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

Radioactive half-life is the time required for half of a radioisotope to decay, and it's a constant value that does not depend on the initial concentration of the substance. A radioisotope, or nuclide, is an unstable isotope that emits radiation during its decay, which can include alpha decay, beta decay, electron capture, or positron emission. The half-life can be calculated using the equation:

$t_{\frac{1}{2}}=\frac{\ln \left(2\right)}{k}$

where ln(2) is approximately 0.693 and k is the decay constant. This equation highlights that half-life is inversely proportional to the decay constant and remains constant throughout the reaction, as demonstrated by a flat line when plotted against time. Understanding half-life is crucial for applications such as radiometric dating and nuclear medicine.

Intro to Radioactive Half-Life

Here we can say that radioactive half lives, which is T_1/2. It's just the amount of time required for half of a radioisotope to decay. Now a radioisotope or nuclide is just an isotope that has an unstable nucleus and emits radiation as it decays. Alpha decay, beta decay, electron capture, positron emission etcetera.

Based on this, if we take a look here, the example says what is the half life of the radioisotope that shows the following data of remaining percentages versus time. So here we have our percentage remaining and here we have number of days. We start out with 100% of our starting material on day zero. Remember, Half Life looks at how much time it takes us to lose half, all right, So 100, half of that would be 50. Half of that would be 25%. Another half of that would be 12.5%.

What are the number of days that separate each one of these percentages? Well, day zero. It takes three days for us to lose half. Looks like it takes another three days to lose half, and it looks like it took another three days to lose half. So based on this chart here, it looks like the half life is equal to 3 days for this particular radioisotope.

Method 1 of Radioactive Half-Life

Now we're going to look at different methods that involve half life in some way. In method one, we're dealing with the direct calculation of half life or rate constant. Now, in method one, you use a radioactive half life equation when dealing with only the half life and the decay constant, which is K.

Now here the formula for the radioactive half life is:

t = ln ⁡ 2 K

Lane two is a constant which is equal, if you plug into your calculator, to approximately 0.693. K Here's our decay constant, which is in times inverse, which shouldn't be mistaken with time, which is T.

Here, when it comes to a plot of half life versus time, we would say that half life does not depend on the initial concentration because if we look, we don't see this variable within the formula and it is constant through the whole reaction. Remember the example that we saw earlier? Every three days we'd lose half of our starting material. It was every three days. It didn't fluctuate where it's 2.5 here and 1.8 here or 7.2 here. It was always three days because of that.

Well, if we were to plot this, we'd say that the half life remains flat. It's constant. So no matter how much time passes, my half life stays the same. So it would just be this flat straight line. O when it comes to the radioactive half life equation. Just remember this is the equation we utilize. It forms a connection between half life and our decay constant K. Half life is a constant idea because we're following first order rate law rules.

Radioactive Half-Life Example

Here we're told if the decay constant of plutonium 244 is 8.66 * 10 to the -9 years at 25°C, what is its half life? All right, so they're asking for half life and all they're giving us is the decay constant. This is just simply utilizing the radioactive half-life equation.

We're half life equals ln2/K plug in. The value that we just got was 8.66×10^-9 year. Inverse should be inverse. That will give me 8.00×10^7 years.

So this would be our final answer. This would be equal to the half life of plutonium 244. It would take this amount of time for us to lose half of our starting material.

Radioactive Half-Life Example

Now method 2 deals with radioactive nuclei concentrations. In method two we utilize both the radioactive half life equation and the radioactive integrated rate law. So if we look, here's our half life equation and here's our integrated rate law. In this case the questions will involve the half life time, your initial concentration or your final concentration for your radioactive nuclei concentration.

If we take a look at this example it says a sample of radon 222 has an initial alpha particle activity. So here a₀. Now this can just be a stand in for our variable of north. It's the same idea here. It has as its concentration disintegrations per second. So we have 8.5 * 10⁴ disintegrations per second. After 7.3 days, its activity now becomes 3.7 * 10⁴ disintegrations per second. What is the half life of radon? 222.

All right, so half life equals ln2k. So for us to know what the half life is, we need to determine what our decay constant will be. From this question, we have what? We have this larger concentration, which is our initial. We have this smaller one after X number of days, which is our final and we have time. Looking at the integrated rate law, we have this variable, this variable and this variable. Knowing those 3 variables we can solve for our K.

Once you solve for K you can plug it into the half life equation. It won't know half life, so here we're going to say ln(A)=-kt+ln(A₀). So here we'd say ln(3.7×10⁴)=-k(7.3)+ln(8.5×10⁴). We're going to subtract ln(8.5×10⁴) from both sides.

When we do that here, we're going to get initially -0.8317333 = -k(7.3) divide from both sides -7.3 and we'll get our K. I'm going to take that K and plug it into my half life equation. The K that I isolate is 0.0113936 days^-1. So when I plug that in, I'm going to get as my number of my half life here as 6.1 days. So 6.1 will be our final answer.

Radioactive Half-Life Example

Now, method three is the easiest one to spot because in this method they're talking about fractions or percentages. In method three we also utilize both the radioactive half-life equation and the radioactive integrated rate law. In this case, the questions will ask for the fraction or percentage remaining while involving half-life. Here we're going to say that the fraction of radioisotope is equal to your final concentration divided by your initial concentration. And remember if we're multiplying a fraction by 100, that will give us a percentage, in this case the percentage remaining of our radioisotope.

Here if we take a look at our examples or one example, it says here the half-life of iodine 131, an isotope used in thyrotherapy, is 8.021 days. What fraction of iodine 131 remains in a sample that is estimated to be 6.25 months old? All right, so we can use this flow chart to help us organize our thoughts. Here we have our starting position. So the first question we ask ourselves is, do you have the decay constant K? Well, if we look here, it doesn't tell us what the K constant K is. So we can't move on to yes. We have to move on to no.

Well, if it's a no, then use the radioactive half-life equation to help us isolate the decay constant K. So here half-life equals ln2 / K. We know what the half-life is. With that information, I could find my K. If I rearrange this equation, K equals ln2 / half-life, so it would be ln2 / 8.021 days. So here that would give me 0.08642 days inverse for K, right? So now I know K, so then I move on. I use the radioactive integrated rate law which is this, and I use it to solve for the initial and final concentration to find our fraction which again is equal to my final divided by my initial.

Now remember we can rearrange this equation for our integrated rate law. Since we're looking for a fraction, it can become lnfinalinitial, and that'll equal -KT. Now here we plug in the information that we know. Our K is -0.08642 days inverse. But remember K and T have to have the same units. Here, K is in days inverse, so time needs to be in days. So I'll have to change these months into days. Now this is going to be an estimate where we're going to say that one month is equal to about 30 days. Of course, some months are 31 days, February is 28 or 29 days. This is just an average, so months here cancel out. I'll get 187.5 days which I plug in here.

When we multiply these two together, that's going to give me -16.20375 which is equal to lnfinalinitial which is my fraction. I need to isolate just that fraction itself. So here, to isolate just that fraction, I'm going to take the inverse of the natural log on both sides. That's going to have the ln drop from my left side and then if I take the inverse of the natural log on the right side it becomes e to this number. And if I were to plug that into my calculator I'll get as my fraction 9 point. And here if we want to do in terms of significant figures, 3 significant figures, so 9.18 times 10 to the -8. So this would be my fraction remaining for this particular example question.

The half-life of arsenic-74 is about 18 days. If a sample initially contains 5.13 x 10⁴ mg arsenic-74, what mass (in mg) would be left after 80 days?

• A. 2.36 x 10³ mg
• B. 7.02 x 10² mg
• C. 1.43 x 10³ mg
• D. 1.14 x 10⁸ mg

What percentage of carbon – 14 ( t_1/2 = 5715 years) remains in a sample estimated to be 18,315 years old? 

• A. 11.09%
• B. 27.96%
• C. 18.72%
• D. 10.85%

Here’s what students ask on this topic:

What is the half-life of a radioactive isotope?

The half-life of a radioactive isotope, also known as a radionuclide, is the time required for half of the atoms in a sample to undergo radioactive decay. During decay, the unstable isotope transforms into a more stable form, which may be another element or a different isotope of the same element. This process emits radiation in the form of particles or electromagnetic waves.

The half-life is a characteristic property of each radioactive isotope and can vary widely, from fractions of a second to millions of years, depending on the isotope in question. For example, the half-life of carbon-14, which is used in radiocarbon dating, is about 5,730 years, whereas the half-life of uranium-238 is about 4.5 billion years. Knowing the half-life allows scientists to determine the age of materials or to understand the timing of processes in nuclear physics and medicine.

How is the half-life of a radioactive parent isotope defined?

The half-life of a radioactive parent isotope is defined as the amount of time it takes for half of the atoms in a sample of the isotope to decay into its daughter isotopes. This decay process happens at a constant rate, which is characteristic of each radioactive isotope. During each half-life, 50% of the remaining radioactive atoms will transform, resulting in a predictable decrease in the amount of the parent isotope over time. It's important to note that the half-life is not affected by external factors such as temperature, pressure, or chemical state; it is a fixed property of the isotope. For example, if the half-life of a particular isotope is 5 years, then after 5 years, only half of the original sample would remain, and after another 5 years (10 years in total), only a quarter of the original sample would be left.

Among the following radioactive parent isotopes, which one has the shortest half-life?

To determine which radioactive parent isotope has the shortest half-life, you would need to compare the specific half-lives of the isotopes in question. The half-life of a radioactive isotope is the time it takes for half of the radioactive atoms present to decay into their daughter isotopes. This can range from fractions of a second to millions of years, depending on the isotope.

However, as the list of isotopes isn't provided in your question, I can't give a specific answer. For example, Polonium-214 has a very short half-life of 164 microseconds, while Uranium-238 has a half-life of about 4.5 billion years. To find out which isotope has the shortest half-life, you would need to look up the half-lives of each isotope mentioned and compare them. The one with the smallest numerical value for its half-life is the one that decays the fastest, meaning it has the shortest half-life.