Question

The rate of radioactive decay is first-order, so we can use our knowledge of first-order kinetics to study it. The half-life for the decay of fermium $-252$ is about $1$ day. The half-life for the decay of ruthenium $-106$ is about $1$ year.
Which of the following statements is the best conclusion from this information?

• A. The rate constant for the decay of fermium $-252$ is equal to the rate constant for the decay of ruthenium$-106$.
• B. The rate constant for the decay of fermium $-252$ is less to the rate constant for the decay of ruthenium$-106$.
• C. The rate constant for the decay of fermium $-252$ is greater than the rate constant for the decay of ruthenium$-106$.
• D. It is not possible to make any comparison of the rate constant for the decay of these two isotopes.

Answer 1

The correct option is C The rate constant for the decay of fermium $-252$ is greater than the rate constant for the decay of ruthenium$-106$.

For a first order reaction,

$t_{1/2}=\frac{0.693}{K}$

For fermium,$\left(t_{1/2}\right)=\frac{0.693}{K_{fermium}}\therefore K_{fermium}=\frac{0.693}{1}{day}^{-1}\therefore K_{fermium}=0693{day}^{-1}$

For Ruthenium, ${\left(t_{1/2}\right)}_{ruthenium}=\frac{0.693}{K_{ruthenium}}1\times 365days=\frac{0.693}{K_{ruthenium}}\therefore K_{ruthenium}=\frac{0.693}{365}=1.9\times {10}^{-3}{day}^{-1}\therefore K_{fermium}>K_{ruthenium}$