Problem 34 - Chemistry: An Atoms-Focused Approach

Question

What percentage of a sample's original radioactivity remains after six half- lives?

Step-by-Step Solution

Answer

Answer: Approximately 1.5625%

Step 1: Understand the exponential decay formula and what it represents

In exponential decay, the remaining amount of a substance after a certain time can be calculated using the formula: A(t) = A₀ * (1/2)^(t/T) where: - A(t) represents the amount of the substance remaining at time t - A₀ is the initial amount of the substance - t is the time that has passed (here, it's six half-lives) - T is the time it takes for half of the substance to decay, called the half-life

Step 2: Plug in the known values into the formula

We know that after six half-lives (t = 6), we want to find the percentage of the initial amount (A₀) remaining. The formula becomes: A(6) = A₀ * (1/2)^(6/T) Since we are looking for the percentage and it's given as a half-life problem, we can set T to 1: A(6) = A₀ * (1/2)^6

Step 3: Calculate the remaining amount

Now we just need to calculate the value of (1/2)^6: A(6) = A₀ * (1/2)^6 A(6) = A₀ * (1/64)

Step 4: Calculate the percentage of the remaining amount

To express the remaining amount as a percentage of the initial amount, we can multiply it by 100: Percentage = (A(6) / A₀) * 100 Percentage = (A₀ * (1/64) / A₀) * 100 The A₀ in the numerator and denominator cancel each other out. Percentage = (1/64) * 100

Step 5: Simplify the percentage

Finally, we can simplify the expression for the percentage: Percentage = 100 * (1/64) Percentage ≈ 1.5625 % Therefore, after six half-lives, approximately 1.5625% of a sample's original radioactivity remains.

Key Concepts

half-life

Half-life is a concept that describes the time required for half of a radioactive substance to decay. It is a constant for each substance and does not change over time. Understanding half-life is essential when studying exponential decay because it gives a straightforward measure of how quickly a substance is losing its radioactivity.
Consider a certain radioactive material with a half-life of 3 years. This means that, after 3 years, only half of the original amount of the substance will remain. After another 3 years (totaling 6 years), only a quarter will be left, because again, only half of what was left after the first period goes away. Through this process of halving, we can observe how the substance decays over time in a predictable manner.
Knowing about half-life is crucial for calculating how much time it will take for a substance to lose a specific percentage of its original radioactivity or determining how much of it remains after a certain period.

radioactivity

Radioactivity is the process by which unstable atomic nuclei lose energy by emitting radiation. This phenomenon is what makes some elements naturally radioactive. It occurs when the nuclei of atoms are unstable and spontaneously break down, emitting particles in the process.
Common forms of radiation include alpha particles, beta particles, and gamma rays. These emissions can be dangerous to human health, which is why understanding radioactivity is essential, especially in fields like nuclear energy and medicine.

• Alpha particles: Heavy and positively charged, they can be stopped by a sheet of paper or the skin.
• Beta particles: Lighter and negatively charged, they require an aluminum sheet to be blocked.
• Gamma rays: Neutral and very penetrating, they require thick lead or concrete shielding.

Recognizing radioactivity's role in radioactive decay helps us understand how substances change over time and why they release radiation.

decay formula

The decay formula is a mathematical representation used to model how the quantity of a radioactive substance decreases over time. It is given by the formula:\[ A(t) = A_0 \times \left(\frac{1}{2}\right)^{t/T} \]Let's dive into each part of this formula:

• A(t): The remaining amount of the substance at time \( t \).
• A₀: The initial amount of the substance before it starts decaying.
  
• t: The specific time period that has passed.
  
• T: The half-life of the substance, which is the time it takes for half of the substance to decay.

This formula allows us to predict how much of a substance remains after a certain number of half-lives. For example, if you want to know what percentage of a substance's original radioactivity remains after 6 half-lives, just substitute in the values:\[ A(6) = A_0 \times \left(\frac{1}{2}\right)^{6} \]Calculating this gives us about 1.5625%, explaining why, after several half-lives, only a small amount of the original substance remains.