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: $6.54\times 10^5years$

Explanation:

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

$t=\frac{2.303}{k}\log\frac{a}{a-x}$

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.

$t_{\frac{1}{2}}=\frac{0.693}{k}$

$k=\frac{0.693}{2.44\times 10^5}=0.284\times 10^{-5}yr^{-1}$

b) for $6.40\times 10^2$ g of the isotope to decay to  $1.00\times 10^2$

$t=\frac{2.303}{0.284\times 10^{-5}}\log\frac{6.40\times 10^2}{1.00\times 10^2}$

$t=6.54\times 10^5years$

The time for $6.40\times 10^2$ g of the isotope to decay to  $1.00\times 10^2$ is $6.54\times 10^5years$