You want to heat a satellite by radioactive decay of 238 plutonium. Assume you need heating of 200 Watts (J/s), how many Becquerel you need in your plutonium sample, if the decay energy of 238 plutonium is 5.593 MeV? (1 joule = 6.24 x 10^18 eV ) When will 1 mg of Pu be converted into energy?
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To calculate the number of Becquerel (Bq) required, we'll need to convert the desired power output to the energy released per unit time by the 238 plutonium decay process.
First, we'll convert the decay energy of 238 plutonium from MeV to Joules:
Decay energy of 238 plutonium = 5.593 MeV = 5.593 x 1.6 x 10^-13 Joules
(1 joule = 6.24 x 10^18 eV, so 1 MeV = 1.6 x 10^-13 Joules)
Next, we'll calculate the decay rate required to produce 200 Watts:
Decay rate = Power output / Decay energy
Decay rate = 200 / (5.593 x 1.6 x 10^-13) Bq = 2.27 x 10^15 Bq
Therefore, you would need a sample of 238 plutonium that has a decay rate of 2.27 x 10^15 Bq in order to generate the desired 200 Watts of heat.
To calculate when 1 mg of Pu will be converted into energy, we need to determine the total energy released and then divide that by the decay energy per unit mass.
The total energy released:
Total energy released = (Decay rate) x (Decay energy per decay) x (time)
Since we're given that 1 Joule = 6.24 x 10^18 eV, we can convert the decay energy from eV to Joules:
Decay energy per decay = 5.593 MeV = 5.593 x 1.6 x 10^-13 Joules
Let's assume the decay process is 100% efficient and all the energy released is converted into heat. The energy released per unit mass can be calculated as follows:
Energy released per unit mass (J/kg) = (Decay rate) x (Decay energy per decay) / (mass of Pu sample)
Assuming a mass of 1 mg (0.001 kg) for the Pu sample, we can calculate the time required for the conversion:
Time (s) = (Energy released per unit mass) / (Decay energy per decay) = (2.27 x 10^15 Bq x 5.593 x 1.6 x 10^-13 J) / (0.001 kg x 6.24 x 10^18 eV)
Now, we can simplify and solve for time:
Time (s) = (2.27 x 5.593 x 1.6) / (0.001 x 6.24) s
Therefore, 1 mg of Pu will be converted into energy in approximately:
Time (s) = 18.43 seconds.
To determine how many Becquerel (Bq) of 238 plutonium you need to provide heating of 200 Watts, you can start by converting the decay energy of 238 plutonium from MeV to joules.
Given that 1 joule is equal to 6.24 x 10^18 electron volts (eV), you can convert the decay energy of 238 plutonium from MeV to joules as follows:
5.593 MeV * (6.24 x 10^18 eV / 1 joule) = 5.593 * 6.24 x 10^18 joules
Next, you need to find the rate of decay of the radioactive plutonium sample in Bq. The rate of decay is given by the equation:
Activity (Bq) = Decay Constant (s^-1) * Number of Atoms
Since you want to determine the number of Bq needed to provide heating of 200 Watts, you need to solve the equation:
200 Watts = Decay Constant (s^-1) * Number of Atoms * Decay Energy (joules)
Rearranging the equation, you can solve for the number of atoms:
Number of Atoms = 200 Watts / (Decay Constant (s^-1) * Decay Energy (joules))
Now, to find the decay constant (s^-1), you can refer to the half-life of 238 plutonium, which is 87.74 years. The decay constant (λ) is given by the equation:
λ = ln(2) / Half-Life
Substituting the value of the half-life, you can calculate the value of λ.
Finally, you can substitute the decay constant, decay energy, and desired heating power into the equation to find the number of atoms needed. Remember to convert the mass of plutonium from milligrams (mg) to kilograms (kg) for accurate calculations.
As for the second part of your question, the conversion of 1 milligram (mg) of plutonium into energy will depend on the rate of decay. Once you determine the rate of decay in Bq (as calculated above), you can use the equation:
Energy (joules) = Decay Energy (joules) * Decay Constant (s^-1) * Time (in seconds) * Number of Atoms
Since you want to know when 1 mg of plutonium will be converted into energy, you can rearrange the equation and solve for time:
Time (in seconds) = 1 mg / (Decay Constant (s^-1) * Number of Atoms)