If 30% of a sample of radioactive isotope decays in 24 hrs, what is the half-life?
How do i know which order to use? and do I convert hr to mins or seconds?
Pls help for my exam tomorrrow
ALL radioactive decay problems are first order so you use ln(No/N) = kt.
If you want the half life in hours, leave it as is. If you want it in min or seconds, then convert the 24 hours to some other unit. Since the problem doesn't specify, I would leave it as 24 hours.
In n=-kt +in,no
To find the half-life of the radioactive isotope, you can use the formula:
t₁/₂ = (ln(2)) / λ
Where:
t₁/₂ is the half-life
ln(2) is the natural logarithm of 2 (approximately 0.693)
λ is the decay constant
In this case, you are given that 30% of the sample decays in 24 hours. To find the decay constant (λ), you need to convert the given time into a rate.
The formula to calculate the decay constant is:
λ = (ln(N₀/N)) / t
Where:
N₀ is the initial quantity of the sample
N is the quantity of the sample after time t
Since you are given that 30% of the sample decays in 24 hours, you can assume that N is 70% (100% - 30%) of the initial quantity, N₀, and t is 24 hours.
Now, you can substitute these values into the decay constant formula:
λ = (ln(N₀/N)) / t
λ = (ln(100%/70%)) / 24
Simplifying further:
λ = (ln(1.43)) / 24
Once you have obtained the value of λ, you can substitute it into the half-life formula:
t₁/₂ = (ln(2)) / λ
t₁/₂ = (ln(2)) / ((ln(1.43)) / 24)
Calculate this expression to find the half-life of the radioactive isotope.
As for converting hours to minutes or seconds, it depends on the units in which the half-life is generally expressed. Typically, half-lives are expressed in minutes or seconds for faster decaying isotopes or in hours for slower ones. However, you should refer to the specific instructions or conventions given in your exam to determine the appropriate unit of time to use.
Remember to double-check any calculations and ensure that units are consistent throughout your work. Good luck with your exam!