Mass Reduction in Combustion: Real or Myth?

[iVenky]

I have studied in high school that all chemical reactions obey conservation of mass, as the atoms are merely re-arranged, but when I read through special relativity, I was reading that you can show an infinitesimal change in mass (based on E=mc2) in combustion that's not noticeable that's being converted to energy. Is that true?

 

[Ibix]

Yes, in principle. If you combust something inside a sealed container its mass afterwards should be a tiny bit lower. As you say, it's an immeasurably tiny difference.

 

[Nugatory]

If you combust something inside a sealed container...

... and some of the energy from the combustion escapes the container, presumably as heat or light. That will be the case for any physically realizable container, as there is no such thing as an ideal sealed container that is completely impervious to all forms of energy.

Ibix knows this of course, presumably considered it too obvious to mention. I'm just putting in a few extra words for the next person to wander into this thread.

 

[PeterDonis]

all chemical reactions obey conservation of mass, as the atoms are merely re-arranged

Chemistry textbooks that make this statement are implicitly using a non-relativistic approximation, where mass is additive (i.e., the mass of a bound system like a molecule is the sum of the masses of the atoms in it, and any change in binding energy does not affect the mass). This is often a useful approximation, but relativity shows us that it is not exactly correct.

 

[vanhees71]

...and it becomes relevant even for practical purposes when you consider nuclear binding, because the binding energies are a considerable fraction of the masses of the nuclei.

 

[hutchphd]

But philosophically it is perhaps disturbing to realize we are awash in energy. $E=mc^2$. What matters is "free energy": that which is available to us within the strictures of entropy and thermodynamics.

 

[PAllen]

But philosophically it is perhaps disturbing to realize we are awash in energy. $E=mc^2$. What matters is "free energy": that which is available to us within the strictures of entropy and thermodynamics.

It would all be too available if antimatter were common ...

 

[anuttarasammyak]

I have studied in high school that all chemical reactions obey conservation of mass, as the atoms are merely re-arranged, but when I read through special relativity, I was reading that you can show an infinitesimal change in mass (based on E=mc2) in combustion that's not noticeable that's being converted to energy. Is that true?

Chemistry states the law of conservation of mass. Physics states the law of conservation of energy.
They were compatible up to the beginning of the 20th century until Einstein discovered special relativity that combines the two as $E=\sqrt{m^2c^4+p^2c^2}$. If the formula were $E=mc^2$, the both conservation formula could stand because they differ by constant coefficient $c^2$. However, kinetic energy part $p^2c^2$ in square root ( and potential energy ) that is usually much less than $m^2c^4$ prevent it. So strictly speaking not mass but energy conserves. Most chemists keep using the law of conservation of mass because they can disregard all the energy other than large $mc^2$ in chemical reaction. In this useful approximation the law of conservation of mass is surviving.

 

[vanhees71]

It's a bit more subtle and is understood in considering the role the continuous part of the spacetime symmetry groups of the space-time models in Newtonian physics and SR play. For Newtonian physics it's the Galilei group, for SR it's the proper orthochronous Poincare group.

In quantum theory you have to find theories that are symmetric under these groups to be compatible with the underlying spacetime model. Since these are Lie groups, i.e., groups with continuous parameters connected smoothly with the identity operation according to Wigner's theorem, these symmetries are represented in QT by unitary ray representations.

You can find them by looking for the unitary ray representations of the corresponding Lie algebras, and there any ray representation can be lifted to a unitary representation of central extensions of these Lie algebras, and for the groups also admitting representations of the covering group.

For SR the issue is simpler, because there the Lie algebra has no nontrivial central extensions, i.e., any unitary ray representation of the Lie algebra can be lifted to a unitary representation, and instead of the original Poincare group you consider the covering group, where the proper orthochronous Lorentz subgroup is substituted by its covering group $\text{SL}(2,\mathbb{C})$, which admits integer and half-integer spin (which obviously is important). The corresponding irreducible representations are characterized by the values of the Casimir operators, which are $m^2$ and $s$ (massive reps.) and $h$ (massless) (the mass squared and the spin or helicity quantum number). As it turns out the physically meaningful representations are for $m^2 \geq 0$, and the $s \in \{0,1/2,1,\ldots \}$ and $h \in \{0,\pm 1/2,\pm 1,\ldots \}$. The details are very well described in Weinberg's quantum theory of fields, vol. 1.

Note that here mass is defined by a Casimir operator of the Lie group/algebra. There are 10 conservation laws from the spacetime symmetries corresponding to the standard basis of observables spanning the Poincare Lie algebra (Energy, momentum, angular momentum, boost generators).

For Newtonian physics fortunately there is a non-trivial extension of the Galilei group since it turns out that the unitary transformations of the Galilei group don't lead to a sensible quantum dynamics (that's shown in a famous paper by Inönü and Wigner). The non-trivial central charge is the mass, extending the classical Galilei group to a central 11-dim. extension of its covering group (which just means to substitute the classical rotation group SO(3) by its covering group SU(2), admitting half-integer representations of the rotation group, i.e., particles with half-integer spin). An immediate consequence of the fact that mass is a central charge of the Gailei group is that there is a superselection rule, i.e., there are no superpositions of states living in representations with different mass, and thus in addition to the 10 space-time conserved quantities given by the generators of the Galilei group (energy, momentum, angular momentum, boost generators) you have an additional independent conservation law of mass.

That's why in Newtonian physics you have an additional independent conservation law for mass, while you have no such additional conservation law for mass in SR. In SR you can simply define the mass by defining at as total energy divided by $c^2$ as measured in the rest frame of the center of momentum of the system, and for composite systems this invariant mass depends on the energy in the excitations and thus this mass is not necessarily conserved in SR. E.g., take some solid body, which has some mass in the rest frame of its center of energy and heat it up. Then its mass gets higher by $\Delta Q/c^2$, because that heat is added to the total energy content of the body as measured in the center-of-momentum frame (rest frame of total momentum). This analysis also illustrates that $m$ is not an additional independent conserved quantity because as a Casimir operator of the proper orthochronous Poincare group it can be calculated from the values of the 10 conserved quantities from the application of Noether's theorem to the proper orthochronous Poincare group.

 

[DrStupid]

[...] until Einstein discovered special relativity that combines the two as $E=\sqrt{m^2c^4+p^2c^2}$. [...] So strictly speaking not mass but energy conserves. [...]

That would mean that momentum is not conserved. But it is. Therefore mass must be conserved too. Nugatory already mentioned in #3 that the mass is only reduced by combustion if the released energy escapes from the system (which is usually the case in reality). The mass of an isolated system doesn't change - no matter if there are chemical reactions or not. The mass of a system at rest that loses energy also loses the mass equivalent of this energy - no matter if there are chemical reactions or not. It is not the chemical reaction (or any other internal process) that changes the mass of a system but the exchange of energy and/or momentum over the system boundaries.

 

[vanhees71]

For an isolated system total energy and momentum are conserved and the center of energy (not mass!) moves with constant velocity. From the point of view of an observer in the corresponding center-momentum frame, the total momentum is 0 and thus $E^*=m c^2$ (where $m$ is the so defined (!)) invariant mass of the system. Thus in SR the "conservation of invariant mass" in this case is just a consequence of energy conservation (via Noether from time-translation invariance of Minkowski space) and the fact that the center of energy of a closed system moves with constant velocity against any inertial frame (via Noether from Lorentz-boost invariance of Minkowski space). Though there is indeed a "mass conservation law" in this sense, it's not an extra conservation law beyond those from Noether's theorem applied to the 10-parameter Poincare Lie group.

As mentioned above that's different in Newetonian physics, where the symmetry group in Newtonian QT is an central extension of the 10-parameter Galilei Lie group of classical mechanics with mass as the additional non-trivial central charge, leading to a superselection rule for mass. There's no such non-trivial central extension of the Poincare group and thus there's no additional independent mass-conservation law in the SRT.

 

[Sagittarius A-Star]

not mass but energy conserves.

That is wrong. The four-momentum of an isolated system conserves and therefore also it's mass conserves.

In the system of natural units where c = 1, for systems of particles (whether bound or unbound) the total system invariant mass is given equivalently by the following:
$m^2 = (\sum E)^2 - ||\sum\vec p||^2$
...
For such a system, in the special center of momentum frame where momenta sum to zero, again the system mass (called the invariant mass) corresponds to the total system energy or, in units where c = 1, is identical to it.
...
Note that the invariant mass of an isolated system (i.e., one closed to both mass and energy) is also independent of observer or inertial frame, and is a constant, conserved quantity for isolated systems and single observers, even during chemical and nuclear reactions.

Source:
https://en.wikipedia.org/wiki/Mass_in_special_relativity#The_mass_of_composite_systems

 

[PeterDonis]

The four-momentum of an isolated system conserves and therefore also it's mass conserves.

The invariant mass of the total system is conserved, but invariant mass is not additive; the invariant mass of the system is not the sum of the invariant masses of its constituents. In the non-relativistic approximation, however, invariant mass is additive, which is why chemistry textbooks typically treat it as such. See my post #4.