Chapter 24: Q24.42 P (page 1104)

The half-life of radium- $226$ is $1.60 \times 10^{3}$ yr. How many hours will it take for a $2.50$ g sample to decay to the point where $0.185$ g of the isotope remains?

Short Answer

Expert verified

The time taken is $5.267 \times 10^{7} hrs$ _.

Step by step solution

Half-life

Half-life is the time taken by the radioactive substance to decay to half of its amount. The radioactive decay follows the first order kinetics.

Calculation

The half-life of radium226 is $t_{1/2} {=1.60×10}^{3} yr$

The initial amount of radium-226 is $N_{0} =2.50g$

The amount of radium-226 after time (t) is $N_{t} =0.185g$

We have to calculate the elapsed time in hours (t).

First, let us calculate the decay constant of radium226

$\begin{matrix} k= \frac{ln(2)}{t_{1/2}} \\ = \frac{0.693}{{1.60×10}^{3} yr} \\ {=4.33×10}^{-4} {yr}^{-1} \end{matrix}$

Therefore, the time elapsed is

$\begin{matrix} ln (\frac{N_{t}}{N_{0}}) =-kt \\ t= \frac{-ln (\frac{N_{t}}{N_{0}})}{k} \end{matrix}$

$\begin{matrix} t= \frac{-ln (\frac{0.185g}{2.50g})}{{4.33×10}^{-4} yr} \\ {=6.013×10}^{3} yr \\ {=6.013×10}^{3} yr× \frac{365d}{1yr} × \frac{24hrs}{1d} \\ {=5.267×10}^{7} hrs \end{matrix}$

Hence,It would take $5.267 \times 10^{7} hrs$

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The half-life of radium- $226$ is $1.60 \times 10^{3}$ yr. How many hours will it take for a $2.50$ g sample to decay to the point where $0.185$ g of the isotope remains?

Short Answer

Step by step solution

Half-life

Calculation

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The half-life of radium- 226is 1.60×103yr. How many hours will it take for a 2.50g sample to decay to the point where0.185 g of the isotope remains?

Short Answer

Step by step solution

Half-life

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Chemistry Textbooks

Kinetics

Chemical Reactions

Organic Chemistry

Chemistry Branches

Inorganic Chemistry

Making Measurements

Study anywhere. Anytime. Across all devices.

The half-life of radium- $226$ is $1.60 \times 10^{3}$ yr. How many hours will it take for a $2.50$ g sample to decay to the point where $0.185$ g of the isotope remains?