Problem 62 - Calculus for Biology and Medicine

Question

After 5 days, a particular radioactive substance decays to \(37 \%\) of its original amount. Find the half-life of this substance.

Step-by-Step Solution

Answer

The half-life of the substance is approximately 3.78 days.

Step 1: Understanding the Decay Formula

Radioactive decay often follows an exponential pattern. The decay of a substance can be described using the formula \( N(t) = N_0 \cdot e^{-kt} \), where \( N(t) \) is the amount of substance at time \( t \), \( N_0 \) is the original amount, \( k \) is the decay constant, and \( e \) is Euler's number.

Step 2: Setting Up the Equation

Given that after 5 days the substance is at \( 37\% \) of its original amount, we can write \( 0.37N_0 = N_0 \cdot e^{-5k} \). After simplifying, we have \( e^{-5k} = 0.37 \).

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

By taking the natural logarithm of both sides, we find \( -5k = \ln(0.37) \). Hence, \( k = -\frac{\ln(0.37)}{5} \).

Step 4: Understanding Half-life

The half-life \( T_{1/2} \) is the time required for a substance to reduce to half its original amount. The formula relating half-life and decay constant is \( T_{1/2} = \frac{\ln(2)}{k} \).

Step 5: Calculate the Half-life

Substitute \( k = -\frac{\ln(0.37)}{5} \) into the half-life formula: \[ T_{1/2} = \frac{\ln(2)}{-\frac{\ln(0.37)}{5}} = \frac{5\ln(2)}{-\ln(0.37)}. \] Calculate this to find \( T_{1/2} \approx 3.78 \text{ days}. \)

Key Concepts

Half-life Calculation

The concept of half-life is crucial for understanding how radioactive substances decay over time. The half-life of a substance is the period required for it to decrease to half its original quantity. This measurement is essential when dealing with radioactive materials, as it provides insight into the longevity and stability of the material.
To calculate the half-life, you can use the formula:

• \( T_{1/2} = \frac{\ln(2)}{k} \)

Here, \( T_{1/2} \) represents the half-life, and \( k \) is the decay constant.
By using this relationship, you can determine how long a radioactive substance takes to reach half its quantity. In the case of our exercise, after calculating the decay constant, we found that the half-life of the substance is approximately 3.78 days.

Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. It's a random process at the atomic level, but it can be described statistically. This decay process transforms the original nucleus called the parent into a different nucleus, a daughter.
The decay can happen through various means, but it often follows an exponential decay model represented by the equation:

• \( N(t) = N_0 \cdot e^{-kt} \)

Where:

• \( N(t) \) = the quantity of substance remaining at time \( t \)
• \( N_0 \) = the initial quantity of substance
• \( k \) = the decay constant

This formula helps in predicting how much of the substance will remain after a certain period. In our exercise, it's used to find the amount of substance remaining as a percentage after a designated number of days.

Decay Constant

The decay constant, denoted as \( k \), is a fundamental factor in calculating radioactive decay. It defines the rate at which a substance decreases over time. Essentially, \( k \) is what makes the decay process exponential.
To find the decay constant, you need to set up an equation based on given conditions. For example, if you know a substance decays to a certain percentage of its original amount after a specific time, you can find \( k \). The equation is:

• \( e^{-kt} = \text{remaining fraction of the substance} \)

Apply natural logarithms to both sides to solve for \( k \):

• \( -kt = \ln(\text{remaining fraction}) \)
• \( k = -\frac{\ln(\text{remaining fraction})}{t} \)

In our exercise, the initial condition was that the substance decayed to 37% of its original amount over 5 days. Using these figures, we calculated \( k \) to be \(-\frac{\ln(0.37)}{5}\). Understanding \( k \) enables us to predict the future state of the substance and calculate its half-life effectively.