Given A Radioactive Material With A Decay Constant Of 0.02 Per Day And An Initial Mass Of 20 Grams, What Formula Is Used To Calculate The Remaining Mass After 10 Days?

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Understanding radioactive decay is crucial in various scientific fields, from nuclear chemistry to environmental science. This article will delve into the formula used to calculate the remaining mass of a radioactive material after a specific period. We'll use a scenario involving a radioactive sample with a decay constant of 0.02 per day and an initial mass of 20 grams to illustrate the application of the formula.

Understanding Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation in the form of particles or electromagnetic waves. This process transforms the original nuclide, called the parent nuclide, into a different nuclide, or daughter nuclide. The rate at which this decay occurs is characterized by the decay constant, a crucial parameter in determining the half-life and the remaining amount of radioactive material over time.

The decay constant, often denoted by the Greek letter lambda (λ), represents the probability of a nucleus decaying per unit time. A higher decay constant indicates a faster decay rate, meaning the radioactive material will decay more quickly. Conversely, a lower decay constant signifies a slower decay rate, and the material will persist for a longer period. The decay constant is typically expressed in units of inverse time, such as per second (s⁻¹), per minute (min⁻¹), per day (d⁻¹), or per year (y⁻¹), depending on the timescale of the decay process.

The half-life, denoted as t₁/₂, is another essential concept in radioactive decay. It represents the time required for half of the radioactive material to decay. The half-life and the decay constant are inversely related, meaning a shorter half-life corresponds to a larger decay constant and vice versa. The relationship between half-life and decay constant is given by the equation: t₁/₂ = ln(2) / λ, where ln(2) is the natural logarithm of 2, approximately equal to 0.693. This relationship is fundamental in understanding the temporal aspect of radioactive decay and is used to predict the amount of radioactive material remaining after a given time.

The Formula for Radioactive Decay

The amount of radioactive material remaining after a certain time can be calculated using the exponential decay formula. This formula is derived from the first-order kinetics of radioactive decay, which states that the rate of decay is proportional to the amount of radioactive material present. The formula is expressed as follows:

N(t) = N₀ * e^(-λt)

Where:

  • N(t) is the amount of radioactive material remaining after time t.
  • N₀ is the initial amount of radioactive material.
  • λ is the decay constant.
  • t is the time elapsed.
  • e is the base of the natural logarithm, approximately equal to 2.71828.

This formula is a cornerstone in nuclear physics and chemistry, allowing scientists and engineers to predict the behavior of radioactive substances over time accurately. It is particularly useful in applications such as radioactive dating, nuclear medicine, and reactor design, where understanding the decay process is critical. The exponential nature of the decay is evident in the formula, highlighting that the amount of radioactive material decreases exponentially with time. This characteristic decay pattern is a key feature of radioactive processes and has significant implications for the long-term behavior of radioactive materials.

Applying the Formula to Our Scenario

In our scenario, we have a radioactive material with a decay constant (λ) of 0.02 per day. The initial amount of the material (N₀) is 20 grams, and we want to determine how much will remain after 10 days (t). To do this, we simply plug the values into the formula:

N(t) = N₀ * e^(-λt)

N(10) = 20 * e^(-0.02 * 10)

Now, we calculate the exponent:

-0.02 * 10 = -0.2

Next, we calculate e^(-0.2):

e^(-0.2) ≈ 0.8187

Finally, we multiply this result by the initial amount:

N(10) = 20 * 0.8187

N(10) ≈ 16.37 grams

Therefore, after 10 days, approximately 16.37 grams of the radioactive material will remain. This calculation demonstrates the practical application of the radioactive decay formula in predicting the amount of material left after a specific period. The exponential decay process is evident in the reduction from 20 grams to 16.37 grams, showcasing the continuous decrease in the material's mass due to radioactive decay. Understanding how to apply this formula is essential in various fields, including nuclear medicine, environmental science, and nuclear engineering, where the behavior of radioactive substances needs to be accurately predicted and managed.

Conclusion

The formula N(t) = N₀ * e^(-λt) is the correct formula to use in this scenario. By substituting the given values, we can accurately determine the amount of radioactive material remaining after a certain time. This formula is a fundamental tool in understanding and predicting radioactive decay, with applications spanning various scientific and industrial fields. The ability to calculate the remaining mass of a radioactive substance is crucial for safety, environmental protection, and the advancement of scientific knowledge. The exponential nature of the decay process, as captured by this formula, highlights the importance of understanding the underlying principles of radioactive decay in various practical applications.

By mastering this concept, you gain a valuable tool for analyzing and predicting the behavior of radioactive materials. Whether you are a student, a scientist, or simply someone curious about the world around you, understanding radioactive decay is a crucial step in appreciating the fundamental processes that govern our universe. The application of this formula extends beyond theoretical calculations, playing a vital role in real-world scenarios, such as determining the age of artifacts through carbon dating or ensuring the safe handling and disposal of radioactive waste. The exponential decay formula, therefore, is not just a mathematical expression but a key to understanding the dynamics of radioactive materials and their impact on our world.