Question

Radioactive plutonium-239 (t½ = 2.44 x 105 yr) is used in nuclear reactors and atomic bombs. If there are 4.4 x 102 g of the isotope in a small atomic bomb, how long will it take (in yr) for the substance to decay to 1.0 x 102 g, too small an amount for an effective bomb? This radioactive decay follows first order kinetics.

Answer 1

Answer: The time time taken for the sample to decay from
`4.4* 10^2g` to
`1.0* 10^2` is
`5.29* 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* 10^5year)=0.28* 10^(-5)year^(-1)`

b) time taken for the sample to decay from
`4.4* 10^2g` to
`1.0* 10^2`

`t=(2.303)/(0.28* 10^(-5))\log(4.4* 10^2)/(1.0* 10^2)`

`t=5.29* 10^5years`

The time time taken for the sample to decay from
`4.4* 10^2g` to
`1.0* 10^2` is
`5.29* 10^5years`