First slide

Radio activity

Question

A and B are two radioactive substances whose half lives are 1 and 2 years respectively. Initially 10 gm of A and 1gm of B is taken. The time (approximate) after which they will have same quantity remaining is

Moderate

A

6.62 years
B

5 years
C

3.2 years
D

7 years

Solution

$N = N_{0} {(\frac{1}{2})}^{t / T_{1 / 2}} \Rightarrow N_{A} = 10 {(\frac{1}{2})}^{t / 1} and N_{B} = 1 {(\frac{1}{2})}^{t / 2}$
$Given N_{A} = N_{B} \Rightarrow 10 {(\frac{1}{2})}^{t} = {(\frac{1}{2})}^{t / 2}$
$\Rightarrow 10 = {(\frac{1}{2})}^{- t / 2} \Rightarrow 10 = 2^{t / 2} .$ Taking log both the sides.
$\log_{10} = \frac{t}{2} \log_{10} 2 \Rightarrow 1 = \frac{t}{2} x 0.3010 \Rightarrow t = 6.62 years .$

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