Radioactive Half-Life - Online Tutor, Practice Problems & Exam Prep

Half-life (t_1/2) is the time required for half of a radioisotope to decay, crucial for understanding radioactive decay. For example, starting with 10 grams of a radioisotope with a half-life of one day, after one day, 5 grams remain; after two days, 2.5 grams; and after three days, 1.25 grams. The fraction remaining can be calculated using the formula 0.5ⁿ, where n is the number of half-lives. This allows for determining either the fraction or the actual amount of the substance remaining over time.

Radioactive Half-Life Concept 1

Here we're going to say that half-life, which uses the variable \( t_{\text{half}} \), is the amount of time required for half or 50% of a radioisotope to decay. A radioisotope is just the isotopic version of an element that has an unstable nucleus and emits radiation as it decays.

Let's consider the radioactive decay of an unknown radioisotope with a half-life of one day. A half-life of one day means that after one day, half of the substance will be gone. Let's say that we're starting out with some unknown isotope. Remember, our isotope here uses \( x \) as the symbol, \( a \) as the mass number, and \( z \) as the atomic number. Assume we begin with 10 grams of it. Nothing has happened yet, so 0 half-lives have occurred. Initially, I'm starting out with 100% of my substance.

Now one day is allowed to pass, and a half-life, which is equal to a day in this scenario, results in the loss of half of my substance. Now I only have 5 grams left, which is 50%. Another day passes, and I lose another half, leaving me with only 2.5 grams. This means 2 half-lives have passed, and now only 25% remains. Remember, 100% was 10 grams, and now I have only 2.5 grams left.

Another half-life passes, so I lose another half of this 2.5 grams, leaving me with 1.25 grams. Thus, 3 half-lives have passed, and I now only have 12.5% remaining of the original amount I started with. This process can continue, demonstrating how half-life works—it's the amount of time it takes to lose half or 50% of the starting material. This decay continues potentially to a fourth half-life, where I would lose another half of the remaining substance.

Radioactive Half-Life Concept 2

With half-life, we can also determine the amount of radioisotope remaining. Here we're going to say the fraction percentage and final amount of radioisotope after each half-life can be calculated by different means. Alright. So first, we have our fraction formula. This just tells us the fraction remaining of my initial amount. Here, that would just be 0.5n. The 0.5 comes from the fact that a half-life is a half. And when you punch a half into your calculator, it gives you back 0.5. N is just the number of half-lives. One important thing to remember about this is if you also multiply the fraction remaining value by 100%, that will give you the percentage remaining of the radioisotope. So keep that in mind. You can calculate either the fraction remaining or the percentage remaining. Now, we also have our final amount formula. This tells me how much of my final amount is actually remaining. Not a fraction, not a percent, but an actual amount. Let's say you started out with a 100g of a substance. You could use this formula to tell you how much of that 100g is remaining. So here, this would just be brackets with I, this would be your initial amount, times what we just saw our fractional remaining, our fraction remaining, which is 0.5n. So you can use either formula if you're trying to find either the fraction, percentage, or final amount remaining for a radioisotope.

Radioactive Half-Life Example 1

Here I noted that sodium-24, a radioisotope used in examining human circulation, has a half-life of about 15 days. Approximately, what percentage of a sample will remain after 3 months if a month is considered 30 days? Alright. So they're asking for the percentage remaining. The percentage remaining is just your fraction remaining, so 0.5∧n×100%. Here we have to figure out how many half-lives have occurred. So here we're going to say 3 months have passed, and each month is 30 days. That's 3 times 30; that's a total of 90 days have occurred. Each half-life is 15 days. So if I do 90 divided by 15, it gives me back 6. That is the number of half-lives that have occurred. So then all we do here is we're going to say 0.5∧6×100% equals 1.5625%. This is the percentage remaining of my radioisotope.

The half-life of arsenic-74 is about 18 days. If a sample initially contains 100.00 mg arsenic-74, what mass (in mg) would be left after 72 days?

• A. 0.0625 mg
• B. 6.25 mg
• C. 25 mg
• D. 12.5 mg
• E. 50 mg

The half-life of iodine-131, an isotope used in thyroid therapy, is 8.021 days. What percentage of iodine-131 remains in a sample that is estimated to be 40.11 days old?

• A. 3.125%
• B. 0.03125%
• C. 96.87%
• D. 87.06%
• E. 2.236%

What is the concentration of a CO₂ in a container after 4 half-lives if 0.325 mol of CO2 is initially placed into a 5.0 L reaction vessel?

• A. 0.020 M
• B. 4.06×10^–3 M
• C. 8.1×10^–3 M
• D. 0.325 M

What is the half-life of the radioisotope that shows the following plot of remaining percentage vs. time?

• A. 3 days
• B. 5 days
• C. 4 days
• D. 10 days

Here’s what students ask on this topic:

What is the definition of radioactive half-life?

Radioactive half-life, denoted as t_1/2, is the amount of time required for half of a given quantity of a radioisotope to decay. This means that after one half-life, 50% of the original radioactive material will have transformed into a different element or isotope. The concept is crucial for understanding the rate at which radioactive substances decay and is used in various fields such as nuclear physics, medicine, and environmental science.

How do you calculate the remaining amount of a radioisotope after a certain number of half-lives?

The remaining amount of a radioisotope after a certain number of half-lives can be calculated using the formula: ${Final\; Amount}_{}={Initial\; Amount}_{}\times \; (0.5)n$, where n is the number of half-lives. For example, if you start with 100 grams of a substance and 3 half-lives have passed, the remaining amount would be 100 × (0.5)³ = 12.5 grams.

What is the formula to calculate the fraction of a radioisotope remaining after n half-lives?

The fraction of a radioisotope remaining after n half-lives can be calculated using the formula: $Fraction\; Remaining\; =\; (0.5)n$. Here, 0.5 represents the fraction remaining after one half-life, and n is the number of half-lives that have passed. For example, after 2 half-lives, the fraction remaining would be (0.5)² = 0.25.

How does the concept of half-life apply to real-world scenarios?

The concept of half-life is widely used in various real-world scenarios. In medicine, it helps determine the dosage and timing of radioactive tracers in diagnostic imaging. In environmental science, it aids in assessing the long-term impact of radioactive contamination. In archaeology, carbon-14 dating uses the half-life of carbon-14 to estimate the age of ancient artifacts. Understanding half-life is essential for managing nuclear waste, as it helps predict how long a radioactive material will remain hazardous.

What is the significance of knowing the half-life of a radioisotope?

Knowing the half-life of a radioisotope is significant for several reasons. It allows scientists to predict how long a radioactive substance will remain active and hazardous. This information is crucial for handling and disposing of radioactive materials safely. In medicine, it helps in planning treatments involving radioactive isotopes. In environmental monitoring, it aids in assessing the persistence of radioactive contamination. Overall, understanding half-life is essential for making informed decisions in various scientific and practical applications.