Question

In a sample of pitchblende, the atomic ratio of $P{{b}^{206}}:{{U}^{238}}$ is 0.23:1. Calculate the age of the mineral if half life of uranium is $4.5\times {{10}^{9}}$year. All lead originated from uranium.

Answer 1

Hint : Nuclear chemistry is the chemistry involving radioactive substances. Nuclear reactions are usually first order reactions. Decay constant or radioactive constant is the ratio of no. of decay per minute of radioactive substance in the sample to the total number of particles present in the sample.

Complete step by step solution :
The radioactive properties of uranium can be used to determine the age of minerals and rocks.
The equation to find half life of a radioactive material is
${{t}_{{}^{1}/{}_{2}}}=\dfrac{\log 2}{\lambda }=\dfrac{0.693}{\lambda }$, where $\lambda$ is the disintegration constant.
In the question half life of the reaction is given. So, we can find the value of disintegration constant$\lambda$ from the equation given above. Half life of the reaction is ${{t}_{\dfrac{1}{2}}}=4.5\times {{10}^{9}}years$.
Let us substitute these values in the above equation.
$\lambda =\dfrac{0.693}{4.5\times {{10}^{9}}}=0.154\times {{10}^{-9}}yea{{r}^{-1}}$
Now consider the first order equation for the disintegration constant, which is
$\lambda =\dfrac{2.303}{t}\log \left( \dfrac{{{N}_{0}}}{N} \right)$--(i), where ${{N}_{0}}$ is the initial amount of uranium present in the reaction and $N$is the amount of uranium present at time t.
It is given in the question, that the ratio of initial amount of uranium${{N}_{0}}$ (which is ${{U}^{238}}$) and the amount at time t $N$( which is$P{{b}^{206}}$) is : $\dfrac{{{N}_{0}}}{N}=\dfrac{{{U}^{238}}}{P{{b}^{206}}}=\dfrac{1}{0.23}$
So, let us substitute all the values in equation (i) and we get
$t=\dfrac{2.303}{0.154\times {{10}^{-9}}}\log \left( \dfrac{1}{0.23} \right)$
$t=0.6381\times 14.954\times {{10}^{9}}=9.541\times {{10}^{9}}years$
Therefore, the age of the mineral is $9.541\times {{10}^{9}}years$.

Additional Information: Uranium disintegrated to give a stable Lead. The half life of Uranium is greater than all the other radioactive elements. We know radioactive decay is a first order reaction, but the rate of the reaction depends on the concentration of one reactant. It is not affected by the factors which affect chemical reactions, i.e. the rate of the reaction does not depend on the temperature, pressure or any other physical determinants. We can also say that radioactive decay is an exponential decay function, which means that if more numbers of atoms are present, the decay rate increases.

Note : The ratio given in the question is in the form amount of uranium present a time t to the initial concentration of uranium. So while substituting the values we need to check it properly.