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In the chemical analysis of a rock the mass ratio of two radio

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In the chemical analysis of a rock the mass ratio of two radioactive isotopes is found to be $100 : 1$ . The mean lives of the two isotopes are $4 \times 1 0^{9}$ years and $2 \times 1 0^{9}$ years, respectively. If it is assumed that at the time of formation the atoms of both the isotopes were in equal proportional, calculate the age of the rock. Ratio of the atomic weights of the two isotopes is $1.02 : 1$ .

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Optics And Modern Physics (DC Pandey)

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Samuel Discussed

In the chemical analysis of a rock the mass ratio of two radioactive isotopes is found to be

100 : 1

. The mean lives of the two isotopes are

4 \times 1 0^{9}

years and

2 \times 1 0^{9}

years, respectively. If it is assumed that at the time of formation the atoms of both the isotopes were in equal proportional, calculate the age of the rock. Ratio of the atomic weights of the two isotopes is

1.02 : 1

10 mins ago

Discuss this question LIVE

10 mins ago

Text solutionVerified

At the time of observation

(t = t)

\frac{m _{1}}{m _{2}} = \frac{100}{1}

(given)
Further it is given that

\frac{A _{1}}{A _{2}} = \frac{1.02}{1}

Number of atoms,

N = \frac{m}{A}

\frac{N _{1}}{N _{2}} = \frac{m _{1}}{m _{2}} \times \frac{A _{2}}{A _{1}} = \frac{100}{1.02}

Let

N_{0}

be the number of atoms of both the isotopes at the time of formation, then

\frac{N _{1}}{N _{2}} = \frac{N _{0} e ^{- λ_{1} t}}{N _{0} e ^{- λ_{2} t}} = e^{(λ_{2} - λ_{1}) t}

Eq. (i) and Eq. (ii), we have

e^{(λ_{2} - λ_{1}) t} = \frac{100}{1.02}

(λ_{2} - λ_{1}) t = ln 100 - ln 1.02

t = \frac{l n 100 - l n 1.02}{( \frac{1}{2 \times 1 0 ^{9}} - \frac{1}{4 \times 1 0 ^{9}} )}

Substituting the values, we have

t = 1.834 \times 1 0^{10} yr

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Question Text	In the chemical analysis of a rock the mass ratio of two radioactive isotopes is found to be $100 : 1$ . The mean lives of the two isotopes are $4 \times 1 0^{9}$ years and $2 \times 1 0^{9}$ years, respectively. If it is assumed that at the time of formation the atoms of both the isotopes were in equal proportional, calculate the age of the rock. Ratio of the atomic weights of the two isotopes is $1.02 : 1$ .
Topic	Nuclei
Subject	Physics
Class	Class 12
Answer Type	Text solution:1
Upvotes	131

Text solutionVerified

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