write the decay function

The decay function can be specified as \( N(t) = N_0 * e^{-kt} \), where \(N_0\) = Initial Quantity = 200 and \( k }\) is the decay constant derived in the first step.

Step 3: Calculate the remaining amount

Substitute \( t = 72 \) days in the decay function derived in the previous step to calculate the remaining amount of iodine-131.

Key Concepts

Half-Life

The concept of half-life is fundamental in understanding exponential decay, particularly with radioactive substances like iodine-131. Half-life refers to the amount of time it takes for half of a given sample of a radioactive isotope to decay. In simpler terms, it measures the rate at which a radioactive substance transforms into another element or isotope.
This concept is crucial when dealing with substances that naturally break down over time, such as in medical treatments or in industrial applications. In the case of iodine-131, the half-life is 8.14 days, which means that every 8.14 days, only half of the remaining iodine-131 will remain active.
This predictable pattern allows us to calculate how much of the substance will be left after any given period if we know its initial amount.

Decay Constant

The decay constant, often denoted as \( k \), is a critical parameter in the exponential decay function. It represents the probability per unit time that a particle will decay. The decay constant is closely related to the half-life of a substance.
To find the decay constant, we use the formula \( k = \frac{\ln(2)}{t_{1/2}} \), where \( t_{1/2} \) is the half-life. For iodine-131:

• The half-life \( t_{1/2} \) is given as 8.14 days.
• The natural logarithm of 2, \( \ln(2) \), approximately equals 0.693.

This formula shows the inverse relationship between the half-life and decay constant: a short half-life means a high decay constant, indicating rapid decay, and vice versa. The decay constant allows us to determine how quickly a substance reduces its intensity over time.

Radioactive Isotopes

Radioactive isotopes, or radioisotopes, are variants of chemical elements that have unstable nuclei and emit radiation as they decay into a stable form. This decay process is crucial in various fields, such as medicine, archaeology, and environmental science.
In medical treatments, isotopes like iodine-131 are used for diagnostics and therapy because their decay can be tracked precisely, and they have suitable half-lives. Radioactive isotopes can help in tracing chemical and biological processes, as well as in dating geological and archaeological samples.
When working with isotopes, understanding their decay behavior through half-life and decay constant is essential. This knowledge enables scientists and practitioners to predict how long the isotope will remain active and effective for its intended use.

Exponential Functions

Exponential functions are a type of mathematical function characterized by a constant rate of growth or decay. They are broadly represented in the form \( y(t) = y_0 \cdot e^{kt} \), where \( y_0 \) is the initial quantity, \( e \) is the base of natural logarithms, \( k \) is the growth or decay constant, and \( t \) is time.
In the context of radioactive decay, exponential functions describe how the quantity of a radioactive substance decreases over time. The specific formula for decay is \( N(t) = N_0 \cdot e^{-kt} \), where:

• \( N(t) \) is the amount of substance remaining at time \( t \).
• \( N_0 \) is the initial amount.
• \( k \) is the decay constant.

This model is beneficial in predicting how quickly a radioactive isotope decays, allowing us to compute the remaining amount after any given period, as demonstrated in the original exercise when determining the iodine-131 remaining after 72 days.