Natural uranium is a mixture of three isotopes 92234U,92235U and 92238U with mass percentage 0.01%,0.71% and 99.28% respectively. The half-life of three isotopes are 2.5×105yr,7.1×108yr and 4.5×109yr respectively.Determine the share of radioactivity of each isotope into the total activity of the natural uranium.

Let R1,R2 and R3 be the activities of U234,U235 and U238 respectively. Total activity,Share of U234,R=R1+R2+R3RR1=λ1N1+λ2N2+λ3N3λ1N1Let m be the total mass of natural uranium.Then,Now,m1=1000.01m,m2=1000.71m and m3=10099.28mN1=M1m1,N2=M2m2 and N3=M3m3 where M1,M2 and M3 are atomic weights.∴ RR1=M1m1T11+M2m2⋅T21+M3m3⋅T31(M1m1)T11

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Natural uranium is a mixture of three isotopes { }_{92}^{234}

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Natural uranium is a mixture of three isotopes $_{92}^{234} U,_{92}^{235} U$ and $_{92}^{238} U$ with mass percentage $0.01%, 0.71%$ and $99.28%$ respectively. The half-life of three isotopes are $2.5 \times 1 0^{5} yr, 7.1 \times 1 0^{8} yr$ and $4.5 \times 1 0^{9} yr$ respectively.
Determine the share of radioactivity of each isotope into the total activity of the natural uranium.

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Optics And Modern Physics (DC Pandey)

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Charlotte Discussed

Natural uranium is a mixture of three isotopes

_{92}^{234} U,_{92}^{235} U

and

_{92}^{238} U

with mass percentage

0.01%, 0.71%

and

99.28%

respectively. The half-life of three isotopes are

2.5 \times 1 0^{5} yr, 7.1 \times 1 0^{8} yr

and

4.5 \times 1 0^{9} yr

respectively.
Determine the share of radioactivity of each isotope into the total activity of the natural uranium.

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Text solutionVerified

Let

R_{1}, R_{2}

and

R_{3}

be the activities of

U^{234}, U^{235}

and

U^{238}

respectively. Total activity,
Share of

U^{234}

R = R_{1} + R_{2} + R_{3}

\frac{R _{1}}{R} = \frac{λ _{1} N _{1}}{λ _{1} N _{1} + λ _{2} N _{2} + λ _{3} N _{3}}

Let

m

be the total mass of natural uranium.
Then,
Now,

m_{1} = \frac{0.01}{100} m, m_{2} = \frac{0.71}{100} m and m_{3} = \frac{99.28}{100} m

N_{1} = \frac{m _{1}}{M _{1}}, N_{2} = \frac{m _{2}}{M _{2}}

and

N_{3} = \frac{m _{3}}{M _{3}}

where

M_{1}, M_{2}

and

M_{3}

are atomic weights.

∴

\frac{R _{1}}{R} = \frac{( \frac{m _{1}}{M _{1}} ) \frac{1}{T _{1}}}{\frac{m _{1}}{M _{1}} \frac{1}{T _{1}} + \frac{m _{2}}{M _{2}} \cdot \frac{1}{T _{2}} + \frac{m _{3}}{M _{3}} \cdot \frac{1}{T _{3}}}

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Question Text	Natural uranium is a mixture of three isotopes $_{92}^{234} U,_{92}^{235} U$ and $_{92}^{238} U$ with mass percentage $0.01%, 0.71%$ and $99.28%$ respectively. The half-life of three isotopes are $2.5 \times 1 0^{5} yr, 7.1 \times 1 0^{8} yr$ and $4.5 \times 1 0^{9} yr$ respectively. Determine the share of radioactivity of each isotope into the total activity of the natural uranium.
Topic	Nuclei
Subject	Physics
Class	Class 12
Answer Type	Text solution:1
Upvotes	113

Text solutionVerified

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