Niobium-91 Decay Calculation Remaining Mass After 2040 Years

In the realm of nuclear chemistry, understanding the concept of radioactive decay is crucial. This article delves into a specific example, focusing on the decay of Niobium-91 (Nb-91). Niobium-91 (Nb-91) is a radioactive isotope with a half-life of 680 years, this means that every 680 years, half of the initial amount of Nb-91 decays into another element. This article will explore a common problem encountered in nuclear chemistry which is calculating the remaining amount of a radioactive substance after a certain period, given its half-life and initial mass. Specifically, we'll address the question: After 2,040 years, how much niobium-91 will remain from a 300.0-g sample? By understanding the principles of half-life and applying a straightforward calculation, we can accurately determine the remaining mass.

The half-life of a radioactive isotope is the time it takes for half of the atoms in a sample to decay. This is a fundamental concept in nuclear chemistry and is essential for understanding the rate at which radioactive materials transform. In the case of niobium-91, the half-life is 680 years. This means that if you start with a certain amount of niobium-91, after 680 years, only half of that amount will remain. After another 680 years (a total of 1,360 years), half of the remaining amount will decay, leaving you with one-quarter of the original amount. This process continues exponentially, with the amount of niobium-91 decreasing by half for every 680-year interval.

Understanding half-life is crucial in various applications, including radioactive dating, nuclear medicine, and nuclear waste management. For instance, carbon-14 dating uses the half-life of carbon-14 (approximately 5,730 years) to determine the age of ancient artifacts and fossils. In medicine, radioactive isotopes with short half-lives are used for imaging and therapeutic purposes, minimizing the exposure of patients to radiation. In nuclear waste management, the long half-lives of certain radioactive isotopes pose a significant challenge, requiring long-term storage solutions.

To accurately calculate the amount of a radioactive substance remaining after a certain time, we need to consider the number of half-lives that have elapsed. This is done by dividing the total time by the half-life of the isotope. The formula for calculating the remaining amount is:

N(t) = N₀ * (1/2)^(t / T)

Where:

• N(t) is the amount remaining after time t.
• N₀ is the initial amount.
• t is the elapsed time.
• T is the half-life.

This formula allows us to precisely determine the decay of a radioactive substance over time, which is essential for various scientific and practical applications. In the following sections, we will apply this formula to the specific problem of determining the remaining amount of niobium-91 after 2,040 years.

Before we can calculate the remaining amount of niobium-91, we need to determine how many half-lives have passed in the given time period. Given that the half-life of niobium-91 is 680 years, and the time period we're considering is 2,040 years, we can calculate the number of half-lives by dividing the total time by the half-life:

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

Number of half-lives = 2,040 years / 680 years

Number of half-lives = 3

This calculation shows that 3 half-lives of niobium-91 have passed in 2,040 years. Each half-life represents a halving of the remaining amount of the substance. Therefore, after 3 half-lives, the initial amount will have been halved three times. Understanding the number of half-lives is crucial for accurately determining the final amount of niobium-91 remaining.

The number of half-lives directly affects the amount of the radioactive substance remaining. After one half-life, 50% of the original substance remains. After two half-lives, 25% remains (50% of 50%). After three half-lives, 12.5% remains (50% of 25%). This exponential decay is a characteristic feature of radioactive substances and is described by the equation we introduced earlier:

N(t) = N₀ * (1/2)^(t / T)

In our case, with 3 half-lives, we can expect the remaining amount of niobium-91 to be significantly less than the initial amount. The next step is to apply this information to the initial sample size to calculate the exact remaining mass. This will involve using the initial mass of 300.0 g and the understanding that the substance has undergone three half-lives.

By calculating the number of half-lives, we have laid the groundwork for determining the final amount of niobium-91 remaining after 2,040 years. This step is essential for understanding the extent of radioactive decay and its implications in various applications, from nuclear waste disposal to radioactive dating. In the next section, we will use this information to calculate the exact mass of niobium-91 that remains.

Now that we know the number of half-lives that have passed (3 half-lives), we can calculate the remaining amount of niobium-91 from the initial 300.0-g sample. To do this, we can use the concept that after each half-life, the amount of the substance is halved.

Initial amount: 300.0 g

• After 1 half-life (680 years): 300.0 g / 2 = 150.0 g
• After 2 half-lives (1,360 years): 150.0 g / 2 = 75.0 g
• After 3 half-lives (2,040 years): 75.0 g / 2 = 37.5 g

Alternatively, we can use the formula for radioactive decay:

N(t) = N₀ * (1/2)^(t / T)

Where:

• N(t) is the amount remaining after time t.
• N₀ is the initial amount (300.0 g).
• t is the elapsed time (2,040 years).
• T is the half-life (680 years).

Plugging in the values, we get:

N(2040) = 300.0 g * (1/2)^(2040 / 680)

N(2040) = 300.0 g * (1/2)^3

N(2040) = 300.0 g * (1/8)

N(2040) = 37.5 g

Both methods yield the same result: after 2,040 years, 37.5 g of the niobium-91 sample will remain. This calculation demonstrates the exponential nature of radioactive decay and how the amount of a radioactive substance decreases over time.

Understanding how to calculate the remaining amount of a radioactive substance after a certain period is essential in various fields, including nuclear chemistry, environmental science, and medicine. For example, in nuclear waste management, it is crucial to know how long radioactive waste will remain hazardous. In nuclear medicine, the decay rate of radioactive isotopes used in diagnostic and therapeutic procedures must be carefully considered to ensure patient safety and treatment efficacy. By accurately calculating the decay of radioactive substances, scientists and practitioners can make informed decisions and manage the risks associated with radioactivity.

In conclusion, starting with a 300.0-g sample of niobium-91, after 2,040 years, 37.5 g will remain. This result was obtained by understanding the concept of half-life and applying it to the specific scenario. The calculations involved determining the number of half-lives that have passed and then using this information to find the remaining amount of the substance.

We utilized two methods to arrive at the same conclusion. The first method involved halving the initial amount successively for each half-life that passed. The second method used the radioactive decay formula, which provides a more direct and generalized approach to calculating the remaining amount of a radioactive substance. Both methods underscore the exponential decay pattern characteristic of radioactive materials.

The significance of understanding half-life extends beyond theoretical calculations. It has practical implications in various fields such as nuclear medicine, environmental science, and nuclear waste management. In nuclear medicine, the half-life of radioactive isotopes used in imaging and therapy is a critical factor in determining the dosage and duration of treatment. In environmental science, understanding the decay rates of radioactive contaminants helps in assessing and mitigating environmental risks. In nuclear waste management, the long half-lives of certain radioactive isotopes necessitate long-term storage solutions to ensure public safety.

By mastering the concepts and calculations related to half-life, scientists and professionals can make informed decisions and effectively manage the challenges associated with radioactive materials. The example of niobium-91 decay illustrates the importance of these skills in addressing real-world problems and ensuring the safe and responsible use of nuclear technology. Therefore, the correct answer to the question "After 2,040 years, how much niobium-91 will remain from a 300.0-g sample?" is C. 37.5 g. This comprehensive exploration of radioactive decay highlights the crucial role of understanding half-life in various scientific and practical applications. By accurately calculating the decay of radioactive substances, we can better manage their use and ensure safety in a world increasingly reliant on nuclear technology and its applications.