Question

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

Short 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-by-step solution

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, \( N = 1.5g * e^{(-0.693 * 1000 / 1599)} \)

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.

Complete the table for the isotope

Finally, we complete the table with the calculated values.