Calculate the half-lives of each of the followingradioisotopes.a It takes 180 days for 80 mg of iron-59 to decayto 5 mg.b It takes 2.1 × 10 12 years for 60 mg ofplatimum-190 to decay to 7.5 mg.
Question
Calculate the half-lives of each of the followingradioisotopes.a It takes 180 days for 80 mg of iron-59 to decayto 5 mg.b It takes 2.1 × 10 12 years for 60 mg ofplatimum-190 to decay to 7.5 mg.
Solution
To calculate the half-life of a radioactive substance, we use the formula:
t1/2 = (t * ln(2)) / ln(N0/N)
where:
- t1/2 is the half-life
- t is the time elapsed
- N0 is the initial quantity of the substance
- N is the final quantity of the substance
- ln is the natural logarithm
Let's calculate for each isotope:
a) Iron-59
- t = 180 days
- N0 = 80 mg
- N = 5 mg
t1/2 = (180 * ln(2)) / ln(80/5) = 39.3 days
b) Platinum-190
- t = 2.1 × 10^12 years
- N0 = 60 mg
- N = 7.5 mg
t1/2 = (2.1 × 10^12 * ln(2)) / ln(60/7.5) = 4.6 x 10^11 years
So, the half-life of Iron-59 is approximately 39.3 days and the half-life of Platinum-190 is approximately 4.6 x 10^11 years.
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