Respuesta :
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,
[tex]k=\frac{2.303}{t}\log\frac{a}{a-x}[/tex]
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.
[tex]k=\frac{2.303}{600}\log\frac{400}{50}=3.466\times 10^{-3}min^{-1}[/tex]
Now we have to calculate the half life of a radioisotope.
Formula used : [tex]t_{1/2}=\frac{0.693}{k}[/tex]
Putting value of 'k' in this formula, we get the half life.
[tex]t_{1/2}=\frac{0.693}{3.466\times 10^{-3}min^{-1}}=199.94min=200min[/tex]
Therefore, the half-life of a radioactive isotope is, 200 min.
