Problem 13 - Chemistry for Allied Health

Question

\(2.86 \mathrm{~g}\) of a certain radioisotope decays to \(0.358 \mathrm{~g}\) over a period of \(22.8\) minutes. What is the half-life of the radioisotope?

Step-by-Step Solution

Answer

The half-life of the radioisotope is approximately 7.45 minutes.

Step 1: Understand the Problem

We need to determine the half-life of a radioisotope given its decay from an initial mass of \(2.86 \, \text{g}\) to \(0.358 \, \text{g}\) over \(22.8\) minutes.

Step 2: Use the Exponential Decay Formula

Radioactive decay follows an exponential decay model, which is given by the formula: \( N(t) = N_0 e^{-\lambda t} \), where \( N(t) \) is the remaining quantity of the substance after time \( t \), \( N_0 \) is the initial quantity, and \( \lambda \) is the decay constant.

Step 3: Set Up the Equation

Substitute the given values into the decay formula: \( 0.358 = 2.86 e^{-\lambda \times 22.8} \).

Step 4: Solve for the Decay Constant \( \lambda \)

Divide both sides by \(2.86\) to isolate \(e^{-\lambda \times 22.8}\): \( e^{-\lambda \times 22.8} = \frac{0.358}{2.86} \approx 0.125 \).

Step 5: Take the Natural Logarithm

Take the natural logarithm of both sides to solve for \(\lambda\): \( -\lambda \times 22.8 = \ln(0.125) \). Thus, \( \lambda = -\frac{\ln(0.125)}{22.8} \approx 0.09307 \, \text{min}^{-1} \).

Step 6: Calculate the Half-Life

The half-life is defined as the time required for the amount of substance to reduce to half its initial quantity. The formula for half-life is \( t_{1/2} = \frac{\ln(2)}{\lambda} \). Use the calculated \(\lambda\): \( t_{1/2} = \frac{\ln(2)}{0.09307} \approx 7.45 \text{ minutes} \).

Key Concepts

Exponential Decay

Exponential decay is a type of mathematical function that describes the process in which a quantity diminishes over time at a rate proportional to its current value. This means the larger the quantity, the faster it decreases. In the context of radioactive decay, this exponential behavior is a result of the constant probability of a radioactive atom decaying at any given moment.

The general formula for exponential decay is given by:

• \[ N(t) = N_0 e^{-\lambda t} \]

Where:

• \(N(t)\): the amount of substance at time \(t\)
• \(N_0\): the initial amount of substance
• \(\lambda\): the decay constant
• \(t\): time

In radioactive materials, the number of particles decreases continuously and predictably due to this nature, allowing us to calculate the remaining quantity at any given time using this formula.
Understanding exponential decay is crucial for fields such as nuclear physics and radiocarbon dating, where the past and future quantities of radioactive materials need to be accurately predicted.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. It is a random but statistically predictable process, as it occurs with a specific probability per unit time, which is characteristic of the particular nuclear species.

There are different types of radioactive decay, including:

• Alpha decay
• Beta decay
• Gamma decay

In each case, the parent nucleus transforms into a daughter nucleus by shedding part of its matter and energy. Over time, as more of the nuclei decay, the overall mass of the radioactive substance diminishes, following the exponential decay model.
Radioactive decay is essential in various applications, such as medical treatments, carbon dating in archaeology, and nuclear energy generation. The fact that decay proceeds predictably allows scientists to use it as a reliable clock to measure absolute timescale durations, like geological epochs or artifact ages.

Decay Constant

The decay constant \(\lambda\) is a parameter crucial in the study of exponential decay, particularly in radioactive substances. It signifies the probability per unit time that a single atom in a sample will decay. A higher decay constant means that the substance decays more rapidly.

In the exponential decay equation:

• \[ N(t) = N_0 e^{-\lambda t} \]

The decay constant determines the slope of the decay curve. Finding \(\lambda\) helps in computing important time-related properties like the half-life, which is calculated using:

• \[ t_{1/2} = \frac{\ln(2)}{\lambda} \]

Since \(\lambda\) reflects how quickly a radioactive atom decays, it's an invaluable parameter for predicting how long a sample will remain active. By knowing the decay constant, one can understand how fast the initial quantity of a material will reduce, aiding in planning and safety measures in nuclear physics and related fields.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a fundamental mathematical function with the base \(e\), where \(e\) is approximately equal to 2.71828. This logarithmic function is particularly important when dealing with exponential growth and decay processes.

In the context of solving exponential decay problems, the natural logarithm allows us to invert exponential functions to determine unknown variables such as time or the decay constant. For example, when you have an equation like:

• \[ e^{-\lambda t} = 0.125 \]

Taking the natural logarithm on both sides, like so:

• \(-\lambda t = \ln(0.125)\)

This operation allows us to solve for the decay constant \(\lambda\) or time \(t\).
Using the natural logarithm simplifies these types of equations by transforming them into linear form, making the algebra manageable. It's an essential tool for scientists and engineers working with growth and decay scenarios, as it plays a key role in making calculations more straightforward.