When Does Mass Conservation Fail?

The law of conservation of mass states that mass within a closed system remains constant over time. In other words, mass can neither be created nor destroyed, only rearranged. However, this law does not always apply. Albert Einstein's special theory of relativity showed that mass and energy are equivalent and can be converted into each other. While the difference in mass is usually too small to measure in ordinary chemical reactions, nuclear reactions involve a significant enough conversion of mass into energy to invalidate the law.

Nuclear reactions

The law of conservation of mass, or principle of mass conservation, states that in a system closed to all transfers of matter, the mass of the system must remain constant over time. This implies that mass can neither be created nor destroyed, though it may change form.

However, this law does not hold for very energetic systems, such as nuclear reactions and particle-antiparticle annihilation in particle physics. Nuclear reactions involve the interaction of two nuclear particles (two nuclei or a nucleus and a nucleon) to produce two or more nuclear particles or gamma rays.

In nuclear reactions, the general law of conservation of mass energy applies, which states that mass and energy are equivalent and convertible. This is one of the striking results of Einstein's theory of relativity, described by his famous formula, E = mc^2.

In both chemical and nuclear reactions, there is some conversion between rest mass and energy, so the products may have a smaller or greater mass than the reactants. The total (relativistic) energy must be conserved, so the "missing" rest mass reappears as kinetic energy released in the reaction. This difference is a measure of the nuclear binding energy, which is enormous—a million times greater than the electron-binding energies of atoms.

While some conservation principles are empirical relationships, any reaction not expressly forbidden by the conservation laws will generally occur, albeit at a slow rate. This is based on quantum mechanics, which states that unless the barrier between the initial and final states is infinitely high, there is always a non-zero probability of a system transitioning between them.

Particle-antiparticle annihilation

The law of conservation of mass states that mass within a closed system remains constant over time. However, this law does not hold in certain cases, such as particle-antiparticle annihilation in particle physics.

However, particle-antiparticle annihilation does not violate the law of conservation of mass because mass and energy are interchangeable. This relationship is described by Einstein's famous equation, E=mc^2, which shows that mass can be converted into energy and vice versa. Therefore, the total mass before and after the annihilation remains the same, as the energy produced has an equivalent mass.

This process can occur naturally in high-energy environments, such as in stars or during cosmic ray collisions, or it can be artificially produced in particle accelerators. An example of particle-antiparticle annihilation is the collision of an electron and a positron (anti-electron), resulting in the creation of two photons with the combined energy of the original particles.

While the term "annihilation" may be confusing as it implies the reduction to nothing, it is used because the particles cease to exist in their original form. The use of the term is a matter of perspective, as from the viewpoint of the particle, it encounters its antiparticle and ceases to exist.

Open systems

The law of conservation of mass, also known as the principle of mass conservation, states that the mass of a closed system must remain constant over time. In other words, mass can neither be created nor destroyed. However, this law does not hold for open systems, where energy or matter is allowed to enter or exit the system.

In an open system, the amount of energy or matter entering or exiting the system can cause a change in the system's mass. This is because the law of conservation of mass assumes that there are no transfers of matter into or out of the system. When this assumption is violated, as in the case of open systems, the law no longer applies.

For example, consider a burning candle. The candle's wax is converted into heat, light, and gaseous products during combustion. If we were to measure the mass of the candle before and after burning, we would find that the mass has decreased. This decrease in mass is due to the escape of gaseous products, such as carbon dioxide and water vapour, from the open system.

Another example of an open system is a plant undergoing photosynthesis. The plant takes in carbon dioxide from the atmosphere and, through a series of biochemical reactions, converts it into glucose and oxygen. The plant then releases the oxygen back into the atmosphere. In this case, the mass of the system decreases due to the escape of oxygen from the plant.

In summary, the law of conservation of mass does not apply to open systems because they allow for the transfer of matter or energy across their boundaries. This transfer of mass or energy can result in a change in the total mass of the system, violating the fundamental assumption of the law of conservation of mass.

Special relativity

The law of conservation of mass states that the mass of a system remains constant over time, provided it is closed to all transfers of matter. However, this law does not hold in certain situations, such as in nuclear reactions and particle-antiparticle annihilation in particle physics. Special relativity is one framework in which the law of conservation of mass is modified.

In special relativity, the word "mass" has two meanings: invariant mass (or rest mass) and relativistic mass. Invariant mass is an invariant quantity that remains the same for all observers in all reference frames, while relativistic mass depends on the velocity of the observer. The concept of mass–energy equivalence states that invariant mass is equivalent to rest energy, and relativistic mass is equivalent to relativistic energy (or total energy).

In special relativity, the rest mass of a system is not simply the sum of the rest masses of its individual components. This is because the total energy of a composite system includes the kinetic energy and field energy in the system. For example, a box containing moving particles will have a larger invariant mass than the sum of the rest masses of the particles it contains. This is because the total energy of all particles and fields in a system must be summed, and this quantity, as seen in the center of momentum frame and divided by the speed of light squared, gives the system's invariant mass.

While the rest mass of a system may not be conserved in special relativity, the invariant mass of an isolated system is conserved. This is because the invariant mass is calculated using the total energy of the system in the center of momentum frame, where the system has no net momentum. This is a closed system, meaning that no mass or energy is allowed in or out. As long as this condition is met, the invariant mass of the system will remain constant.

In summary, while the law of conservation of mass as typically understood does not hold in special relativity, the concept of invariant mass provides a way to understand how mass is conserved in certain situations within this framework.

Quantum mechanics

The law of conservation of mass, which states that mass can neither be created nor destroyed, holds true in most cases. However, there are certain scenarios where this law does not apply, and one of them is in the realm of quantum mechanics.

In quantum mechanics, the concept of mass conservation becomes more complex and needs to be modified to align with the principles of quantum mechanics and special relativity. This is because, in quantum mechanics, the state of a system is described by a wave function, which is a probability distribution of possible outcomes rather than definite values. The behaviour of particles at this scale is governed by probabilities, and their positions and momenta are not well-defined until measured.

The law of conservation of mass, in its classical form, assumes that mass is a well-defined characteristic of a system, which is not the case in quantum mechanics. The modification required is to account for the probabilistic nature of quantum mechanics and the fact that mass and energy are interchangeable as per Einstein's famous equation, E=mc^2.

In quantum mechanics, the concept of "average energy" becomes important. While the total energy of a system may fluctuate due to measurements, the average energy over time remains constant if the system obeys the Schrödinger equation. This notion of "energy conservation" is what applies in quantum mechanics, and it differs from the classical understanding of conservation laws.

Furthermore, in quantum mechanics, the standard definition of conservation laws is statistical in nature and applies to an ensemble of repeated identical experiments. However, this statistical definition has been challenged and extended to address individual cases. For example, in certain experiments, it has been shown that energy may not be conserved in individual instances, even though it is conserved statistically. This has led to a re-evaluation of the role of the preparation stage of the initial state of a particle and the interplay of conservation laws with frames of reference.

While the law of conservation of mass does not apply in the traditional sense in quantum mechanics, the concept of conservation evolves to incorporate the probabilistic and wave-like nature of quantum systems. The average energy, or expectation value, becomes the relevant quantity to consider when discussing conservation in this context.

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