Question

The half life of radioactive isotope ‘X’ is 20 years. It decays to another element Y which is stable. The two elements X and Y were found to be in the ratio 1:7 in a sample of given rock. The age of rock is estimated to be
A. 40 Years
B. 60 years
C. 80 years
D. 100 years

Answer 1

Hint: In this question we have been asked to calculate the age of rock having two radioactive elements as given. We have been given the half life of element X and the ratio of the elements X and Y found in rock. Therefore, we shall use the formula for activity of radioactive elements, since it deals with the ratio of decaying elements and the half life and the age of rock.

Formula used:
\[N={{N}_{o}}{{\left( \dfrac{1}{2} \right)}^{{}^{t}/{}_{T}}}\]
Where,
T is the half life of element
t is the age of element
N is the amount of element decayed
\[{{N}_{o}}\] is the initial amount of the element

Complete step-by-step answer:
We know that the activity of radioactive elements is given by
\[N={{N}_{o}}{{\left( \dfrac{1}{2} \right)}^{{}^{t}/{}_{T}}}\]
Therefore, we can say that
\[\dfrac{N}{{{N}_{o}}}={{\left( \dfrac{1}{2} \right)}^{{}^{t}/{}_{T}}}\]
Now, we know that ratio of elements found in the given rock is,
\[\dfrac{{{N}_{x}}}{{{N}_{y}}}=\dfrac{1}{7}\]
Therefore, the rock initially contained
\[{{N}_{o}}={{N}_{x}}+{{N}_{y}}\]
Therefore,
\[{{N}_{o}}=1+7=8\]
Therefore, we can say that,
\[\dfrac{1}{8}={{\left( \dfrac{1}{2} \right)}^{{}^{t}/{}_{20}}}\] …………. (substitute values)
We can say that,
\[{{\left( \dfrac{1}{2} \right)}^{3}}={{\left( \dfrac{1}{2} \right)}^{{}^{t}/{}_{20}}}\]
Therefore,
\[3=\dfrac{t}{20}\]
Therefore,
\[t=60\] years.

So, the correct answer is “Option B”.

Note: Radioactivity is the emission of particles from the nucleus due to instability. The radioactive elements undergo decay as they are unstable. The elements lose energy and settle at a stable element. The instability in the radioactive elements is due to the imbalance in protons and electrons in their atomic nucleus. The geologists find the age of rock by measuring the amount of radioactive elements in the rock.