Question

Element X is a radioactive isotope such that every 22 years, its mass decreases by
half. Given that the initial mass of a sample of Element X is 50 grams, how long
would it be until the mass of the sample reached 41 grams, to the nearest tenth of a
year?

Answer 1

Answer: The time taken for the mass to reduce from 50 g to 41 g is 6 years

Step-by-step 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.

$t_{\frac{1}{2}}=22 years$

$k=\frac{0.693}{t_{\frac{1}{2}}}=\frac{0.693}{22}=0.0315years^{-1}$  

b) for mass to reduce from 50 g to 41 g

$t=\frac{2.303}{0.0315}\log\frac{50}{41}$

$t=6years$  

The time taken for the mass to reduce from 50 g to 41 g is 6 years