Problem 35 - Calculus Early Transcendentals

Question

A scientist wants to determine the half-life of a certain radioactive substance. She determines that in exactly 5 days a 10.0 -milligram sample of the substance decays to 3.5 milligrams. Based on these data, what is the half- life?

Step-by-Step Solution

Answer

The half-life is approximately 3.31 days.

Step 1: Understand the Problem

To solve this problem, we need to calculate the half-life of a substance. We know that 10.0 mg of the substance decayed to 3.5 mg in 5 days.

Step 2: Define the Exponential Decay Formula

The decay of a substance over time can be described using the exponential decay formula: \[ N(t) = N_0 \cdot e^{-kt} \] where \( N(t) \) is the remaining quantity at time \( t \), \( N_0 \) is the initial quantity, and \( k \) is the decay constant.

Step 3: Identify Known Values

From the problem, we have:- \( N_0 = 10.0 \) mg (initial quantity)- \( N(t) = 3.5 \) mg (remaining quantity after 5 days)- \( t = 5 \) days

Step 4: Solve for the Decay Constant (k)

Substitute the known values into the decay formula:\[ 3.5 = 10.0 \cdot e^{-k \times 5} \]Divide both sides by 10:\[ 0.35 = e^{-5k} \]Take the natural logarithm (ln) of both sides:\[ \ln(0.35) = -5k \]Solve for \( k \):\[ k = -\frac{\ln(0.35)}{5} \]Calculate \( k \).

Step 5: Calculate the Half-Life

The half-life \( T_{1/2} \) is given by the formula:\[ T_{1/2} = \frac{\ln(2)}{k} \]Substitute the calculated \( k \) value from Step 4 to find \( T_{1/2} \).

Step 6: Perform the Final Calculation

By calculating \( k \) and substituting it in the half-life formula, we find:1. \( k = -\frac{\ln(0.35)}{5} \approx 0.2092 \)2. \( T_{1/2} = \frac{\ln(2)}{0.2092} \approx 3.31 \) days.

Key Concepts

Exponential Decay

Exponential decay is a mathematical concept used to describe the process by which a quantity decreases at a rate proportional to its current value. In the context of radioactive substances, it explains how a material diminishes over time due to radioactive decay. The amount of substance decreases exponentially rather than linearly, meaning it decreases more rapidly at first and then slows down over time.
The formula for exponential decay is expressed as: \[ N(t) = N_0 \cdot e^{-kt} \]Where:

• \( N(t) \) is the quantity remaining at time \( t \)
• \( N_0 \) is the initial quantity of the substance
• \( k \) is the decay constant
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828

This formula helps us model how substances decay over time, assisting scientists in predicting how long it takes for a substance to reduce to a certain level.

Decay Constant

The decay constant, denoted as \( k \), is a critical factor in the exponential decay formula that determines the rate of decay of a radioactive substance. It is a measure of how quickly a substance undergoes decay processes. Understanding the decay constant helps in predicting how fast a radioactive material will reduce to half its initial amount, known as the half-life.
To find the decay constant, we use the relationship:\[ k = -\frac{\ln(N(t)/N_0)}{t} \]
Where:

• \( N(t) \) is the remaining quantity after time \( t \)
• \( N_0 \) is the original quantity
• \( t \) is the elapsed time

A higher decay constant means a more rapid decay, and different substances have different decay constants based on their physical and chemical properties. In our exercise, solving for \( k \) allowed us to understand the rapidity of decay to compute the half-life.

Radioactive Substance

A radioactive substance is a material composed of atoms that are unstable and can emit radiation during their decay process. These substances are intrinsic in fields such as nuclear medicine, archeology (carbon dating), and energy production. Understanding these substances requires a grasp of how they behave over time, particularly as they undergo radioactive decay, emitting particles and energy.
This decay process follows the principles of exponential decay, described by formulas that include the decay constant \( k \). This constant is unique to each substance and dictates how quickly the substance will decay.
Radioactive substances have various applications but also pose potential hazards if not managed properly, due to their potential to emit harmful radiation. Safe handling and understanding of each substance's half-life and decay characteristics are crucial in optimizing their use while minimizing risks.