Problem 63 - Calculus for Biology and Medicine

Question

Polonium \(210\left(\mathrm{Po}^{210}\right)\) has a half-life of 140 days. (a) If a sample of \(\mathrm{Po}^{210}\) has a mass of 100 micrograms, find a formula for the mass after \(t\) days. (b) How long would it take this sample to decay to \(10 \%\) of its original amount? (c) Sketch the graph of the amount of mass left after \(t\) days.

Step-by-Step Solution

Answer

(a) Formula: \( M(t) = 100 \times \left(\frac{1}{2}\right)^{t/140} \); (b) \( t \approx 465.76 \) days; (c) Exponential decay graph.

Step 1: Understand the Decay Formula

Radioactive decay can be modeled using the formula: \[ M(t) = M_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]where \( M(t) \) is the mass at time \( t \), \( M_0 \) is the initial mass, and \( T_{1/2} \) is the half-life.

Step 2: Plug Values into the Formula for Part (a)

Given that the half-life \( T_{1/2} \) is 140 days and the initial mass is 100 micrograms, we substitute these into the decay formula:\[ M(t) = 100 \times \left(\frac{1}{2}\right)^{\frac{t}{140}} \]This is the formula for the mass after \( t \) days.

Step 3: Solve for Time to Decay to 10% for Part (b)

For 10% of the initial amount, \( M(t) = 10 \) micrograms. Use the formula:\[ 10 = 100 \times \left(\frac{1}{2}\right)^{\frac{t}{140}} \]Dividing both sides by 100 gives:\[ 0.1 = \left(\frac{1}{2}\right)^{\frac{t}{140}} \]

Step 4: Apply Logarithms to Solve for t

Take the natural logarithm of both sides:\[ \ln(0.1) = \ln \left(\left(\frac{1}{2}\right)^{\frac{t}{140}}\right) \]Using the power rule for logarithms gives:\[ \ln(0.1) = \frac{t}{140} \cdot \ln\left(\frac{1}{2}\right) \]Solve by isolating \( t \):\[ t = \frac{140 \times \ln(0.1)}{\ln(0.5)} \approx 465.76 \text{ days}\]

Step 5: Sketch the Decay Graph for Part (c)

To sketch the graph, plot \( M(t) \) against \( t \) starting from \( t = 0 \) with \( M(t) = 100 \) micrograms. The curve will decrease exponentially and approach zero but never actually reach it. Key points are at \( t = 0 \) and \( t = 140 \) (where \( M(t) = 50 \) micrograms).

Key Concepts

Half-life

Half-life represents the time it takes for half of a radioactive sample to decay, reducing its mass by half. It's a crucial concept in understanding radioactive decay as it gives a consistent method to predict how the sample diminishes over time. For any radioactive substance, the half-life is constant.
In the case of Polonium-210, its half-life is 140 days. This means that after 140 days, only half of the original mass will remain. If you start with 100 micrograms, after 140 days, you will have 50 micrograms left.

• This process continues, meaning every subsequent period equal to the half-life will again result in half of the remaining mass decaying away.
• It’s a key parameter used in the exponential decay formula.

Understanding half-life helps us predict the decay pattern and calculate how long it takes for a substance to decrease to a certain level.

Exponential Decay Formula

The exponential decay formula is central to modeling how substances like Polonium-210 decay over time. The formula used here is:\[M(t) = M_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}\]where:

• \(M(t)\) is the remaining mass at time \(t\),
• \(M_0\) is the initial mass,
• \(T_{1/2}\) is the half-life of the substance.

This formula reflects how the mass decreases over successive periods equal to the half-life.
As \(t\) increases, the fraction \(\left(\frac{1}{2}\right)\) raised to the \(\left(\frac{t}{T_{1/2}}\right)\) power decreases exponentially, causing the mass to decrease. For our exercise:\[M(t) = 100 \times \left(\frac{1}{2}\right)^{\frac{t}{140}}\]This helps predict the mass at any given time \(t\), showing the exponential nature of the decay process.

Logarithms in Decay Calculations

Logarithms play a vital role when you need to solve for time in decay processes. They help us handle exponential equations, which are not straightforward to solve using basic algebra.
In finding the time it takes for the mass of a sample to decay to a certain percentage, such as 10% in our example, the formula:\[0.1 = \left(\frac{1}{2}\right)^{\frac{t}{140}}\]requires taking the log of both sides to isolate \(t\). Using the power rule of logarithms, \[\ln(0.1) = \frac{t}{140} \cdot \ln\left(\frac{1}{2}\right)\]allows us to solve for \(t\) by dividing both sides by \(\ln(0.5)\):\[t = \frac{140 \times \ln(0.1)}{\ln(0.5)}\]This results in approximately 465.76 days for the sample to decay to 10% of its original mass.
Using logarithms in such calculations transforms complex exponentials into manageable linear equations.

Graphing Exponential Functions

Graphs of exponential decay functions provide a visual understanding of how a substance reduces over time. When graphing the mass \(M(t)\) against time \(t\), you'll observe a curve that starts at the initial mass and rapidly decreases, getting closer to zero without ever reaching it.

• Always begin your graph at \(t = 0\), where \(M(t) = M_0\).
• A key point is the half-life \(T_{1/2}\); at \(t = 140\) days for Polonium-210, the mass is reduced to 50% of \(M_0\).
• As \(t\) increases, the curve depicts the exponential decay pattern, creating a smooth, downward-sloping function.

This visual representation helps reinforce the concept of half-life and the constant rate of exponential decay. The graph's distinct shape, steep at first and flattening over time, illustrates how the sample diminishes quickly at the start and slows down as time passes.