Nuclear fusion problem (2nd semester physics)

[Rockstar47]

Homework Statement

The first step in nuclear fusion on the sun involves the collison of two protons, which form a deutron. Consider two protons far apart on a collison course with equal but opposite velocity. Their average kinetic energy is given by K = 1/2mv^2 = 3/2KbT where Kb is Boltzmann's constant and T is the Kelvin temperature. The reaction can only occur if the protons come into contact. The radius of a proton is rp = 10^-15 meters. What temperature is necessary for this to take place?

Homework Equations

I believe that the most useful equations here are K.E. + P.E. = Total Energy
and Ea = Eb (energy at a is equal to energy at b)

The Attempt at a Solution

I have a strong feeling here that this problem is best solved using the laws of the conservation of energy. As a result I feel I can set the energy of the two protons equal to one another: KE1 + PE1 = KE2 + PE2.

The kinetic energy is given by 3/2KbT. So, I believe I can set
3/2KbT + PE1 = 3/2KbT + PE2. The problem then is that I need to have my potential. That's sort of where I'm stuck. My feeling is that if I had the potentials, I would be able to solve for T. But then again, will everything just cancel out? I sort of get the feeling that the radius of the proton is extra and unnecessary information. My research into this suggests the temperature I'm trying to derive is somewhere in the range of 10 to 15 million K.

Help!

 

[anathema]

Thank you for your post. Your thinking is on the right track, but there are a few things that can be clarified to help you solve this problem.

Firstly, the potential energy in this case is the Coulomb potential energy, which is given by the equation PE = kq1q2/r, where k is the Coulomb constant, q1 and q2 are the charges of the two protons, and r is the distance between them. In this case, since the two protons are positively charged, their potential energy will be positive.

Secondly, you are correct in thinking that the radius of the proton is not necessary for solving this problem. It is given in the problem as a reference for the distance between the two protons, but it is not needed in the calculations.

To solve for the temperature, you can set the initial kinetic energy of the protons equal to the potential energy at the moment of collision. This is because at the moment of collision, the kinetic energy of the protons will be zero, and all of their energy will be in the form of potential energy. So you can set 3/2KbT = kq1q2/r.

Finally, you can use the fact that the reaction can only occur if the protons come into contact to solve for T. This means that the distance between the protons at the moment of collision will be equal to the sum of their radii (2rp). So you can substitute this value for r in the equation and solve for T.

I hope this helps you solve the problem. Let me know if you have any further questions. Good luck!




Scientist

 

[m2003]

Thank you for sharing your thoughts on this problem. It seems like you have a good understanding of the concepts involved and are on the right track. As you mentioned, the conservation of energy is a key principle to use in solving this problem. The potential energy in this case is the electrostatic potential energy, which can be calculated using Coulomb's law. The radius of the proton is indeed necessary for this calculation, as it is a factor in determining the distance between the two protons and therefore the strength of the electrostatic force between them.

To continue with your solution, you can set the potential energy at the starting point (PE1) to zero, as the protons are far apart and have not yet interacted. Then, using the equation for Coulomb's law, you can calculate the potential energy at the point of contact (PE2) and solve for the temperature. This should give you a temperature in the range you mentioned, which is indeed the temperature necessary for nuclear fusion to occur on the sun.

I hope this helps and good luck with your studies! Remember to always keep an open mind and don't be afraid to ask for help when needed. Science is all about collaboration and learning from others.