# Nuclear fusion reactions violate law of conservation of mass, but it has not yet been taken into account. Why?

According to the law of conservation of mass, during any physical or chemical changes, the total mass of the products is equal to the total mass of the reactants. But in nuclear fusion reaction if considered an example, energy that sun emits in its core is due to collision of hydrogen nuclei and formation of helium nuclei. Here conservation of mass is not obeyed as certain part of mass is converted into energy. So, law of conservation of mass is violated here. Then law of conservation must be redefined as: during any physical or chemical change, the total mass of the products is equal to the total mass of reactants provided mass has not undergone conversion into energy.

• If you consider the mass-energy equivalence that Einstein discoverd there is no contradiction: en.wikipedia.org/wiki/Mass-energy_equivalence – Philipp Sep 18 '13 at 15:29
• This law apilicable only chemical reactions. If gases are products ,reactants shuld be take in closed system. – user4748 Mar 7 '14 at 11:38
• Since nuclear physics is physical the first sentence should read; "According to the law of conservation of mass, during any mechanical or chemical changes,..." – Taemyr Jul 7 '14 at 9:07
• The mass loss/gain in chemical reactions is extremely small, but it exists according to relativity. For example, the complete detonation of $100\ \mathrm{g}$ of TNT produces approximately $99.99999999\ \mathrm{g}$ of chemical products, with the remaining $0.00000001\ g$ of the initial mass transformed into energy. In chemical reactions, the mass difference between all the reactants and all the products is on the order of 0.1 parts per billion or less, so mass may seem conserved, but it is not. Compare this to nuclear fusion, where the mass difference can reach the order of 1%. – Nicolau Saker Neto Jun 21 '15 at 15:56

Nuclear reactions appear to violate both the Laws of Conservation of Mass and Energy because mass is converted into energy or vice versa.

However, the concept of mass-energy equivalence that emerges as a consequence of Einstein's General Theory of Relativity makes the Laws of Conservation of Mass and Energy special limiting cases of the Law of Conservation of Mass-Energy. Since mass is now a form of energy, we can convert back and forth as long as there is no net loss between the two.

The equivalence of mass and energy comes from Einstein's famous $E=mc^2$, where $c$ is the speed of light (in a vacuum).

• The Law of Conservation of Mass is no longer valid in its original formulation under the mass-energy equivalence given by $E=mc^2$ - not only is it not violated, it no longer even applies. – Ben Norris Sep 19 '13 at 1:40
• ... However, Conservation of Mass is still taught in elementary chemistry, because chemical reactions do not result in the changing of atoms from one element to another (or even one isotope to another) as nuclear reactions do. As such, CoM and CoE hold for chemical reactions, and the laws remain useful teaching tools in this capacity. – KeithS Sep 19 '13 at 18:22
• @KeithS I'd say that in mass is actually transformed in energy in chemical reactions (and vice-versa). However, the amount of mass and energy involved is so tiny that it can be neglected for most of practical purposes and CoM and CoE are still very useful in elementary chemistry. It's the same as situation in physics: Newton's laws don't hold, but they are still very useful when we can neglect relativistic effects. – Pere Jan 30 '18 at 0:44

Since this thread seems to have been bumped by @Community a better answer is in order. Physicists don't typically make a distinction between rest mass $m_0$ and moving mass $m$ but rather consider mass to be the invariant of the energy-momentum $4$-vector: $mc^2=\sqrt{E^2-\boldsymbol{p}^2c^2}$ where $m$ is the invariant mass of the system, $E$ its total energy and $\boldsymbol p$ its total momentum. $\boldsymbol p=\sum\boldsymbol p_i$, the vector sum of the momenta of the system's constituents. For a massless particle, $\boldsymbol p_i=E_i\hat{\boldsymbol T}_i/c$, where $\hat{\boldsymbol T}_i$ is the unit vector in the direction of propagation of the particle, while for a particle with mass $m_i$ its momentum is $\boldsymbol p_i=\gamma_im_i\boldsymbol v_i$ where $\boldsymbol v_i$ is the particle's velocity and $\gamma_i=1/\sqrt{1-\boldsymbol v_i^2/c^2}$ is its Lorentz factor.

The mass is invariant under Lorentz transformations which is to say that's it's the same in all inertial frames, so even when moving we consider the mass of the system to be the same. As an extreme example of a reaction that seems to convert mass into energy, consider $e^+e^-$ annihilation. In the rest frame of the $e^+e^-$ pair, $\boldsymbol p_+=\boldsymbol p_-=\boldsymbol0$ so $E_+=E_-=m_ec^2\approx0.511MeV$. So the total energy is $E=E_++E_-=2m_ec^2$ and the total momentum is $\boldsymbol p=\boldsymbol p_++\boldsymbol p_-=\boldsymbol0$ and so the invariant mass of the system works out to $m=2m_e$.

After annihilation, we get a pair of photons and from momentum conservation we get that $\boldsymbol p_1=-\boldsymbol p_2$ and energy conservation tells us that $E_1=|\boldsymbol p_1|c=E_2=|\boldsymbol p_2|c=m_ec^2$ but what happened to the mass? Doesn't a photon have zero mass? A photon does in fact have zero mass, but a system of two photons has nonzero mass unless the photons are moving in the same direction. Take the $x$-direction to be the direction of travel of the first photon and let the other photon travel in the $xy$-plane with non-negative $y$-velocity. Then \begin{align}\boldsymbol p^2&=p_x^2+p_y^2=(p_{1x}+p_{2x})^2+p_{2y}^2=\left(\frac{E_1}c+\frac{E_2}c\cos\theta\right)^2+\left(\frac{E_2}c\sin\theta\right)^2\\ &=\frac{E_1^2}{c^2}+\frac{E_2^2}{c^2}+2\frac{E_1E_2}{c^2}\cos\theta\end{align} where $\theta$ is the angle between the velocities of the to photons. So for the system of two photons, $$m^2c^4=E^2-\boldsymbol p^2c^2=\left(E_1+E_2\right)^2-E_1^2-E_2^2-2E_1E_2\cos\theta=4E_1E_2\sin^2\frac{\theta}2$$ So the mass of the system of two photons is $$m=2\frac{\sqrt{E_1E_2}}{c^2}\sin\frac{\theta}2$$ In our example, $E_1=E_2=m_ec^2$ and $\theta=180°$ so we have $m=2m_e$, so the mass didn't change nor did the energy in the reaction of $e^+e^-$ annihilation. This is emphasized the the Wikipedia section cited earlier: mass and energy don't get interconverted but rather are separately conserved.

If this seems like sort of a theoretical definition of mass, where do you think most of the mass in your body is concentrated? I have had students who, after a lifetime of body shaming answer, "In my butt?", but the right answer is of course: "In your atomic nuclei!" and the quarks that make up your atomic nuclei are moving at a healthy fraction of the speed of light, so if we didn't use the relativistic $4$-vector definition of mass to compute your mass we wouldn't come up with a number that was all that close to your mass as measured by a balance or your resistance to acceleration.

After atomic nuclei or molecules react, the mass of the reactants is identical to the mass of the products, and the same holds for their momentum and energy. A well-insulated reaction vessel would have the same mass after reaction as before, it's only later as heat of reaction leaks out, carrying with it energy and also mass that the mass changes, immeasurably for a chemical reaction but significantly for a nuclear reaction. The mass may be almost unrecoverable if it's carried away by neutrinos in a beta decay or by gravitational waves in collisions of compact objects, but it's still out there in the universe someplace.

The masses of the individual components of the system in general won't be the same for reactants as they were for products, but masses aren't what add, it's the components of the energy-momentum $4$-vector that add like $4$-vectors and from the $4$-vector we can determine the mass of the system.

By considering that mass and energy are the two faces of the same coin then the conservation is valid according to Einstein concept.

I dont really think it violates the concepts of mass and energy conservation as dE=dmc2 and the speed of alpha beta and gamma rays are almost as equivalent as the speed of light hence taking in consideration that einstein stated that energy can be converted from mass if the body is in speed of light or nearly the speed of light ,the statement is correct. .To be short and non-lengthy i can state that the basic concepts of mass conservation and energy convervation are violated but the einstein's theory of relativity proves the above statement.

• Could you reformat this to not be two run-on sentences? It's almost unreadable as-is. – pentavalentcarbon Mar 11 '18 at 21:50