Solved: PHYSICS Question for NEET | Sushrut

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NEET PHYSICSMedium

The half-life of a 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 a given rock. The age of the rock is estimated to be:

A

60 years

B

80 years

C

100 years

D

40 years

Step-by-Step Solution

Understand the Decay Process: The radioactive isotope $X$ decays into the stable element $Y$ . The amount of $Y$ present represents the amount of $X$ that has decayed (assuming no $Y$ was present initially).
Determine Initial Amount ( $N_0$ ):

Ratio of remaining parent ( $X$ ) to daughter ( $Y$ ) nuclei = $1:7$ .
Let the number of nuclei of $X$ be $x$ and $Y$ be $7x$ .
Total initial nuclei $N_0 = N_X + N_Y = x + 7x = 8x$ .

Calculate Fraction Remaining:

Fraction of $X$ remaining = $\frac{N_X}{N_0} = \frac{x}{8x} = \frac{1}{8}$ .

Relate to Half-Lives:

The fraction remaining is given by $(\frac{1}{2})^n$ , where $n$ is the number of half-lives passed.
$\frac{1}{8} = (\frac{1}{2})^3$ , so $n = 3$ half-lives.

Calculate Age:

Age = $n \times T_{1/2} = 3 \times 20 \text{ years} = 60 \text{ years}$ .

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