Radioactive Half-Life: Videos & Practice Problems

Half-life, denoted as \( t_{\text{half}} \), is a crucial concept in understanding radioactive decay. It represents the time required for half of a given quantity of a radioactive substance, known as a radioisotope, to decay. A radioisotope is an unstable isotopic form of an element that emits radiation as it transforms into a more stable state.

For instance, consider a radioisotope with a half-life of one day. If we start with 10 grams of this substance, we can track its decay over time. Initially, at \( t = 0 \) days, we have 100% of the substance, which is 10 grams. After one day, or one half-life, half of the substance decays, leaving us with 5 grams, which is 50% of the original amount.

Continuing this process, after another day (two half-lives), we again lose half of the remaining substance. Starting with 5 grams, we are left with 2.5 grams, representing 25% of the original quantity. After a third day (three half-lives), we lose half of 2.5 grams, resulting in 1.25 grams, which is 12.5% of the initial amount.

This pattern illustrates that with each passing half-life, the quantity of the substance decreases by half. The decay process can continue indefinitely, with each half-life reducing the remaining amount of the radioisotope. Understanding half-life is essential for applications in fields such as nuclear medicine, radiometric dating, and understanding the behavior of radioactive materials.

Understanding the decay of radioisotopes involves calculating the remaining quantity after a series of half-lives. The concept of half-life refers to the time required for half of a radioactive substance to decay. To determine the fraction of the radioisotope remaining after a certain number of half-lives, we can use the formula:

Fraction Remaining = 0.5^n

In this equation, 0.5 represents the halving of the substance, and n is the number of half-lives that have passed. To convert this fraction into a percentage, simply multiply the fraction by 100%:

Percentage Remaining = (0.5^n) × 100%

Additionally, if you want to find the actual amount of the radioisotope remaining, you can use the final amount formula. This formula calculates the remaining mass based on the initial amount:

Final Amount = I × (0.5^n)

Here, I represents the initial amount of the substance. For example, if you start with 100 grams of a radioisotope, you can apply this formula to determine how much remains after a specified number of half-lives. By utilizing these formulas, you can effectively calculate either the fraction, percentage, or final amount of a radioisotope remaining after decay.

Sodium-24 is a radioisotope commonly used in medical applications, particularly for examining human circulation. It has a half-life of approximately 15 days, which is the time required for half of the substance to decay. To determine the percentage of Sodium-24 remaining after a period of time, we can use the formula for percentage remaining, which is expressed as:

Percentage Remaining = (0.5ⁿ) × 100%

In this context, n represents the number of half-lives that have elapsed. To find n, we first calculate the total time that has passed. Given that 3 months is equivalent to 90 days (3 months × 30 days/month), we can then determine how many half-lives fit into this time frame:

n = Total Time / Half-Life Duration = 90 days / 15 days = 6

Now that we know 6 half-lives have occurred, we can substitute this value back into the percentage remaining formula:

Percentage Remaining = (0.5⁶) × 100% = 1.5625%

This calculation indicates that after 3 months, approximately 1.5625% of the original Sodium-24 sample will remain. Understanding the decay of radioisotopes like Sodium-24 is crucial in fields such as nuclear medicine, where precise measurements can impact patient care and treatment outcomes.