The energy of nuclear reactions
masses are listed here. From this table we can see that most
of the nuclear energy produced in the universe comes from the fusion
of hydrogen into helium in the main sequences of stars.
The fact that deuterium is much less abundant than helium, means
that it undergoes further reactions soon after it is produced. Thus
the fusion of hydrogen into deuterium appears the slowest process in
which thus determines its overall speed. But this is also the least
energetical reaction: it gives away "only" about 0.077% of its mass
into energy, and part of this energy (a half ?) is lost in the form
of neutrinos. Thus the energy it gives to the star is equal to the
energy E=mv²/2 the same quantity of matter undergoing this reaction
would have if sent at the speed v = c √
0.00077 = 0.0277
c = 8318 km/s.
Then, the fusion of deuterium into helium is the most energetical
reaction. It converts 0,64% of the mass into energy, that is the
energy it would have at the speed of c √2*
0.0064 = 34000
The 3 next elements (Li, Be, B) have a higher mass per nucleus than
He, and thus are not produced in significant quantities (they easily
disappear by further reactions into other elements). Next is carbon
: the fusion of helium into carbon, in the giant phase of stars,
releases 0.065% of the mass into energy, corresponding to the energy
of a speed of 11000 km/s.
The next important reaction is from carbon to oxygen. The rest of
elements only have a small place in the composition
of the universe.
See the binding
energy curve for a global picture of the difference of mass
per nucleus between the different elements.
The fact that chemical energies are much smaller per mass, is the
reason why we don't speak about the difference of mass in a
reaction : this difference would be too small to be measurable in
The radius of nuclei
The nuclei have a roughly constant density of protons and
neutrons : their radius R as function of the atomic mass number A
satisfies R=r0 A1/3 where r0 = 1.25 fm
(femtometre) = 1.25 × 10−15 m, so that each
individual nuclus roughly takes the volume of a sphere with radius
In fact the law that determines the volume taken is the Pauli exclusion principle,
saying that no two particles can have the same quantum state. But
in fact 4 nuclei can take the same space-time location : a pair of
protons and a pair of neutrons (as they are spin 1/2 particles,
each one has 2 possible spin states). The total number of states
of movement taken by the nucleus is thus A/4.
Each pack of 4 nuclei takes the volume of a sphere with radius r0*41/3=
To be compared with the range L of the interaction between nuclei
(such that the intensity of interaction depending on distance x
undergoes a decrease by the factor exp(-x/L)), deduced from the
mass 135 MeV/c2 of pions
L= ℏc/135 MeV= 1.46 fm.
As quantum physics gives each state of movement a volume of h3
in the 6-dimensional
phase space (3 coordinates of position, 3 coordinates of
momentum), the volume it takes in this phase space is h3A/4=VV'
where V is the space volume V=4πR3/3, and V' is the
momenta volume V'=4πp3/3 where p is the maximum momentum
taken by a nucleon, p=Mv where v
is the maximum speed and M=1.67×10−27 kg is
the mass of a nucleon.
This equation thus says (R/r0)3h3/4=(4π/3)2(Rp)3
It simplifies into v=((3/π)-2/3/4)h/Mr0=85,000
It is no coincidence that this result has the same order of
magnitude as those calculated in the previous section.
Now if quantum mechanics determines the nuclei to be so small, it
also determines, from the same Pauli exclusion principle, the atoms
themselves to be quite bigger. There are 2 reasons for this : one is
that electrons have smaller mass, the other is that they are kept in
place by the electric force instead of the nuclear force. Let us now
measure the electric force.
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Theory and foundations of mathematics and physics : An exploration
of physics by dimensional analysis
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