78.78 g out of 250 g of Thorium-234 would remain after 40 days.
Explanation:
All radioactive samples are unstable in nature and so they will be decaying with time. So the equation of disintegration of the radioactive element is given as
![N=N_{0}e^{-kt}](https://tex.z-dn.net/?f=N%3DN_%7B0%7De%5E%7B-kt%7D)
So N is the mass of the radioactive element at time t and
is the mass of the radioactive element at initial stage. Here k is the disintegration constant and t is the time.
We can determine the disintegration constant from the half life term of any element.
![Half life = \frac{0.693}{k}](https://tex.z-dn.net/?f=Half%20life%20%3D%20%5Cfrac%7B0.693%7D%7Bk%7D)
So, ![k = \frac{0.693}{Half life}](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7B0.693%7D%7BHalf%20life%7D)
Since, the half life time of thorium-234 is known to be 24 days. Then, the disintegration constant is
![days^{-1}](https://tex.z-dn.net/?f=days%5E%7B-1%7D)
As the given questions have the initial mass
as 250 g and t is given as 40 days, then the mass after 40 days will be
![N = 250 * e^{-0.02887*40}= 250 * e^{-1.1548}=78.78 g.](https://tex.z-dn.net/?f=N%20%3D%20250%20%2A%20e%5E%7B-0.02887%2A40%7D%3D%20250%20%2A%20e%5E%7B-1.1548%7D%3D78.78%20g.)
Thus, 78.78 g of thorium-234 would remain after 40 days.