Conservation of mass

<h2 id="formulation-and-examples">Formulation and examples</h2>
The law of conservation of mass can only be formulated in <a href="/facts/Classical_mechanics/ykG8upsN">classical mechanics</a>, in which the energy scales associated with an isolated system are much smaller than 
 
 
 
 m
 
 c
 
 2
 
 
 
 
 {\displaystyle mc^{2}}
 
, where 
 
 
 
 m
 
 
 {\displaystyle m}
 
 is the mass of a typical object in the system, measured in the <a href="/facts/Frame_of_reference/15yYfB9M">frame of reference</a> where the object is at rest, and 
 
 
 
 c
 
 
 {\displaystyle c}
 
 is the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a>.
The law can be formulated mathematically in the fields of <a href="/facts/Fluid_mechanics/NIy4tRfE">fluid mechanics</a> and <a href="/facts/Continuum_mechanics/vI7LpFDk">continuum mechanics</a>, where the conservation of mass is usually expressed using the <a href="/facts/Continuity_equation/GVav2Vet">continuity equation</a>, given in <a href="/facts/Differential_equation/9MGbE3Ca">differential form</a> as 
 
 
 
 
 
 
 ∂
 ρ
 
 
 ∂
 t
 
 
 
 +
 ∇
 ⋅
 (
 ρ
 
 v
 
 )
 =
 0
 ,
 
 
 {\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {v} )=0,}
 
 where 
 
 
 
 ρ
 
 
 {\textstyle \rho }
 
 is the <a href="/facts/Density/92p6hh9I">density</a> (mass per unit volume), 
 
 
 
 t
 
 
 {\textstyle t}
 
 is the time, 
 
 
 
 ∇
 ⋅
 
 
 {\textstyle \nabla \cdot }
 
 is the <a href="/facts/Divergence/zpwwkFoU">divergence</a>, and 
 
 
 
 
 v
 
 
 
 {\textstyle \mathbf {v} }
 
 is the <a href="/facts/Flow_velocity/amIT4Ht8">flow velocity</a> field.
The interpretation of the continuity equation for mass is the following: For a given closed surface in the system, the change, over any time interval, of the mass enclosed by the surface is equal to the mass that traverses the surface during that time interval: positive if the matter goes in and negative if the matter goes out. For the whole isolated system, this condition implies that the total mass 
 
 
 
 M
 
 
 {\textstyle M}
 
, the sum of the masses of all components in the system, does not change over time, i.e. 
 
 
 
 
 
 
 
 d
 
 M
 
 
 
 d
 
 t
 
 
 
 =
 
 
 d
 
 
 d
 
 t
 
 
 
 ∫
 ρ
 
 
 d
 
 V
 =
 0
 ,
 
 
 {\displaystyle {\frac {{\text{d}}M}{{\text{d}}t}}={\frac {\text{d}}{{\text{d}}t}}\int \rho \,{\text{d}}V=0,}
 
 where 
 
 
 
 
 d
 
 V
 
 
 {\textstyle {\text{d}}V}
 
 is the <a href="/facts/Differential_(infinitesimal)/3dxU2Wy2">differential</a> that defines the <a href="/facts/Integral/4PeM0mJc">integral</a> over the whole volume of the system.
The continuity equation for the mass is part of the <a href="/facts/Euler_equations_(fluid_dynamics)/EYW5lerl">Euler equations</a> of fluid dynamics. Many other <a href="/facts/Convection%25E2%2580%2593diffusion_equation/jHoHG5Xj">convection–diffusion equations</a> describe the conservation and flow of mass and matter in a given system.
In chemistry, the calculation of the amount of <a href="/facts/Reagent/c3GLInCu">reactant</a> and <a href="/facts/Product_(chemistry)/8uCoDUdv">products</a> in a chemical reaction, or <a href="/facts/Stoichiometry/I5p9cJzV">stoichiometry</a>, is founded on the principle of conservation of mass. The principle implies that during a chemical reaction the total mass of the reactants is equal to the total mass of the products. For example, in the following reaction

CH4 + 2 O2 → CO2 + 2 H2O,
where one <a href="/facts/Molecule/N9ljSfFq">molecule</a> of <a href="/facts/Methane/kBHneYnh">methane</a> (CH4) and two <a href="/facts/Oxygen/acyn6o8r">oxygen</a> molecules O2 are converted into one molecule of <a href="/facts/Carbon_dioxide/xGzpppEp">carbon dioxide</a> (CO2) and two of <a href="/facts/Water/0lkXnJPK">water</a> (H2O). The number of molecules resulting from the reaction can be derived from the principle of conservation of mass, as initially four <a href="/facts/Hydrogen/BoYHjl0J">hydrogen</a> atoms, 4 oxygen atoms and one carbon atom are present (as well as in the final state); thus the number water molecules produced must be exactly two per molecule of carbon dioxide produced.
Many <a href="/facts/Engineering/RFVg8H3I">engineering</a> problems are solved by following the mass distribution of a given system over time; this methodology is known as <a href="/facts/Mass_balance/NTAEvs9E">mass balance</a>.

<h2 id="history">History</h2>

As early as 520 BCE, <a href="/facts/Jain_philosophy/RWQtyapu">Jain philosophy</a>, a <a href="/facts/Jainism_and_non-creationism/4FuF7W1X">non-creationist philosophy</a> based on the teachings of <a href="/facts/Mahavira/46A5AGc3">Mahavira</a>,<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> stated that the universe and its constituents such as matter cannot be destroyed or created. The <a href="/facts/Jain_text/gw4tubm3">Jain text</a> <a href="/facts/Tattvarthasutra/qP00GMtN">Tattvarthasutra</a> (2nd century CE) states that a substance is permanent, but its modes are characterised by creation and destruction.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
An important idea in <a href="/facts/Ancient_Greek_philosophy/LWEkp8JX">ancient Greek philosophy</a> was that "<a href="/facts/Creatio_ex_materia/IhY0hoJn">Nothing comes from nothing</a>", so that what exists now has always existed: no new matter can come into existence where there was none before. An explicit statement of this, along with the further principle that nothing can pass away into nothing, is found in <a href="/facts/Empedocles/C6sl6Lz6">Empedocles</a> (c. 4th century BCE): "For it is impossible for anything to come to be from what is not, and it cannot be brought about or heard of that what is should be utterly destroyed."<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
A further principle of conservation was stated by <a href="/facts/Epicurus/ww2teSaQ">Epicurus</a> around the 3rd century BCE, who wrote in describing the nature of the Universe that "the totality of things was always such as it is now, and always will be".<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h3>Discoveries in chemistry</h3>
By the 18th century the principle of conservation of mass during chemical reactions was widely used and was an important assumption during experiments, even before a definition was widely established,<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> though an expression of the law can be dated back to Hero of Alexandria's time,<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> as can be seen in the works of <a href="/facts/Joseph_Black/0HD6fdYR">Joseph Black</a>, <a href="/facts/Henry_Cavendish/UsIDwP3F">Henry Cavendish</a>, and <a href="/facts/Jean_Rey_(physician)/XGNkNKrC">Jean Rey</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> One of the first to outline the principle was <a href="/facts/Mikhail_Lomonosov/8Hdq5JEm">Mikhail Lomonosov</a> in 1756. He may have demonstrated it by experiments and certainly had discussed the principle in 1748 in correspondence with <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a>,<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> though his claim on the subject is sometimes challenged.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a><a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> According to the Soviet physicist Yakov Dorfman:<blockquote>The universal law was formulated by Lomonosov on the basis of general philosophical materialistic considerations, it was never questioned or tested by him, but on the contrary, served him as a solid starting position in all research throughout his life.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> </blockquote>A more refined series of experiments were later carried out by <a href="/facts/Antoine_Lavoisier/UheVQrhu">Antoine Lavoisier</a> who expressed his conclusion in 1773 and popularized the principle of conservation of mass.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> The demonstrations of the principle disproved the then popular <a href="/facts/Phlogiston_theory/6PDveBmv">phlogiston theory</a> that said that mass could be gained or lost in <a href="/facts/Combustion/ohQoWj8z">combustion</a> and heat processes.
The conservation of mass was obscure for millennia because of the buoyancy effect of the Earth's atmosphere on the weight of gases. For example, a piece of wood weighs less after burning;<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> this seemed to suggest that some of its mass disappears, or is transformed or lost. Careful experiments were performed in which chemical reactions such as rusting were allowed to take place in sealed glass ampoules; it was found that the chemical reaction did not change the weight of the sealed container and its contents. Weighing of gases using scales was not possible until the invention of the <a href="/facts/Vacuum_pump/QTkV5dXr">vacuum pump</a> in the 17th century.
Once understood, the conservation of mass was of great importance in progressing from <a href="/facts/Alchemy/CEx0U1Em">alchemy</a> to modern chemistry. Once early chemists realized that chemical substances never disappeared but were only transformed into other substances with the same weight, these scientists could for the first time embark on quantitative studies of the transformations of substances. The idea of mass conservation plus a surmise that certain "elemental substances" also could not be transformed into others by chemical reactions, in turn led to an understanding of <a href="/facts/Chemical_element/0jducgWD">chemical elements</a>, as well as the idea that all chemical processes and transformations (such as burning and metabolic reactions) are reactions between invariant amounts or weights of these chemical elements.
Following the pioneering work of Lavoisier, the exhaustive experiments of <a href="/facts/Jean_Stas/6oTfCaxr">Jean Stas</a> supported the consistency of this law in chemical reactions,<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> even though they were carried out with other intentions. His research<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a><a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> indicated that in certain reactions the loss or gain could not have been more than 2 to 4 parts in 100,000.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> The difference in the accuracy aimed at and attained by Lavoisier on the one hand, and by <a href="/facts/Edward_W._Morley/yNp2YFUe">Edward W. Morley</a> and Stas on the other, is enormous.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>

<h3>Modern physics</h3>
Main articles: <a href="/facts/Mass%25E2%2580%2593energy_equivalence/ExXawGcc">Mass–energy equivalence</a> and <a href="/facts/Mass_in_general_relativity/7IycC0jq">Mass in general relativity</a>
The law of conservation of mass was challenged with the advent of special relativity. In one of the <a href="/facts/Annus_Mirabilis_papers/8LhL5hm7">Annus Mirabilis papers</a> of <a href="/facts/Albert_Einstein/CNTet8qu">Albert Einstein</a> in 1905, he suggested an equivalence between mass and energy. This theory implied several assertions, like the idea that internal energy of a system could contribute to the mass of the whole system, or that mass could be converted into <a href="/facts/Electromagnetic_radiation/Ynx7ujCW">electromagnetic radiation</a>. However, as <a href="/facts/Max_Planck/KKk0698y">Max Planck</a> pointed out, a change in mass as a result of extraction or addition of chemical energy, as predicted by Einstein's theory, is so small that it could not be measured with the available instruments and could not be presented as a test of special relativity. Einstein speculated that the energies associated with newly discovered <a href="/facts/Radioactivity/Ya6FVjt4">radioactivity</a> were significant enough, compared with the mass of systems producing them, to enable their change of mass to be measured, once the energy of the reaction had been removed from the system. This later indeed proved to be possible, although it was eventually to be the first artificial <a href="/facts/Nuclear_transmutation/MjlvjsV6">nuclear transmutation</a> reaction in 1932, demonstrated by <a href="/facts/Cockcroft-Walton_generator/2zIWAtRV">Cockcroft and Walton</a>, that proved the first successful test of Einstein's theory regarding mass loss with energy gain.
The law of conservation of mass and the analogous law of <a href="/facts/Conservation_of_energy/12CdN4Ik">conservation of energy</a> were finally generalized and unified into the principle of <a href="/facts/Mass%25E2%2580%2593energy_equivalence/ExXawGcc">mass–energy equivalence</a>, described by <a href="/facts/Albert_Einstein/CNTet8qu">Albert Einstein</a>'s equation E = m c 2 {\displaystyle E=mc^{2}}
. Special relativity also redefines the concept of mass and energy, which can be used interchangeably and are defined relative to the frame of reference. Several quantities had to be defined for consistency, such as the <a href="/facts/Invariant_mass/wNGXNBW6">rest mass</a> of a particle (mass in the rest frame of the particle) and the relativistic mass (in another frame). The latter term is usually less frequently used.
In <a href="/facts/General_relativity/jVsoIg8X">general relativity</a>, conservation of both mass and energy is not globally conserved and its definition is more complicated.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Charge_conservation/WGQXmoE8">Charge conservation</a></li>
<li><a href="/facts/Conservation_law/6g1Ay2FR">Conservation law</a></li>
<li><a href="/facts/Fick%2527s_laws_of_diffusion/2guvKD4T">Fick's laws of diffusion</a></li>
<li><a href="/facts/Law_of_definite_proportions/spaNgzcc">Law of definite proportions</a></li>
<li><a href="/facts/Law_of_multiple_proportions/e69PdPwl">Law of multiple proportions</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">John Olmsted; Gregory M. Williams (1997). Chemistry: The Molecular Science (illustrated ed.). Jones & Bartlett Learning. p. 69. ISBN 978-0-8151-8450-8. Extract of page 69 <a href="978-0-8151-8450-8" target="_blank">978-0-8151-8450-8</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Lavoisier's Method <a href="http://historyofscience.free.fr/Lavoisier-Friends/a_chap2_lavoisier.html#method" target="_blank">http://historyofscience.free.fr/Lavoisier-Friends/a_chap2_lavoisier.html#method</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Mahavira is dated 598 BC - 526 BC. See: Dundas, Paul (2002). The Jains. book series, Library of Religious Beliefs and Practices, edited by John R. Hinnels & Ninian Smart. London: Routledge. ISBN 978-0-415-26606-2. p. 24 <a href="978-0-415-26606-2" target="_blank">978-0-415-26606-2</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Devendra (Muni.), T. G. Kalghatgi, T. S. Devadoss (1983) A source-book in Jaina philosophy Udaipur:Sri Tarak Guru Jain Gran. p.57. Also see Tattvarthasutra verses 5.29 and 5.37 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Fr. 12; see pp.291–2 of Kirk, G. S.; J. E. Raven; Malcolm Schofield (1983). The Presocratic Philosophers (2 ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-27455-5. <a href="978-0-521-27455-5" target="_blank">978-0-521-27455-5</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Long, A. A.; D. N. Sedley (1987). "Epicureanism: The principals of conservation". The Hellenistic Philosophers. Vol 1: Translations of the principal sources with philosophical commentary. Cambridge: Cambridge University Press. pp. 25–26. ISBN 978-0-521-27556-9. <a href="978-0-521-27556-9" target="_blank">978-0-521-27556-9</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Whitaker, Robert D. (1975-10-01). "An historical note on the conservation of mass". Journal of Chemical Education. 52 (10): 658. Bibcode:1975JChEd..52..658W. doi:10.1021/ed052p658. ISSN 0021-9584. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Tanner, R. I.; Walters, K. (1998). Rheology: An Historical Perspective. Elsevier. ISBN 9780444829467. <a href="9780444829467" target="_blank">9780444829467</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Robert D. Whitaker, "An Historical Note on the Conservation of Mass", Journal of Chemical Education, 52, 10, 658-659, Oct 75 <a href="https://eric.ed.gov/?id=EJ128341" target="_blank">https://eric.ed.gov/?id=EJ128341</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Pismen, Len (2018). The Swings of Science: From Complexity to Simplicity and Back. Springer. p. 41. ISBN 978-3-319-99777-3. <a href="978-3-319-99777-3" target="_blank">978-3-319-99777-3</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Pomper, Philip (October 1962). "Lomonosov and the Discovery of the Law of the Conservation of Matter in Chemical Transformations". Ambix. 10 (3): 119–127. doi:10.1179/amb.1962.10.3.119. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Lomonosov, Mikhail Vasil’evich (1970). Mikhail Vasil'evich Lomonosov on the Corpuscular Theory. Henry M. Leicester (trans.). Cambridge, Mass.: Harvard University Press. Introduction, p. 25. <a href="/wiki/Henry_M._Leicester" target="_blank">/wiki/Henry_M._Leicester</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Дорфман, Яков (1961). Закон сохранения массы при химических реакциях и физические воззрения Ломоносова // Ломоносов М.В. Сборник статей и материалов, T.5. М.-Л.: Издательство АН СССР. p. 193. <a href="http://gidropraktikum.narod.ru/Lomonosov-Dorfman.djvu" target="_blank">http://gidropraktikum.narod.ru/Lomonosov-Dorfman.djvu</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Agnew, Henry; Alviar-Agnew, Marisa (7 January 2020). "3.7 Conservation of Mass - There is No New Matter". LibreTexts™ Chemistry. Retrieved 10 January 2024. <a href="https://chem.libretexts.org/Courses/Arkansas_Northeastern_College/CH14133%3A_Chemistry_for_General_Education/03%3A_Matter_and_Energy/3.07%3A_Conservation_of_Mass_-_There_is_No_New_Matter#:~:text=If%20you%20witness%20a%20300%20kg%20tree%20burn%20to%20the%20ground,%20there%20are%20only%20ashes%20left%20after%20the%20burn,%20and%20all%20of%20them%20together%20weigh%2010%20kg.%20It%20may%20make%20you%20wonder%20where%20the%20other%20290%20kg%20went.%20The%20missing%20290%20kg%20was%20released%20into%20the%20atmosphere%20as%20smoke,%20so%20the%20only%20thing%20left%20that%20you%20can%20see%20is%20the%2010%20kg%20of%20ash." target="_blank">https://chem.libretexts.org/Courses/Arkansas_Northeastern_College/CH14133%3A_Chemistry_for_General_Education/03%3A_Matter_and_Energy/3.07%3A_Conservation_of_Mass_-_There_is_No_New_Matter#:~:text=If%20you%20witness%20a%20300%20kg%20tree%20burn%20to%20the%20ground,%20there%20are%20only%20ashes%20left%20after%20the%20burn,%20and%20all%20of%20them%20together%20weigh%2010%20kg.%20It%20may%20make%20you%20wonder%20where%20the%20other%20290%20kg%20went.%20The%20missing%20290%20kg%20was%20released%20into%20the%20atmosphere%20as%20smoke,%20so%20the%20only%20thing%20left%20that%20you%20can%20see%20is%20the%2010%20kg%20of%20ash.</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Agnew, Henry; Alviar-Agnew, Marisa (7 January 2020). "3.7 Conservation of Mass - There is No New Matter". LibreTexts™ Chemistry. Retrieved 10 January 2024. <a href="https://chem.libretexts.org/Courses/Arkansas_Northeastern_College/CH14133%3A_Chemistry_for_General_Education/03%3A_Matter_and_Energy/3.07%3A_Conservation_of_Mass_-_There_is_No_New_Matter#:~:text=If%20you%20witness%20a%20300%20kg%20tree%20burn%20to%20the%20ground,%20there%20are%20only%20ashes%20left%20after%20the%20burn,%20and%20all%20of%20them%20together%20weigh%2010%20kg.%20It%20may%20make%20you%20wonder%20where%20the%20other%20290%20kg%20went.%20The%20missing%20290%20kg%20was%20released%20into%20the%20atmosphere%20as%20smoke,%20so%20the%20only%20thing%20left%20that%20you%20can%20see%20is%20the%2010%20kg%20of%20ash." target="_blank">https://chem.libretexts.org/Courses/Arkansas_Northeastern_College/CH14133%3A_Chemistry_for_General_Education/03%3A_Matter_and_Energy/3.07%3A_Conservation_of_Mass_-_There_is_No_New_Matter#:~:text=If%20you%20witness%20a%20300%20kg%20tree%20burn%20to%20the%20ground,%20there%20are%20only%20ashes%20left%20after%20the%20burn,%20and%20all%20of%20them%20together%20weigh%2010%20kg.%20It%20may%20make%20you%20wonder%20where%20the%20other%20290%20kg%20went.%20The%20missing%20290%20kg%20was%20released%20into%20the%20atmosphere%20as%20smoke,%20so%20the%20only%20thing%20left%20that%20you%20can%20see%20is%20the%2010%20kg%20of%20ash.</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Matthew Moncrieff Pattison Muir, The Elements of Chemistry (1904) <a href="https://books.google.com/books?id=EwZZAAAAYAAJ" target="_blank">https://books.google.com/books?id=EwZZAAAAYAAJ</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Nouv. Recherches sur les lois des proportions chimiques (1865) 152, 171, 189 <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">"Conservation of Mass in Chemical Changes"Journal - Chemical Society, London, Vol.64, Part 2 Chemical Society (Great Britain) <a href="https://books.google.com/books?id=bhQ3AAAAYAAJ" target="_blank">https://books.google.com/books?id=bhQ3AAAAYAAJ</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">William Edwards Henderson, A Course in General Chemistry (1921) <a href="https://archive.org/details/acourseingenera00hendgoog" target="_blank">https://archive.org/details/acourseingenera00hendgoog</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Ida Freund, The study of Chemical Composition: an account of its method and historical development, with illustrative quotations (1904) <a href="/wiki/Ida_Freund" target="_blank">/wiki/Ida_Freund</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
</ol>

Conservation of mass open-in-new

Conservation of mass