Problem 20 - Calculus for Biology and Medicine

Question

Assume that \(W(t)\) denotes the amount of radioactive material in a substance at time \(t\). Radioactive decay is then described by the differential equation $$\frac{d W}{d t}=-\lambda W(t) \quad \text { with } W(0)=W_{0} $$ where \(\lambda\) is a positive constant called the decay constant. (a) Solve \((8.42)\). (b) Assume that \(W(0)=123 \mathrm{gr}\) and \(W(5)=20 \mathrm{gr}\) and that time is measured in minutes. Find the decay constant \(\lambda\) and determine the half-life of the radioactive substance.

Step-by-Step Solution

Answer

(a) \( W(t) = W_0 e^{-\lambda t} \). (b) \( \lambda = 0.238 \text{ min}^{-1} \), half-life \( \approx 2.91 \text{ minutes} \).

Step 1: Identify the Differential Equation

The given differential equation is \( \frac{d W}{d t}=-\lambda W(t) \). Our goal is to solve this equation given the initial condition \( W(0) = W_0 \).

Step 2: Separate the Variables

To solve for \( W(t) \), we separate the variables by rearranging the terms to obtain \( \frac{dW}{W} = -\lambda \, dt \). This allows us to integrate both sides.

Step 3: Integrate Both Sides

Integrate both sides of the separation equation to find \( \int \frac{dW}{W} = \int -\lambda \, dt \). This results in \( \ln |W| = -\lambda t + C \), where \( C \) is the integration constant.

Step 4: Solve for W(t)

To express \( W(t) \) explicitly, exponentiate both sides of the equation: \( W(t) = e^{C}e^{-\lambda t} \). Using the initial condition \( W(0) = W_0 \), we find \( W_0 = e^{C} \), thus giving \( W(t) = W_0 e^{-\lambda t} \).

Step 5: Solve for Lambda Using Given Conditions

To find \( \lambda \), use the conditions \( W(0) = 123 \) gr and \( W(5) = 20 \) gr. Substitute these into the solution equation: \( 20 = 123 e^{-5\lambda} \). Solve for \( \lambda \) to get \( \lambda = 0.238 \text{ min}^{-1} \).

Step 6: Find the Half-Life

The half-life \( t_{1/2} \) is the time when \( W(t_{1/2}) = \frac{W_0}{2} \). Using \( \frac{W_0}{2} = W_0 e^{-\lambda t_{1/2}} \), solve for \( t_{1/2} \): \( t_{1/2} = \frac{\ln(2)}{\lambda} \). Substitute \( \lambda = 0.238 \text{ min}^{-1} \) to find the half-life \( t_{1/2} \approx 2.91 \text{ minutes} \).

Key Concepts

Radioactive Decay

Radioactive decay is a natural process in which unstable atomic nuclei lose energy by emitting radiation. This is a spontaneous process and happens without any external influence. The materials undergoing decay are commonly known as "radioactive materials." Radioactive decay is essential in understanding the behavior of matter over time and is modeled using differential equations. These equations help predict how the quantity of radioactive material changes as time progresses.

In the case of radioactive decay, the rate of change of the amount of substance, represented by the function \(W(t)\) for time \(t\), is proportional to its current amount. The equation used to model this phenomenon is a first-order linear differential equation given by:

\[ \frac{dW}{dt} = -\lambda W(t) \]

Here, the parameter \(\lambda\) is the decay constant, and the negative sign indicates a decrease over time. Initial conditions such as \(W(0) = W_0\) indicate the initial amount of radioactive material present when the observation starts.

Radioactive decay models are vital in fields such as archaeology for radiocarbon dating, medical applications like radiation therapy, and understanding environmental radioactivity.

Decay Constant

The decay constant, denoted by \(\lambda\), is a key term in the differentiation equation that models radioactive decay. It is defined as the proportionality constant in the equation:

\[ \frac{dW}{dt} = -\lambda W(t) \]

This constant determines how quickly a radioactive substance decays over time. A larger \(\lambda\) value implies a faster decay rate. The units of the decay constant are reciprocal time units, matching the time units used in the model, such as \(\text{min}^{-1}\).

To determine \(\lambda\), one can rearrange and solve the equation using known values from an experiment. For instance, in practical calculations, where specifics like initial and observed amounts are given, \(\lambda\) can be solved from the equation after substituting the known values.

Understanding the decay constant is crucial for accurately predicting the decay behaviors in different scenarios. It also helps in applications like calculating the dosage of radioactive materials in medicine and understanding the longevity of nuclear fuels.

Half-Life

Half-life is a critical concept in the study of radioactive decay. It refers to the time required for half of the original amount of a radioactive substance to decay. The half-life provides a convenient measure to compare different radioactive materials, as it remains constant over time for a given isotope.

The mathematical expression for half-life \(t_{1/2}\) is derived using the decay constant \(\lambda\):

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

This formula is helpful because it links the continuous decay process to a discrete time frame that is often easier to measure and conceptualize.

For instance, if \(\lambda = 0.238 \text{ min}^{-1}\), the half-life can be quickly calculated to be approximately 2.91 minutes, which means in 2.91 minutes, half of the radioactive material will have decayed.

The half-life is used extensively in dating techniques like radiocarbon dating, where scientists determine the age of archaeological samples, as well as in nuclear medicine to predict the safe disposal times for radioactive waste.