First slide

Radio activity

Question

For a substance the average life for $α$ - emission is 1620 years and for $β$ -emission is 405 years. After how much time the 1/4 of the material remains after $α and β$ emission?

Moderate

A

1500 years
B

300 years
C

449 years
D

810 years

Solution

$λ_{α} = \frac{1}{1620} per year and λ_{β} = \frac{1}{405}$ per year and it is given that the fraction of the remained activity $\frac{A}{A_{0}} = \frac{1}{4}$
Total decay constant
$λ = λ_{α} + λ_{β} = \frac{1}{1620} + \frac{1}{405} = \frac{1}{324} per year We know that A = A_{0} e^{- λt} \Rightarrow t = \frac{1}{λ} \log_{e} \frac{A_{0}}{A}$
$\Rightarrow t = \frac{1}{λ} \log_{e} 4 = \frac{2}{λ} \log_{e} 2 = 324 x 2 x 0.693 = 449 years .$

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