Question

An unknown radioactive element decays into non-radioactive substances. In 340
days the radioactivity of a sample decreases by 61 percent.

(a) What is the half-life of the element?
half-life: ? (days)

(b) How long will it take for a sample of 100 mg to decay to 81 mg?
time needed: ? (days)

Answer 1

(a) The half-life of the unknown radioactive element is 250.28 days.

(b) The time taken for a sample of 100 mg to decay to 81 mg is 76.1 days.

Half life of the  unknown radioactive element

N(t) = N₀(0.5)^t/h

where;

• t is time of decay
• h is half life
• N₀ is initial mass
• N(t) remaining mass at time, t

in 340 days; N(340) = N₀(0.39);

1 - 0.61 = 0.39

N(340) = N₀(0.5)^340/h

N(340)/N₀ = 0.5^340/h

0.39 = 0.5^340/h

log(0.39) = 340/h x log(0.5)

log(0.39) /log(0.5) = 340/h

1.358 = 340/h

h = 340/1.358

h = 250.28 days

Time taken for the sample to decay 81 mg

81 = 100(0.5)^t/250.28

81/100 = (0.5)^t/250.28

0.81 = (0.5)^t/250.28

log(0.81) = t/250.28 x log(0.5)

log(0.81) / log(0.5) = t/250.28

0.304 = t/250.28

0.304(250.28) = t

76.1 days = t

Thus, the half-life of the unknown radioactive element is 250.28 days and the time taken for a sample of 100 mg to decay to 81 mg is 76.1 days.

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