In a radioactive sample, _ 10 ^a K nuclei either decay into stabl… - Vidyadip

MCQ

In a radioactive sample, ${ }_{10}^a K$ nuclei either decay into stable ${ }_{20}^{* 0} Ca$ nuclei with decay constant $4.5 \times 10^{-10}$ per year or into stable ${ }_{18}^{40}$ Ar muclei with decay constant $0.5 \times 10^{-10}$ per year. Given that in this sample all the stable ${ }_{20}^{\infty 0} Ca$ and ${ }_{15}^{20} Ar$ nuclei are produced by the ${ }_{19}^{* 0} K$ muclei only. In time $t \times 10^{\circ}$ years, if the ratio of the sum of stable ${ }_{30}^{40} Ca$ and ${ }_{15} \operatorname{An}$ nuclei to the radioactive ${ }_{19} K$ muclei is $99$ , the ralue of $t$ will be : [Given $\ln 10=2.3]$

✓
$9.2$
B
$1.15$
C
$4.6$
D
$2.3$

✓

Answer

Correct option: A.

$9.2$

a
$\lambda=\lambda_1+\lambda_2=5 \times 10^{-10} \text { per year }$

$N = N _0 e ^{-\lambda . t}$

$N _0- N = N _{\text {sable }}$

$N = N _{\text {rrdiooctire }}$

$\frac{ N _0}{ N }-1=99$

$\frac{ N _0}{ N }=100$

$\frac{ N }{ N _0}= e ^{-\lambda . t }=\frac{1}{100}$

$\Rightarrow \lambda t =2 \text { en10 }$

$=4.6$

$t =9.2 \times 10^9 \text { years }$

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