Problem 4 - Active Calculus

Question

An unknown radioactive element decays into non-radioactive substances. In 320 days the radioactivity of a sample decreases by 58 percent. (a) What is the half-life of the element? half-life: (b) How long will it take for a sample of \(100 \mathrm{mg}\) to decay to \(88 \mathrm{mg} ?\) time needed:

Step-by-Step Solution

Answer

The half-life is approximately 226.3 days. It takes about 37.5 days for the sample to decay to 88 mg.

Step 1 - Understand the Decay Formula

Radioactive decay can be described with the formula: \( N(t) = N_0 e^{-\frac{t}{\tau}} \), where \( N(t) \) is the amount of radioactive substance at time \( t \), \( N_0 \) is the initial amount, and \( \tau \) is the mean lifetime of the substance. For half-life calculations, use the formula: \( N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \),where \( T_{1/2} \) is the half-life.

Step 2 - Calculate the Decay Constant

Given that the radioactivity decreases by 58% in 320 days, 42% of the original amount remains. Thus, \( N(320) = 0.42 N_0 \). Using the decay formula: \( 0.42 N_0 = N_0 e^{-\frac{320}{\tau}} \), we get \( 0.42 = e^{-\frac{320}{\tau}} \). Taking the natural logarithm of both sides: \( \ln(0.42) = -\frac{320}{\tau} \). Solving for \( \tau \): \( \tau = -320 / \ln(0.42) \).

Step 3 - Calculate the Half-Life

With the mean lifetime \( \tau \) calculated, the half-life \( T_{1/2} \) can be found using the relation: \( T_{1/2} = \tau \ln(2) \). Substitute the value of \( \tau \) from Step 2 to find \( T_{1/2} \).

Step 4 - Find the Time for Specific Decay

To find the time needed for the sample to decay from \( 100 \, \mathrm{mg} \) to \( 88 \, \mathrm{mg} \), apply the same decay formula: \( 88 = 100 \left( \frac{1}{2} \right)^\frac{t}{T_{1/2}} \). Solving for \( t \): \( \left( \frac{88}{100} \right) = \left( \frac{1}{2} \right)^\frac{t}{T_{1/2}} \). Taking the natural logarithm of both sides: \( \ln\left( \frac{88}{100} \right) = \frac{t}{T_{1/2}} \ln \left( \frac{1}{2} \right) \). Finally, solve for \( t \) using the known value of \( T_{1/2} \).

Key Concepts

Half-life calculations

Half-life is a crucial concept in radioactive decay. It is the time required for half of a sample of a radioactive substance to decay into non-radioactive material. The half-life remains constant, meaning it doesn't depend on the amount of substance you have. The formula to calculate the half-life, denoted as \(T_{1/2}\), is derived from the decay formula. The key equation we'll use here is: \[N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}\]
This equation helps us understand how much of the substance remains at any time \(t\). Here’s how to identify the half-life step-by-step:

- Given: In 320 days, the substance’s radioactivity decreases by 58%, so 42% remains.
- Set up the equation: \(N(320) = 0.42 N_0\)
- Use the formula: \(0.42 = e^{-\frac{320}{\tau}}\), then solve for \(\tau\)
- Convert the mean lifetime \(\tau\) to half-life using: \(T_{1/2} = \tau ln(2)\)

With these steps, you can find the exact half-life of the substance in question.

Decay constant

The decay constant, denoted as \(\lambda\), is another essential parameter in radioactive decay. It represents the probability per unit time that a given nucleus will decay. This is related to the mean lifetime \(\tau\) of the radioactive substance via: \[\lambda = \frac{1}{\tau}\]

The smaller the decay constant, the longer it takes for the substance to decay. To find \(\lambda\), use the relationship between the decay constant and the half-life:
- Firstly, calculate the mean lifetime \(\tau\) from the given data using \(\tau = -\frac{320}{ln(0.42)}\)
- Then, use \(\lambda = \frac{1}{\tau}\)

By knowing \(\lambda\), we can better understand the decay process and calculate how quickly or slowly it occurs. This constant gives a more intuitive grasp of the radioactive decay rate.

Exponential decay

Exponential decay describes the process where the quantity of a substance decreases at a rate proportional to its current value. This is a fundamental principle observed in radioactive decay. The underlying equation reflects an exponential relationship:

\[N(t) = N_0 e^{-\lambda t}\]

Here, \(N_0\) is the initial amount, \(\lambda\) is the decay constant, and \(t\) is time. This equation implies that the substance decays faster at first, and then the rate slows down over time. To understand it better:

- Assume we start with 100 mg. If it decays to 88 mg over time, we'd use the equation in a rearranged form:
\(88 = 100 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}\)
- Take the natural logarithm of both sides, solve for \(t\), and with the known \(T_{1/2}\), get the exact time.

Exponential decay covers various real-world scenarios beyond radioactivity, highlighting its importance in both academics and research.