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Exercise:
A radioactive isotope with a decay constant caO decays o a second radioative isotope with a decay constant cbO. In the ning there are mO of each nuclide. abcliste abc Calculate the remaining masses after tO. abc Graph the masses as a function of time. abcliste

Solution:
abcliste abc The system is defined by the matrix bf A leftmatrix-c_ & c_ & -c_matrixright with eigenvalues lambda_c_ and lambda_c_ and eigenvectors bf v_ leftmatrix fracc_c_-c_matrixright bf v_ leftmatrix matrixright where c_ and c_ are the decay constants of the two isotopes see notes. For the general solution we have to solve the leftmatrixm_ m_matrixright a_ bf v_+a_ bf v_ leftmatrixa_ fraca_ c_c_-c_+a_matrixright The obvious solution is a_ m_ a_ m_left-fracc_c_-c_right so the general solution for the decay can be written as m_t maF m_t mbF The numerical results are m_ mtimes e^-catimes t resultmaS m_ mtimes fraccacb-ca e^-ca times t+mtimesleft-fraccacb-caright e^-cbtimes t resultmbP abc The graph confirms the results in a. center includegraphicswidthtextwidth#image_path:two-stagdecay-chain# center abcliste
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Exercise:
A radioactive isotope with a decay constant caO decays o a second radioative isotope with a decay constant cbO. In the ning there are mO of each nuclide. abcliste abc Calculate the remaining masses after tO. abc Graph the masses as a function of time. abcliste

Solution:
abcliste abc The system is defined by the matrix bf A leftmatrix-c_ & c_ & -c_matrixright with eigenvalues lambda_c_ and lambda_c_ and eigenvectors bf v_ leftmatrix fracc_c_-c_matrixright bf v_ leftmatrix matrixright where c_ and c_ are the decay constants of the two isotopes see notes. For the general solution we have to solve the leftmatrixm_ m_matrixright a_ bf v_+a_ bf v_ leftmatrixa_ fraca_ c_c_-c_+a_matrixright The obvious solution is a_ m_ a_ m_left-fracc_c_-c_right so the general solution for the decay can be written as m_t maF m_t mbF The numerical results are m_ mtimes e^-catimes t resultmaS m_ mtimes fraccacb-ca e^-ca times t+mtimesleft-fraccacb-caright e^-cbtimes t resultmbP abc The graph confirms the results in a. center includegraphicswidthtextwidth#image_path:two-stagdecay-chain# center abcliste
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Attributes & Decorations
Branches
Differential equations
Tags
decay, radioactivity
Content image
Difficulty
(2, default)
Points
0 (default)
Language
ENG (English)
Type
Calculative / Quantity
Creator by
Decoration