Question

Radioactive plutonium−239 (t1/2 = 2.44 × 105 yr) is used in nuclear reactors and atomic bombs. If there are 6.40 × 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.)

Answer 1

Answer:
`6.54* 10^5years`

Step-by-step explanation:

Expression for rate law for first order kinetics is given by:

`t=(2.303)/(k)\log(a)/(a-x)`

where,

k = rate constant

t = age of sample

a = let initial amount of the reactant

a - x = amount left after decay process

a) for completion of half life:

Half life is the amount of time taken by a radioactive material to decay to half of its original value.

`t_{(1)/(2)}=(0.693)/(k)`

`k=(0.693)/(2.44* 10^5)=0.284* 10^(-5)yr^(-1)`

b) for
`6.40* 10^2` g of the isotope to decay to
`1.00* 10^2`

`t=(2.303)/(0.284* 10^(-5))\log(6.40* 10^2)/(1.00* 10^2)`

`t=6.54* 10^5years`

The time for
`6.40* 10^2` g of the isotope to decay to
`1.00* 10^2` is
`6.54* 10^5years`