Answer :
To determine the time required to produce 7 kg of Pu-239, we need to consider the rate of production and the number of fission events. We know that for every three U-235 fission events, two Pu-239 nuclei are produced. We also have the energy released per fission event, which is 30pJ.
First, we need to calculate the total energy required to produce 7 kg of Pu-239. Since 1mg of Pu-239 contains 2×10^18 nuclei, 7 kg (or 7,000,000 mg) would contain[tex]1.4×10^25[/tex] nuclei. So the total energy required can be calculated as[tex]1.4×10^25[/tex] nuclei × 30pJ per fission [tex]= 4.2×10^25[/tex]pJ. Next, we can calculate the number of fission events required to produce this energy. Since each fission event generates 30pJ, the total number of fission events needed is 4.2×10^25 pJ / 30pJ per fission = 1.4×10^24 fission events.
To find the time needed, we need to divide the number of fission events by the power output of the nuclear power plant. The power output is given as 1,145 MW, which is equivalent to [tex]1.145×10^9[/tex]Watts. Using the formula time = number of fission events / power, we get the time as [tex]1.4×10^24 / 1.145×10^9 = 1.22×10^15[/tex] seconds. Therefore, it would take approximately[tex]1.22×10^15[/tex]seconds to produce 7 kg of Pu-239 in a 1,145MW nuclear power plant.
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It takes approximately 2.5 x 10^31 hours to produce 7 kg of Pu-239 in the given nuclear power plant.
To calculate the time it takes to produce 7 kg of Pu-239, we need to consider the following information:
Each U-235 fission event produces two Pu-239 nuclei for every three U-235 fission events.
Each U-235 fission event generates 30 pJ of energy.
1 mg of Pu-239 contains 2x10^18 Pu-239 nuclei.
First, let's find out how many Pu-239 nuclei are produced per joule of energy generated:
Number of Pu-239 nuclei produced per joule = (2 nuclei / 3 fission events) / 30 pJ
Number of Pu-239 nuclei produced per joule = (2/3) / (30 * 10^-12)
Number of Pu-239 nuclei produced per joule ≈ 2.22 x 10^10 nuclei/J
Next, let's find out how many joules are produced per kg of Pu-239:
Energy produced per kg of Pu-239 = 7 kg * (2 x 10^18 nuclei/mg) * (2.22 x 10^10 nuclei/J)
Energy produced per kg of Pu-239 ≈ 3.11 x 10^29 J
Now, we know that each fission event generates 30 pJ of energy. Let's find out how many fission events are required to produce the energy needed for 7 kg of Pu-239:
Number of fission events = Energy produced per kg of Pu-239 / Energy per fission event
Number of fission events = 3.11 x 10^29 J / 30 x 10^-12 J
Number of fission events ≈ 1.037 x 10^41 fission events
Finally, let's calculate the time it takes to produce this many fission events at the given power of the nuclear power plant:
Time (in seconds) = Number of fission events / Power of the nuclear power plant
Time (in seconds) = 1.037 x 10^41 / 1,145 x 10^6 W
Time (in seconds) ≈ 9.05 x 10^34 seconds
Now, let's convert the time to hours:
Time (in hours) = 9.05 x 10^34 seconds / (60 seconds * 60 minutes)
Time (in hours) ≈ 2.514 x 10^31 hours
Rounded to one decimal place, it takes approximately 2.5 x 10^31 hours to produce 7 kg of Pu-239 in the given nuclear power plant.
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