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Answer :
It would take 493,150 seconds to produce 6 kg of Pu-239 in the given nuclear power plant.
To determine how long it takes to produce 6 kg of Pu-239, we need to calculate the number of Pu-239 nuclei that will be produced and then convert it into the mass of Pu-239.
Given that each U-235 fission event produces two Pu-239 nuclei for every three U-235 fission events, we can calculate the ratio of Pu-239 nuclei produced per U-235 fission event as follows:
2 Pu-239 nuclei / 3 U-235 fission events
Now, we need to find the number of U-235 fission events required to produce 6 kg of Pu-239. To do this, we'll need to calculate the number of Pu-239 nuclei in 6 kg:
Number of Pu-239 nuclei = (6 kg) / (1 mg) * (2×[tex]10^18[/tex] Pu-239 nuclei/mg)
Substituting the given values, we get:
Number of Pu-239 nuclei = 6 * [tex]10^6[/tex]* 2 * 10^18 = 12 * [tex]10^24[/tex] Pu-239 nuclei
Next, we can find the number of U-235 fission events required to produce this many Pu-239 nuclei:
Number of U-235 fission events = Number of Pu-239 nuclei / (2 Pu-239 nuclei / 3 U-235 fission events)
Substituting the values, we get:
Number of U-235 fission events = (12 * [tex]10^24[/tex] Pu-239 nuclei) / (2 Pu-239 nuclei / 3 U-235 fission events)
Simplifying the expression, we find:
Number of U-235 fission events = 18 * [tex]10^24[/tex] U-235 fission events
Now, we need to calculate the amount of energy produced by these fission events. Given that each U-235 fission generates 30 pJ of energy, we can calculate the total energy produced as follows:
Energy = Number of U-235 fission events * Energy per fission
Substituting the values, we get:
Energy = (18 * [tex]10^24[/tex] U-235 fission events) * (30 pJ/fission)
Simplifying the expression, we find:
Energy = 540 *[tex]10^24[/tex] pJ
Finally, we need to determine the time it takes to produce this amount of energy. Since power is defined as the energy produced per unit time, we can use the power formula to find the time:
Power = Energy / Time
Rearranging the formula, we get:
Time = Energy / Power
Substituting the values, we get:
Time = (540 * [tex]10^24[/tex]pJ) / (1,095 MW) = (540 * [tex]10^24[/tex] * [tex]10^-12[/tex] J) / (1,095 * [tex]10^6[/tex] W)
Simplifying the expression, we find:
Time = (540 * [tex]10^12[/tex] J) / (1,095 * [tex]10^6[/tex]W) = (540 / 1,095) * [tex]10^6[/tex] s
Calculating the value, we find:
Time ≈ 493,150 s
Therefore, it would take approximately 493,150 seconds to produce 6 kg of Pu-239 in the given nuclear power plant.
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