Question

Element X is a radioactive isotope such that every 82 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 8900 grams, how long would it be until the mass of the sample reached 7700 grams, to the nearest tenth of a year

Answer 1

Answer: 17 years

Explanation:

Expression for rate law for first order kinetics  for radioactive substance 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.

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

$k=\frac{0.693}{82years}=8.4\times 10^{-3}years^{-1}$

b) for 8900 g of the mass of the sample to reach  7700 grams

$t=\frac{2.303}{8.4\times 10^{-3}}\log\frac{8900}{7700}$

$t=17years$

Thus it will take 17 years