Radioactive isotopes are unstable elements that emit particles and energy from their nucleus to become stable. The half-life of a radioisotope is the amount of time it takes for half of the amount of the isotope to decay. As the original, or parent, isotope decays, a new, or daughter, isotope is formed.
The half-lives of five radioactive isotopes are shown in the table.
Radioactive Isotope Half-Life
Phosphorus-32 14 days
Chromium-51 28 days
Iodine-125 60 days
Cesium-137 30 years
Radium-226 1,600 years
32.
Use the drop-down menu to identify the number of half-lives that the sample has undergone.
A 20-gram sample of radium-226 undergoes nuclear decay. If 2.5 grams of the original isotope is remaining after 4,800 years, then the sample has undergone (Select) half-lives.
To determine the number of half-lives that the sample has undergone, we can divide the remaining mass by the initial mass and compare it to the fraction present after one half-life.
The formula for radioactive decay is given by:
A = A₀ * (1/2)^(t / T)
Where:
A is the remaining amount of the isotope,
A₀ is the initial amount of the isotope,
t is the time passed,
T is the half-life of the isotope.
From the given information, we have:
A = 2.5 grams (remaining amount)
A₀ = 20 grams (initial amount)
T = 1,600 years (half-life)
We can rearrange the formula to solve for t:
t = T * log₂(A / A₀)
t = 1,600 years * log₂(2.5 / 20)
t ≈ 1,600 years * (-2.3219)
t ≈ -3,715 years
Since time cannot be negative, we must have made an error in our calculations, potentially in the logarithm. Let's try again:
t = T * log₂(A / A₀)
t = 1,600 years * log₂(20 / 2.5)
t ≈ 1,600 years * (2.3219)
t ≈ 3,715 years
Now, we can calculate the number of half-lives:
Number of half-lives = t / T
Number of half-lives = 3,715 years / 1,600 years
Number of half-lives ≈ 2.32
Since we cannot have fractional half-lives, we would round down to the nearest whole number.
Therefore, the sample has undergone 2 half-lives.