Concept explainers
Interpretation: Half life of 238U and 235U and their relative abundances is given. Their relative abundances when earth was formed 4.5 billion years ago are to be calculated.
Concept introduction: A process through which an unstable nuclide looses its energy due to excess of protons or neutrons is known as radioactive decay. Decay constant is the quantity that expresses the rate of decrease of number of atoms of a radioactive element per second. Half life of radioactive sample is defined as the time required for the number of nuclides to reach half of the original value.
To determine: The relative abundances of 238U and 235U when earth was formed.
Answer to Problem 79CP
Answer
The relative abundance of 238U when earth was formed is 76%_.
The relative abundance of 235U when earth was formed is 24%_.
Explanation of Solution
Explanation
The decay constant is calculated by the formula given below.
λ=0.693t1/2
Where
- t1/2 is the half life of nuclide.
- λ is decay constant.
The value of t1/2 for 238U is 4.5×109 years.
Substitute the value of half life in the above formula.
λ=0.6934.5×109 years-1
The decay constant is 0.6934.5×109 years-1.
The decay constant is calculated by the formula given below.
λ=0.693t1/2
Where
- t1/2 is the half life of nuclide.
- λ is decay constant.
The value of t1/2 for 235U is 7.1×108 years.
Substitute the value of half life in the above formula.
λ=0.6937.1×108 years-1.
The decay constant is 0.6937.1×108 years-1.
The relative abundance of 238U is calculated by the formula,
ln(nn0)=−λ⋅t
Where
- n0 is the amount of 238U when earth was formed.
- n is the amount of 238U after 4.5×109 years.
- t is the decay time.
The value of t is 4.5×109 years.
Substitute the values of t and decay constant in the above equation.
ln(nn0)=−λ⋅tln(nn0)=−0.693×4.5×1094.5×109ln(nn0)=−0.693(nn0)=0.50
The value of (nn0) for 238U is 0.50.
The relative abundance of 235U is calculated by the formula,
ln(nn0)=−λ⋅t
Where
- n0 is the amount of 235U when earth was formed.
- n is the amount of 235U after 4.5×109 years.
- t is the decay time.
The value of t is 7.1×108 years.
Substitute the values of t and decay constant in the above equation.
ln(nn0)=−λ⋅tln(nn0)=−0.693×4.5×1097.1×108(nn0)=e−4.39(nn0)=0.012
The value of (nn0) for 235U is 0.012.
The natural composition of 238U is 99.28%=99281000
Therefore, 9928 nuclei of 238U are present in 10000 uranium nuclei.
Therefore, the value of n for 238U is 9928.
Substitute the value of n in the equation,
(nn0)=0.50n0=n0.50n0=99280.50=1.9×104 238U nuclei
The amount of 238U when earth was formed is 1.9×104 238U nuclei
The natural composition of 235U is 0.72%=721000
Therefore, 72 nuclei of 235U are present in 10000 uranium nuclei.
Therefore, the value of n for 235U is 72.
Substitute the value of n in the equation,
(nn0)=0.012n0=n0.012n0=720.012=6×103 235U nuclei
The amount of 235U when earth was formed is 6×103 235U nuclei.
The composition of 238U from 4.5 billion years ago is calculated by the formula,
Composition of 238U=Amount of 238U when earth was formedAmount of (238U+235U) when earth was formed×100
The composition of
The composition of
The composition of
Conclusion
The relative abundance of
The relative abundance of
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