Question

Radioactive plutonium−239 (t1/2 = 2.44 × 105 yr) is used in nuclear reactors and atomic bombs. If there are 3.50 × 102 g of the isotope in a small atomic bomb, how long will it take for the substance to decay to 1.00 × 102 g, too small an amount for an effective bomb? (Hint: Radioactive decays follow first-order kinetics.) .0000441 yr

Answer 1

Answer:

t = 4.41 10⁻⁴ years

Explanation:

For this exercise we must use the concept of average life time, which is the time in which the quantity and substance decays in half

    $T_{1/2}$  = ln2 / λ

Let's calculate the decay constant of plutonium

    λ  = ln2 /  $T_{1/2}$

    λ  = ln 2 / 2.44 10⁵  

    λ  = 2.84 10⁻⁶  s⁻¹

Radioactive decay is a first order process

     N = No e (-λ   t)

Where N is the number of nuclei, the mass is this by molecular weight

    m = N PM

    m / PM = m₀ / PM e (- λ  t)

    m / m₀ = e (- λ  t)

    -λ  t = ln (m / m₀)

     t = -1 /λ   ln (m/m₀)

     t = - 1 / 2.84 10⁻⁶   ln (0.1 / 0.35)

     t = 4.41 10⁻⁴ years