# Velocity of high temperature hydrogen atom

1. Sep 11, 2009

### timbot

What is the relative velocity of two hydrogen atoms colliding head on at a temperature of 10 million Kelvin?

How do you calculate this?

2. Sep 11, 2009

### tiny-tim

timmy-welcome to pf!

hi timbot! timmy-welcome to pf!
You find the actual velocity of one atom, and then double it (or use the formula 2v/(1 + v2) if v is near the speed of light).

3. Sep 11, 2009

### Bob S

To convert temperature into electron volts, use the Boltzmann constant k = 8.617 x 10-5 eV per degree Kelvin.

4. Sep 11, 2009

### hamster143

people often memorize it the other way around: 1 eV = 11605 Kelvin.

5. Sep 11, 2009

### timbot

Many thanks folks. But what is, say, 1000 ev in terms of a relative speed of two head on hydrogen atoms in kilometers per hour?

Maybe this sounds a stupid question, but it seems that nobody has asked this question before. My guess this velocity is not very high.

6. Sep 12, 2009

### hamster143

Mass of a hydrogen atom is roughly 1 GeV. 1000 eV is much less than 1 GeV, so we know that the atom is non-relativistic and we can use a non-relativistic formula for kinetic energy

mv^2/2 = 1000 eV
mc^2 = 1 GeV

-> v^2 / c^2 = 2*10^-6

v = 0.0014*c ~ 400 km/s

Relative velocity of one atom wrt the other is double that.

7. Sep 12, 2009

### timbot

Fantastic!!

Lets go over this again.

The temperature of hydrogen at 10 million Kelvin is when it begins to fuse into helium in a Tokomak.

1ev is 11605 kelvin.

10 million Kelvin is 861.7 electron volts

For 1000 eV (slightly higher) the relative velocity for head on fusion is 800 km/s.

This is 48000 km/minute or 2,880,000 km/hour.

That'a a shame. If that figure was three zeros less I was thinking that plasma fusion would be possible by firing two jets of hydrogen plasma directly into each other.

Maybe there is a probabilistic "fix" on this? All we need is a small proportion to fuse.

8. Sep 12, 2009

### Bob S

Hi Timbot-
Two deuterons might fuse, but two protons will never fuse. There is no bound state of two protons.
Bob S

9. Sep 12, 2009

### timbot

This shows my lack of knowledge of physics. I suppose with all this talk of fusing hydrogen, the term they should really use is fusing deutronium?

At the expense of straining everyones patience, I have an additional question.

What then is the energy required (no doubt measured in electron volts) to fuse deutronium nuclei into helium?

From this we can go back to the exercise of calculating the head-on relative velocity of two deutronium atoms in kilometers per hour necessary for them to fuse into hydrogen.

I am just hoping that with twice the mass of the nucleus this velocity is in the feasible region.

10. Sep 13, 2009

### timbot

Correction to what I wrote above. ".....fuse into helium." is what I should have written in the second last paragraph.

And yes, deuterium, with a proton and neutron nucleus, has twice the mass of a hydrogen nucleus.