Problem 16 - Precalculus

Question

We list some radioactive isotopes and their associated half-lives. Assume that each decays according to the formula \(A(t)=A_{0} e^{k t}\) where \(A_{0}\) is the initial amount of the material and \(k\) is the decay constant. For each isotope: \- Find the decay constant \(k\). Round your answer to four decimal places. \- Find a function which gives the amount of isotope \(A\) which remains after time \(t\). (Keep the units of \(A\) and \(t\) the same as the given data.) \- Determine how long it takes for \(90 \%\) of the material to decay. Round your answer to two decimal places. (HINT: If \(90 \%\) of the material decays, how much is left?) Chromium 51, used to track red blood cells, initial amount 75 milligrams, half-life 27.7 days.

Step-by-Step Solution

Answer

The decay constant is approximately \(-0.0250\), the decay function is \(A(t) = 75e^{-0.0250t}\), and it takes around 92.07 days for 90% of Chromium 51 to decay.

Step 1: Write the half-life equation

The equation for half-life can be expressed using the formula: \[ A(t) = A_0 e^{kt} \]Since after half-life, half of the initial amount is left, we have:\[ \frac{A_0}{2} = A_0 e^{k t_{1/2}} \]where \( t_{1/2} \) is the half-life of the isotope.

Step 2: Solve for the decay constant \(k\)

Using the half-life equation derivation from Step 1, we solve for \(k\):\[ e^{k t_{1/2}} = \frac{1}{2} \]Taking the natural logarithm of both sides, we get:\[ k t_{1/2} = \ln\left(\frac{1}{2}\right) \]Therefore, \( k = \frac{\ln\left(\frac{1}{2}\right)}{t_{1/2}} \)Substitute \( t_{1/2} = 27.7 \) days for Chromium 51:\[ k = \frac{\ln\left(\frac{1}{2}\right)}{27.7} \approx -0.0250 \]

Step 3: Write the decay function

Now that we have \(k\), we can write the decay function:\[ A(t) = 75 e^{-0.0250 t} \]This function expresses the amount of Chromium 51 remaining after \(t\) days.

Step 4: Determine the time for 90% decay

If 90% decays, only 10% of the original amount is left, so:\[ A(t) = 0.1 \times 75 \]Using the decay equation, we have:\[ 0.1 \times 75 = 75 e^{-0.0250 t} \]Simplifying gives:\[ 0.1 = e^{-0.0250 t} \]Taking the natural log of both sides:\[ \ln(0.1) = -0.0250 t \]Solving for \(t\), we find:\[ t \approx \frac{\ln(0.1)}{-0.0250} \approx 92.07 \]

Key Concepts

Radioactive Isotopes

Radioactive isotopes are a fascinating group of substances that naturally emit radiation as they undergo a process called radioactive decay. Each isotope is unique in the type of particles it releases and the speed at which it decays, a characteristic defined by its half-life. These isotopes are used in a variety of applications, such as medical imaging and treatment, carbon dating, and energy generation. For instance, Chromium-51 is a radioactive isotope employed in medical tests to trace red blood cells.

During radioactive decay, isotopes transform into other elements or isotopes. Importantly, this transformation follows a predictable pattern over time, which can be modeled using mathematical functions. Such models help scientists calculate how much of an isotope remains after a certain period, or to ascertain how long a specific percentage of the isotope will persist. A typical model for such analyses is the exponential decay formula:

• Determining the decay constant helps in predicting the behavior and lifespan of the radioactive material.
• Understanding decay patterns is crucial in areas such as radiation safety and nuclear medicine.

Half-life

The half-life of a radioactive isotope is a central concept in understanding its decay behavior. It is defined as the time required for half of the isotope's nuclei to decay into its end product, effectively reducing its original quantity by 50%. The half-life is crucial for calculating how long an isotope will remain active or hazardous.

Each isotope has a distinctive half-life, ranging from fractions of a second to billions of years, allowing for various practical applications. In the case of Chromium-51, its half-life is approximately 27.7 days. This means that after this duration, only half of the initial amount would remain.
Using the half-life, we can establish relationships between time, initial quantity, and remaining quantity by applying the exponential decay formula. The half-life is tied to the decay constant through the equation

• \[ k = \frac{\ln\left(\frac{1}{2}\right)}{t_{1/2}} \]
• These calculations are indispensable in fields such as archaeology, medicine, and environmental science.

Decay Constant

The decay constant, denoted as \( k \), is a vital parameter in the study of radioactive decay. It measures the probability of a single nucleus decaying per unit time. Essentially, it provides a rate of decay for the radioactive isotope. A higher decay constant implies a faster decay rate - meaning the isotope will become less active more quickly.

The decay constant is inversely proportional to the half-life. Once the half-life of an isotope is known, as with Chromium-51, which is 27.7 days, we can calculate the decay constant using the equation:

• \[ k = \frac{\ln\left(\frac{1}{2}\right)}{t_{1/2}} \]
• For Chromium-51, this results in a value of approximately \( k = -0.0250 \) per day.

This constant is integral to forming the exponential decay model and predicting the behavior of radioactive substances.

Understanding and calculating the decay constant allows scientists to describe how quickly a nucleus will decay and help in managing the use and disposal of radioactive substances.

Exponential Functions

Exponential functions are powerful mathematical tools used to model situations where quantities grow or decay at constant rates. In the context of radioactive decay, the exponential decay formula \( A(t) = A_0 e^{kt} \) is employed. Here, \( A(t) \) is the amount remaining at time \( t \), \( A_0 \) is the initial quantity, \( e \) is the base of natural logarithms, and \( k \) is the decay constant. This formula helps predict how much of a radioactive isotope remains after a given time period.

Exponential functions describe processes that reduce over time, making them perfect for modeling radioactive decay. They help solve practical problems such as determining the required storage time for nuclear waste or finding the best isotopes for short-lived medical studies. For instance, using the defined decay constant for Chromium-51 of \(-0.0250\), we can create the decay function:

• \[ A(t) = 75 e^{-0.0250 t} \]
• This illustrates how much of the 75 mg initial quantity will remain after \( t \) days.

Ultimately, these calculations and models are essential in numerous scientific and industrial applications, providing insights into both the duration and safety measures needed for handling radioactive materials.