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 $4.4\times 10^2g$ to $1.0\times 10^2$ is $5.29\times 10^5years$

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.

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

$k=\frac{0.693}{2.44\times 10^5year}=0.28\times 10^{-5}year^{-1}$

b) time taken for the sample to decay from $4.4\times 10^2g$ to $1.0\times 10^2$

$t=\frac{2.303}{0.28\times 10^{-5}}\log\frac{4.4\times 10^2}{1.0\times 10^2}$

$t=5.29\times 10^5years$

The time time taken for the sample to decay from $4.4\times 10^2g$ to $1.0\times 10^2$ is $5.29\times 10^5years$