Understanding Radioactive Half-Life and Decay

The question asks about the time it takes for a radioactive substance to disintegrate to a certain amount, given its half-life. The half-life of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay or disintegrate.

When a substance undergoes radioactive decay, its amount decreases over time. After one half-life, half of the initial amount remains. After two half-lives, half of the remaining half decays, leaving one quarter of the original amount.

Calculating Remaining Substance after Half-Lives

Let the initial amount of the radioactive substance be \(N_0\). After time \(t\), the remaining amount \(N(t)\) is given by the formula:

\[N(t) = N_0 \left(\frac{1}{2}\right)^{t/T}\]

Where:

• \(N(t)\) is the amount of substance remaining after time \(t\)
• \(N_0\) is the initial amount of substance
• \(T\) is the half-life of the substance
• \(t\) is the time elapsed

Solving the Radioactive Decay Problem

We are given that the half-life \(T\) is 10 days. The question asks for the time \(t\) it takes for 3/4 of the initial value to disintegrate. If 3/4 has disintegrated, the amount remaining is:

\[N(t) = N_0 - \frac{3}{4}N_0 = \frac{1}{4}N_0\]

Now we can use the formula for radioactive decay:

\[\frac{1}{4}N_0 = N_0 \left(\frac{1}{2}\right)^{t/T}\]

Divide both sides by \(N_0\):

\[\frac{1}{4} = \left(\frac{1}{2}\right)^{t/T}\]

We can express 1/4 as a power of 1/2:

\[\left(\frac{1}{2}\right)^2 = \left(\frac{1}{2}\right)^{t/T}\]

Since the bases are equal, the exponents must be equal:

\[2 = \frac{t}{T}\]

Now, substitute the given half-life \(T = 10\) days:

\[2 = \frac{t}{10 \text{ days}}\]

Solve for \(t\):

\[t = 2 \times 10 \text{ days} = 20 \text{ days}\]

Step-by-Step Disintegration Analysis

Alternatively, we can think about this in terms of half-lives:

• Initially, at time \(t=0\), the amount is \(N_0\). Fraction remaining is 1.
• After 1 half-life (10 days), the amount remaining is \(\frac{1}{2}N_0\). Fraction remaining is 1/2. Amount disintegrated is \(1 - 1/2 = 1/2\).
• After 2 half-lives (10 + 10 = 20 days), the amount remaining is \(\frac{1}{2} \times \frac{1}{2}N_0 = \frac{1}{4}N_0\). Fraction remaining is 1/4. Amount disintegrated is \(1 - 1/4 = 3/4\).

We see that it takes 2 half-lives for 3/4 of the substance to disintegrate, leaving 1/4 remaining. Since each half-life is 10 days, the total time is \(2 \times 10 = 20\) days.

From the table, we can clearly see that after 20 days (which is two half-lives), the fraction disintegrated is 3/4.

Radioactive Decay Calculation Summary

The problem required finding the time for a specific fraction of radioactive substance to decay. By understanding the definition of half-life and how the amount of substance decreases exponentially, we determined that 3/4 disintegration means 1/4 remaining. This occurs after exactly two half-lives. Given the half-life of 10 days, the total time is 20 days.

Revision Table: Radioactive Decay Concepts

Additional Information on Radioactive Half-Life

Half-life is a fundamental concept in nuclear physics and chemistry. It is used to describe the decay rate of radioactive isotopes. The length of the half-life varies greatly among different isotopes, from fractions of a second to billions of years.

Understanding half-life is crucial in many fields:

• Carbon Dating: Used to determine the age of organic materials based on the half-life of Carbon-14 (approx. 5730 years).
• Medicine: Radioactive isotopes used in medical imaging or treatment have specific half-lives chosen for safety and effectiveness.
• Nuclear Energy: Managing nuclear waste requires considering the very long half-lives of some radioactive byproducts.
• Geology: Used in radiometric dating of rocks and minerals to determine the age of the Earth and geological formations.

While we used a simple fractional approach here, the exponential formula \(N(t) = N_0 e^{-\lambda t}\) is also commonly used, where \(\lambda\) is the decay constant. The decay constant is related to the half-life by the equation \(\lambda = \frac{\ln(2)}{T}\).

Both the fractional approach and the exponential decay formula yield the same results and describe the same underlying physical process of radioactive decay based on half-life.