Question

Element X is a radioactive isotope such that every 51 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 7500 grams, how long would it be until the mass of the sample reached 6900 grams, to the nearest tenth of a year?

Answer 1

Answer: It will take 6 years for 7500 grams of X to reach 6900 grams.

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.

$51=\frac{0.69}{k}$

$k=\frac{0.69}{51}=0.0135years^{-1}$

b) for  7500 g to reach to 6900 g

$t=\frac{2.303}{0.0135}\log\frac{7500}{6900}$

$t=6years$

It will take 6 years for 7500 grams of X to reach 6900 grams.