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Zorluk: ZorRadioactive Decay Law and Half-life

A sample of a radioactive substance has a decay constant of 0.0154 h−10.0154\text{ h}^{-1}. What is the elapsed time, in hours, required for 87.5%87.5\% of the original sample to decay? (Take ln⁡2=0.693\ln 2 = 0.693)

Cevap: 135 hours

Cevap

135 hours
First, calculate the half-life of the substance using the relation T1/2=ln⁡2λ=0.6930.0154 h−1=45 hoursT_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{0.0154\text{ h}^{-1}} = 45\text{ hours}. Since 87.5%87.5\% of the sample has decayed, the remaining fraction of the sample is 100%−87.5%=12.5%=18100\% - 87.5\% = 12.5\% = \frac{1}{8}. Expressing 18\frac{1}{8} as a power of 12\frac{1}{2} gives (12)3\left(\frac{1}{2}\right)^3, indicating that 33 half-lives have passed. The total elapsed time is therefore 3×45 hours=135 hours3 \times 45\text{ hours} = 135\text{ hours}.

Adım Adım Çözüm

1
Calculate the half-life from the given decay constant
Half-life T1/2=45 hoursT_{1/2} = 45\text{ hours}
The decay constant λ\lambda and half-life T1/2T_{1/2} are related by the formula T1/2=ln⁡2λT_{1/2} = \frac{\ln 2}{\lambda}
2
Find the remaining percentage and fraction of the sample
Remaining fraction is 12.5%12.5\% or 18\frac{1}{8}
Radioactive decay equations use the undecayed remaining amount, which is 100%−87.5%=12.5%100\% - 87.5\% = 12.5\%
3
Calculate the number of half-lives elapsed
Number of half-lives n=3n = 3
Since (12)n=18\left(\frac{1}{2}\right)^n = \frac{1}{8}, solving for nn gives n=3n = 3
4
Compute the total elapsed time
Total elapsed time t=135 hourst = 135\text{ hours}
Total elapsed time is the product of the number of half-lives and the half-life duration (3×45 hours3 \times 45\text{ hours})

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Radioactive Decay Law and Half-life Relationship
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Radioactive Decay Law and Half-life

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