Anybody know enough nuclear physics to calculate out how hot (i.e. fast)
the helium/alpha exhaust would be going? (clearly, you have to dump the
electrons as well - pointing that stream could be interesting as well.)
Cheers!
Rich
Greetings,
Since this post is fairly long the answer is in the last two paragraphs for
those that might be too put off by length to read the whole thing. That
said...
The first step in making this calculation would be to find the "Q" value for
this reaction. Of course this reaction (four protons coming together to
form an alpha particle) doesn't really occur in real life, so this is just a
theoretical exercise...
The Q value tells us how much energy is released by this reaction. We use
Einstein's E=mc^2 formula along with the mass defect (difference in masses
between the four individual separate protons and the mass of the single
alpha particle).
mass proton = 1.0072764669 amu
mass neutron = 1.0086649233 amu
mass electron = 5.48579911E-4 amu
Helium 4 atom = 4.0026032497 amu
Approximate alpha particle mass = mass(He-4 atom) - 2*mass(electron)
Approximate alpha particle mass = 4.001506089878 amu
Note: MeV stands for mega electron volts.
So mass defect between four protons and one alpha particle ~= 4*1.0072764669
amu - 4.001506089878 amu = 0.02759977772 amu. To find the Q value simply
multiply the mass defect by the conversion factor 931.5MeV/amu.
Theoretical Q for this reaction = 0.02759977772 amu * 931.5 MeV/amu = 25.71
MeV.
This is how much energy would theoretically be liberated by this reaction.
A small amount of mass was converted into this much energy.
For comparisons sake a single fission of some atom such as U-235 releases
approximately 200MeV, while the oxidation of say carbon liberates only a few
electron volts.
Okay, so now how fast is the alpha moving? Well in theory all that 25.71
MeV is transferred into kinetic energy of the product(s) (in this case an
alpha particle). There is a slight problem here in that an alpha particle
is composed of 2 neutrons and 2 protons, not 4 protons. From the
information given above we see a neutron is actually slightly more massive
than a proton. In practice the kinetic energy of the alpha particle in this
theoretical reaction would probably be approximately 1.2MeV less than the
calculated 25.71MeV figure since some of that energy would have to get used
up creating something like a couple of positrons or something so that a
couple of the protons can convert into neutrons. So as a decent
approximation the alpha particle's kinetic energy would be around 24.5MeV.
How fast is that?
Well use the formula 0.5mv^2 = kinetic energy. Where m is mass in kg, and v
is the velocity of the particle in meters per second. You have to be
careful in this step to make sure you don't use classical mechanics if the
particle is moving at relativistic velocities. In this case 24.5MeV for an
alpha particle is quite "slow", so classical mechanics will yield a fairly
accurate result.
So first convert MeV into joules. 24.5MeV (1.6022E-19 J/1 eV) = 3.93E-12 J.
Now find the mass of an alpha particle in kilograms.
4.001506089878 amu * (1.6605387E-27 kg / amu) = 6.6446557205E-27 kg
Solving for velocity v = sqrt(2*kinetic energy / mass) = 34373216
meters/second
Or in other words 3.44E7 m/s
So this "exhaust" for this theoretical reaction is boogieing along with
pretty high velocity. For comparisons sake room temperature (ie: 300 K)
particles have about 0.025eV of kinetic energy. Of course I want to
reemphasize that this reaction does not occur in real life. Even single
proton-proton --> deuterium fusion reactions are exceedingly "slow" at
occurring, much less four protons all perfectly coming together all at once.
