Examkin

Question

Difficulty: MediumRadioactive Decay Law and Half-life

A sample of a radioactive element has a half-life of 3 hours3\text{ hours}. If the initial mass of the sample is 48 g48\text{ g}, what mass of the element has decayed after a period of 9 hours9\text{ hours}?

  1. 42 g42\text{ g}Answer
  2. B
    6 g6\text{ g}
  3. C
    12 g12\text{ g}
  4. D
    24 g24\text{ g}

Answer

The mass of the element that has decayed after 9 hours is 42 g42\text{ g}.
In a period of 9 hours, exactly 3 half-lives pass (9/3=39/3 = 3). The fraction of the radioactive substance remaining is (1/2)3=1/8(1/2)^3 = 1/8, which corresponds to 48 g×(1/8)=6 g48\text{ g} \times (1/8) = 6\text{ g}. Therefore, the mass that has decayed is 48 g−6 g=42 g48\text{ g} - 6\text{ g} = 42\text{ g}.

Step-by-Step Solution

1
Determine the number of half-lives (nn) that have elapsed.
n=tT1/2=9 hours3 hours=3n = \frac{t}{T_{1/2}} = \frac{9\text{ hours}}{3\text{ hours}} = 3
Dividing the total elapsed time by the half-life period gives the number of decay cycles.
2
Calculate the mass of the sample remaining (NN) after 3 half-lives.
N=N0(12)n=48 g×(12)3=48 g×18=6 gN = N_0 \left(\frac{1}{2}\right)^n = 48\text{ g} \times \left(\frac{1}{2}\right)^3 = 48\text{ g} \times \frac{1}{8} = 6\text{ g}
The remaining quantity decreases by a factor of 2 for each half-life.
3
Calculate the mass of the sample that has decayed.
Mass decayed=N0−N=48 g−6 g=42 g\text{Mass decayed} = N_0 - N = 48\text{ g} - 6\text{ g} = 42\text{ g}
The amount decayed is equal to the initial mass minus the remaining mass.

Key Concept

Radioactive Decay Law and Half-life
Estimated Time:1m 30s
Rate this question

Topics

Radioactive Decay Law and Half-life

More questions from this topic