Question

If a sample of radioactive isotopes takes 600 minutes to decay from 400 grams to 50 grams, what is the half-life of the isotope?

Answer 1

Answer : The half-life of the radioactive isotope is, 200 min.

Solution :

As we know that the radioactive isotopes decays follow first order kinetics.

So, the expression for rate law for first order kinetics is given by,

`k=(2.303)/(t)\log(a)/(a-x)`

where,

k = rate constant

t = time taken for decay process = 600 min

a = initial amount of the reactant = 400 g

a - x = amount left after decay process = 50 g

Now put the all given values in above equation, we get the value of rate constant.

`k=(2.303)/(600)\log(400)/(50)=3.466* 10^(-3)min^(-1)`

Now we have to calculate the half life of a radioisotope.

Formula used :
`t_(1/2)=(0.693)/(k)`

Putting value of 'k' in this formula, we get the half life.

`t_(1/2)=(0.693)/(3.466* 10^(-3)min^(-1))=199.94min=200min`

Therefore, the half-life of a radioactive isotope is, 200 min.