<u>Answer:</u> The energy released for the decay of 3 grams of 230-Thorium is ![2.728\times 10^{-15}J](https://tex.z-dn.net/?f=2.728%5Ctimes%2010%5E%7B-15%7DJ)
<u>Explanation:</u>
First we have to calculate the mass defect
.
The equation for the alpha decay of thorium nucleus follows:
![_{90}^{230}\textrm{Th}\rightarrow _{88}^{226}\textrm{Ra}+_2^{4}\textrm{He}](https://tex.z-dn.net/?f=_%7B90%7D%5E%7B230%7D%5Ctextrm%7BTh%7D%5Crightarrow%20_%7B88%7D%5E%7B226%7D%5Ctextrm%7BRa%7D%2B_2%5E%7B4%7D%5Ctextrm%7BHe%7D)
To calculate the mass defect, we use the equation:
Mass defect = Sum of mass of product - Sum of mass of reactant
![\Delta m=(m_{Ra}+m_{He})-(m_{Th})](https://tex.z-dn.net/?f=%5CDelta%20m%3D%28m_%7BRa%7D%2Bm_%7BHe%7D%29-%28m_%7BTh%7D%29)
![\Delta m=(225.9771+4.008)-(229.9837)=1.4\times 10^{-3}amu=2.324\times 10^{-30}kg](https://tex.z-dn.net/?f=%5CDelta%20m%3D%28225.9771%2B4.008%29-%28229.9837%29%3D1.4%5Ctimes%2010%5E%7B-3%7Damu%3D2.324%5Ctimes%2010%5E%7B-30%7Dkg)
(Conversion factor:
)
To calculate the energy released, we use Einstein equation, which is:
![E=\Delta mc^2](https://tex.z-dn.net/?f=E%3D%5CDelta%20mc%5E2)
![E=(2.324\times 10^{-30}kg)\times (3\times 10^8m/s)^2](https://tex.z-dn.net/?f=E%3D%282.324%5Ctimes%2010%5E%7B-30%7Dkg%29%5Ctimes%20%283%5Ctimes%2010%5E8m%2Fs%29%5E2)
![E=2.0916\times 10^{-13}J](https://tex.z-dn.net/?f=E%3D2.0916%5Ctimes%2010%5E%7B-13%7DJ)
The energy released for 230 grams of decay of thorium is ![2.0916\times 10^{-13}J](https://tex.z-dn.net/?f=2.0916%5Ctimes%2010%5E%7B-13%7DJ)
We need to calculate the energy released for the decay of 3 grams of thorium. By applying unitary method, we get:
As, 230 grams of Th release energy of = ![2.0916\times 10^{-13}J](https://tex.z-dn.net/?f=2.0916%5Ctimes%2010%5E%7B-13%7DJ)
Then, 3 grams of Th will release energy of = ![\frac{2.0916\times 10^{-13}}{230}\times 3=2.728\times 10^{-15}J](https://tex.z-dn.net/?f=%5Cfrac%7B2.0916%5Ctimes%2010%5E%7B-13%7D%7D%7B230%7D%5Ctimes%203%3D2.728%5Ctimes%2010%5E%7B-15%7DJ)
Hence, the energy released for the decay of 3 grams of 230-Thorium is ![2.728\times 10^{-15}J](https://tex.z-dn.net/?f=2.728%5Ctimes%2010%5E%7B-15%7DJ)