A newly prepared radioisotope has activity 3.7x10^11 Bq. After 4 hours, its activity is reduced to 2.96x10^11.calculate
a) decay constant
b) its half life
To calculate the decay constant and half-life of a radioactive substance, we can use the decay equation:
Activity(t) = Activity(0) * e^(-λt)
Where:
- Activity(t) is the activity at time t
- Activity(0) is the initial activity
- λ (lambda) is the decay constant
- t is the time
Let's solve for the decay constant:
a) Decay Constant (λ):
Using the given information, we have:
Activity(0) = 3.7x10^11 Bq (initial activity)
Activity(4) = 2.96x10^11 Bq (activity after 4 hours)
t = 4 hours
We can rearrange the equation to solve for λ:
λ = -1/t * ln(Activity(t) / Activity(0))
Substituting the values:
λ = -1/4 * ln(2.96x10^11 Bq / 3.7x10^11 Bq)
λ = -1/4 * ln(0.8)
Using a calculator, we find:
λ ≈ 0.277 Bq/hr
b) Half-life (T):
The half-life (T) is the time it takes for half of the radioactive substance to decay. It is related to the decay constant by the equation:
T = ln(2) / λ
Substituting the value of λ we found:
T ≈ ln(2) / 0.277 Bq/hr
Using a calculator, we find:
T ≈ 2.510 hours
Therefore, the decay constant is approximately 0.277 Bq/hr, and the half-life is approximately 2.510 hours.