A sample initially contains only U-238 isotope of uranium. With time, some of the U-238 radioactively decays into Pb -206 while the rest of it remains undisintegrated. When the age of the sample is P 10^8 years, the ratio of mass of Pb -206 to that of U -238 in the sample is found to be 7 . The value of P is ______ [Given: Half-life of U-238 is 4.5 10^9 years; _c 2=0.693 ]

Step-by-step solution

₉₂ U ²³⁸ ₈₂ ~Pb ²⁰⁶+8 ₂ He ^4+6_ -1 ^0 t =0 ~N ₀= ( 1 238 + 7 206 ) moles t = tN _ t = 1 238 moles x = 7 206 moles Life of sample ( t ) years ([A]₀ ) Initial mole of ( U -238 ) ([A]]_t ) Final mole of U-238 ( aligned & |A|₀ |A|_t = 1 228 + 7 206 1 238 & = 0.0042+0.0340 0.0042 & =9.1 & = 2.303 2 t 4.5 10^9 =2.303 9.1 & t=14.27 10^9 years & =142.7 10^8 years & P=142.7 & P=143 aligned )