Radioactive waste is difficult to dispose of due to the long time periods needed before the radiation effectively disappears. The US protocol is to store the waste for 10 half-lives of the material after which the waste is no longer considered radioactive. After this time, very little of the original chemical remains. Assume you conducted an experiment with 1.00 g of carbon-14 that has a half-life of 5730 years. How much radioactive carbon-14 will remain when your waste is considered not radioactive? Report the answer in milligrams.
1/2^10 = ?
To determine how much radioactive carbon-14 remains when the waste is considered not radioactive, we first need to find out the number of half-lives that need to pass before the waste is no longer considered radioactive.
The US protocol states that the waste needs to be stored for 10 half-lives. Therefore, we need to calculate how many years correspond to 10 half-lives of carbon-14.
Since the half-life of carbon-14 is 5730 years, we can calculate the total number of years for 10 half-lives by multiplying the half-life by 10:
5730 years x 10 half-lives = 57,300 years
After 57,300 years, the waste will no longer be considered radioactive.
Now, let's calculate how much radioactive carbon-14 will remain after this time period.
To do this, we'll use the radioactive decay formula, which calculates the amount of remaining radioactive material based on the initial amount, the decay constant, and the time passed:
N_final = N_initial * (1/2)^(t / t_half)
Where:
N_final is the final amount of the radioactive material
N_initial is the initial amount of the radioactive material
t is the time passed
t_half is the half-life of the radioactive material
In this case, we know the initial amount (1.00 g of carbon-14) and the time passed (57,300 years). We also know the half-life of carbon-14 is 5730 years.
Substituting these values into the formula:
N_final = 1.00 g * (1/2)^(57,300 years / 5730 years)
N_final = 1.00 g * (0.5)^(10)
N_final ≈ 1.00 g * 0.09765625
N_final ≈ 0.09765625 g
To convert this to milligrams, we multiply by 1000:
0.09765625 g * 1000 = 97.65625 mg
Therefore, approximately 97.66 milligrams of radioactive carbon-14 will remain when your waste is considered not radioactive.