Amount of Energy to Fuse Two Atoms

JordanGo

I was wondering if there was a simplified equation to determine the amount of energy required to fuse two atoms together (for example a sodium atom with a hydrogen atom to form a magnesium atom).

QuantumPion

Yes. You simply find the difference in mass * c² between the starting products and ending products.

M_Na-23*c² + M_H-1*c² - M_Mg-24*c² = Energy released

There is a simple calculator located at this link which does it for you: http://www.nndc.bnl.gov/qcalc/

JordanGo

Thank you, I wasn't sure if it was as simply as E=mc2 or there was something else more to it. But I guess not! Thanks again, highly appreciate you time!

Actually, I thought of something...

Does it not depend on the energy required to hold the atoms together instead of the energy of rest mass?

Drakkith

It does not require energy to hold the nucleus together, that is done by the strong nuclear force. When you fuse two nuclei together, if the end product of the reaction is less massive than the combined mass of the nuclei prior to fusion, the reaction is exothermic and will release energy. If it is more massive then it is endothermic and will require energy to react. Do you know what energy is?

JordanGo

Not sure... Can you describe this energy? Don't hold back on description, I'm in my third year university physics, so the more complex the better!

Drakkith

To my knowledge energy is the ability of one system to perform work on another. If a proton and a neutron are being held together inside the nucleus, is work being done? No. This is similar to a book being left on a table. Is work being performed to keep the book on the table? No, there is no change in the system so no work is being done. Since no work is being done then energy isn't a useful way of describing anything in the system. AKA it takes zero energy to keep a book on a table or to keep two nucleons bound together.

AdrianTheRock

@JordanGo - are you, perhaps, referring to the kinetic energy required to overcome the Coulomb repulsion between the two nuclei in order to get them close enough together for fusion to be able to happen?

For nuclei whose combinations are below Iron, fusion is either exothermic or just doesn't occur (an example of the latter would be ⁴He + ⁴He → ⁸Be). But before the reaction can happen you do need to provide enough energy to allow the nuclei to overcome the electromagnetic force and get to within the (fm-scale) range of one another where the nuclear force kicks in. Once the latter happens, though, it releases further energy, while the original KE remains.

JordanGo

Yes, that is what I was wondering about. Thank you Drakkith, that was also a very good answer, its interesting!

Now, I've never taking a particle class yet, I'm just being curious! So can someone do a worked example for me: let's say you want to fuse a hydrogen atom to a sodium atom to make magnesium (there's no isotopes). How much energy is required to do this?

QuantumPion

Use the link I provided in my first post.

QuantumPion

Oh, were you asking how to calculate the coulomb barrier? In that case the equation for a single particle pair is [itex] E = \frac{k Z_1 Z_2 e^2}{r}[/itex] where k is the coulomb constant, the Z's are the atomic numbers, and r is the interaction radius.

Calculating the temperature for fusion for a collection of particles is a bit more complicated as there are quantum mechanical and statistical effects to consider. It is lower than the required energy for a single particle.