Radioactive Decay Calculation Remaining Substance After 3 Years

Radioactive decay is a fundamental process in nuclear chemistry, describing the spontaneous breakdown of an unstable atomic nucleus, resulting in the release of energy and matter. Each radioactive isotope decays at a unique rate, characterized by its half-life, which is the time required for half of the radioactive material to decay. This decay process follows first-order kinetics, meaning the decay rate is proportional to the amount of radioactive substance present. In simpler terms, if we start with a certain amount of a radioactive substance, after one half-life, half of it will have decayed into a different element or isotope. After another half-life, half of the remaining substance will decay, and so on. This exponential decay continues indefinitely, theoretically never reaching zero, but practically becoming negligible after a certain number of half-lives.

Understanding half-life is crucial in various fields, including nuclear medicine, where radioactive isotopes are used for diagnostic imaging and cancer treatment, and in environmental science, where radioactive decay is a key factor in assessing the long-term impact of nuclear waste. In geology, radioactive dating techniques rely on the known half-lives of certain isotopes, such as carbon-14 and uranium-238, to determine the age of rocks and fossils. The concept of half-life also extends beyond radioactive decay and can be applied to other first-order processes, such as the elimination of drugs from the body or the cooling of an object.

The mathematical representation of radioactive decay is essential for quantitative analysis. The equation that governs this process is:

$N(t) = N_0 * (1/2)^{(t/T)}$

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 elapsed time.
• $T$ is the half-life of the substance.

This equation highlights the exponential nature of radioactive decay. The fraction remaining after each half-life is always 1/2, and the exponent $(t/T)$ indicates how many half-lives have passed during the elapsed time. This formula allows us to calculate the amount of radioactive substance remaining after any given time, provided we know the initial amount and the half-life. In our example, we'll use this equation to determine the remaining amount of a radioactive substance after 3 years, given its half-life and initial quantity.

Let's tackle a specific problem to illustrate how to calculate the remaining amount of a radioactive substance after a certain period. Suppose we have a radioactive substance with a half-life of 1 year. This means that every year, half of the substance decays. We start with an initial amount of 50 grams of this substance. The question we aim to answer is: how much of the substance will remain after 3 years?

This problem is a classic example of radioactive decay calculations, and it underscores the importance of understanding the concept of half-life. The key to solving this problem is recognizing the exponential decay pattern. After one year, half of the 50 grams will decay, leaving 25 grams. After two years, half of the 25 grams will decay, leaving 12.5 grams. And after three years, half of the 12.5 grams will decay. However, to get an accurate answer, especially for non-integer multiples of the half-life, it's best to use the radioactive decay formula we introduced earlier.

The problem also sets up the initial framework for the solution by providing a partially filled equation:

Remaining Amount $= 50 * (1 - [?])$

This equation suggests that we need to determine a decay factor that represents the fraction of the substance lost over a single time period. However, the correct approach involves using the half-life formula directly, which accounts for the exponential decay over time. The provided equation hints at a linear decay model, which is not accurate for radioactive decay. Therefore, we need to replace the bracketed term with the correct exponential decay factor derived from the half-life formula. By using the correct formula, we can accurately calculate the remaining amount of the radioactive substance after 3 years. This problem demonstrates the practical application of the half-life concept and the importance of using the correct mathematical model for radioactive decay.

To accurately calculate the remaining amount of the radioactive substance, we'll employ the radioactive decay formula:

$N(t) = N_0 * (1/2)^{(t/T)}$

Where:

• $N(t)$ is the amount remaining after time $t$.
• $N_0$ is the initial amount (50 grams).
• $t$ is the elapsed time (3 years).
• $T$ is the half-life (1 year).

Plugging in the given values, we get:

$N(3) = 50 * (1/2)^{(3/1)}$

Simplifying the exponent:

$N(3) = 50 * (1/2)^3$

Now, we calculate $(1/2)^3$:

$(1/2)^3 = 1/2 * 1/2 * 1/2 = 1/8$

Substitute this back into the equation:

$N(3) = 50 * (1/8)$

Finally, calculate the remaining amount:

$N(3) = 50/8 = 6.25 grams$

Therefore, after 3 years, 6.25 grams of the radioactive substance would remain. This result demonstrates the exponential decay process, where the substance decreases by half for every half-life that passes. The initial equation provided, Remaining Amount $= 50 * (1 - [?])$, was a misleading setup as it suggests a linear decay. The correct approach using the half-life formula reveals the true exponential nature of radioactive decay. The accurate calculation highlights the importance of using the appropriate formula for these types of problems, ensuring we correctly account for the exponential decrease in the radioactive substance over time. This method provides a precise and reliable way to determine the remaining amount of a radioactive material after any given period.

Another formula presented in the problem is:

Remaining Amount $= I(1 - r)^t$

Let's break down this formula and understand its components in the context of radioactive decay. In this equation:

• $I$ represents the initial amount of the substance.
• $r$ represents the decay rate per time period.
• $t$ represents the number of time periods that have elapsed.

This formula is a general form for exponential decay, applicable to various scenarios beyond radioactive decay. It essentially states that the remaining amount is equal to the initial amount multiplied by the factor $(1 - r)$ raised to the power of $t$. The term $(1 - r)$ represents the fraction of the substance that remains after each time period. However, the crucial point here is determining the correct value for $r$ in the context of radioactive decay.

In our problem, the half-life is 1 year. This means that after each year, half of the substance decays. Therefore, the decay rate $r$ is not simply 1/2. Instead, we need to relate this formula to the half-life formula we used earlier:

$N(t) = N_0 * (1/2)^{(t/T)}$

To connect the two formulas, we can rewrite the half-life formula as:

$N(t) = N_0 * ( (1/2)^{(1/T)} )^t$

Comparing this with $I(1 - r)^t$, we can see that:

$(1 - r) = (1/2)^{(1/T)}$

In our case, $T = 1$ year, so:

$(1 - r) = (1/2)^{(1/1)} = 1/2$

Thus, $r = 1 - 1/2 = 1/2$. However, this value of $r$ is specific to the half-life scenario. If the decay was characterized by a different fraction remaining each year (e.g., one-third decaying each year), the value of $r$ would change accordingly. The key takeaway is that the formula $I(1 - r)^t$ is a versatile representation of exponential decay, but the value of $r$ must be carefully determined based on the specific decay process.

Applying this understanding to our problem, we can rewrite the formula as:

Remaining Amount $= 50 * (1 - 0.5)^3 = 50 * (0.5)^3 = 50 * (1/8) = 6.25 grams$

This confirms our previous result obtained using the half-life formula directly. The formula $I(1 - r)^t$ provides an alternative approach to solving radioactive decay problems, emphasizing the concept of a decay rate per time period. However, it's crucial to correctly interpret and calculate the decay rate based on the given information, such as the half-life of the substance.

In conclusion, we successfully calculated the remaining amount of a radioactive substance after 3 years, given its half-life of 1 year and an initial amount of 50 grams. The calculation revealed that 6.25 grams of the substance would remain. We utilized the radioactive decay formula:

$N(t) = N_0 * (1/2)^{(t/T)}$

and the alternative exponential decay formula:

Remaining Amount $= I(1 - r)^t$

to arrive at the same answer.

The problem highlighted the importance of understanding radioactive decay as an exponential process, where the substance decreases by half with each passing half-life. It also underscored the significance of using the correct formula to accurately model this decay. The initial partially filled equation provided, Remaining Amount $= 50 * (1 - [?])$, was shown to be misleading as it suggested a linear decay, which is not the case for radioactive substances.

Furthermore, we explored the connection between the half-life formula and the general exponential decay formula, demonstrating how the decay rate $r$ can be derived from the half-life. This understanding allows for flexibility in approaching radioactive decay problems, as one can choose the formula that best suits the given information and the desired approach.

Radioactive decay has numerous real-world applications, from medical imaging and cancer treatment to geological dating and environmental monitoring. A solid grasp of the principles and formulas discussed here is essential for anyone working in these fields. The ability to accurately calculate the remaining amount of a radioactive substance is crucial for safety protocols, dosage calculations, and understanding the long-term impact of radioactive materials. This problem serves as a valuable example of how these concepts are applied in practice, emphasizing the importance of a thorough understanding of radioactive decay and its mathematical representation.