Class 12 Physics Modern Physics Nuclei

In an ore containing uranium, the ratio of $U_{238}$ to $Pb_{206}$ is $3$. Calculate the age of the ore, assuming that all the leas present in the ore is the final stable product of $U_{238}$. Take the half-life of $U_{238}$ to be $4.5×10_{9}$ yr.

(a)

$1.6×19_{3}yr$

(b)

$1.5×10_{4}yr$

(c)

$1.867×10_{9}yr$

(d)

$2×10_{5}yr$

Correct answer: (c)

Solution: Let the initial mass of uranium be $M_{0}$

Final mass of uranium after time $t,M=43 M_{0}$

According to the law of radioactive disintegration.

$M_{0}M =(21 )_{t/T}⇒MM_{0} =(2)_{t/T}$

$∴g_{10}(MM_{0} )=Tt g_{10}(2)$

$t=Tg_{10}(2)g_{10}(MM_{0} ) =g_{10}(2)Tg_{10}(34 ) $

$=g_{10}(2)Tg_{10}(1.333) =4.5×10_{9}(0.30100.1249 )$

$⇒t=1.867×10_{9}yr$.

Final mass of uranium after time $t,M=43 M_{0}$

According to the law of radioactive disintegration.

$M_{0}M =(21 )_{t/T}⇒MM_{0} =(2)_{t/T}$

$∴g_{10}(MM_{0} )=Tt g_{10}(2)$

$t=Tg_{10}(2)g_{10}(MM_{0} ) =g_{10}(2)Tg_{10}(34 ) $

$=g_{10}(2)Tg_{10}(1.333) =4.5×10_{9}(0.30100.1249 )$

$⇒t=1.867×10_{9}yr$.

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