Problem 33 - Chemistry The Science in Context

Question

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

Step-by-Step Solution

Answer

Answer: 25%

Step 1: Identify the number of half-lives

In this problem, we are given that there are two half-lives.

Step 2: Calculate the amount of radioactivity remaining after each half-life

Since the radioactivity of a sample is reduced by half after each half-life, we multiply the original radioactivity by 0.5 (50%) for each half-life. In this case, since there are two half-lives, we will perform this multiplication twice. Amount of radioactivity remaining = Original radioactivity * 0.5 * 0.5

Step 3: Express the remaining radioactivity as a percentage

To convert the fraction of remaining radioactivity into a percentage, simply multiply it by 100%. Percentage of remaining radioactivity = (Amount of remaining radioactivity / Original radioactivity) * 100%

Step 4: Calculate the percentage

Using the formula derived from step 3, we can now calculate the percentage of remaining radioactivity after two half-lives: Percentage of remaining radioactivity = (Original radioactivity * 0.5 * 0.5 / Original radioactivity) * 100% = 0.25 * 100% = 25% After two half-lives, 25% of a sample's original radioactivity remains.

Key Concepts

Understanding Half-Life

The concept of half-life is crucial in understanding radioactive decay. It's the time required for half of the radioactive atoms in a sample to decay. In other words, after one half-life, you'll have 50% of the original radioactivity left. Imagine you start with a radioactive substance. After one half-life, only half of it remains radioactive. After another half-life, just a quarter is left. This sequential halving continues, creating a distinct decay pattern. Half-life does not depend on the amount of the substance or its condition. It's a constant that is unique to each radioactive isotope.

• If you know the half-life, you can predict how much of your substance will remain radioactive after a certain time.
• You can also calculate how old a radioactive sample is based on its current activity and its half-life.

Understanding half-life is vital in fields like geology, archaeology, and medicine, where it helps in dating samples and ensuring safety around radioactive materials.

Radioactivity Calculation

Calculating radioactivity over time involves determining how much radioactivity remains after certain periods, usually measured in half-lives.To find out how much radioactivity is left after any number of half-lives, you repeatedly halve the initial value. For instance, if you're told something undergoes two half-lives, you take the initial radioactivity, halve it once for the first half-life, and then halve that result again for the second half-life.Expression:The formula looks like this:\[ \text{Remaining Radioactivity} = \text{Initial Radioactivity} \times \left( \frac{1}{2} \right)^{\text{number of half-lives}} \]This formula lets you quickly compute remaining radioactivity without the need for manual step-by-step calculations.Such calculations play a critical role in monitoring nuclear health and environmental safety.

Percentage Calculation in Radioactivity

After calculating the remaining radioactivity, it's often useful to express this as a percentage of the original radioactivity. To find out how much of the initial radioactivity still remains as a percentage, take the remaining fraction of radioactivity and convert it into a percentage. Steps:

• First, determine the remaining fraction of the original sample after a given number of half-lives.
• Multiply this fraction by 100 to convert it into a percentage.

For instance, if you have 0.25 of the original radioactivity left, that's equivalent to 25%. This method is handy when you're dealing with larger data sets, or when communicating results is more effective in percentages. Knowing the percentage helps in assessing how much radioactivity has decreased over time and making informed decisions based on this data. In our example problem, after two half-lives, 25% of the original radioactivity remains, offering a clear view of the decay process.