Problem 26 - Essential Calculus

Question

Complete the table for the radioactive isotope. $$\begin{array}{llll} & & & \text { Amount } & \text { Amount } \\\& \text { Half-Life } & \text { Initial } & \text { After } & \text { After } \\ \text { Isotope } & \text { (in years) } & \text { Quantity } & \text { 1000 Years } & 10,000 \text { Years } \\\\\end{array}$$ $$ { }^{226} \mathrm{Ra} \quad 1599 \quad \quad 1.5 \mathrm{~g} $$

Step-by-Step Solution

Answer

The quantity of the \( ^{226}Ra \) isotope after 1000 years is approximately 1.08g, and after 10,000 years it is approximately 0.0234g.

Step 1: Calculate the amount after 1000 years

The half-life of \( ^{226}Ra \) is 1599 years. This means after 1599 years, half of the initial amount will remain. To find out how much of the radiactive isotope is left after 1000 years, we need to keep in mind that 1000 years is less than one half-life of the isotope. Hence, by the end of 1000 years, not even half of the isotope has decayed yet. To estimate the value, one could use a formula for the decay of radioactive isotopes: \( N = N0 * e^{(-0.693 * t / T)} \), where: - \( N \) is the final amount, - \( N0 \) is the initial amount, - \( t \) is the time evolved, - \( T \) is the half-life of the isotope.So, \( N = 1.5g * e^{(-0.693 * 1000 / 1599)} \)

Step 2: Calculate the amount after 10000 years

Now, let's calculate the amount after 10000 years. 10000 years is more than six half-lives of \( ^{226}Ra \). Therefore, we can simply divide the initial amount by 2 for each observed half-life. As there are six half-lives in 10000 years, we will divide the initial quantity by 2, six times. It simplifies calculations as compared to earlier step.

Step 3: Complete the table for the isotope

Finally, we complete the table with the calculated values.

Key Concepts

Half-Life Calculation

When discussing radioactive decay, the concept of half-life is fundamental. The half-life of an isotope is the time it takes for half of the radioactive material to decay. In simpler terms, if you start with a certain amount of radioactive substance, after one half-life, only half of the original amount will remain.

To calculate the amount of an isotope that remains after a certain period, knowing the half-life is crucial. For the isotope \(^ {226} \text{Ra}\) in this exercise, the half-life is 1599 years. This means if you start with 1.5 grams, after 1599 years, you would have 0.75 grams left.

Half-life calculations help predict the behavior of isotopes over time and are essential in fields such as nuclear medicine, archeology, and environmental science. Always remember that the shorter the half-life, the faster the isotope decays, and the faster it reaches safe levels.

Exponential Decay Formula

Exponential decay is a mathematical concept used to describe how quantities decrease over time. In radioactive decay, it is used to estimate the amount of substance remaining as isotopes decay.

The formula used for calculating this decay is \( N = N_0 \cdot e^{(-0.693 \cdot t / T)} \), where:

• \( N \) is the remaining amount of the substance.
• \( N_0 \) is the initial quantity.
• \( t \) is the time that has passed.
• \( T \) represents the half-life of the isotope.

Using the provided formula, you can calculate how much of a radioactive isotope remains after 1000 years or any given period. This formula accounts for the gradual decrease instead of assuming a strict half-life step, offering a more precise decay estimation when time periods do not align perfectly with multiple half-lives.

For instance, after 1000 years (less than one half-life), the decay is calculated as \( N = 1.5 \cdot e^{(-0.693 \cdot 1000 / 1599)} \). Exponential decay approaches allow us to understand and predict how concentrations of isotopes change over time more accurately.

Isotope Table Completion

Completing isotope tables involves calculating and filling in the values that depict the changes in amount over designated periods. It helps track the decay progress of isotopic material like \(^ {226} \text{Ra}\).

For our exercise, the initial quantity is provided as 1.5 grams and the half-life of \(^ {226} \text{Ra}\) is known to be 1599 years. By using both the exponential decay formula and understanding the concept of multiple half-lives, one can complete the isotope table.

For instance:

• After 1000 years, using the exponential decay formula if the exact calculation is needed, the value is found.
• After 10,000 years, more than six half-lives, the approximation is done through repeated halving: divide 1.5 by 2 six times.

The process of table completion is iterative, requiring careful calculation to ensure each value reflects the decay correctly. This procedure aids in building a comprehensive understanding of radioactive decay processes, forming the basis for educated predictions and safety evaluations in handling isotopes.