9. Thallium-208 has a half-life of 3.053 min. How long will it take for 120 g of it to decay to 7.5 g?

Question

9. Thallium-208 has a half-life of 3.053 min. How long will it take for 120 g of it to decay
to 7.5 g?

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Khang Minh 4 years 2021-08-23T03:50:41+00:00 1 Answers 89 views 0

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  1. thienan
    0
    2021-08-23T03:51:51+00:00
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    Answer:

    12.213 minutes will be taken for 120 g-Thalium-208 to decay to 75 grams.

    Explanation:

    Radioactive isotopes decay exponentially in time, the mass of the isotope (m(t)), in grams, is described by the formula in time (t), in minutes:

    m(t) = m_{o}\cdot e^{-\frac{t}{\tau} } (1)

    Where:

    m_{o} – Initial mass of the isotope, in grams.

    \tau – Time constant, in minutes.

    In addition, the time constant associated with the isotope decay can be described in terms of half-life (t_{1/2}), in minutes:

    \tau = \frac{t_{1/2}}{\ln 2} (2)

    If we know that m(t) = 7.5\,g, m_{o} = 120\,g and t_{1/2} = 3.053\,min, then the time taken by the isotope is:

    \tau = \frac{t_{1/2}}{\ln 2}

    \tau = \frac{3.053\,min}{\ln 2}

    \tau \approx 4.405\,min

    t = -\tau \cdot \ln \frac{m(t)}{m_{o}}

    t = -(4.405\,min)\cdot \ln \left(\frac{7.5\,g}{120\,g} \right)

    t \approx 12.213\,min

    12.213 minutes will be taken for 120 g-Thalium-208 to decay to 75 grams.

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