Absolutely, let’s break down the steps to solve the problem of calculating the energy released in a fission reaction of a plutonium nucleus.
Step 1: Understand the Problem and Given Data
We are provided with:
- The original mass of the plutonium nucleus, [tex]\( 4.986 \times 10^{-27} \)[/tex] kg.
- The total mass of the resulting fragments after fission, [tex]\( 4.198 \times 10^{-27} \)[/tex] kg.
- The speed of light, [tex]\( c = 3.0 \times 10^8 \)[/tex] m/s (a known constant required for energy calculations).
Step 2: Calculate the Mass Difference
The mass difference, [tex]\( \Delta m \)[/tex], can be found by subtracting the total mass of the fragments from the original mass of the plutonium nucleus:
[tex]\[ \Delta m = \text{original mass} - \text{fragmented mass} \][/tex]
Plugging in the values:
[tex]\[ \Delta m = 4.986 \times 10^{-27} \, \text{kg} - 4.198 \times 10^{-27} \, \text{kg} \][/tex]
[tex]\[ \Delta m = 7.88 \times 10^{-28} \, \text{kg} \][/tex]
(Note that the precise value here is [tex]\( 7.879999999999999 \times 10^{-28} \)[/tex] kg.)
Step 3: Calculate the Energy Released
The energy released during the fission reaction can be calculated using Einstein’s mass-energy equivalence formula [tex]\( E = mc^2 \)[/tex]:
[tex]\[ E = \Delta m \times c^2 \][/tex]
Where:
- [tex]\( \Delta m \)[/tex] is the mass difference calculated above.
- [tex]\( c \)[/tex] is the speed of light.
Using the values:
[tex]\[ E = 7.88 \times 10^{-28} \, \text{kg} \times (3.0 \times 10^8 \, \text{m/s})^2 \][/tex]
Perform the calculations:
[tex]\[ E = 7.88 \times 10^{-28} \, \text{kg} \times 9.0 \times 10^{16} \, \text{m}^2/\text{s}^2 \][/tex]
[tex]\[ E = 7.09 \times 10^{-11} \, \text{J} \][/tex]
(Note that the precise value here is [tex]\( 7.091999999999999 \times 10^{-11} \)[/tex] J.)
Step 4: Conclusion
Thus, the energy released in the fission reaction of the plutonium nucleus is approximately [tex]\( 7.09 \times 10^{-11} \)[/tex] joules.
To summarize the results:
- Original mass: [tex]\( 4.986 \times 10^{-27} \)[/tex] kg
- Fragmented mass: [tex]\( 4.198 \times 10^{-27} \)[/tex] kg
- Mass difference: [tex]\( 7.88 \times 10^{-28} \)[/tex] kg
- Energy released: [tex]\( 7.09 \times 10^{-11} \)[/tex] J
This concludes the step-by-step solution for the given problem.
