Question

I've just discovered a new radioactive element. At 1:00 pm I measure 10 grams of the element, but at 1:24 pm I measure only 1.25 grams of the element (with 8.75 grams of daughter product). How long is each half life, in minutes? (Just enter the number).

Answer 1

Answer: Each half life is 8 minutes long.

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  = 24 minutes

a = initial amount of the reactant  = 10 grams

a - x = amount left after decay process  = 1.25 grams

$24min=\frac{2.303}{k}\log\frac{10}{1.25}$

$k=\frac{2.303}{24}\log\frac{10}{1.25}$

$k=0.0866min^{-1}$

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.

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

$t_{\frac{1}{2}}=\frac{0.693}{0.0866min^{-1}}=8.00min$

Thus half life is 8 minutes long