The Hydrogen Molecule and Deuterium Atom
in the Light of de Broglie’s Theory*
Pavle I. Premović
Laboratory for Geochemistry,
Cosmochemistry&Astrochemistry,
University of Niš, pavleipremovic@yahoo.com, Niš, Serbia
General
chemistry textbooks state that hydrogen molecule (H2) consists of
two protons and two electrons. According to this
model, protons exist in its nucleus as two independent particles.
However, this picture does not just agree
with the picture of this molecule based on de Broglie's hypothesis of the wave
nature of particles.
The de
Broglie equation can be used to describe the wave nature of the proton and the
H2 nucleus in motion. Usually, this expression is expressed in the
following form
λ = h/mυ ... (1)
where λ is the
wavelength, h (= 6.63 × 1034 J sec) is Planck's constant and m
is the mass of a particle moving at a speed υ.
If H2
travels with a non-relativistic speed υ its wavelength is then
λ(H 2) = h/m(H2)υ ... (2)
where
m(H2) is the rest mass
of this molecule.
The
wavelength of protons in this moving H2 is
λ(H+) = h/m(H+)υ ... (3).
As the rest mass of the electron is negligible compared to the rest
mass of the proton m(H+) then m(H+) =
1/2m(H2). Plugging this term into eqn. (3) and taking into account
eqn. (2) we get
λ(H+) = 2λ(H2) ... (4).
The experimental bond (equilibrium) bond distance of H2
is about 7.4 × 10-11 m.
Taking into account that the proton is about 0.85 × 10-15 m,
this distance is equal to about 43500 proton diameters. Thus, one concludes
that H2 consists of two separate protons having the same wavelength λ(H+).
Let us now
assume that the proton waves of H2 interact with each other. The superposition of these waves can result in two types of
interference depending
on whether
their waves are in phase or out of phase: constructive and destructive
interference.
In their constructive interference, the
resultant wave would have twice large an amplitude as the proton but its
wavelength would be the same as the proton wavelength. In the destructive
interference, the two proton waves would cancel each other so there would be no
resultant wave. Since they are associated with H2 their speed is identical then,
according to eqn. (4), the wavelength of H+ is twice the wavelength of H2.
Most astronomers and
cosmologists believe Universe’s formation started with the Bing Bang about 13.8
Gy ago. Atomic hydrogen (H) comprises about 90 % of the current Universe by
number density or about 75 % of the Universe by mass. It was created in the
early Universe after the Big Bang event.
In
the early Universe, protons were produced abundantly. They were exposed to
enough high temperatures (or having very high kinetic energy) to fuse to form
the diproton nucleus. This nucleus is unstable and one proton converts into a
neutron so that a deuteron (deuterium nucleus) D+ (or 21D)
results, releasing a positron e+ and a neutrino νe. The
diproton nucleus is, however,
unstable and one proton converts into the neutron so that a deuteron (deuterium
nucleus) D+ results, in releasing a positron e+ and
a neutrino νe.
In the equation form
p + p = D+
+ e+ + νe … (5).
This reaction is extremely slow because it is endothermic
as neutrinos released carrying energies of 0.42
MeV or 6.7 × 10-14
J.
The diproton formation followed by a production of deuterium
occurs also in the Sun and other similar stars. The deuterium continues in
further fusion reactions fueling the Sun.
In the early Universe, neutrons were also abundantly present.
It is a widely held view that in this Universe the fusion reaction of proton and
neutron creates a composite stable D+. In the equation form
p + n → D+ ...
(6).
The formation of D+ is from proton and neutron
would be expected to have a mass
m p + mn = 3.3476 × 10-27 kg
where mp and mn are the mass of
the proton and neutron at rest. The
observed mass of D+ is 3.3436 × 10-27 kg and the mass defect
for the D+ formation process is 4 × 10-30 kg which
is the equivalent of about 2.24 MeV or 3.6 × 10-13J. The fusion reaction (6) is thus exothermic and thus
more probable in the early Universe than the reaction (5).
Therefore, the nucleus of D+ consists of the “fused”
proton and neutron containing the total mass equal to the mass of the proton
and neutron combined.
Let
us assume that the proton and neutron of D+ attract each other by a
force FG described by Newton’s gravitation force equation
FG = G(mpmn)/R2 ... (7)
where FG is the gravitational force, G is the
gravitational constant (equal to 6.67 × 10-11 m3 kg-1
sec-2) and R is the distance between their centers. Let us first assume that this distance is equal
approximately to the bond distance of H2 about 7.4
× 10-11 m.
The mp and mn masses
are approximately the same: ca. 1.7 × 10-27 kg. Thus, we can
write mp = mn = m = 1.7 × 10-27 kg and
eqn. (7) can be written as
FG = Gm2/R2 ... (8).
We know
that
aG = FG/m
where aG is Newton’s
gravitational acceleration. Combining this equation and eqn. (8) we get
aG = Gm/R2 … (9).
Plugging into this equation the above
values of G, m and R (= 1.7 × 10-11 m) we obtain aG
≈ 4 × 10-16 m sec-2. This acceleration would be much higher when the proton and
neutron approach each other just
before forming a deuteron [see (6)].
The strong attractive (nuclear) force of gravity FΓ acts between a proton and neutron during their
collision (or fusion) reaction generating deuteron
FΓ = Γm2/R2
or
aΓ = Γm/R2 … (10)
The strong gravitational
constant Γ was assessed for several various cases
ranging from 1025 to 1032 N m2
kg-2 [(see
Wikiversity: Strong gravitational constant)]. For example, its value obtained
from Fermi’s weak coupling constant is equal to approximately 6.94 × 1031
N m2 kg-2.
It is reasonable to assume that the distance
between proton and neutron just before collision (or fusion) is equal
approximately to the charge radius of deuteron about 2.13 × 10-15 m.
Plugging into eqn. (9)
the above values for Γ (= 6.94
× 1031 N m2 kg-2), m (= 1.7
× 10-27 kg) and R (= 2.13 × 10-15 m) we find
aΓ ≈ 3 × 1034
m sec-2.
It means that between 10-14 m and about
2 ×10-15 m
proton (or neutron) would be accelerated about 1040 times.
This acceleration would also be when the proton and
neutron approach each other just before forming a deuteron [see (6)]. With this
average acceleration, the proton (or neutron) would reach the speed of light c
(≈ 3 × 108 m sec-1) for Δt ≈ 10-26
sec.
The quantum
mechanical expression for the energy-time uncertainty is
ΔEΔt ≈ h ... (11).
This relates the minimum
uncertainty in energy ΔE of a particle and its change during a time interval Δt.
Solving for ΔE and substituting Δt ≈ 10-26 sec gives ΔE
≈ 7 × 10-8 J. This uncertainty in energy is about 2 × 105 times higher than the internal
energy of deuteron of about 3.6 ×10-13 J. It is reasonable in this case to
assume that ΔE
is about this energy then the time interval Δt has to be 2 × 10-21 sec [eqn. (11)]. By that time the speed of a proton (or neutron) just
before collision (or fusion) would be about 2 × 10-21 sec × 1034 m sec-2 ≈ 2 × 1013 m
sec-1 or about 70000 times higher than the speed of light.
The
strong (nuclear) acceleration can be presented as aΓ = dυ/dt where dυ and dt represent the corresponding
changes in speed and time. Special relativity limits dυ with the speed of light then
aΓ < c/dt or
dt < 10-26 sec.
It is
generally accepted that the Planck time tp (≈ 10-43 sec)
is the smallest time interval that has a physical meaning. However, the
smallest time interval measured to date is about 10-21
sec.
Therefore, dt can be from 10-43 sec to 10-26 sec.
The question is now: Which value of dt is right?
Let us
first assume that the proton and neutron are moving with a non-relativistic
speed υ just
before collision (or “fusion”). After this event, the proton and neutron move
together as deuteron. Their total non-relativistic kinetic energy Ek
= mυ2. A part of this energy
would be transformed into the internal energy of deuteron and a part into its
kinetic energy. As we noted above, this internal energy is 3.6 × 10-13 J.
Therefore, Ek > 3.6 × 10-13 J or after a bit of algebra and
calculation we find that υ > 1.4 × 107 m sec-1.
Consider now a relativistic case. The formula for the total
relativistic kinetic energy of proton and neutron is
Erk = 2m(γ – 1)c2
where γ = 1/√(1– υ2/c2) is a relativistic
factor. Plugging into this equation the above values
of m and c and having in mind that Erk
> 3.6 × 10-13 J we find that the speed of a
proton (or neutron) just before a collision (or fusion) is close to the speed
of light. In this case, dt is about 10-26 sec.
Quantum mechanics allows us to write a general expression
for the energy-time principle:
ΔEΔt ≈ h.
This expression relates the minimum uncertainty in energy
dE of any particle system changing during a time interval Δt. Solving for ΔE and substituting the above values for Δt gives
dE ≈ 10-7
J – 1010 J.
Something
is wrong but the question is: what?
Reference
{1} R. Yang, B. Chen, H. Zhao et al., Test
of conformal gravity with astrophysical observations. arXiv: 1311.2800v1
[gr-qc] 12 Nov 2013.
The attraction of
each electron by the two protons generates a force that pulls the protons
toward each other and balances the repulsive force between protons and the repulsive
force between electrons.
Note here that all possible mechanisms in the generation of deuterium are still incomplete {1}.
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