Calculation on spontaneous fission and effect from alpha decay

Luxdot
Summary:: Help needed on how to calculate on spontaneous fission and effect from alpha decay

Heat from alpha decay from Pu-238 is used to generate direct current. At the beginning (1977) it generated 470W, how large is the effect now? And if the efficiency between the electricity and heat transfer is 6%, how much did the fuel weight at the start?
Another question that I have is if a small fraction (7*10^-9 %) of the decay from U-235 occurs through spontaneous fission, how many spontaneous fissions occur per hour in 1 kg U-235. The half life time is 7,038*10^8 years.

Staff Emeritus
One can take the initial power of 470 W and divide by the specific energy of Pu-238 to give the mass required to generate to total thermal energy. That will give the activity, A, and the activity is simply the product of the decay constant, λ, and the number of atoms present, N, at a given time.

hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/halfli2.html

From a mass of U-235, one can calculate the number of atoms present and multiply that number by the decay constant to get the activity, or decay rate, as explained in the Hyperphysics link.

Mentor
Welcome to PF.
Summary:: Help needed on how to calculate on spontaneous fission and effect from alpha decay

And if the efficiency between the electricity and heat transfer is 6%, how much did the fuel weight at the start?
This sounds like it could be a homework problem. Is this for schoolwork?

Luxdot
One can take the initial power of 470 W and divide by the specific energy of Pu-238 to give the mass required to generate to total thermal energy. That will give the activity, A, and the activity is simply the product of the decay constant, λ, and the number of atoms present, N, at a given time.

hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/halfli2.html

From a mass of U-235, one can calculate the number of atoms present and multiply that number by the decay constant to get the activity, or decay rate, as explained in the Hyperphysics link.
Thank you! It’s a bit clearer now! However, I still don’t understand how I get the current effect….

Luxdot
Welcome to PF.

This sounds like it could be a homework problem. Is this for schoolwork?
Thanks! Kind of, I’m trying to learn a bit about nuclear power and found some old course material without solutions….. and I’m struggling a bit

Mentor
Okay, I'll move this to the schoolwork forums then. Even self-study problems go there.

Luxdot
Staff Emeritus
Thank you! It’s a bit clearer now! However, I still don’t understand how I get the current effect….
If the 470 W is the electrical rather than the thermal power, then one should realize that electrical power (Pe) = thermal power (Pt) * efficiency, or conversely, Pt = Pe/efficiency.

Luxdot
If the 470 W is the electrical rather than the thermal power, then one should realize that electrical power (Pe) = thermal power (Pt) * efficiency, or conversely, Pt = Pe/efficiency.
The 470 W comes from the radioisotope thermoelectric generator that uses the heat from the alpha decay of Pu-238. According to the task the first part is to calculate the current effect (470W was in 1977) and the second part is to calculate the weight of the fuel if the efficiency rate was 6%. So for the first part I shouldn’t use the mass of the fuel, right? Sorry if I am a bit confused, this is new to me….

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The 470 W comes from the radioisotope thermoelectric generator that uses the heat from the alpha decay of Pu-238. According to the task the first part is to calculate the current effect (470W was in 1977) and the second part is to calculate the weight of the fuel if the efficiency rate was 6%. So for the first part I shouldn’t use the mass of the fuel, right? Sorry if I am a bit confused, this is new to me….
Yes, for the first part you do not need the efficiency, you just have to assume it does not change.
Had you posted in the homework forum initially, you would have seen a template that invited you to state some Relevant Equations. Do you know or can you find any for radioactive decay? E.g. if you start with N atoms of a known half life, what is the rate of decay (atoms per unit time) and how many will remain undecayed after time t?

Luxdot
Yes, for the first part you do not need the efficiency, you just have to assume it does not change.
Had you posted in the homework forum initially, you would have seen a template that invited you to state some Relevant Equations. Do you know or can you find any for radioactive decay? E.g. if you start with N atoms of a known half life, what is the rate of decay (atoms per unit time) and how many will remain undecayed after time t?
I see! I know the formula for radioactive decay, but the initial number of atoms, N is not stated. Can this number be found or calculated for PU-235? The only known factors I have is the time, effect and that Pu-235 is used.

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I see! I know the formula for radioactive decay, but the initial number of atoms, N is not stated. Can this number be found or calculated for PU-235? The only known factors I have is the time, effect and that Pu-235 is used.
For the first part you do not need to know N. Just use it as an unknown as necessary and it will cancel out.
For the next part, you can calculate N from the amount of heat energy being released in total.

Luxdot
For the first part you do not need to know N. Just use it as an unknown as necessary and it will cancel out.
For the next part, you can calculate N from the amount of heat energy being released in total.
Okey, so N(t)=N(0)*e-lambda*t and lambda=ln(2)/T1/2. This gives me that lambda=0,0079. So N(t)=N(0)*e-0,0079*44. How do get rid of N(0) and I still don't see how I will get the current power?

Luxdot
I solved the current power part! But I'm struggling a bit with the initial weight of the fuel. The way I've done it is by taking the electrical power of 470W and divided that by the efficiency to get the thermal power, this gives me 470/0,06=7834W. I then multiply this by the time (44 years in seconds) to get the Ws. 7834*(44*365*24*60*60) and then convert this to Joule, I get 1,087*1013 J. I then use E=mc2 and solve it for m. This gives me that m=1,087*1013/(3*108)2 = 1,2*10-4 kg. After some googling I found that the initial weight of the plutonium was around 2,4 kg, what have I done wrong?

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I then multiply this by the time (44 years in seconds)
But, as you calculated, the power has been diminishing those 44 years.
Anyway, you don't need to worry with how much total energy has been generated. If initially you got 7834W of thermal power, how many atoms were decaying per second? How many atoms does that imply in total at that time?
I then use E=mc2 and solve it for m.
And what will that m be the mass of?

Luxdot
But, as you calculated, the power has been diminishing those 44 years.
Anyway, you don't need to worry with how much total energy has been generated. If initially you got 7834W of thermal power, how many atoms were decaying per second? How many atoms does that imply in total at that time?

And what will that m be the mass of?
That makes sense, my problem is how I go from knowing the thermal power to how many atoms are decaying per second. The m would be the mass of the plutonium?

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The m would be the mass of the plutonium?
No, it is whatever mass is completely converted to energy. What is the mass of one Pu-238 atom and what is the sum of masses of the decay products?
How much mass has disappeared?

Luxdot
No, it is whatever mass is completely converted to energy. What is the mass of one Pu-238 atom and what is the sum of masses of the decay products?
How much mass has disappeared?
I see! The mass of 1 Pu-238 atom is 238/6,022*1023 = 3,95*10-22. How do I fid the total weight of the initial amount of plutonium from this?

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I see! The mass of 1 Pu-238 atom is 238/6,022*1023 = 3,95*10-22. How do I fid the total weight of the initial amount of plutonium from this?
What units?
As I wrote, you also need the masses of all the decay products. One is an alpha particle. What else?

Luxdot
What units?
As I wrote, you also need the masses of all the decay products. One is an alpha particle. What else?
Grams? It is not stated what the decay products are... Is this something that is the same for decay with plutonium? In the task it is stated that it uses the heat from the alpha decay. The question is about the Voyager 1 and how much plutonium it had in the beginning.

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Grams? It is not stated what the decay products are... Is this something that is the same for decay with plutonium? In the task it is stated that it uses the heat from the alpha decay. The question is about the Voyager 1 and how much plutonium it had in the beginning.
I then use E=mc2 and solve it for m. This gives me that m=1,087*1013/(3*108)2 = 1,2*10-4 kg.
This is the mass loss, which is a very small fraction of the Pu-238 remaining.

Okey, so N(t)=N(0)*e-lambda*t and lambda=ln(2)/T1/2. This gives me that lambda=0,0079. So N(t)=N(0)*e-0,0079*44. How do get rid of N(0)
One does not get rid of N(0)! One is trying to solve for N(0) in the first part of the problem.

N(0) is related to the total mass of the Pu-238.

The thermal power at any given time is related to the decay activity of the active isotope, which in this case is Pu-238. The total thermal power (Pt (t)) = e * A(t), where e is the energy per decay, and A(t) is the activity or decay rate (decays/time). A(t) = λ * N(t), where N(t) is the number of atoms of radioisotope of interest.

Focus on the Pu-238, which decays to U-234 by alpha decay. The half-life of Pu-238 is 87.7 y. The half-life of U-234 is 2.455×10+5 y, so it does not contribute much energy (heat) to the system, one because there is very little, and secondly because it has a relatively long half-life.

Luxdot
This is the mass loss, which is a very small fraction of the Pu-238 remaining.

One does not get rid of N(0)! One is trying to solve for N(0) in the first part of the problem.

N(0) is related to the total mass of the Pu-238.

The thermal power at any given time is related to the decay activity of the active isotope, which in this case is Pu-238. The total thermal power (Pt (t)) = e * A(t), where e is the energy per decay, and A(t) is the activity or decay rate (decays/time). A(t) = λ * N(t), where N(t) is the number of atoms of radioisotope of interest.

Focus on the Pu-238, which decays to U-234 by alpha decay. The half-life of Pu-238 is 87.7 y. The half-life of U-234 is 2.455×10+5 y, so it does not contribute much energy (heat) to the system, one because there is very little, and secondly because it has a relatively long half-life.
I still don't quite get it... Could you please elaborate a bit?

Staff Emeritus
I still don't quite get it... Could you please elaborate a bit?
What don't you get? I provided sufficient detail to solve the problem, and even provided a link to the Hyperphysics discussing the topic of radioactive decay.

For the current problem, if one determines the total energy (power integrated over time) and then calculates the mass producing that energy, one only calculates the mass of material consumed, not the initial mass, or the mass remaining at a given time.

Edit/update: After giving this problem some more thought, I realized that the time frame is 44 years, which is approximately half of one half-life (87.7 years), which would make it easier to solve.

Last edited:
Luxdot
Solved it! Could you please help me to understand how I calculate the number of spontaneous fissions that occur every hour in 1 kg of U-235, knowing that 7*10-9% of the decay from U-235 occurs spontaneously.

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