Question

A sample of uraninite, a uranium containing mineral, was found in analysis to contain 0.214g of lead -206 for every g uranium. If all the lead came from the radioactive disintegration of the uranium and assume that all isotopes of uranium other than ${U^{238}}$ can be neglected , estimated the date when the mineral was formed in the earth’s crust ?(the half-life of ${U^{238}}$ is $4.5*{10^9}years$).
USE, $1 + \left[ {Pb} \right]/\left[ U \right] = {2^Y}$ ,Where $y = t/{T_{50}}$ .

Answer 1

Hint:Radioactive elements are those having unstable nuclei and have high molecular mass. They are rare to find in nature. There also most commonly decay alpha decay, beta decay and gamma decay. For solving this question, one should have knowledge of carbon dating methods.

Complete step by step answer:
As we know radioactive elements are those having unstable nuclei and heavier molecular mass .There are some common terms in radioactivity which are decay and half-life of radioactivity element.
Radioactive decay: it is defined as the breakdown of atomic nucleus resulting in the release of energy and matter from the nucleus.
Half-life: it may be defined as the time period required to decompose one half of the radioactive sample.
Radioactive decay and half-life are related to each other, and this relation may be given as –
${T_{1/2}} = 0.693/\lambda $ , where $\lambda = $ decay constant and ${T_{1/2}} = $ half-life.
Now, comes to the solution part:
Sample containing $0.214g$ of lead-206 and 1g of ${U^{238}}$ is also present.
$\left[ {Pb} \right] = 0.214/206 = 0.00104mol$ And,
$\left[ U \right] = 1/238 = 0.0042mol$
Then ratio $\left[ {Pb} \right]/\left[ U \right] = 0.247$
For rock dating method, we use $1 + \left[ {Pb} \right]/\left[ U \right] = {2^y}$ where, $y = t/{T_{50}}$
$ \Rightarrow \left[ {1 + 0.247} \right] = {2^y}$
From solving above equation we get $y = 0.3188$
$t/{T_{50}} = 0.3188$
$ \Rightarrow t = 0.3188*{T_{50}}$
$\therefore t = 0.3188*4.5*{10^9}$
$ \Rightarrow t = 1.4344*{10^9}years$

Hence, the required time is $1.4344*{10^9}$ years.

Note: All radioactive reactions follow first order reaction .Carbon dating and rock dating methods are used by archaeologists to search or get the approximate years how many years old the compound is.