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Question
Disintegration rate of a sample is 1010 per hour at 20 hours from the start. It reduces to 6.3 x 109 per hour after 30 hours. Calculate its half-life and the initial number of radioactive atoms in the sample.
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Solution
Data: A(t1) = 1010 per hour, where t1 = 20 h,
A(t1) = 6.3 × 1010 per hour, where t2 = 30 h
A(t) = A0e-λt ∴ A(t1) = `"A"_0"e"^(-λ"t"_1)` and
A(t2) = `"A"_0"e"^(-λ"t"_2)`
∴ `("A"("t"_1))/("A"("t"_2)) = ("e"^(-lambda"t"_1)/"e"^(-lambda"t"_2)) = "e"^(lambda("t"_2 - "t"_1))`
∴ `10^10/(6.3 xx 10^9) = "e"^(lambda(30 - 20)) = "e"^(10lambda)`
∴ 1.587 = e10λ
∴ 10λ = 2.303 log10(1.587)
∴ λ = (0.2303)(0.2007) = 0.04622 per hour
The half life of the material, T1/2 = `0.693/lambda = 0.693/0.04622`
= 14.99 hours
Now, `"A"_0 = "A"("t"_1)"e"^(lambda"t"_1) = 10^10"e"^((0.04622)(20))`
= `10^10 "e"^0.9244`
Let x = `"e"^0.9244`
∴ 2.303 log10x = 0.9244
∴ log10x = `0.9244/2.303 = 0.4014`
∴ x = antilog 0.4014 = 2.52
∴ A0 = 2.52 x 1010 per hour
Now A0 = N0λ
∴ `"N"_0 = "A"_0/lambda = (2.52 xx 10^10)/0.04622`
= 5.452 × 1011
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