Nuclear fusion question -- Calculations for Hydrogen fusing into Helium

[luckis11]

I read in 2 books that 4 atoms of Hydrogen fuse and give 1 atom of Helium and 2 electrons, and these 2 electrons convert to light. And that the mass of the Helium is less than the mass of the 4 atoms of Hydrogen, thus that the mass lost converted to light too. But I sum up the masses of protons, neutrons, and electrons before and after the fusion, and the mass of the Helium is larger than the mass of the 4 Hydrogens. Note that I did not count the mass of the 2 electrons that converted to light. Where am I mistaken?

 

[willem2]

I read in 2 books that 4 atoms of Hydrogen fuse and give 1 atom of Helium and 2 electrons, and these 2 electrons convert to light. And that the mass of the Helium is less than the mass of the 4 atoms of Hydrogen, thus that the mass lost converted to light too. But I sum up the masses of protons, neutrons, and electrons before and after the fusion, and the mass of the Helium is larger than the mass of the 4 Hydrogens. Note that I did not count the mass of the 2 electrons that converted to light. Where am I mistaken?

You probably took the mass of 2 protons and 2 neutrons, instead of the mass of an helium atom or an alpha particle. The mass of neutron really shouldn't appear in the calculation.

mass of proton	1.00727 amu
mass of alpha particle	4.00151 amu
mass of electron	0.00055 amu

4 times the proton mass plus 2 electrons is heavier than the mass of an alpha particle.

 

[luckis11]

e=electron mass=9.10939*10^-31 kg
n=neutron mass =1.67493*10^-27 kg
p=proton mass =1.67262*10^-27 kg
2p+2n+2e-(4p+4e)=
2*1.67262*10^-27+2*1.67493*10^-27+2*9.10939*10^-31-
(4*1.67262*10^-27+4*9.10939*10^-31)=
2.798122 × 10^-30>0 instead of a negative value
I took as Helion mass the mass of 2 protons + 2 neutrons + 2 electrons. Why I should not do that. I just found it,¨¨ it is claimed that
"Mass defect (also called "mass deficit") is the difference between the mass of an object and the sum of the masses of its constituent particles. Discovered by Albert Einstein in 1905, it can be explained using his formula E = mc2, which describes the equivalence of energy and mass. The decrease in mass is equal to the energy given off in the reaction of an atom's creation divided by c2.[7] By this formula, adding energy also increases mass (both weight and inertia), whereas removing energy decreases mass. For example, a helium atom containing four nucleons has a mass about 0.8% less than the total mass of four hydrogen nuclei (which contain one nucleon each). The helium nucleus has four nucleons bound together, and the binding energy which holds them together is, in effect, the missing 0.8% of mass".
(Wikipedia)

 

[Vanadium 50]

Why I should not do that

Because the mass of the helium atom is less than the mass of the constituents. Indeed, that's the energy source you are trying to tap, so if you don't include it you won't get the right answer.

 

[mfb]

I read in 2 books that 4 atoms of Hydrogen fuse and give 1 atom of Helium and 2 electrons, and these 2 electrons convert to light.

If a book writes that, throw it away. But I don't think the books wrote that. The overall reaction produces two positrons, these annihilate with two electrons.
4 p -> He-4 + 2 e⁺ or, if you include the annihilation reactions, 4 p + 2 e^- -> He-4

He-4 has less mass than the sum of protons and neutrons. That's the point. Otherwise it could not form.

 

[luckis11]

And the explanation they give based on the assumption that the protons and the neutrons of helium are the same particles with the protons of hydrogen and the free neutrons and yet they have smaller mass when they are in helium, is? Some which lies at «the binding energy which holds them together is, in effect, the missing 0.8% of mass.»? I.e.? This means that the mass lost exists as binding energy? Then how it is lost outwardly as explosion energy.

 

[mfb]

The protons and neutrons don't have a smaller mass when in helium. The mass of a composite object is not the sum of masses of the individual components. The difference is called the binding energy - it's released when the object forms. It can be released as radiation or as kinetic energy of particles, for example.

You can verify that helium has 2 protons and 2 neutrons e.g. by looking what reacts to form helium, with scattering experiments, or simply by comparing the combined nucleus to the prediction for the bound state of two protons and two neutrons and realizing there is nothing else that comes anywhere close to it.

 

[luckis11]

Still, I don't think you answered my last question. E.g., you imply that they don't have smaller mass when in helium because the missing mass is tranformed into binding energy. Then how the missing mass was given out as explosion energy.

 

[vanhees71]

As with any reactions also in a fusion process energy and momentum are conserved. The "missing energy/mass" goes into the kinetic energy of the products.

https://en.wikipedia.org/wiki/Nuclear_fusion

 

[luckis11]

Kinetic energy of the helium atom, or vibrational speed of the helium's nucleons inside the helium atom, or light that is emitted outside helium atoms?

 

[vanhees71]

I'm talking simply about the motion of the particles. Look at this example from the Wikipedia article

https://en.wikipedia.org/wiki/File:Deuterium-tritium_fusion.svg

As in any reaction also here in the reaction $\text{d}+\text{t} \rightarrow ^{4}\text{He}+\text{n}$ energy and momentum are conserved. Since the reaction is "exothermic" there's some extra energy going into the motion of the helium nucleus and the neutron.

 

[luckis11]

So you are saying that protons transform their missing mass ΔΜ into
(mass of the 1 helium)(speed of the 1 helium)^2-
(mass of the 4 hydrogens)(speed of the 4 hydrogens)^2? In this case:
-The light emitted comes from annihilation of the 2 missing electrons alone?
-"In physics and chemistry, binding energy is the smallest amount of energy required to remove a particle from a system of particles or to disassemble a system of particles into individual parts.[1] In the former meaning the term is predominantly used in condensed matter physics, atomic physics, and chemistry, whereas in nuclear physics the term separation energy is used. A bound system is typically at a lower energy level than its unbound constituents. According to relativity theory, a ΔE decrease in the total energy of a system is accompanied by a decrease ΔM in the total mass, where ΔM⋅c2=ΔE.[2]"
https://en.wikipedia.org/wiki/Binding_energy
Not the "total " but The REST energy was decreased, the total i.e. rest+kinetic energy stayed the same, am I correct? And the statement "the binding energy which holds them together is, in effect, the missing 0.8% of mass" is misleading as "binding energy is the amount of energy required to transform the 1 helium into 4 hydrogens again" which amount is equal to ΔΜ, and the missing mass ΔΜ was not transformed into binding energy (statement which implies an energy which was created and takes place inside the helium's space), but to kinetic energy. Am I correct?
"Exothermic chemical reactions in closed systems do not change mass, but do become less massive once the heat of reaction is removed, though this mass change is too small to measure with standard equipment."
But I just saw that they DO change (i.e. lose) mass which was transformed into heat equal to (the sum of the after the reaction kinetic energies)-(the sum of the before the reaction kinetic energies).
"In nuclear reactions, the fraction of mass that may be removed as light or heat, i.e. binding energy, is often a much larger fraction of the system mass".
This implies that ΔΜ=(the sum of the after the reaction kinetic energies of nucleons and electrons)+(energy of the after the reaction born light)-(the sum of the before the reaction kinetic energies of nucleons and electrons), I don't see any of my above conclusions to become invalid because of it, but does it happen?

 

[vanhees71]

I don't know what you mean by "missing" mass. (Asymptotic) free particles have a mass, an energy, and momentum related by
$$E=c \sqrt{(m c)^2+\vec{p}^2}.$$
Energy and momentum together are the components of a four-vector,
$$(p^{\mu})=\begin{pmatrix} E/c \\ \vec{p} \end{pmatrix}.$$
These "on-shell momenta" obey the on-shell condition
$$p_{\mu} p^{\mu} = (E/c)^2-\vec{p}^2=m^2 c^2.$$
In collisions like the described one energy-momentum conservation holds, i.e., the total four-momentum is conserved,
$$p_1+p_2=p_1'+p_2'.$$
This together with the mass-shell conditions determining the energies and momenta of the outgoing particles given those of the incoming. Usually it's most simple to calculate this "kinematics" in the center-momentum frame, where $\vec{p}_1+\vec{p}_2=\vec{p}_1'+\vec{p}_2'=0$.

For details, see

https://pdg.lbl.gov/2020/reviews/rpp2020-rev-kinematics.pdf

Now you can try this analysis for the above quoted example for a nuclear fusion reaction.

 

[Astronuc]

Then how the missing mass was given out as explosion energy.

Please elaborate on 'explosion energy'. What explosion?

In p+p fusion, one is missing a number of steps. http://hyperphysics.phy-astr.gsu.edu/hbase/Astro/procyc.html
In the first step, p+p => d + e⁺ + ν. The process is usually steady state, and in fact it could take hundreds of millions of years for a single proton to fuse with another proton (reaction has a very low cross section, or probability). The binding energy goes to the kinetic energy of the deuteron and positron, which being charged particles interact with other charged particles to disperse their energy. Heat is basically kinetic energy of the atoms (or molecules) in a gas, or nuclei and electrons in a plasma.

Deuterons can also form by the capture of a neutron by a proton in which case a gamma ray (light) is emitted.

The interaction of nuclei (including protons) with electrons results in Bremsstrahlung radiation (photons, or gamma rays), and in some cases, electrons and protons combine as hydrogen atoms producing X-rays.

Photons (X-rays and gamma rays) scatter off electrons transferring energy to electrons.

 

[luckis11]

By explosion I mean of the hydrogen bomb. What happens in the sun is more uncertain. Please do not complicate things, it is enough for me if you verify that my last post is correct, i.e. according to the theory.

 

[Astronuc]

By explosion I mean of the hydrogen bomb. What happens in the sun is more uncertain. Please do not complicate things, it is enough for me if you verify that my last post is correct, i.e. according to the theory.

I suspected that was the meaning of 'explosion'.

Your question in the original post,

I read in 2 books that 4 atoms of Hydrogen fuse and give 1 atom of Helium and 2 electrons, and these 2 electrons convert to light. And that the mass of the Helium is less than the mass of the 4 atoms of Hydrogen, thus that the mass lost converted to light too. But I sum up the masses of protons, neutrons, and electrons before and after the fusion, and the mass of the Helium is larger than the mass of the 4 Hydrogens. Note that I did not count the mass of the 2 electrons that converted to light. Where am I mistaken?

has nothing to do with hydrogen bombs (thermonuclear weapons), the basis of which are fission based trigger to get the high temperatures and a fusion component based on deuterium and tritium fusion. in d+t fusion, the deuteron and tritium fuse and reconfigure into an alpha particle and fast (14.1 MeV) neutron as vanhees71 explained in post #11.

I'm talking simply about the motion of the particles. Look at this example from the Wikipedia article

https://en.wikipedia.org/wiki/File:Deuterium-tritium_fusion.svg

As in any reaction also here in the reaction $\text{d}+\text{t} \rightarrow ^{4}\text{He}+\text{n}$ energy and momentum are conserved. Since the reaction is "exothermic" there's some extra energy going into the motion of the helium nucleus and the neutron.

The fast neutrons produced can induce fission in tampers that surround the d+t, which increases the yield.

The concept of binding energy applies to all fusion reactions. The excess (binding) energy is manifest in the kinetic energy of the products, the nuclei and possible neutron, which is basically thermal energy. High temperatures (i.e., millions K) cause atoms to fully ionize and the recombination causes a spectrum of X-rays, UV and visible light. Gamma rays occur from rapidly decaying fission product and radiative capture of neutrons.

The 4 protons fusing into He represent a net reaction in the sun and stars.

 

[snorkack]

"Exothermic chemical reactions in closed systems do not change mass, but do become less massive once the heat of reaction is removed, though this mass change is too small to measure with standard equipment."
But I just saw that they DO change (i.e. lose) mass which was transformed into heat equal to (the sum of the after the reaction kinetic energies)-(the sum of the before the reaction kinetic energies).

The point is that heat has mass. Objects gain mass when heated, and lose mass when cooled. The point is that when mass is transformed into heat, mass is unchanged because heat still has the mass that was transformed into heat - the mass is decreased when the heat escapes.

 

[luckis11]

Thanks for the info. So I guess my last post is totally according to the theory. But you did not answer this question:
ΔΜ of nucleons=m-μ,
m=mass of the nucleons before the reaction,
μ=mass of the nucleons after the reaction,
u=speed of the nucleons before the reaction,
U=speed of the nucleons after the reaction,
i.e. ignoring the electrons for the moment
(m-μ)c^2=(1/2)μU^2-(1/2)mu^2, or
(m-μ)c^2=(1/2)μU^2-(1/2)mu^2+light?
I.e. the born light comes from the dissolution of electrons alone, or also from the ΔΜ of nucleons?

 

[Astronuc]

Thanks for the info. So I guess my last post is totally according to the theory. But you did not answer this question:
ΔΜ of nucleons=m-μ,
m=mass of the nucleons before the reaction,
μ=mass of the nucleons after the reaction,
u=speed of the nucleons before the reaction,
U=speed of the nucleons after the reaction,
i.e. ignoring the electrons for the moment
(m-μ)c^2=(1/2)μU^2-(1/2)mu^2, or
(m-μ)c^2=(1/2)μU^2-(1/2)mu^2+light?
I.e. the born light comes from the dissolution of electrons alone, or also from the ΔΜ of nucleons?

One is headed in the right direction, but not quite there.

Firstly, it's not the mass of the nucleons, but the nuclei. If it was 4 protons (p) combining to one alpha (₂He⁴), it would be the mass of 4 protons - mass of an alpha particle (He nucleus). Proton-proton fusion, which is one of the reactions in stars, is part of a complicated chain of reactions. Four protons would never spontaneous fuse to form an alpha particle.

In the case of d+t => α+n, then one looks at the masses: (m_d + m_t - m_α - m_n)c² = Q. If Q > 0, i.e., the reaction is exothermic, the increase in energy manifest in the increase in kinetic energy of the products (α, n) over the kinetic energies of the reactants (d, t). The total energy is conserved, and one must also consider the conservation of momentum.

In a thermonuclear device, fission trigger produces a prodigious amount of X-rays as the electrons around the fission products reconfigure to ground states (in addition to the kinetic energy of the fission products = a lot of thermal energy = high temperature), while other electrons experience bremsstrahlung effects. The quantity of X-rays heats the fusion system, which then causes fusion of light isotopes, e.g., the aforementioned d,t, which also gets very hot. Very hot = high temperatures on the order of many keV (many millions K; 1 keV = 11.6 million K). Very hot masses produce light from the very excited electrons and as electrons recombine with nuclei (or ions). Some excited nuclei can emit gamma rays as they decay to a lower energy state.

For nuclei to undergo fusion, they have to have some initial kinetic energy so that the nuclei can approach each other in order to 'fuse', or rather, undergo the nuclear reaction. Nuclei of different masses do not necessarily have the same kinetic energy, especially if they are not in thermal equilibrium, as in the case of a very rapid transient.

In more general terms of a binary nuclear reaction:
(T + E₀)_a + (T + E₀)_b = (T + E₀)_c + (T + E₀)_d, where T is kinetic energy, E₀ is rest mass, and a,b;c,d indicate reactant nuclei (a,b) and product nuclei, or nucleon (c,d). Protons and neutrons can be products of some fusion reactions.

See pages 2, 3 and 4 of
https://www.lehigh.edu/~eus204/teaching/ME362/lectures/lecture01.pdf

 

[snorkack]

The whole mass difference will NOT be converted into light.
One prevalent reaction for protons is:
1) p+p→d+e⁺+ν_e
The energy of this reaction is shared around equally between kinetic energy of e⁺ and neutrino.
The kinetic energy of positron and its rest mass plus electron rest mass are converted to light when positron annihilates. The energy of neutrino is never converted into light.
But another competing reaction is:
2) p+p+e→d+ν_e
Nearly the whole energy of this reaction escapes with the neutrino. Only the tiny recoil of deuteron eventually is converted into light.
The rest mass of reactants and products is identical. The fraction of energy given to the neutrino is very different. And the branching ratio between the two reactions differs depending on conditions.
In Sun, 1) happens to be the more common.

 

[luckis11]

Ok, I correct my phrase "of the nucleons after the reaction" to "of the nuclei after the reaction".
So, you are saying that at some reactions
(m-μ)c^2=(1/2)μU^2-(1/2)mu^2, and at some reactions
(m-μ)c^2=(1/2)μU^2-(1/2)mu^2+electromagnetic waves?