Conservation of mass
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The law of conservation of mass/matter (The Lomonosov-Lavoisier law) states that the mass of a system of substances will remain constant, regardless of the processes acting inside the system. An equivalent statement is that matter changes form, but cannot be created or destroyed. This implies that for any chemical process in a closed system, the mass of the reactants must equal the mass of the products.
Conservation of mass does not always hold true: its conservation depends on your definition of mass/matter. In modern physics, mass and matter are equivalent to energy, and thus the conservation of energy (which always holds) encompasses matter and energy.
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Mass conservation in the theory of special relativity
"Matter" (not including many types of energy) is generally not conserved in special relativity.
Whether mass is conserved in special relativity depends on which kind of mass is being referred to, and how it is measured and whether or not just one observer measures it for a system.
Relativistic mass is equivalent to relativistic energy, so this type of mass is always conserved in all processes, because total energy is conserved. The caveat is that this type of mass is frame-dependent, so it may not be conserved in a process for an accelerated observer. However, for inertial observers, relativistic mass is conserved.
Invariant mass is observer and frame invariant, which means all observers measure it the same. Invariant mass is sometimes also known as system rest mass even when the masses which make up a system under consideration-- such as vibrating atoms-- are not at rest. It is conserved also, for all single inertial observers, so long as the system is closed.
Sometimes a form of mass is not considered to be conserved in relativistic processes in systems. This can happen for one of two reasons:
1) When invariant mass (as various forms of active energy) has been allowed to escape the system, and this escape has not been kept track of. (See section The mass of systems in mass in special relativity). Complete system closure (including closure to heat and radiation) is needed for system mass to be conserved. When the mass of a system is measured only at standard temperatures, for example, this allows for the escape of mass and energy, as heat.
2) Sometimes a form of mass is not conserved when the mass of a system is found by adding the rest masses of its components. However, for massive particles this amounts to using the measurements of many different observers, and this sort of bookkeeping it is not allowed in special relativity. Even for photons, a single observer and a closed system is required for mass conservation, since photons as considered singly have zero mass, where as pairs or systems of photons moving in different directions will in general exhibit an invariant mass which is associated with the system of photons, but not with any single photon.
Sometimes either of the above processes are equivalently at work. For example, after an energy-releasing transformation, the sum of rest masses of the resulting particles may said to be different from the sum of the rest masses of the particles which began the reaction. But how was this sum measured? If these rest masses were determined with the system closed, this requires that varying frames of references have been used (one for each massive particle, and no mass ascribed to photons). If, however, the rest mass of the system after a reaction has been measured as a whole by a single observer, and found to be changed, this must occur because energy released from the reaction has been allowed to leave the system, as heat or light or other radiation. In the latter case, it will be found that this energy is the missing mass (i.e., it would exhibit the missing mass if captured, confined, and weighed).
Another way of expressing this idea is that if released energy is allowed to remain in a system (for example, as heat, or even trapped radiation), this energy will be measured as, and included in, the ordinary "rest" mass of the system (that is, this energy still contributes to the inertia of the system and its gravitational field). In that case, the mass of the total system will not change in special relativity, for any transformation (including nuclear processes). Only if released energy is allowed to escape the system, will any "defect" in mass appear, as seen by a single observer.
Approximate conservation vs. serious violation
Even when energy (as heat or light) is allowed to enter a system, or escape it, the law of conservation of mass holds approximately in cases where energies are small, and relativistic effects can be neglected. In particular, mass is conserved to a high precision in mechanical processes involving macroscopic objects, even if heat is allowed to enter or escape, because the energy and mass associated with this amount of heat, is small. Similarly, deviations from the conservation of mass are larger, but usually still negligible, in most chemical reactions, even if heat conservation is neglected.
On the other hand, drastic (apparent) violations of the conservation of mass can occur for relativistic processes involving very high speeds or very strong fields, such as nuclear and subnuclear reactions and very large astronomical objects. In these situations, whenever a system loses potential energy, and this energy is allowed to escape the system as radiation or heat, the system also loses the corresponding amount of mass with an appropriate factor of c2. Essentially, this loss in mass is ponderable in such systems, because a very great amount of radiation or heat is involved (enough to have appreciable mass). Even here, however, it must be again emphasized that the loss or gain in mass would not appear if the energy associated with it were not allowed to enter or escape the system.
From the non-system view, large violations of conservation of mass in nuclear reactions occur when the total mass of the reaction products is derived in Newtonian fashion from the sum of their rest masses. Such derivation results either from allowing energy to escape (i.e., measuring products at "rest," or with zero momentum, as a whole system); or else is equivalent to changing observers (i.e., the rest mass of products is calculated by using the value for the mass of each product in its own frame of rest). None of these operations are correct in special relativity, although they result in no great errors in Newtonian mechanics.
Quantitative example in chemistry
Conservation of mass is to a very high accuracy applicable in chemical reactions, since in every case the mass-energy of the reactants is huge in comparison to the energy absorbed or released when they react. By way of example, a gram of TNT releases 4.61 kJ of energy when exploded. However, the rest-energy of a gram of TNT (or anything else) is 90 TJ, or about 20 billion times as much. This means that even if the products of a TNT explosion were stopped and allowed to cool to the original temperature, they would only lose 1 part in 20 billion in weight. This amount would be very difficult to measure.
Historical development and importance
The law of conservation of mass (which is effectively conservation of weight when weights are properly taken) was clearly formulated by Antoine Lavoisier in 1789, who is often for this reason (see below) referred to as the father of modern chemistry. However Mikhail Lomonosov (1748) had previously expressed similar ideas and proved them in experiments. Historically, the conservation of mass and weight was kept obscure for millennia by the buoyant effect of the Earth's atmosphere on quantities of gasses, an effect not understood until the vacuum pump first allowed the effective weighing of gasses using scales. Once understood, conservation of mass was of key importance in changing alchemy to modern chemistry. When scientists realized that substances never disappeared from measurement with the scales (once buoyancy had been accounted for), they could for the first time embark on quantitative studies of the transformations of substances. This in turn led to ideas of chemical elements, and the idea of all chemical processes and transformations (including both fire and metabolism) as simple reactions between invariant amounts/weights of these elements.
