100g of radioactive element X are placed in a sealed box. The half-life of this isotope of X is 2 days. After 4 days have passed, what is the mass of element X in the box?
b) If the box is completely evacuated (perfect vacuum), then N atoms of X are added. After 4 days have passed, how many atoms are in the box?
Please help!! Thanks
4 days is 2 half-lives, so 1/4 is left
For b) would the answer be the same number of N atoms of X, because the box is in a perfect vacuum?
the vacuum does not matter. After 4 days, there will be 1/4 of the starting amount left.
I guess the vacuum is provided to be sure there are no residual atoms of X hiding there.
To answer the questions, we need to understand the concept of half-life and use the decay formula. The half-life of a radioactive isotope is the time it takes for half of the original sample to decay.
a) The half-life of element X is 2 days, which means that after every 2 days, half of the remaining mass will decay. Since 4 days have passed, two half-lives have occurred.
To find the mass of element X remaining after 4 days, we can use the following formula:
Final mass = Initial mass * (1/2)^(number of half-lives)
In this case, the initial mass is 100g. As two half-lives have occurred, the number of half-lives is 2. Let's substitute these values into the formula:
Final mass = 100g * (1/2)^(2)
Final mass = 100g * (1/4)
Final mass = 25g
Therefore, the mass of element X in the box after 4 days would be 25 grams.
b) In a perfect vacuum, the number of atoms present in the box will remain constant, as there are no external influences affecting the decay of the radioactive material.
After 4 days, we need to determine the number of atoms in the box. To do this, we'll use the following formula:
Final number of atoms = Initial number of atoms + (N * e^(-λt))
In this case, N represents the number of additional atoms added, and λ is the decay constant. The decay constant is related to the half-life as follows: λ = ln(2) / half-life. As the half-life is 2 days, we can calculate the decay constant:
λ = ln(2) / 2
Substituting this value and the time (4 days) and the additional atoms (N) into the formula, we can calculate the final number of atoms.
Final number of atoms = Initial number of atoms + (N * e^(-(ln(2)/2)*4))
Since the initial number of atoms is not given in the question, we can only calculate the change in the number of atoms.
Final change = N * e^(-(ln(2)/2)*4)
Therefore, the number of atoms in the box after 4 days would be the sum of the initial number of atoms (if provided) and the calculated final change.