Problem 43 - Physics for Scientists and Engineers with Modern Physics

Question

(1) What fraction of a sample is left after exactly 6 half-lives?

Step-by-Step Solution

Answer

After 6 half-lives, \(\frac{1}{64}\) of the sample remains.

Step 1: Understanding Half-Lives

In the context of radioactive decay, a 'half-life' is the amount of time it takes for half of a sample to decay. After one half-life, 50% of the original sample remains.

Step 2: Determine Fraction Per Half-Life

Each half-life reduces the remaining sample by half. This means after one half-life, 1/2 of the sample remains; after two half-lives, (1/2)^2 remains, and so on.

Step 3: Calculate After 6 Half-Lives

To find out how much of the sample is left after 6 half-lives, use the formula for the fraction remaining: \((1/2)^n\), where \(n\) is the number of half-lives. Substitute \(n = 6\).

Step 4: Simplify the Calculation

After substituting, the calculation becomes \((1/2)^6 = 1/64\). Therefore, the fraction of the sample left after 6 half-lives is \(\frac{1}{64}\).

Key Concepts

half-life

Half-life is a key concept in understanding radioactive decay. It represents the period of time required for half of a radioactive substance to decay. Imagine you have a radioactive sample. After one half-life, half of the initial amount remains. After another half-life, only half of that remaining amount stays. It continues to reduce by half each half-life period. This predictable pattern allows us to calculate how long it will take for a certain percentage of the substance to remain or decay completely.
Understanding the half-life is crucial in various fields like archaeology, medicine, and nuclear physics, where knowing the rate of decay helps in dating artifacts, determining the dosage of radioactive treatments, and managing nuclear waste respectively.
- Each half-life reduces the sample to half of its previous size. - Knowing the half-life helps to predict the decay of radioactive materials over time.

radioactive decay calculation

Radioactive decay calculation helps us understand how a radioactive substance decreases over time. Using the concept of half-life, we can calculate what fraction of a sample remains after several half-lives have passed. This involves using a simple exponential formula.
The formula to determine the remaining fraction of a sample after a certain number of half-lives is \[(1/2)^n\] where \(n\) is the number of half-lives elapsed. This formula uses exponentiation to reflect how the remaining portion decreases exponentially as time progresses.
For example, if you want to know how much of a sample is left after 6 half-lives, you would calculate \((1/2)^6\).
Each application of half-life in decay calculations is straightforward as it just involves reducing by half again and again. - Exponential decay allows us to calculate remaining sample sizes at any point.- The formula is easy to use and applies consistently across all radioactive materials.

fraction of remaining sample

The fraction of the remaining sample is what’s left of a radioactive substance after a certain number of half-lives. This fraction indicates how much of the initial amount is still present.
Every time a half-life passes, the fraction reduces by half. This results in a geometric sequence where each term represents the remaining fraction after each half-life.
To calculate the fraction after multiple half-lives, use the exponent notation on the base fraction of one half. For example, after 6 half-lives, this becomes:\[(1/2)^6 = \frac{1}{64}\].
Hence, after six half-lives, only 1/64 of the original sample remains showing how quickly a material can decay.
Understanding this fraction is important because it provides insight into how long a substance remains active and potentially hazardous. - Geometric sequence in decay provides clear visualization of remaining matter.- Calculations show the diminishing amount of unchanged substance.