Problem 61 - Calculus for Biology and Medicine

Question

After 7 days, a particular radioactive substance decays to half of its original amount. Find the decay rate of this substance.

Step-by-Step Solution

Answer

The decay rate is approximately 0.099 per day.

Step 1: Understand the Problem

The problem asks us to find the decay rate of a radioactive substance where half of the substance decays in 7 days. This implies we need to find the decay constant, often denoted by the letter 'k' in the exponential decay formula.

Step 2: Use the Exponential Decay Formula

The formula for exponential decay is: \[ N(t) = N_0 e^{-kt} \] where \( N(t) \) is the amount of substance at time \( t \), \( N_0 \) is the initial amount, \( k \) is the decay constant, and \( e \) is the base of the natural logarithm.

Step 3: Substitute Known Values

In this problem, after 7 days, the amount is half of its original amount. Thus, we have:\[ \frac{1}{2}N_0 = N_0 e^{-k imes 7} \]We can cancel \( N_0 \) from both sides of the equation because it's non-zero, resulting in:\[ \frac{1}{2} = e^{-7k} \]

Step 4: Solve for the Decay Constant 'k'

To isolate \( k \), take the natural logarithm of both sides of the equation:\[ \ln\left(\frac{1}{2}\right) = -7k \times \ln(e) \]Since \( \ln(e) = 1 \), the equation simplifies to:\[ \ln\left(\frac{1}{2}\right) = -7k \]Solve for \( k \):\[ k = -\frac{\ln\left(\frac{1}{2}\right)}{7} \]

Step 5: Compute the Value of k

Calculate the natural logarithm of \( \frac{1}{2} \):\[ \ln\left(\frac{1}{2}\right) = -0.693147 \]Substitute this value back into the equation for \( k \):\[ k = \frac{0.693147}{7} \approx 0.099 \]

Key Concepts

Exponential Decay Formula

Exponential decay is a fundamental concept in understanding how quantities decrease over time at a rate proportional to their current value. In the context of radioactive substances, this is particularly relevant. The exponential decay formula captures this behavior and can be expressed as: \[ N(t) = N_0 e^{-kt} \]Where:

• \(N(t)\) is the quantity of the substance remaining after time \(t\).
• \(N_0\) represents the initial amount of the substance.
• \(k\) is the decay constant, which is a specific rate for each substance.
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.

This equation highlights that as time \(t\) increases, the remaining amount \(N(t)\) exponentially decreases, leading to a smaller quantity of the substance. It is a continuous model, meaning it applies smoothly over any period. The exponential decay equation is powerful for calculating how long it takes for a substance to reduce to a certain amount, predicting future quantities, or examining rates of decay with the decay constant.

Decay Constant

The decay constant, symbolized by \(k\), is an essential value in the study of radioactive decay. It denotes the probability per unit time that a single atom will decay. In practical terms, it offers a measure of how quickly a radioactive substance reduces its presence. For a specific radioactive material, \(k\) remains constant, providing a measurable and predictable way to analyze decay.You can determine the decay constant from experimental or given data using the rearranged exponential decay formula:First, rearrange the exponential decay equation to find \(k\):\[ \ln\left(\frac{N(t)}{N_0}\right) = -kt\]By knowing how much of the substance is left after a certain time period \(t\), \(k\) can be accurately calculated:\[k = -\frac{\ln\left(\frac{N(t)}{N_0}\right)}{t}\]This understanding helps scientists and engineers calculate the characteristics of radioactive substances effectively and plan for safety measures.

Half-Life Calculation

The half-life of a substance is an important concept because it symbolizes the time required for half of any given quantity of a substance to decay. In the context of radioactive substances, half-life provides insight into how fast the material loses its radioactive property.Mathematically, we link half-life, typically denoted as \(t_{1/2}\), to the decay constant \(k\):By setting \(N(t)\) to half its initial value \(N_0\), we derive the equation:\[ \frac{1}{2} N_0 = N_0 e^{-kt_{1/2}}\]Cancelling \(N_0\) and solving for \(t_{1/2}\) yields:\[ t_{1/2} = \frac{\ln(2)}{k}\]Here, \(\ln(2)\) represents the natural logarithm of 2, approximately equal to 0.693. Therefore, the half-life calculation allows you to predict behaviors of radioactive materials over set time segments.Knowing the half-life enables professionals to estimate when a substance will become substantially less radioactive, helping guide decisions in fields like nuclear medicine, radiology, and environmental science.