A radioactive element x has half life of 30 days and will decay to become another new element Y..
a) calculate decay constant
b) how long it will take for 90% of the no.Of atom in radioisotope x to decay?
A radioactive waste products of a reactor has half life of 250 years
A) determine the decay constant ,what is its unit and explain what it is?
B) what fraction (%) of a given sample of this product will remain after 1000 years?
a) To calculate the decay constant (λ) of a radioactive element, we can use the formula:
λ = ln(2) / T
Where:
λ is the decay constant
ln(2) is the natural logarithm of 2 (approximately 0.693)
T is the half-life of the radioactive element
In this case, the half-life (T) of element X is 30 days. Substituting this value into the formula:
λ = ln(2) / 30
Using a scientific calculator or software, we can calculate:
λ ≈ 0.0231 (rounded to four decimal places)
So, the decay constant of element X is approximately 0.0231.
b) To determine how long it will take for 90% of the atoms in radioisotope X to decay, we can use the formula for radioactive decay:
N(t) = N₀ * e^(-λt)
Where:
N(t) is the remaining number of atoms at time t
N₀ is the initial number of atoms
e is the base of the natural logarithm (approximately 2.718)
λ is the decay constant
t is the time elapsed
In this case, we want to find the time (t) when 90% of the atoms decay, so N(t) will be 10% (or 0.1) of the initial number of atoms. Let's assume the initial number of atoms is N₀.
0.1 * N₀ = N₀ * e^(-λt)
Cancelling N₀ from both sides:
0.1 = e^(-λt)
Taking the natural logarithm of both sides:
ln(0.1) = -λt
Solving for t:
t = - (ln(0.1) / λ)
Substituting the value of λ (0.0231) we calculated earlier:
t ≈ - (ln(0.1) / 0.0231)
Using a scientific calculator or software, we can calculate:
t ≈ 74.06 (rounded to two decimal places)
Therefore, it will take approximately 74.06 days for 90% of the atoms in radioisotope X to decay.