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23 January, 12:45

In an experiment on the transport of nutrients in a plant's root structure, two radioactive nuclides X and Y are used. Initially, 2.70 times more nuclei of type X are present than of type Y. At a time 2.50 d later, there are 4.60 times more nuclei of type X than of type Y. Isotope Y has a half-life of 1.50 d. What is the half-life of isotope X?

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  1. 23 January, 13:19
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    From radioactive equation, it is know that,

    N = No•exp (-λt)

    Where λ is the decay constant

    N is is the material remaining after time t

    No is the original material at the beginning I. e. at t = 0

    Half-life is given as

    T½ = In (2) / λ

    Now

    Given that,

    The initial number of nuclides of type X to that of type Y is

    No (X) / No (Y) = 2.7

    The final number of nuclides of type X to that of type Y is

    N (X) / N (Y) = 4.7

    The time elapsed between this period is

    t = 2.5d

    the half life of isotope Y is

    T½ (Y) = 1.5d

    What is the half life of isotope X

    From first equation at t = 2.5d

    For type X

    N = No•exp (-λ•t)

    N (X) = No (X) • exp (-2.5•λx)

    For Type Y

    N = No•exp (-λ•t)

    N (Y) = No (Y) • exp (-2.5•λy)

    Divide N (X) by N (Y)

    Then, we have

    N (X) / N (Y) = [No (X) / No (Y) ]•exp (-2.5λx) / exp (-2.5λy)

    Let substitute the given ratio

    4.7 = 2.7 exp (-2.5λx+2.5λy)

    4.7 / 2.7 = exp (-2.5λx+2.5λy)

    1.741 = exp (-2.5λx+2.5λy)

    Take In of both sides

    In (1.741) = - 2.5λx+2.5λy

    0.5543 = - 2.5 (λx-λy)

    (λx-λy) = 0.5543/-2.5

    (λx-λy) = - 0.2217 equation 1

    From the second equation

    T½ = In2 / λ

    For Type Y

    T½ (Y) = In2 / λy

    λy = In2 / T½ (Y)

    λy = In2 / 1.5d

    λy = 0.4621 / d

    From equation 1

    λx-λy = - 0.2217

    λx - 0.4621 = - 0.2217

    λx = - 0.2217 + 0.4621

    λx = 02204 / d

    Finally,

    T½ (X) = In2 / λx

    T½ (X) = In2 / 0.2204 / d

    T½ (X) = 2.8833 d

    So the half life of Element X is 2.8833d
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