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Zorluk: KolayRadioactive Decay Law and Half-life

A radioactive isotope has a half-life of 10 hours10\text{ hours}. If a sample initially contains 64 g64\text{ g} of the isotope, what mass of the isotope has decayed after an elapsed time of 30 hours30\text{ hours}?

  1. A
    8 g8\text{ g}
  2. B
    16 g16\text{ g}
  3. C
    48 g48\text{ g}
  4. 56 g56\text{ g}Cevap

Cevap

The mass of the isotope that has decayed after 30 hours30\text{ hours} is 56 g56\text{ g}.
The correct answer is 56 g56\text{ g}. With a half-life of 10 hours10\text{ hours}, an elapsed time of 30 hours30\text{ hours} represents 33 half-lives. After 33 half-lives, the mass remaining undecayed is 64 g×(1/2)3=8 g64\text{ g} \times (1/2)^3 = 8\text{ g}. Therefore, the mass that has decayed is 64 g−8 g=56 g64\text{ g} - 8\text{ g} = 56\text{ g}.

Adım Adım Çözüm

1
Calculate the number of half-lives (nn) that have elapsed.
n=tT1/2=30 hours10 hours=3 half-livesn = \frac{t}{T_{1/2}} = \frac{30\text{ hours}}{10\text{ hours}} = 3\text{ half-lives}
Determining how many half-life intervals occur during the total elapsed time.
2
Calculate the mass of the sample remaining undecayed (NN).
N=N0(12)n=64 g×(12)3=64 g×18=8 gN = N_0 \left(\frac{1}{2}\right)^n = 64\text{ g} \times \left(\frac{1}{2}\right)^3 = 64\text{ g} \times \frac{1}{8} = 8\text{ g}
Using the radioactive decay formula to find the remaining undecayed mass.
3
Calculate the mass of the sample that has decayed (NdecayedN_{\text{decayed}}).
Ndecayed=N0−N=64 g−8 g=56 gN_{\text{decayed}} = N_0 - N = 64\text{ g} - 8\text{ g} = 56\text{ g}
Subtracting the remaining mass from the initial mass to find the amount decayed.

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Radioactive Decay Law and Half-life
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