Based on their half life, which of the following radioisotopes is most dangerous to handle?

Question

Which of the following does not form a chelate?

Answer 1

Use the idea that activity is proportional to $1/t_{1/2}$.

The decay constant is $\lambda = \dfrac{0.693}{t_{1/2}}$, and activity $A = \lambda N$, so a smaller half life directly means a larger decay constant and higher activity for the same number of atoms.
Ranking the four half lives from shortest to longest: $0.01$ minute, $13$ days, $100$ years, $3$ billion years.
The shortest half life, $0.01$ minute, gives the largest $\lambda$, hence the highest activity and the most intense radiation output per second.
An isotope releasing intense radiation almost instantly is the hardest to handle safely, since exposure happens in a burst rather than spread over years.
So the isotope with half life $0.01$ minute, option C, is the most dangerous to handle.

Answer 2

Use the idea that activity is proportional to $1/t_{1/2}$.

The decay constant is $\lambda = \dfrac{0.693}{t_{1/2}}$, and activity $A = \lambda N$, so a smaller half life directly means a larger decay constant and higher activity for the same number of atoms.
Ranking the four half lives from shortest to longest: $0.01$ minute, $13$ days, $100$ years, $3$ billion years.
The shortest half life, $0.01$ minute, gives the largest $\lambda$, hence the highest activity and the most intense radiation output per second.
An isotope releasing intense radiation almost instantly is the hardest to handle safely, since exposure happens in a burst rather than spread over years.
So the isotope with half life $0.01$ minute, option C, is the most dangerous to handle.

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