Answer : The correct answer is 96.68 yrs
Radioactivity Decay :
it is a process in which a nucleus of unstable atom emit energy in form of radiations like alpha particle , beta particle etc .
Radioactive decay follows first order kinetics , so its rate , rate constant , amount o isotopes can be calculated using first order equations .
The first order equation for radioactive decay can be expressed as :
----------- equation (1)
Where : N = amount of radioisotope after time "t"
N₀ = Initial amount of radioisotope
k = decay constant and t = time
Following steps can be used to find time :
1) To find deacy constant :
Decay constant can be calculated using half life . Decay constant and half life can be related as :
---------equation (2)
Given : Half life of Strontium -90 = 28.8 years
Plugging value of
in above formula (equation 2) :
![28.8 yrs = \frac{ln 2}{ k }](https://tex.z-dn.net/?f=%2028.8%20yrs%20%3D%20%5Cfrac%7Bln%202%7D%7B%20k%20%7D%20%20)
Multiply both side by k
![28.8 yrs * k = \frac{ln 2 }{k} * k](https://tex.z-dn.net/?f=%2028.8%20yrs%20%2A%20k%20%3D%20%5Cfrac%7Bln%202%20%7D%7Bk%7D%20%2A%20k%20)
Dividing both side by 28.8 yrs
![\frac{28.8 yrs * k}{28.8 yrs} = \frac{ln 2}{28.8 yrs}](https://tex.z-dn.net/?f=%20%5Cfrac%7B28.8%20yrs%20%2A%20k%7D%7B28.8%20yrs%7D%20%3D%20%5Cfrac%7Bln%202%7D%7B28.8%20yrs%7D%20%20)
(ln 2 = 0.693 )
k = 0.0241 yrs⁻¹
Step 2 : To find time :
Given : N₀ = 10.3 ppm N = 1.0 ppm k = 0.0241 yrs⁻¹
Plugging these value in equation (1) as :
![ln (\frac{1.0 ppm}{10.3 ppm} ) = - 0.0241 yrs^-^1 * t](https://tex.z-dn.net/?f=%20ln%20%28%5Cfrac%7B1.0%20ppm%7D%7B10.3%20ppm%7D%20%29%20%3D%20-%200.0241%20yrs%5E-%5E1%20%2A%20t%20%20)
![ln (0.0971 ) = -0.0241 yrs ^-^1 * t](https://tex.z-dn.net/?f=%20ln%20%280.0971%20%29%20%3D%20-0.0241%20yrs%20%5E-%5E1%20%2A%20t%20%20)
(ln 0.0971 = - 2.33 )
Dividing both side by - 0.0241 yrs⁻¹
![\frac{-2.33}{-0.0241 yrs^-^1} = \frac{-0.0241 yrs^-^1 * t}{-0.0241 yrs^-^1}](https://tex.z-dn.net/?f=%20%5Cfrac%7B-2.33%7D%7B-0.0241%20yrs%5E-%5E1%7D%20%3D%20%5Cfrac%7B-0.0241%20yrs%5E-%5E1%20%2A%20t%7D%7B-0.0241%20yrs%5E-%5E1%7D%20%20%20)
t = 96.68 yrs
Hence the concentration of Strontium-90 will drop from 10.3 ppm to 1.0 ppm is 96.68 yrs