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Difficulty: MediumRadioactive Decay Law and Half-life

A sample of a radioactive nuclide with a half-life of 6 hours6\text{ hours} initially has a mass of 200 mg200\text{ mg}. What mass of the nuclide has decayed after an elapsed time of 24 hours24\text{ hours}?

  1. 187.5 mg187.5\text{ mg}Answer
  2. B
    12.5 mg12.5\text{ mg}
  3. C
    50.0 mg50.0\text{ mg}
  4. D
    150.0 mg150.0\text{ mg}

Answer

The mass of the nuclide that has decayed after 24 hours24\text{ hours} is 187.5 mg187.5\text{ mg}.
After 24 hours24\text{ hours}, exactly 44 half-lives (24/6=424 / 6 = 4) have passed. The remaining radioactive mass is 200 mg/24=12.5 mg200\text{ mg} / 2^4 = 12.5\text{ mg}. Subtracting the remaining mass from the initial mass (200 mg−12.5 mg200\text{ mg} - 12.5\text{ mg}) yields 187.5 mg187.5\text{ mg} as the decayed mass.

Step-by-Step Solution

1
Determine the number of half-lives (nn) that have elapsed.
n=tT1/2=24 hours6 hours=4 half-livesn = \frac{t}{T_{1/2}} = \frac{24\text{ hours}}{6\text{ hours}} = 4\text{ half-lives}
The total elapsed time divided by the half-life duration gives the total number of decay cycles.
2
Calculate the remaining mass (NN) after 44 half-lives.
N=N0(12)n=200 mg×(12)4=200 mg16=12.5 mgN = N_0 \left(\frac{1}{2}\right)^n = 200\text{ mg} \times \left(\frac{1}{2}\right)^4 = \frac{200\text{ mg}}{16} = 12.5\text{ mg}
The radioactive decay law states that after nn half-lives, the initial quantity reduces by a factor of 2n2^n.
3
Calculate the mass of the nuclide that has decayed (NdecayedN_{\text{decayed}}).
Ndecayed=N0−N=200 mg−12.5 mg=187.5 mgN_{\text{decayed}} = N_0 - N = 200\text{ mg} - 12.5\text{ mg} = 187.5\text{ mg}
The amount decayed is the initial mass minus the mass remaining.

Key Concept

Radioactive Decay Law and Half-life Calculation
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Topics

Radioactive Decay Law and Half-life

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