Concept explainers
Interpretation: List of isotopes of krypton is given. The most stable and the hottest among them is to be stated. Time of decay of 87.5% of each isotope is to be stated.
Concept introduction: 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.
The nuclides having longer half life are more stable while nuclides having shorter half life are less stable.
To determine: The most stable and the hottest isotope among the given isotopes of krypton; the time of decay for 73Kr; the time of decay for 74Kr; the time of decay for 76Kr and the time of decay for 81Kr.
Answer to Problem 34E
Answer
The most stable isotope is 81Kr and the hottest one is 73Kr
The time of decay for 87.5% of 73Kr is 81.428 s_ .
The time of decay for 87.5% of 74Kr is. 34.51 s_
The time of decay for 87.5% of 76Kr is 44.41 h_
The time of decay for 87.5% of 81Kr is 6.302×105years_
Explanation of Solution
Explanation
The most stable isotope is 81Kr and the hottest one is 73Kr
The nuclides having longer half life are more stable while nuclides having shorter half life are less stable. Thus the most stable isotope is 81Kr and the hottest isotope is 73Kr.
The time of decay for 73Kris 81.428 s_
Explanation
The decay constant can be calculated by the formula given below.
λ=0.693t1/2
Where
- t1/2 is the half life of nuclide.
- λ is the decay constant.
Substitute the value of λ in the above expression.
λ=0.69327s−1=0.0256 s−1
The fraction of isotope decayed is 87.5100.
The fraction remaining =1−87.5100=12.5100
The time of decay can be calculated by the formula,
t=2.303λlogn0n
Where
- n0 is the number of atoms initially present.
- n is the number of atoms after time “t”.
Substitute the values of λ, n0 and n in the above expression.
t=2.303λlogn0nt=2.303λlog10012.5t=2.3030.0256log10012.5t=81.428 s_.
The time of decay for 74Kris 34.51 s_.
Explanation
The decay constant is calculated by the formula,
λ=0.693t1/2
Where
- t1/2 is the half life of nuclide.
- λ is the decay constant.
Substitute the value of t1/2 in the above expression.
λ=0.69311.5 min−1
The fraction of isotope decayed is 87.5100.
The fraction remaining =1−87.5100=12.5100
The time of decay can be calculated by the formula,
t=2.303λlogn0n
Where
- n0 is the number of atoms initially present.
- n is the number of atoms after time “t”.
Substitute the values of λ, n0 and n in the above expression.
t=2.303λlogn0nt=2.303λlog10012.5t=2.303×11.50.693log8t=34.51 min_
The time of decay for 76Kr is 44.41 h_
Explanation
The decay constant can be calculated by the formula given below.
λ=0.693t1/2
Where
- t1/2 is the half life of nuclide.
- λ is the decay constant.
Substitute the value of t1/2 in the above expression.
λ=0.69314.8 h−1
The fraction of isotope decayed is 87.5100.
The fraction remaining =1−87.5100=12.5100
The time of decay can be calculated by the formula,
t=2.303λlogn0n
Where
- n0 is the number of atoms initially present.
- n is the number of atoms after time “t”.
Substitute the values of λ, n0 and n in the above expression.
t=2.303λlogn0nt=2.303λlog10012.5t=2.303×14.80.693log8t=44.41 h_
The time of decay for 81Kr is 6.302×105years_.
Explanation
The decay constant can be calculated by the formula given below.
λ=0.693t1/2
Where
- t1/2 is the half life of nuclide.
- λ is the decay constant.
Substitute the value of t1/2 in the above expression.
λ=0.6932.1×105 year
The fraction of isotope decayed is 87.5100.
The fraction remaining =1−87.5100=12.5100
The time of decay can be calculated by the formula,
t=2.303λlogn0n
Where
- n0 is the number of atoms initially present.
- n is the number of atoms after time “t”.
Substitute the values of λ, n0 and n in the above expression.
t=2.303λlogn0nt=2.303×2.1×1050.693log10012.5t =2.303×2.1×1050.693log8t=6.302×105years_
Conclusion
The most stable isotope is 81Kr and the hottest one is 73Kr
The time of decay for 87.5% of 73Kr is 81.428 s_ .
The time of decay for 87.5% of 74Kr is. 34.51 s_
The time of decay for 87.5% of 76Kr is 44.41 h_
The time of decay for 87.5% of 81Kr is 6.302×105years_
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Chapter 19 Solutions
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