A) \[1.54\times {{10}^{-10}}\] per year
B) \[2.54\times {{10}^{-10}}\] per year
C) \[0.54\times {{10}^{-10}}\] per year
D) none of these
Correct Answer: A
Solution :
Let n is number of half-lives, the \[\frac{A}{{{A}_{0}}}={{\left( \frac{1}{2} \right)}^{n}}\] or \[\left( \frac{1}{32} \right)={{\left( \frac{1}{2} \right)}^{n}}\] or \[{{\left( \frac{1}{2} \right)}^{5}}={{\left( \frac{1}{2} \right)}^{n}}\] So, \[n=5\] So, number of half lives n = 5 or \[n=\frac{t}{{{T}_{12}}}\] or \[{{T}_{1/2}}=\frac{t}{n}=\frac{22.5\times {{10}^{9}}}{5}\] \[=4.5\times {{10}^{9}}\] year Hence, disintegration constant is given by \[\lambda =\frac{0.693}{{{T}_{12}}}=\frac{0.693}{4.5\times {{10}^{9}}}\] \[=1.54\times {{10}^{-10}}\] per year
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