Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

An unknown amount of a radioactive substance is being studied. After two days, the mass is \(15.231\) grams. After eight days, the mass is \(9.086\) grams. How much was there initially? What is the half-life of this substance?

Step-by-Step Solution

Answer

Initial mass: 16.87 grams; Half-life: 7.55 days.

Step 1: Understand the Exponential Decay Formula

The exponential decay formula is given by \( m(t) = m_0 e^{-kt} \), where \( m(t) \) is the mass at time \( t \), \( m_0 \) is the initial mass, \( k \) is the decay constant, and \( t \) is time elapsed. Our task is to find \( m_0 \) and the half-life of the substance.

Step 2: Set Up Equations Using Given Data

We are given the mass after 2 days as \( 15.231 \) grams and after 8 days as \( 9.086 \) grams. We can set up the following equations based on the decay formula: 1. \( 15.231 = m_0 e^{-2k} \) 2. \( 9.086 = m_0 e^{-8k} \).

Step 3: Solve the Equations for Decay Constant \( k \)

Divide the second equation by the first to eliminate \( m_0 \):\[ \frac{9.086}{15.231} = \frac{m_0 e^{-8k}}{m_0 e^{-2k}} = e^{-6k} \]Calculate \( \frac{9.086}{15.231} \) to get \( 0.596 \).Thus, \( 0.596 = e^{-6k} \).Take the natural logarithm of both sides to solve for \( k \):\[ -6k = \ln(0.596)\]\[ k \approx \frac{-\ln(0.596)}{6} \approx 0.0918\].

Step 4: Calculate Initial Mass \( m_0 \)

Substitute the value of \( k \) back into one of the original equations to find \( m_0 \):\[ 15.231 = m_0 e^{-2(0.0918)} \]Thus, \( m_0 \approx \frac{15.231}{e^{-0.1836}} \approx 16.87 \) grams.

Step 5: Calculate Half-Life

The half-life \( T_{1/2} \) can be found using the formula \( T_{1/2} = \frac{\ln(2)}{k} \).Substitute \( k \approx 0.0918 \) into the formula:\[ T_{1/2} = \frac{\ln(2)}{0.0918} \approx 7.55 \text{ days} \].

Key Concepts

Radioactive Decay

Radioactive decay is a process by which an unstable atomic nucleus loses energy. This process happens over time and results in the transformation of one element into another. The rate at which this transformation takes place is not constant, but rather exponential.
This means that the decay can be described mathematically using the exponential decay formula:

• \( m(t) = m_0 e^{-kt} \)

Here:

• \( m(t) \) represents the mass at a given time \( t \)
• \( m_0 \) is the initial mass
• \( k \) is the decay constant, a measure of the rate of decay
• \( t \) is time elapsed since the beginning of the observation

When considering radioactive decay, items often refer to how long it takes for half of the substance to decay, which is further explained in the concept of half-life.

Half-Life

Half-life is a critical concept when studying radioactive materials. It signifies the time required for half of the radioactive atoms in a sample to decay. The half-life is important because it helps scientists understand how quickly a substance decays.
The formula used to determine the half-life \( T_{1/2} \) of a substance is:

• \( T_{1/2} = \frac{\ln(2)}{k} \)

In this formula:

• \( \ln(2) \) is the natural logarithm of 2, approximately equal to 0.693
• \( k \) represents the decay constant, which measures how fast the decay occurs

To compute the half-life of a substance, you simply need to know the decay constant. For instance, using the data provided in the exercise, the half-life of the studied substance is roughly 7.55 days.

Decay Constant

The decay constant \( k \) is a vital parameter in the study of radioactive decay. It quantifies how rapidly the substance undergoes decay. A larger \( k \) value means the substance decays faster, while a smaller \( k \) value indicates a slower decay process.
To find \( k \), you can use measurements of the substance’s mass at different times and set up the exponential decay equation. For example, using the masses recorded at two different times given:

• \( m(t_1) = m_0 e^{-kt_1} \)
• \( m(t_2) = m_0 e^{-kt_2} \)

By dividing these two equations, you can solve for \( k \). In the exercise, solving these gives \( k \approx 0.0918 \). Knowing \( k \) allows further calculations like determining the half-life.

Initial Mass

Initial mass \( m_0 \) is the quantity of a radioactive substance present at the start of observation. When calculating radioactive decay, it's important to determine this value to understand the scale of decay that will occur over time.
In the context of the given problem, once the decay constant \( k \) is known, finding the initial mass becomes straightforward. By substituting \( k \) back into one of the decay equations you set up originally, you can isolate \( m_0 \). Here is the typical rearrangement to obtain \( m_0 \):

• \( m_0 = \frac{m(t)}{e^{-kt}} \)

For the exercise, calculating with the known values gives \( m_0 \approx 16.87 \) grams, indicating the initial amount of the radioactive substance before it started decaying.