Question

Complete the table for the radioactive isotope. $$\begin{array}{llll} & & & \text { Amount } & \text { Amount } \\\& \text { Half-Life } & \text { Initial } & \text { After } & \text { After } \\ \text { Isotope } & \text { (in years) } & \text { Quantity } & \text { 1000 Years } & 10,000 \text { Years } \\\\\end{array}$$ $$ \begin{array}{llll} { }^{14} \mathrm{C} & 5715 & 5 \mathrm{~g} \end{array} $$

Short answer

After 1000 years, about 4.323 grams of Carbon-14 remains and after 10000 years, approximately 0.156 grams remains in the sample.

Step-by-step solution

Understand the half-life

The half-life of Carbon-14 has been given as 5715 years. This means, after every 5715 years, only half of the initial amount of this isotope remains while the rest decays.

Calculate the amount after 1000 years

The half-life of Carbon-14 is greater than 1000 years, thus after 1000 years, more than half of its initial quantity will still remain. The exact quantity can be calculated using the formula for exponential decay which is \(M = M_0 \times (1/2)^{t/T}\), where \(M\) is the remaining amount after time \(t\), \(M_0\) is the initial amount, and \(T\) is the half-life. Substituting the given values: \(M = 5 \times (1/2)^{1000/5715}\), we find that the remaining quantity after 1000 years is approximately 4.323 grams.

Calculate the amount after 10000 years

Using the same formula as in step 2, but this time with \(t = 10000\) years, we get \(M = 5 \times (1/2)^{10000/5715}\). The remaining quantity is much less due to the long time period and approximately equals 0.156 grams.