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Conservation of mass

The law of conservation of mass, a fundamental principle in chemistry and physics, states that the total mass of all substances involved in a closed system remains constant before and after any physical or chemical change, implying that matter cannot be created or destroyed.^[1] This principle, first quantitatively established through experiments, asserts that in chemical reactions, the mass of the reactants equals the mass of the products.^[2] Formulated by French chemist Antoine Lavoisier in the late 18th century—specifically around 1785 during his studies of combustion and calcination—this law marked a pivotal shift in scientific thought by disproving the prevailing phlogiston theory, which posited that a weightless substance called phlogiston was released during burning, inconsistently explaining observed mass changes.^[3] Lavoisier's meticulous measurements, often conducted with his wife Marie-Anne as a collaborator, demonstrated that combustion involved the combination with oxygen (then called "dephlogisticated air"), and that mass was preserved, laying the groundwork for stoichiometry and modern quantitative chemistry.^[4] His work, published in Traité élémentaire de chimie in 1789, emphasized precise weighing and sealed reaction vessels to ensure no matter escaped, revolutionizing chemical analysis.^[2] In chemistry, the law underpins balancing chemical equations and predicting reaction outcomes, enabling advancements in fields from industrial synthesis to environmental monitoring, while remaining valid for all ordinary chemical processes where energy-mass conversions are negligible.^[5] In physics, it manifests as the continuity equation in fluid dynamics, describing how mass flow remains constant in incompressible flows, essential for engineering applications like aerodynamics and hydraulics.^[6] However, in modern relativistic physics, as articulated by Albert Einstein in 1905, the principle extends to the conservation of mass-energy, where mass can convert to energy via E = mc^2, as seen in nuclear reactions, though the total mass-energy remains invariant.^[7] This unified conservation law integrates the classical principle with special relativity, confirming its enduring relevance across scales from atomic to cosmic.^[8]

Core Concepts

Definition and Statement

The conservation of mass is a fundamental principle in classical physics and chemistry stating that, in a closed system, the total mass remains constant over time, as matter cannot be created or destroyed through ordinary chemical reactions or physical transformations. This law underscores the invariance of mass in processes where no matter enters or leaves the system, forming a cornerstone for understanding material transformations.^[9] A closed system is defined as a collection of matter that does not exchange mass with its surroundings (though energy may be exchanged), ensuring that any changes occur internally without external mass influence. The total mass in such a system represents the aggregate of all individual masses of its components, which stays unchanged regardless of internal rearrangements or conversions between forms of matter.^[10] Intuitively, this principle aligns with everyday observations; for example, when water boils, the liquid seems to vanish, but its mass is preserved in the form of water vapor that disperses into the air if the system is not fully enclosed. Likewise, burning a piece of wood appears to reduce its mass to ash, yet including the mass of released gases like carbon dioxide and water vapor, along with the consumed oxygen, yields the same total as the original wood. These examples illustrate how mass conservation holds even when products are not immediately visible. The empirical basis of the conservation of mass derives from reproducible experiments in closed setups, where precise measurements consistently show mass before and after processes to be equal.

Mathematical Formulation

The conservation of mass principle in classical mechanics and thermodynamics is quantitatively expressed for a closed system—where no matter crosses the boundaries—as the total mass remaining constant over time: m_{\text{initial}} = m_{\text{final}}, or equivalently, \frac{dm}{dt} = 0, with m denoting the total mass of the system.^[11] For discrete systems, such as a chemical reaction involving distinct reactants and products, the principle derives from summing the masses on both sides of the process, ensuring balance without creation or destruction of matter. Consider reactants with masses m_A and m_B transforming into products with masses m_C and m_D; the conservation equation is m_A + m_B = m_C + m_D, which extends to any finite number of components where the total initial mass equals the total final mass.^[11] In continuous systems, such as those analyzed in fluid dynamics or thermodynamics, the principle manifests as a mass balance equation. For a closed system (one with impermeable boundaries and no internal sources or sinks of mass), the rate of change of mass is zero: \frac{dm}{dt} = 0. More generally, the differential form, known as the continuity equation, accounts for mass flow across boundaries: \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0, where \rho is the mass density and \mathbf{u} is the velocity field, implying that local density changes are balanced by the net flux of mass.^[12]^[13] This formulation relies on key assumptions valid in classical contexts: non-relativistic speeds (where velocities are much less than the speed of light), absence of nuclear reactions (which involve mass-energy conversion), and closed or isolated boundaries preventing external mass transfer.^[14]^[15]

Historical Development

Early Observations and Precursors

The concept of mass conservation has roots in ancient Greek philosophy, where Democritus (c. 460–370 BCE) developed atomism, positing that all matter consists of eternal, indestructible atoms that undergo rearrangement but never creation or destruction during natural changes.^[16] This view implied a fundamental constancy of matter, contrasting with earlier ideas of elemental transformation and providing an early intuitive precursor to quantitative preservation principles. During the medieval and early modern periods, European alchemists from the 13th to 17th centuries documented puzzling mass discrepancies in transmutation experiments, such as the apparent weight gain of metals during calcination when heated in air, which contradicted expectations of matter loss through fire and remained unexplained within their qualitative frameworks.^[17] These observations, often recorded in treatises on metallic refinement and elixir production, highlighted inconsistencies in matter's behavior without resolving them, as alchemists attributed changes to mystical principles like the philosopher's stone rather than measurable constancy. In the mid-18th century, Russian polymath Mikhail Lomonosov conducted unpublished experiments around 1748, sealing metals like lead in glass vessels and heating them to induce combustion or reaction with air, consistently finding that the total mass before and after remained unchanged, thus demonstrating preservation in a controlled setting.^[18] Underpinning these empirical insights were philosophical developments in Cartesian mechanics, where René Descartes (1596–1650) described matter as an immutable quantity of extension filling space, with its total "quantity of motion" (size times velocity) conserved by divine immutability across all interactions.^[19] This framework treated matter as inherently persistent, influencing later thinkers to view it as a conserved entity rather than subject to arbitrary alteration. These precursors collectively built toward more rigorous formulations in the late 18th century.

Formulation in Chemistry

In the 1770s, Antoine Lavoisier conducted pivotal experiments on combustion and calcination using sealed glass vessels to precisely measure mass changes. In 1774, he heated tin and lead in closed containers filled with air, observing that while the volume of air decreased and a calx (oxide) formed, the total mass of the vessel and contents remained unchanged before and after the reaction.^[20] These results indicated that a constituent of the air, which Lavoisier later identified as oxygen (or "dephlogisticated air"), combined with the metal, rather than any substance being lost or gained from the system.^[2] Building on these findings, Lavoisier published his oxygen theory of combustion in 1777, proposing that burning involved the combination of substances with oxygen, a process that conserved mass overall.^[21] This directly challenged the prevailing phlogiston theory, developed by Georg Ernst Stahl in the early 18th century, which posited that combustible materials contained phlogiston—a hypothetical fire-like principle—released during combustion or calcination.^[17] Stahl's model struggled to explain the observed weight gain in calx formation, as it implied phlogiston had negative mass to account for the increase; Lavoisier resolved this by demonstrating that the gain resulted from oxygen absorption, aligning with quantitative mass measurements.^[2] Lavoisier's ideas culminated in his seminal 1789 work, Traité Élémentaire de Chimie, where he explicitly articulated the law of conservation of mass, stating that "in every operation an equal quantity of matter exists both before and after the operation."^[2] This text not only named and formalized the principle but also redefined chemical elements and rejected alchemical notions of transmutation. Earlier precursors, such as Mikhail Lomonosov's 18th-century observations on mass preservation in reactions, provided an independent demonstration without the widespread recognition or experimental rigor that Lavoisier later employed.^[1] The immediate impact of Lavoisier's formulation was profound, establishing precise gravimetric analysis as the standard for chemical investigations and eliminating concepts of matter creation or destruction in reactions.^[2] Chemists thereafter prioritized accurate weighing and closed-system experiments, fostering a shift toward empirical, reproducible science that underpinned modern stoichiometry and reaction analysis.^[4]

Developments in Physics

In the 19th century, physicists integrated the conservation of mass into Newtonian mechanics, treating mass as an invariant property essential to the laws of motion. Newton's second law, F = ma, posits mass as constant for a given body, enabling derivations of momentum conservation in isolated systems, such as elastic collisions where total mass remains unchanged despite velocity alterations. This framework assumed no creation or destruction of mass, aligning with empirical observations in mechanical experiments and providing a foundation for analyzing dynamic systems without mass variability.^[22] Hermann von Helmholtz advanced this integration in his 1847 paper Über die Erhaltung der Kraft (On the Conservation of Force), which established energy conservation across mechanical, thermal, and electrical domains while relying on mass constancy in processes like collisions and motion. Helmholtz demonstrated that force (energy) transformations preserve quantitative relations only if mass remains fixed, linking mass invariance to broader physical invariances in classical mechanics.^[23] John Dalton's 1808 atomic theory reinforced mass conservation by modeling matter as composed of indestructible atoms that combine in fixed proportions, enabling the calculation of relative atomic weights from conserved total mass in combinations. This physical model explained why mass ratios in compounds remain constant, bridging atomic structure with Newtonian principles of invariance.^[24] During the 1850s, thermodynamic developments by Rudolf Clausius and William Thomson (Lord Kelvin) affirmed mass constancy amid energy transformations and entropy considerations. Clausius's mechanical theory of heat treated mass as fixed in closed systems, ensuring that heat-to-work conversions obey conservation without mass alteration. Thomson's parallel work on entropy as a dissipative measure similarly presupposed unchanging mass, solidifying the principle in analyses of irreversible processes. By the late 1800s, Henri Becquerel's 1896 discovery of radioactivity in uranium compounds hinted at potential exceptions to strict mass conservation, as salts emitted penetrating rays spontaneously without accounting for mass loss in classical terms, though these anomalies lacked resolution until subsequent atomic insights.^[25]

Applications

In Chemical Reactions

The principle of conservation of mass is fundamental to stoichiometry in chemical reactions, where balanced equations ensure that the total mass of reactants equals the total mass of products. For instance, in the formation of water from hydrogen and oxygen, the balanced equation is

$2\mathrm{H_2} + \mathrm{O_2} \rightarrow 2\mathrm{H_2O}

Using atomic masses of hydrogen (1 g/mol) and oxygen (16 g/mol), 4 g of hydrogen reacts with 32 g of oxygen to produce 36 g of water, demonstrating exact mass equality.^[26] This balancing process relies on the conservation law to maintain atomic proportions, allowing chemists to predict reaction outcomes quantitatively.^[27] In closed systems, where no matter enters or leaves, conservation of mass is directly observable, as in laboratory setups like sealed flasks for combustion reactions. For example, burning magnesium in a sealed container with oxygen yields magnesium oxide, with the system's total mass remaining unchanged before and after the reaction.^[28] In contrast, open systems can create apparent violations; during rusting of iron in air, oxygen from the environment combines with the metal to form iron oxide, increasing the mass of the rust but dispersing the added oxygen mass if not accounted for in a closed setup.^[29] Antoine Lavoisier's 18th-century experiments with sealed vessels first rigorously demonstrated this distinction in combustion.^[1] The conservation principle underpins practical applications in chemistry, such as determining relative molecular weights through stoichiometric ratios and calculating reaction yields. In the 19th century, John Dalton used mass conservation alongside the law of definite proportions to assign relative atomic weights, forming the basis for modern atomic mass tables; for example, analyzing compound formation ratios allowed him to propose hydrogen's weight as 1 relative to oxygen's 7 (later revised).^[30] Today, it enables quantitative analysis by comparing actual product masses to theoretical yields from balanced equations, as in the reduction of iron oxide with carbon monoxide: the expected yield informs efficiency assessments in industrial processes.^[31] A common misconception arises in reactions producing gases, such as the baking soda and vinegar demonstration, where sodium bicarbonate reacts with acetic acid to form carbon dioxide, water, and sodium acetate, leading some to believe mass has vanished as the solid "disappears" and fizzing occurs. However, in a closed container like a sealed soda bottle, the total mass remains constant, with the gas contributing to the unchanged weight.^[32] This illustrates that mass is conserved across phases, countering the illusion of loss in open setups.^[33]

In Physical Processes

In physical processes, the principle of conservation of mass asserts that the total mass of a closed system remains constant, even as matter undergoes changes in form, motion, or state without nuclear reactions. This invariance is fundamental to classical mechanics, fluid dynamics, and thermodynamics, enabling predictive models for systems ranging from colliding objects to flowing fluids and phase transitions. Unlike energy, which may convert between forms, mass itself is neither created nor destroyed in these non-chemical contexts, providing a stable quantity for analysis. In mechanical interactions, such as elastic and inelastic collisions, the total mass of the involved bodies stays constant, allowing momentum conservation to rely on the unchanging product of mass and velocity. For example, in an elastic collision between two billiard balls of equal mass, the balls exchange velocities while the combined mass before and after impact remains identical, preserving the system's total momentum as m_1 \mathbf{v}_1 + m_2 \mathbf{v}_2 = m_1 \mathbf{v}_1' + m_2 \mathbf{v}_2', where primes denote post-collision values. In inelastic collisions, like a bullet embedding in a block, kinetic energy dissipates as heat or deformation, but the fused mass equals the initial sum, underscoring mass invariance as a prerequisite for momentum analysis in isolated systems. This principle, rooted in Newtonian mechanics, applies universally to macroscopic collisions where relativistic effects are negligible.^[34]^[35] Fluid dynamics exemplifies mass conservation through the continuity equation, derived from the requirement that mass inflow equals outflow in a control volume. For steady, incompressible flows—common in engineering pipes or channels—the equation simplifies to \rho A v = \constant, where \rho is fluid density, A is cross-sectional area, and v is flow velocity. This relation ensures preserved mass flow rate; for instance, in a constricted pipe, velocity increases inversely with area to compensate, preventing mass accumulation. The general form, \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0, extends to compressible and unsteady cases, forming the basis for Navier-Stokes equations in modeling phenomena like river flows or ventilation systems.^[36]^[15] Thermodynamic processes uphold mass conservation during energy transfers and state changes, particularly in open systems where fluid enters or exits. In phase transitions, such as ice melting at 0°C, the mass of solid ice precisely equals the mass of resulting liquid water, with latent heat absorbed without altering total mass. This equality holds because the process involves molecular rearrangement rather than matter addition or removal. In heat engines, like steam cycles, mass balance requires that the working fluid's inflow mass rate matches outflow, ensuring steady operation; for a control volume, \dot{m}_{in} = \dot{m}_{out}, where \dot{m} is mass flow rate, preventing imbalances that could disrupt efficiency. Such balances are critical in analyzing cycles like the Rankine engine, where phase changes (evaporation and condensation) occur without net mass loss.^[37]^[38]^[39] Engineering designs leverage mass conservation for reliability and efficiency in fluid transport and propulsion. In pipeline networks, the continuity principle dictates flow distribution to maintain pressure stability; for a junction, sum of incoming mass flows equals outgoing, modeled as \sum \rho A v = 0, guiding sizing of diameters to avoid surges in water or oil systems. In classical rocket propulsion, treated as a variable-mass system, the rocket's initial mass (structure plus propellant) equals the final mass plus ejected exhaust mass, conserving total mass for the isolated rocket-exhaust system. This underpins the Tsiolkovsky rocket equation, \Delta v = v_e \ln \left( \frac{m_0}{m_f} \right), where v_e is exhaust velocity, m_0 initial mass, and m_f final mass, emphasizing propellant mass as the driver of velocity change without relativistic mass variation.^[6]^[40]

Modern Interpretations

Relativistic Framework

In the framework of special relativity, the classical notion of mass conservation is extended and modified by the principle of mass-energy equivalence, recognizing that mass is a concentrated form of energy. Albert Einstein's seminal 1905 paper, "Does the Inertia of a Body Depend Upon Its Energy Content?", derived this relation through analysis of energy emission in different inertial frames, culminating in the equation E = mc^2, where E is the rest energy, m is the rest mass, and c is the speed of light in vacuum. This equivalence implies that isolated systems conserve total energy, including contributions from mass, rather than mass alone; any conversion between mass and other energy forms, such as kinetic or radiant energy, preserves the overall invariant.^[41]^[7] Einstein employed thought experiments to illustrate this, such as a body at rest emitting two equal pulses of light in opposite directions, resulting in a momentum balance but a net loss of energy L that reduces the body's mass by \Delta m = L / c^2. A concrete manifestation occurs in atomic and nuclear processes where photons are emitted; for example, the release of nuclear binding energy in reactions like fission or fusion leads to a measurable mass defect, where the total mass of the products is slightly less than the reactants, with the difference accounting for the emitted energy via E = \Delta m c^2. This mass decrease, though minuscule on atomic scales (e.g., about 0.1% in uranium fission), underscores how energy emission directly alters rest mass.^[41]^[42] For closed systems in relativity, conservation is governed by the four-momentum vector p^\mu = (E/c, \mathbf{p}), where \mathbf{p} is three-momentum and the invariant mass is given by m = \sqrt{(E/c)^2 - p^2}/c; this four-vector is conserved in interactions, ensuring that rest mass can vary with internal energy configuration or velocity while total mass-energy remains frame-invariant. In relativistic mechanics, the effective or relativistic mass m_{\text{rel}} = \gamma m (with \gamma = 1/\sqrt{1 - v^2/c^2}) increases with speed, reflecting added kinetic energy, but modern formulations emphasize invariant rest mass, with variations arising from energy exchanges.^[43] Post-1905 experimental confirmations in particle accelerators have robustly validated these principles, observing direct mass-to-energy conversions such as pair production (where gamma rays create electron-positron pairs) and annihilation (where particles convert to photons). A high-precision 2005 experiment at the Institut Laue-Langevin (ILL) and National Institute of Standards and Technology (NIST), measuring gamma-ray wavelengths from excited nuclei of silicon-28 and sulfur-32, verified E = mc^2 to within 0.0004% accuracy, aligning emitted energy precisely with predicted mass differences. Such results from accelerators like CERN's Large Hadron Collider further demonstrate conservation through the routine creation of particles from collision energies exceeding rest masses.^[44]^[45]

Quantum and Particle Physics

In quantum field theory (QFT), the classical notion of mass conservation is superseded by the framework of fields and particles, where virtual particles—intermediate states in interactions—temporarily violate the on-shell condition E^2 = p^2 c^2 + m^2 c^4, allowing their effective masses to deviate from observed rest masses without contradicting overall energy-momentum conservation.^[46] These virtual particles, governed by the Heisenberg uncertainty principle, contribute to processes like vacuum fluctuations but do not lead to net mass creation or destruction; instead, conservation is enforced through global symmetries corresponding to quantum numbers such as baryon number B and lepton number L.^[47] For instance, in electron-positron annihilation, the rest masses of the electron and positron (m_e \approx 0.511 MeV/c^2 each) are fully converted into the energy of two or more photons, preserving total four-momentum while the total rest mass becomes zero. A classic example of how quantum effects challenge apparent mass conservation arises in beta decay, where a neutron (m_n \approx 939.57 MeV/c^2) decays into a proton (m_p \approx 938.27 MeV/c^2), electron, and antineutrino, resulting in a rest mass deficit of about 0.78 MeV/c^2 that manifests as kinetic energy of the products.^[48] This process initially appeared to violate energy conservation due to the continuous electron energy spectrum observed in the 1920s, prompting Wolfgang Pauli in 1930 to hypothesize a massless, neutral "neutrino" to carry away the missing energy and momentum, a proposal formalized by Enrico Fermi in 1934 as part of weak interaction theory.^[49] The neutrino's subsequent detection in 1956 confirmed that total energy-momentum is conserved, with the apparent mass change attributable to the creation of new particles rather than a loss of mass.^[50] Within the Standard Model of particle physics, mass conservation applies strictly to stable particles but is relaxed for unstable ones through decays and interactions mediated by gauge bosons, where additional conserved quantum numbers like B and L prevent processes such as proton decay in perturbation theory.^[51] Baryon number, assigned as B = 1/3 for quarks and thus B = 1 for baryons like protons and neutrons, remains conserved in all known strong, electromagnetic, and weak interactions, while lepton number (L = 1 for electrons and neutrinos, L = -1 for positrons and antineutrinos) similarly safeguards against unphysical flavor-changing processes.^[52] However, non-perturbative effects, such as sphaleron processes at high temperatures, can violate B + L but conserve B - L, illustrating that mass-energy equivalence from relativity integrates with these quantum symmetries to maintain overall balance.^[53] High-energy experiments at CERN's Large Hadron Collider (LHC) further illuminate this by revealing that fundamental particle masses originate not from intrinsic properties but from couplings to the Higgs field, as evidenced by the 2012 discovery of the Higgs boson with mass m_H \approx 125 GeV/c^2. In proton-proton collisions reaching \sqrt{s} = 13 TeV, processes like Higgs production via gluon fusion create new particles whose total rest mass exceeds that of the initial protons, yet energy conservation holds through the field's vacuum expectation value (v \approx 246 GeV) that "breaks" electroweak symmetry and generates masses dynamically.^[54] ATLAS and CMS detectors have measured Higgs decays, such as to bottom quarks or W/Z bosons, confirming that interaction rates align with predictions where rest mass is not conserved individually but emerges from field interactions, underscoring QFT's resolution of mass conservation at subatomic scales.^[55]

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1930 Pauli suggested that the continuum spectra might be due to one more “invisible” light neutral particle (later to be named neutrino) involved in the β- ...
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[PDF] arXiv:1210.3065v1 [hep-ph] 10 Oct 2012
Oct 10, 2012 · The hypothesis of the neutrino allowed Fermi to build the first theory of the β-decay which he considered as a process of a quantum transition ...
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The Early Days of Experimental Neutrino Physics - Science
The neutrino hypothesis, put forward by Pauli to account for the apparent loss of energy and momentum in beta decay, was verified by a series of ...
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Baryon number violation in the Standard Model - Book chapter
The baryon number is a classically conserved quantity in the Standard Model as can be demonstrated by Noether's theorem or even a cursory look at the ...
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Particle Interactions and Conservation Laws - HyperPhysics Concepts
Strong overall conservation laws are the conservation of baryon number and the conservation of lepton number. Specific quantum numbers have been assigned to ...
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[PDF] Baryon and lepton numbers in particle physics beyond the standard ...
The standard model has non-perturbative violation of baryon number (B). This source of baryon number non-conservation also violates lepton number (L), however, ...
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Higgs Boson | CMS Experiment - CERN
The CMS experiment was built as a general purpose detector also for this purpose: to be sensitive to any possible Higgs mass.
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The Higgs boson: the hunt, the discovery, the study and some future ...
Jul 4, 2018 · The “Higgs mechanism,” which consists of the Higgs field and its corresponding Higgs boson, is said to give mass to elementary particles.Missing: conservation | Show results with:conservation