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Uranium ores on the earth at the present time typically have a

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Solving time: 4 mins

Uranium ores on the earth at the present time typically have a composition consisting of $99.3%$ of the isotope $_{92} U^{238}$ and $0.7%$ of the isotope $_{92} U^{235}$ . The half lives of these isotopes are $4.47 \times 1 0^{9} y$ and $7.04 \times 1 0^{8} y$ respectively. If these isotopes were equally abundant when the earth was formed, estimate the age of the earth.

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Master Resource Book in JEE Main Physics (Arihant)

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Uranium ores on the earth at the present time typically have a composition consisting of

99.3%

of the isotope

_{92} U^{238}

and

0.7%

of the isotope

_{92} U^{235}

. The half lives of these isotopes are

4.47 \times 1 0^{9} y

and

7.04 \times 1 0^{8} y

respectively. If these isotopes were equally abundant when the earth was formed, estimate the age of the earth.

6 mins ago

Discuss this question LIVE

6 mins ago

Text solutionVerified

Let

N_{0}

be number of atoms of each isotope at the time of formation of the earth

(t = 0)

and

N_{1}

and

N_{2}

the number of atoms at present

(t = t)

. Then,

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

Further, it is given that

\frac{N _{1}}{N _{2}} = \frac{99.3}{0.7}

Solving, we have

(λ_{2} - λ_{1}) t = ln (\frac{99.3}{0.7})

t = (\frac{1}{λ _{2} - λ _{1}}) ln (\frac{99.3}{0.7})

Substituting the values, we have

t = \frac{1}{\frac{0.693}{7.04 \times 1 0 ^{8}} - \frac{0.693}{4.47 \times 1 0 ^{9}}} ln (\frac{99.3}{0.7})

t = 5.97 \times 1 0^{9} yr

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Question Text	Uranium ores on the earth at the present time typically have a composition consisting of $99.3%$ of the isotope $_{92} U^{238}$ and $0.7%$ of the isotope $_{92} U^{235}$ . The half lives of these isotopes are $4.47 \times 1 0^{9} y$ and $7.04 \times 1 0^{8} y$ respectively. If these isotopes were equally abundant when the earth was formed, estimate the age of the earth.
Topic	Nuclei
Subject	Physics
Class	Class 12
Answer Type	Text solution:1
Upvotes	123

Text solutionVerified

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