Leutium-176 Decay Calculation How Much Remains After 1.155 X 10^11 Years

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Hey guys! Ever wondered how much of a radioactive substance sticks around after, like, a really long time? We're diving into that today with a question about Leutium-176, which has a mind-blowingly long half-life. So, buckle up, and let's get into some chemistry!

The Leutium-176 Half-Life Puzzle

So, the problem states: Leutium-176 has a half-life of $3.85 imes 10^{10}$ years. After $1.155 imes 10^{11}$ years, how much leutium-176 will remain from an original 16.8-g sample?

To tackle this, we need to understand what half-life actually means. In the realm of nuclear chemistry, half-life is the time it takes for half of the atoms in a radioactive sample to decay. Think of it like this: if you start with a room full of 100 people, the half-life is how long it takes for 50 of those people to leave. After another half-life, half of the remaining people will leave, and so on.

In the case of Leutium-176, its half-life is an astounding $3.85 imes 10^{10}$ years. That's 38.5 billion years! This incredibly long half-life indicates that Leutium-176 decays very, very slowly. This also implies that Leutium-176 has important implications for geochronology, a scientific field that uses radioactive decay to date geological formations and materials. The decay of Leutium-176 can act as a geological clock, helping scientists determine the age of ancient rocks and minerals. With its extremely long half-life, Leutium-176 decay is particularly useful for dating extremely old geological samples, adding crucial data points to our understanding of the Earth's long history. This is because more commonly known isotopes, such as Carbon-14, can only be used to date materials up to approximately 50,000 years old due to their relatively short half-lives. Materials billions of years old require isotopes with extremely long half-lives, like Leutium-176.

Now, the big question: We start with 16.8 grams of Leutium-176, and $1.155 imes 10^{11}$ years pass. How much is left? This is like figuring out how many people are left in the room after a few rounds of departures. We need to calculate how many half-lives have elapsed during that time period. To do this, we can divide the total time passed by the half-life of the substance.

Calculating the Number of Half-Lives

First, we need to figure out how many half-lives have gone by during the $1.155 imes 10^{11}$ years. This is just a simple division problem:

Number of half-lives = (Total time) / (Half-life)

Number of half-lives = ($1.155 imes 10^{11}$ years) / ($3.85 imes 10^{10}$ years)

Number of half-lives = 3

Okay, so three half-lives have passed. This is a crucial piece of information! It means that the initial amount of Leutium-176 has been halved three times. Each half-life reduces the amount of the substance by half, so after each subsequent period of $3.85 imes 10^{10}$ years, we have half the amount of Leutium-176 we had before.

Determining the Remaining Amount

We started with 16.8 grams of Leutium-176. After one half-life, half of it will decay. After the second, half of what's left will decay, and so on. To find out the remaining amount, we can perform the following calculation:

  • After 1 half-life: 16.8 g / 2 = 8.4 g
  • After 2 half-lives: 8.4 g / 2 = 4.2 g
  • After 3 half-lives: 4.2 g / 2 = 2.1 g

So, after three half-lives, there will be 2.1 grams of Leutium-176 remaining from the original 16.8-gram sample. Therefore, the answer is A. 2.10 g

Alternative Approach: The Formula

For those who love formulas, we can also use the half-life decay formula to solve this problem. The formula is:

Nt=N0imes(1/2)(t/T)N_t = N_0 imes (1/2)^{(t/T)}

Where:

  • N_t$ is the amount of the substance remaining after time *t*

  • N_0$ is the initial amount of the substance

  • t is the time elapsed
  • T is the half-life of the substance

Plugging in the values we have:

Nt=16.8extgimes(1/2)(1.155imes1011extyears/3.85imes1010extyears)N_t = 16.8 ext{ g} imes (1/2)^{(1.155 imes 10^{11} ext{ years} / 3.85 imes 10^{10} ext{ years})}

Nt=16.8extgimes(1/2)3N_t = 16.8 ext{ g} imes (1/2)^{3}

Nt=16.8extgimes(1/8)N_t = 16.8 ext{ g} imes (1/8)

Nt=2.1extgN_t = 2.1 ext{ g}

Once again, we arrive at the same answer: 2.1 grams of Leutium-176 remaining.

Why This Matters: Applications of Half-Life

Understanding half-life isn't just about solving chemistry problems. It has real-world applications in various fields, such as:

  1. Radiocarbon Dating: This technique uses the half-life of carbon-14 to determine the age of organic materials up to about 50,000 years old. It's widely used in archaeology and paleontology to date fossils and artifacts.
  2. Medical Imaging: Radioactive isotopes with short half-lives are used in medical imaging techniques like PET scans. These isotopes decay quickly, minimizing the patient's exposure to radiation. These isotopes are attached to various compounds that selectively concentrate in specific organs or tissues, allowing doctors to visualize internal structures and detect abnormalities.
  3. Nuclear Medicine: Radioactive isotopes are used in targeted therapies to treat diseases like cancer. These isotopes deliver radiation directly to cancer cells, minimizing damage to healthy tissues. One such treatment is iodine-131 therapy for thyroid cancer, where the thyroid gland absorbs the radioactive iodine, leading to the destruction of cancerous cells.
  4. Geochronology: As mentioned earlier, isotopes with very long half-lives, like Leutium-176, are used to date geological formations and determine the age of the Earth itself. These techniques help scientists understand the history of our planet and the processes that have shaped it over billions of years.
  5. Nuclear Waste Management: Understanding half-lives is crucial for managing nuclear waste. Radioactive waste contains isotopes with varying half-lives. Some isotopes decay quickly, while others remain radioactive for thousands or even millions of years. Scientists and policymakers use this information to develop safe storage and disposal methods for nuclear waste, minimizing the risk of environmental contamination.
  6. Industrial Applications: Radioactive isotopes are used in various industrial applications, such as gauging the thickness of materials, tracing the flow of liquids and gases, and sterilizing medical equipment. The half-life of the isotope is carefully considered to ensure the safety and effectiveness of these applications.

Understanding radioactive decay and half-life is critical in many scientific disciplines and practical applications. The specific rate of decay for an isotope allows for the precise dating of objects, medical procedures, and various industrial processes. The mathematics of half-life, as demonstrated with the Leutium-176 problem, enables us to quantify radioactive decay accurately.

Key Takeaways

  • Half-life is the time it takes for half of a radioactive substance to decay.
  • We can calculate the remaining amount of a radioactive substance after a certain time using the half-life concept or the half-life decay formula.
  • Half-life has important applications in radiocarbon dating, medical imaging, nuclear medicine, and geology.

In conclusion, understanding half-life is crucial for solving problems related to radioactive decay and appreciating its wide range of applications in various scientific fields. Leutium-176, with its incredibly long half-life, serves as a testament to the power of nuclear processes that unfold over vast stretches of time. So, next time you encounter a problem involving half-life, remember the steps we've discussed, and you'll be well-equipped to tackle it!

Repair Input Keyword

Original Question: Leutium-176 has a half-life of $3.85 imes 10^{10}$ years. After $1.155 imes 10^{11}$ years, how much leutium-176 will remain from an original 16.8-g sample?

Rewritten Question: If Leutium-176 has a half-life of $3.85 imes 10^{10}$ years, how many grams of Leutium-176 will remain after $1.155 imes 10^{11}$ years if you start with a 16.8-g sample?