Radioactive Decay Calculation Remaining Substance After 6 Years

Radioactive decay is a fascinating phenomenon where unstable atomic nuclei lose energy by emitting radiation. This process is characterized by the half-life, a crucial concept in nuclear physics and chemistry. The half-life of a radioactive substance is the time it takes for half of its atoms to decay into a different element or isotope. In simpler terms, if you start with a certain amount of a radioactive substance, after one half-life, only half of that amount will remain. After another half-life, only half of that remaining amount will remain, and so on. This decay process follows an exponential pattern, which is described by the formula:

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

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 formula is essential for calculating the remaining amount of a radioactive substance after a certain period, given its initial amount and half-life. It highlights the exponential nature of radioactive decay, where the amount of substance decreases by half with every passing half-life. Understanding this formula is crucial for various applications, including carbon dating, nuclear medicine, and nuclear waste management.

The half-life of a radioactive substance is a constant value, meaning it doesn't change with temperature, pressure, or the amount of substance present. This makes it a reliable measure for radioactive decay rates. Different radioactive isotopes have different half-lives, ranging from fractions of a second to billions of years. For example, some isotopes used in medical imaging have short half-lives to minimize patient exposure to radiation, while isotopes in nuclear waste can have very long half-lives, requiring long-term storage solutions.

The concept of half-life is not just theoretical; it has practical applications in various fields. In medicine, radioactive isotopes are used for diagnostic imaging and cancer treatment. The half-life of the isotope is a critical factor in determining the dosage and timing of these procedures. In archaeology and geology, carbon-14 dating uses the half-life of carbon-14 to estimate the age of organic materials. This technique has revolutionized our understanding of the past, allowing us to date fossils, artifacts, and other materials with remarkable accuracy.

Let's consider a specific problem to illustrate the application of the half-life concept. Suppose we have a radioactive substance with a half-life of 3 years. We start with 60 grams of this substance. The question is: how much of the substance will remain after 6 years? This problem is a classic example of a radioactive decay calculation and provides a practical way to understand the concept of half-life.

To solve this problem, we can use the formula mentioned earlier, but let's first analyze the situation step-by-step. The initial amount of the substance is 60 grams. After one half-life (3 years), half of the substance will decay, leaving us with 30 grams. After another half-life (another 3 years, making a total of 6 years), half of the remaining 30 grams will decay, leaving us with 15 grams. This step-by-step approach provides an intuitive understanding of the decay process. However, for more complex scenarios or to avoid manual calculations, the formula is invaluable.

Alternatively, we can directly apply the formula: $N(t) = N_0 * (1/2)^(t/T)$. Here, $N_0$ is 60 grams, $t$ is 6 years, and $T$ is 3 years. Plugging these values into the formula, we get:

$N(6) = 60 * (1/2)^(6/3) = 60 * (1/2)^2 = 60 * (1/4) = 15$ grams.

This calculation confirms our step-by-step analysis. After 6 years, 15 grams of the radioactive substance will remain. This example demonstrates how the half-life concept and the decay formula can be used to accurately predict the amount of radioactive material remaining after a given time. Understanding these calculations is crucial in various fields, including nuclear medicine, environmental science, and nuclear waste management.

To solve this, we can use the formula for exponential decay, which is derived from the half-life concept. The formula is:

Remaining Amount = Initial Amount * (1/2)^(Time / Half-life)

In this case:

• Initial Amount (I) = 60 grams
• Time (t) = 6 years
• Half-life = 3 years

Plugging these values into the formula, we get:

Remaining Amount = 60 * (1/2)^(6/3)

Now, let's simplify the exponent:

6 / 3 = 2

So the formula becomes:

Remaining Amount = 60 * (1/2)^2

Next, we calculate (1/2)^2:

(1/2)^2 = 1/4

Now, multiply 60 by 1/4:

Remaining Amount = 60 * (1/4) = 15 grams

Therefore, after 6 years, 15 grams of the radioactive substance would remain. This calculation demonstrates the power of the exponential decay formula in predicting the amount of radioactive material left after a certain period. The formula takes into account the half-life of the substance and the elapsed time, providing an accurate estimate of the remaining material. This is essential in various applications, such as determining the safety of radioactive materials and calculating the dosage for medical treatments.

The step-by-step calculation also highlights the nature of exponential decay. With each half-life that passes, the amount of substance decreases by half. This means that the decay rate slows down over time, but the substance never completely disappears. This understanding is critical in managing radioactive waste and assessing the long-term impact of radioactive materials on the environment.

The problem also presents an equation in the form:

Remaining Amount = 60(1 - [?])

This equation is a variation of the exponential decay formula, and we need to determine the missing number inside the parentheses. To do this, we can use the information we already have: the half-life, the initial amount, and the time elapsed.

We know that after 6 years, 15 grams remain. So, we can set up the equation:

15 = 60(1 - [?])

Now, let's solve for the missing number. First, divide both sides of the equation by 60:

15 / 60 = 1 - [?]

Simplify the left side:

1/4 = 1 - [?]

Now, isolate the term with the missing number by subtracting 1 from both sides:

1/4 - 1 = -[?]

Simplify the left side:

-3/4 = -[?]

Finally, multiply both sides by -1 to solve for the missing number:

3/4 = [?]

So, the missing number in the equation is 3/4 or 0.75. This number represents the fraction of the substance that decays in each half-life, expressed as a decimal. The equation highlights another way to represent exponential decay, emphasizing the fraction of the substance that is lost over time.

This equation format is particularly useful in certain contexts, such as modeling decay processes where the decay rate is expressed as a fraction or percentage. It also provides a different perspective on the decay process, focusing on the amount that is lost rather than the amount that remains. Understanding both forms of the decay equation is beneficial for a comprehensive understanding of radioactive decay.

The problem also presents another equation: Remaining Amount = I(1-r)^t. This is yet another way to express the exponential decay formula, where:

• I represents the initial amount
• r represents the decay rate per time period
• t represents the number of time periods

In this context, we need to find the value of 'r', the decay rate. We already know that the Remaining Amount after 6 years is 15 grams, the Initial Amount (I) is 60 grams, and the time (t) can be expressed in terms of half-lives. Since 6 years is two half-lives (6 years / 3 years per half-life = 2), we can substitute these values into the equation:

15 = 60(1 - r)^2

Now, let's solve for 'r'. First, divide both sides by 60:

15 / 60 = (1 - r)^2

Simplify the left side:

1/4 = (1 - r)^2

Now, take the square root of both sides:

√(1/4) = √(1 - r)^2

1/2 = |1 - r|

This gives us two possible equations:

1/2 = 1 - r or -1/2 = 1 - r

Let's solve each equation separately:

For 1/2 = 1 - r:

r = 1 - 1/2

r = 1/2

For -1/2 = 1 - r:

r = 1 + 1/2

r = 3/2

Since the decay rate 'r' must be between 0 and 1 (representing a fraction of decay), the valid solution is r = 1/2 or 0.5. This means that in each half-life, half of the substance decays, which aligns with the definition of half-life. This equation format provides a more general way to represent exponential decay, applicable to various scenarios where a quantity decreases at a constant rate over time.

The use of 'r' as the decay rate is common in many fields, including finance, population dynamics, and pharmacology. Understanding this form of the equation allows for a broader application of the half-life concept beyond just radioactive decay. It emphasizes the rate at which a quantity decreases, providing a different perspective compared to the half-life itself.

In conclusion, we have explored the concept of half-life and its application in calculating the remaining amount of a radioactive substance after a certain time. We used the formula Remaining Amount = Initial Amount * (1/2)^(Time / Half-life) to determine that 15 grams of the substance would remain after 6 years, starting with an initial amount of 60 grams and a half-life of 3 years.

We also solved for the missing number in the equation Remaining Amount = 60(1 - [?]), finding it to be 3/4 or 0.75, representing the fraction of the substance that decays in each half-life. Furthermore, we used the equation Remaining Amount = I(1-r)^t to calculate the decay rate 'r', which we found to be 0.5, consistent with the half-life concept.

These calculations demonstrate the importance of understanding half-life and exponential decay in various scientific and practical applications. From nuclear medicine to archaeology, the ability to predict the amount of radioactive material remaining over time is crucial. The different equation formats we explored provide various perspectives on the decay process, each with its own advantages and applications.

Mastering these calculations requires a solid understanding of exponential functions and the concept of half-life. By applying the formulas and understanding their underlying principles, one can accurately predict the behavior of radioactive substances and utilize this knowledge in diverse fields. The examples discussed in this article serve as a foundation for further exploration of radioactive decay and its applications in science and technology.