Question

The half-life of a radioactive element with mass number 234 g is 2.5 x 105 years. How long after the isolation of a sample of this isotope will only one-six of the original mass be left?

Answer 1

Answer on Question #67069 - Chemistry - Physical Chemistry

Question:

The half-life of a radioactive element with mass number $234\mathrm{g}$ is $2.5 \times 10^{5}$ years. How long after the isolation of a sample of this isotope will only one-six of the original mass be left?

Solution:

The reactions of radioactive decay are related to first-order reactions.

For a first-order reaction, the half-life is defined as: $t_{1/2} = \frac{\ln 2}{k}$.

The kinetic equation for the first-order reaction has the form: $k = \frac{1}{t} \cdot \ln \frac{[X]_0}{[X]}$,

$$t = \frac {\ln \frac {[ X ] _ {0}}{[ X ]}}{k} = \frac {\ln \frac {[ X ] _ {0}}{[ X ]} t _ {1 / 2}}{\ln 2} = \frac {\ln \frac {2 3 4}{3 9} \cdot 2 . 5 \cdot 1 0 ^ {5}}{\ln 2} = \frac {4 4 7 9 3 9 . 8 7}{0 . 6 9} = 6 4 9 1 8 8 \text{ years} \approx 6. 5 \cdot 1 0 ^ {5} \text{ years}.$$

Answer: $6.5 \times 10^{5}$ years.

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