Problem 15 - Chemistry for Allied Health

Question

A radioisotope decays from \(55.9 \mathrm{~g}\) to \(6.99 \mathrm{~g}\) over a period of \(72.5\) hours. What is the half-life of the isotope?

Step-by-Step Solution

Answer

The half-life of the isotope is approximately 24.1 hours.

Step 1: Understanding Decay Formula

The formula for exponential decay is given by \( N(t) = N_0 e^{-kt} \), where \( N(t) \) is the remaining quantity of the substance at time \( t \), \( N_0 \) is the initial quantity, \( k \) is the decay constant, and \( t \) is time.

Step 2: Set Up the Known Values

From the problem, we know that the initial quantity \( N_0 = 55.9 \) g and the remaining quantity \( N(t) = 6.99 \) g after \( t = 72.5 \) hours.

Step 3: Solve for the Decay Constant

Using the decay formula \( N(t) = N_0 e^{-kt} \), substitute the known values: \( 6.99 = 55.9 e^{-72.5k} \). First, solve for \( e^{-72.5k} \) by dividing both sides by 55.9, resulting in \( \frac{6.99}{55.9} = e^{-72.5k} \).

Step 4: Calculate the Decay Constant

Find \( e^{-72.5k} \) by calculating \( \frac{6.99}{55.9} \approx 0.125 \). Now, take the natural logarithm of both sides: \( \ln(0.125) = -72.5k \). Solving for \( k \), we get \( k \approx \frac{\ln(0.125)}{-72.5} \).

Step 5: Find the Natural Logarithm

Calculate \( \ln(0.125) \approx -2.079 \). Thus, \( k \approx \frac{-2.079}{-72.5} \approx 0.0287 \).

Step 6: Calculate the Half-Life

The half-life \( T_{1/2} \) is the time at which half of the initial amount remains, given by \( T_{1/2} = \frac{\ln(2)}{k} \). Since \( k \approx 0.0287 \), calculate \( T_{1/2} \approx \frac{0.693}{0.0287} \approx 24.1 \) hours.

Key Concepts

Radioactive Decay

Radioactive decay is a fundamental concept in nuclear physics and helps us understand why certain isotopes release energy over time. This process happens when the unstable nucleus of a radioisotope loses energy by emitting radiation. It's important to know that decay is random for individual nuclei, but predictable across a large number of them.

• Unlike chemical processes, radioactive decay is not affected by external conditions like temperature or pressure.
• The decay continues until a stable, non-radioactive isotope forms.
• This natural process can be used in carbon dating, medical imaging, and power generation.

By understanding this phenomenon, we better appreciate the changes that occur in nature and how they can be harnessed in various technological applications.

Decay Constant

The decay constant, represented as \( k \), is a crucial parameter that informs us about the rate of radioactive decay of a substance. It's a measure of how quickly or slowly a substance undergoes radioactive decay. Specifically, it defines the probability of a single, unstable nucleus decaying per unit time.

Here's how it works in the decay formula:

• The smaller the decay constant, the slower the isotope decays, meaning a longer half-life.
• The larger the decay constant, the faster the decay, resulting in a shorter half-life.
• The decay constant is inherent to each isotope and can be calculated using the natural logarithm in the exponential decay formula.

Understanding the decay constant helps scientists predict the behavior of radioactive substances and is key in fields such as geology, archaeology, and medicine.

Exponential Decay

Exponential decay describes how the amount of a radioactive substance decreases over time. This process can be visualized using the exponential decay formula \( N(t) = N_0 e^{-kt} \). Here, \( N(t) \) is the amount remaining at time \( t \), \( N_0 \) is the initial amount, and \( k \) is the decay constant.

Main characteristics of exponential decay include:

• It leads to a rapid decrease initially, which gradually slows down.
• The rate of decay is proportional to the quantity remaining; the more substance present, the faster it depletes.
• In a graph of this function, you will see a downward curve approaching zero.

This type of decay is used to model not only radioactive decay but also population decreases, cooling of materials, and elimination of drugs from the body.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a mathematical function essential in the study of exponential processes like radioactive decay. It helps to 'undo' the exponential part of exponential growth or decay equations, making it easier to solve for unknowns.

• It is based on the number \( e \), which is approximately 2.71828.
• In radioactive decay calculations, the natural logarithm helps us find the decay constant when we know the initial and remaining amounts of a substance.
• For half-life calculations, \( \ln(2) \approx 0.693 \) is often used.

By utilizing the natural logarithm, mathematicians and scientists can efficiently work with exponential equations, making them crucial in calculations involving growth and decay scenarios.