Question

In uranium minerals, the atomic ratio ${{N}_{U-238}}/{{N}_{Pb-206}}$ is nearly equal to one. The age (in years) of the mineral is nearly: (half life period of U-238 is $4.5\text{ x 1}{{\text{0}}^{9}}$ years)
(a)- $3.0\text{ x 1}{{\text{0}}^{8}}$
(b)- $4.5\text{ x 1}{{\text{0}}^{9}}$
(c)- $3.0\text{ x 1}{{\text{0}}^{9}}$
(d)- $4.5\text{ x 1}{{\text{0}}^{8}}$

Answer 1

For finding the age of the mineral, we have to use the formula:
$1+\text{Atomic ratio = }{{\text{2}}^{y}}$
From this we can find the y, this y is equal to the ratio of age of the mineral and half life of the element in it.

Complete answer:
We know that there are many radioactive elements in nature that degrade in nature depending on the half-life of the element. Some of the radioactive elements are uranium, thorium, radium, etc.
We are given the uranium mineral, in which the atomic ratio of lead and uranium is given equal to one, i.e., given as ${{N}_{U-238}}/{{N}_{Pb-206}}$.
For finding the age of the mineral, we have to use the formula:
$1+\text{Atomic ratio = }{{\text{2}}^{y}}$
We can write:
$1+\dfrac{[Pb]}{[U]}={{2}^{y}}$
Given the atomic ratio is 1, putting the value, we get:
$1+1={{2}^{y}}$
$2={{2}^{y}}$
From this, we can see that value of y is 1.
From this we can find the y, this y is equal to the ratio of age of the mineral and half life of the element in it.
So, we can write the above as:
$y=\dfrac{t}{{{t}_{1/2}}}$
Where t is the age of the rock and ${{t}_{1/2}}$ is the half-life period. We are given the half-life period of U-238 is $4.5\text{ x 1}{{\text{0}}^{9}}$
Putting this value in the formula, we get:
$1=\dfrac{t}{4.5\text{ x 1}{{\text{0}}^{9}}}$
$t=4.5\text{ x 1}{{\text{0}}^{9}}$ years
So, the age of the mineral is $4.5\text{ x 1}{{\text{0}}^{9}}$ years.

Therefore, the correct answer is option (b)- $4.5\text{ x 1}{{\text{0}}^{9}}$.

Note:
When the radioactive element decays then it converts into smaller nuclei. From the half-life of the element we can find the stability of the element, they are elements whose half-life is in the sec because those elements are very unstable.