Can some one explain why in practice, fusion occurs at a much lower temperature.

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TonyDoe
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Hello guys. I was doing a nuclear physics question and the question is:

For the fusion reaction to occur the separation between the deuterium and tritium nuclei must be less than 10–14 m. This means that the average kinetic energy of these hydrogen nuclei needs to be about 70 keV. The energy released by the fusion reaction is 18 MeV.
(i) Calculate the repulsive electrical force between the deuterium and tritium nuclei at a separation of 10^–14 m.
(ii) Assume that a mixture of these hydrogen nuclei behaves as an ideal gas. Estimate the temperature of the mixture of nuclei required for this fusion reaction.
(iii) In practice, fusion occurs at a much lower temperature. Suggest a reason why.

I did (i) and (ii) but can someone explain (iii)? Thanks a lot!
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TonyDoe
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Hello guys. I was doing a nuclear physics question and the question is:

For the fusion reaction to occur the separation between the deuterium and tritium nuclei must be less than 10–14 m. This means that the average kinetic energy of these hydrogen nuclei needs to be about 70 keV. The energy released by the fusion reaction is 18 MeV.
(i) Calculate the repulsive electrical force between the deuterium and tritium nuclei at a separation of 10^–14 m.
(ii) Assume that a mixture of these hydrogen nuclei behaves as an ideal gas. Estimate the temperature of the mixture of nuclei required for this fusion reaction.
(iii) In practice, fusion occurs at a much lower temperature. Suggest a reason why.

I did (i) and (ii) but can someone explain (iii)? Thanks a lot!
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lordaxil
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I expect that the answer they're looking for is quantum tunneling, which allows fusion between the hydrogen nuclei at lower energies than equivalent classical particles, and hence a lower temperature. If it weren't for laws of quantum mechanics, then the sun wouldn't shine (or indeed any other star).
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Mr Wednesday
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(Original post by TonyDoe)
Hello guys. I was doing a nuclear physics question and the question is:

For the fusion reaction to occur the separation between the deuterium and tritium nuclei must be less than 10–14 m. This means that the average kinetic energy of these hydrogen nuclei needs to be about 70 keV. The energy released by the fusion reaction is 18 MeV.
(i) Calculate the repulsive electrical force between the deuterium and tritium nuclei at a separation of 10^–14 m.
(ii) Assume that a mixture of these hydrogen nuclei behaves as an ideal gas. Estimate the temperature of the mixture of nuclei required for this fusion reaction.
(iii) In practice, fusion occurs at a much lower temperature. Suggest a reason why.

I did (i) and (ii) but can someone explain (iii)? Thanks a lot!
Do all the gas atoms have exactly the same energy ? Remember that temperature is an average property, what you need to look at is the Boltzmann distribution which describes the likelihood of an individual particle being “hot” or “cold”. In lots of situations its the small number of “hot” particles that drive the action.
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TonyDoe
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(Original post by Mr Wednesday)
Do all the gas atoms have exactly the same energy ? Remember that temperature is an average property, what you need to look at is the Boltzmann distribution which describes the likelihood of an individual particle being “hot” or “cold”. In lots of situations its the small number of “hot” particles that drive the action.
Ok, so due to the temperature is the average kinetic energy of the atom, some of the atoms would gain enough Ek even the temperature is lower than the temperature we calculated because the actual Ek gained by the atom is higher than the average Ek and meet the requirement. Is that correct?
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Mr Wednesday
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(Original post by TonyDoe)
Ok, so due to the temperature is the average kinetic energy of the atom, some of the atoms would gain enough Ek even the temperature is lower than the temperature we calculated because the actual Ek gained by the atom is higher than the average Ek and meet the requirement. Is that correct?
Yes, pretty much, you don’t need to get every atom up to the right energy to overcome the coulomb barrier at the same time, have a quick google for the Boltzmann Distribution if you have not come across this yet.
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TonyDoe
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(Original post by Mr Wednesday)
Yes, pretty much, you don’t need to get every atom up to the right energy to overcome the coulomb barrier at the same time, have a quick google for the Boltzmann Distribution if you have not come across this yet.
Thanks a lot!
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lordaxil
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Mr Wednesday is quite right that the Boltzmann distribution plays a role, but this cannot be the reason for the fusion reaction to happen at any reasonable rate. Even at core temperature of sun (15 million K), only a neglible fraction of protons would be able to overcome an energy barrier of 70 keV (around \exp (-36) = 2.16 \times 10^{-16}). Quantum effects are absolutely essential to explain why the reaction occurs at lower temperatures.
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