Editing Boyle's law

{{Short description|Relation between gas pressure and volume}}
[[File:Boyles Law animated.gif|thumb|upright=1.3|An animation showing the relationship between pressure and volume when mass and temperature are held constant]]
{{Continuum mechanics|fluid}}

'''Boyle's law''', also referred to as the '''Boyle–Mariotte law''' or '''Mariotte's law''' (especially in France), is an empirical [[gas laws|gas law]] that describes the relationship between [[pressure]] and [[volume]] of a confined [[gas]]. Boyle's law has been stated as:
<blockquote>
The absolute pressure exerted by a given mass of an [[ideal gas]] is inversely proportional to the volume it occupies if the [[temperature]] and [[amount of substance|amount of gas]] remain unchanged within a [[closed system]].<ref>Levine, Ira N. (1978). ''Physical Chemistry''. [[McGraw-Hill]]</ref><ref name="levine_1">Levine (1978) p. 12 gives the original definition.</ref>
</blockquote>
Mathematically, Boyle's law can be stated as:

{|
|-
| style="padding: 0.2em 0.4em;" | <math>P \propto \frac{1}{V}</math>
| style="padding: 0.2em 0.4em;" | Pressure is inversely proportional to the volume
|}

or

{|
|-
| style="padding: 0.2em 0.4em;" | {{math|''PV'' {{=}} ''k''}}
| style="padding: 0.2em 0.4em;" | The product of pressure and volume is a constant number (here denoted as {{mvar|k}})
|}
where {{mvar|P}} is the pressure of the gas, {{mvar|V}} is the volume of the gas, and {{mvar|k}} is a [[Constant (mathematics)|constant]] for a particular temperature and amount of gas.

Boyle's law states that when the temperature of a given [[mass]] of confined gas is constant, the product of its pressure and volume is also constant. When comparing the same substance under two different sets of conditions, the law can be expressed as:
 
<math display="block">P_1 V_1 = P_2 V_2.</math>

showing that as volume increases, the pressure of a gas decreases proportionally, and vice versa. 

Boyle's law is named after [[Robert Boyle]], who published the original law in 1662.<ref>In 1662, he published a second edition of the 1660 book ''New Experiments Physico-Mechanical, Touching the Spring of the Air, and its Effects'' with an addendum ''Whereunto is Added a Defence of the Authors Explication of the Experiments, Against the Obiections of Franciscus Linus and Thomas Hobbes''; see {{cite journal |title=Robert Boyle's landmark book of 1660 with the first experiments on rarified air |first=John B. |last=West |date=1 January 2005 |doi=10.1152/japplphysiol.00759.2004 |journal=Journal of Applied Physiology |volume=98 |issue=1 |pages=31–39|pmid=15591301 }}</ref> An equivalent law is Mariotte’s law, named after French physicist [[Edme Mariotte]].

==History==
{{main article|History of thermodynamics}}
[[File:Boyles Law.svg|thumb|Graph of Boyle's original data<ref>{{cite book | page=60 | url=https://bvpb.mcu.es/en/catalogo_imagenes/grupo.do?path=11143411 | title=A Defence Of the Doctrine touching the Spring and Weight Of the Air | author=Robert Boyle | year=1662}}</ref> showing the [[Hyperbola#Hyperbola with equation y = A/x|hyperbolic curve]] of the relationship between pressure ({{mvar|P}}) and volume ({{mvar|V}}) of the form {{mvar|1=P = k/V}}. ]] 
The relationship between pressure and volume was first noted by [[Richard Towneley]] and [[Henry Power]] in the 17th century.<ref>See:
*  Henry Power, ''Experimental Philosophy, in Three Books'' (London:  Printed by T. Roycroft for John Martin and James Allestry, 1663), pp. 126–130.  Available online at  [https://quod.lib.umich.edu/e/eebo/a55584.0001.001/155?page=root;size=125;view=text Early English Books Online]. On page 130, Power presents (not very clearly) the relation between the pressure and the volume of a given quantity of air: "That the measure of the Mercurial Standard, and Mercurial Complement, are measured onely by their perpendicular heights, over the Surface of the restagnant Quicksilver in the Vessel: But Ayr, the Ayr's Dilatation, and Ayr Dilated, by the Spaces they fill. So that here is now four Proportionals, and by any three given, you may strike out the fourth, by Conversion, Transposition, and Division of them. So that by these Analogies you may prognosticate the effects, which follow in all Mercurial Experiments, and predemonstrate them, by calculation, before the senses give an Experimental [eviction] thereof." In other words, if one knows the volume V<sub>1</sub> ("Ayr") of a given quantity of air at the pressure p<sub>1</sub> ("Mercurial standard", i.e., atmospheric pressure at a low altitude), then one can predict the volume V<sub>2</sub> ("Ayr dilated") of the same quantity of air at the pressure p<sub>2</sub> ("Mercurial complement", i.e., atmospheric pressure at a higher altitude) by means of a proportion (because p<sub>1</sub> V<sub>1</sub> = p<sub>2</sub> V<sub>2</sub>).
* Charles Webster (1965). "The discovery of Boyle's law, and the concept of the elasticity of air in seventeenth century", ''Archive for the History of Exact Sciences'', '''2''' (6): 441–502; see especially pp. 473–477.
*  Charles Webster (1963). "Richard Towneley and Boyle's Law", ''Nature'', '''197''' (4864):  226–228.
*  Robert Boyle acknowledged his debts to Towneley and Power in: R. Boyle, ''A Defence of the Doctrine Touching the Spring and Weight of the Air'' (London, England:  Thomas Robinson, 1662).  Available online at [http://bvpb.mcu.es/en/consulta/registro.cmd?id=406806 {{lang|es|italic=no|La Biblioteca Virtual de Patrimonio Bibliográfico}}]. On pages 50, 55–56, and 64, Boyle cites experiments by Towneley and Power showing that air expands as the ambient pressure decreases. On p. 63, Boyle acknowledges Towneley's help in interpreting Boyle's data from experiments relating the pressure to the volume of a quantity of air. (Also, on p. 64, Boyle acknowledges that [[William Brouncker, 2nd Viscount Brouncker|Lord Brouncker]] had also investigated the same subject.)</ref><ref name="Holton2001">{{cite book|author=Gerald James Holton|title=Physics, the Human Adventure: From Copernicus to Einstein and Beyond|url=https://books.google.com/books?id=czaGZzR0XOUC&pg=PA270|year=2001|publisher=Rutgers University Press|isbn=978-0-8135-2908-0|pages=270–}}</ref> [[Robert Boyle]] confirmed their discovery through experiments and published the results.<ref>R. Boyle, ''A Defence of the Doctrine Touching the Spring and Weight of the Air'' (London: Thomas Robinson, 1662).  Available online at [http://bvpb.mcu.es/en/consulta/registro.cmd?id=406806 {{lang|es|italic=no|Spain's La Biblioteca Virtual de Patrimonio Bibliográfico}}]. Boyle presents his law in "Chap. V. Two new experiments touching the measure of the force of the spring of air compress'd and dilated", pp. 57–68. On p. 59, Boyle concludes that "the same air being brought to a degree of density about twice as that it had before, obtains a spring twice as strong as formerly". That is, doubling the density of a quantity of air doubles its pressure. Since air's density is proportional to its pressure, then for a fixed quantity of air, the product of its pressure and its volume is constant.  On page 60, he presents his data on the compression of air: "A Table of the Condensation of the Air." The legend (p. 60) accompanying the table states: "E. What the pressure should be according to the ''Hypothesis'', that supposes the pressures and expansions to be in reciprocal relation." On p. 64, Boyle presents his data on the expansion of air: "A Table of the Rarefaction of the Air."</ref> According to [[Robert Gunther]] and other authorities, it was Boyle's assistant, [[Robert Hooke]], who built the experimental apparatus.  Boyle's law is based on experiments with [[air]], which he considered to be a fluid of particles at rest in between small invisible springs. Boyle may have begun experimenting with gases due to an interest in air as an essential element of life;<ref>''The Boyle Papers'' [http://www.bbk.ac.uk/boyle/boyle_papers/bp09_docs/bp9_075v-076r.htm BP 9, fol. 75v–76r]. {{webarchive|url=https://web.archive.org/web/20091122203756/http://www.bbk.ac.uk/boyle/boyle_papers/bp09_docs/bp9_075v-076r.htm |date=2009-11-22 }}</ref> for example, he published works on the growth of plants without air.<ref>''The Boyle Papers'', [http://www.bbk.ac.uk/boyle/boyle_papers/bp10_docs/bp10_138v-139r.htm BP 10, fol. 138v–139r]. {{webarchive|url=https://web.archive.org/web/20091122203809/http://www.bbk.ac.uk/boyle/boyle_papers/bp10_docs/bp10_138v-139r.htm |date=2009-11-22 }}</ref> Boyle used a closed J-shaped tube and after pouring [[Mercury (element)|mercury]] from one side he forced the air on the other side to contract under the pressure of mercury. After repeating the [[experiment]] several times and using different amounts of mercury he found that under controlled conditions, the pressure of a gas is inversely proportional to the volume occupied by it.<ref name="scientists">{{cite book |title=Scientists and Inventors of the Renaissance |date=2012 |publisher=Britannica Educational Publishing |pages=94–96 |url=https://books.google.com/books?id=ndubAAAAQBAJ&pg=PA94|isbn=978-1615308842 }}</ref> 

The French [[physicist]] [[Edme Mariotte]] (1620–1684) discovered the same law independently of Boyle in 1679,<ref>See:
*  Mariotte, ''Essais de Physique, ou mémoires pour servir à la science des choses naturelles'' (Paris, France: E. Michallet, 1679); {{lang|fr|italic=no|"Second essai. De la nature de l'air"}}.
* Mariotte, Edmé, {{lang|fr|Oeuvres de Mr. Mariotte, de l'Académie royale  des sciences}}, vol. 1 (Leiden, Netherlands: P. Vander Aa, 1717); see especially [http://babel.hathitrust.org/cgi/pt?id=nyp.33433084031396;view=1up;seq=181 pp. 151–153].
* Mariotte's essay {{lang|fr|italic=no|"De la nature de l'air"}} was reviewed by the French Royal Academy of Sciences in 1679.  See: Anon. (1733), [https://archive.org/stream/histoiredelacad00acad#page/270/mode/2up {{lang|fr|italic=no|"Sur la nature de l'air"}}], {{lang|fr|Histoire de l'Académie Royale des Sciences}}, '''1''':  270–278.
* Mariotte's essay {{lang|fr|italic=no|"De la nature de l'air"}} was also reviewed in the {{lang|fr|Journal des Sçavans}} (later:  {{lang|fr|Journal des Savants}}) on 20 November 1679. See: Anon. (20 November 1679), [https://books.google.com/books?id=Wsrc_p4pHK8C&pg=PA269 {{lang|fr|italic=no|"Essais de physique"}}], {{lang|fr|Journal des Sçavans}}, pp. 265–269.</ref> after Boyle had published it in 1662.<ref name="scientists"/> Mariotte did, however, discover that air volume changes with temperature.<ref name="ley196606">{{Cite magazine
 |last=Ley
 |first=Willy
 |author=
 |last2=
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 |last3=
 |first3=
 |date=June 1966
 |title=The Re-Designed Solar System
 |department=For Your Information
 |url=https://archive.org/stream/Galaxy_v24n05_1966-06#page/n93/mode/2up
 |magazine=Galaxy Science Fiction
 |pages=94–106
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}}</ref> Thus this law is sometimes referred to as Mariotte's law or the Boyle–Mariotte law. Later, in 1687 in the {{lang|la|[[Philosophiæ Naturalis Principia Mathematica]]}}, [[Isaac Newton|Newton]] showed mathematically that in an [[Elasticity (physics)|elastic]] fluid consisting of particles at rest, between which are repulsive forces inversely proportional to their distance, the density would be directly proportional to the pressure,<ref>{{lang|la|Principia}}, Sec. V, prop. XXI, Theorem XVI</ref> but this mathematical treatise does not involve any Mariott temperature dependance and is not the proper physical explanation for the observed relationship. Instead of a static theory, a [[Kinetic theory of gases|kinetic theory]] is needed, which was developed over the next two centuries by [[Daniel Bernoulli]] (1738) and more fully by [[Rudolf Clausius]] (1857), [[James Clerk Maxwell|Maxwell]] and [[Ludwig Boltzmann|Boltzmann]].

This law was the first physical law to be expressed in the form of an equation describing the dependence of two variable quantities.<ref name="scientists"/>

== Definition ==
[[File:Boyle's Law Demonstrations.webm|thumb|Boyle's law demonstrations]]

The law itself can be stated as follows:

{{quotation|For a fixed mass of an [[ideal gas]] kept at a fixed temperature, pressure and volume are inversely proportional.<ref name="levine_1"/>}}
Boyle's law is a [[Gas laws|gas law]], stating that the pressure and volume of a gas have an inverse relationship. If volume increases, then pressure decreases and vice versa, when the temperature is held constant.

Therefore, when the volume is halved, the pressure is doubled; and if the volume is doubled, the pressure is halved.

=== Relation with kinetic theory and ideal gases ===
As the pressure on a gas increases, the volume of the gas decreases because the gas particles are forced closer together. 

Most gases behave like [[ideal gas]]es at moderate pressures and temperatures. The technology of the 17th century could not produce very high pressures or very low temperatures. Hence, the law was not likely to have deviations at the time of publication. As improvements in technology permitted higher pressures and lower temperatures, deviations from the ideal gas behavior became noticeable, and the relationship between pressure and volume can only be accurately described employing [[real gas]] theory.<ref name="levine_2">Levine, Ira. N. (1978), p. 11 notes that deviations occur with high pressures and temperatures.</ref>  The deviation is expressed as the [[compressibility factor]].

Boyle (and  Mariotte) derived the law solely by experiment. The law can also be derived theoretically based on the presumed existence of [[atom]]s and [[molecule]]s and assumptions about motion and perfectly elastic collisions (see [[kinetic theory of gases]]). These assumptions were met with enormous resistance in the [[positivist]] scientific community at the time, however, as they were seen as purely theoretical constructs for which there was not the slightest observational evidence.

[[Daniel Bernoulli]] (in 1737–1738) derived Boyle's law by applying [[Newton's laws of motion]] at the molecular level. It remained ignored until around 1845, when [[John Waterston]] published a paper building the main precepts of kinetic theory; this was rejected by the [[Royal Society of England]]. Later works of [[James Prescott Joule]], [[Rudolf Clausius]] and in particular [[Ludwig Boltzmann]] firmly established the [[kinetic theory of gases]] and brought attention to both the theories of Bernoulli and Waterston.<ref name="levine_3">Levine, Ira. N. (1978), p. 400 – Historical background of Boyle's law relation to Kinetic Theory</ref>

The debate between proponents of [[Thermodynamics|energetics]] and [[atomism]] led Boltzmann to write a book in 1898, which endured criticism until his suicide in 1906.<ref name="levine_3" /> [[Albert Einstein]] in 1905 showed how kinetic theory applies to the [[Brownian motion]] of a fluid-suspended particle, which was confirmed in 1908 by [[Jean Perrin]].<ref name="levine_3" />

===Equation===
{{Ideal gas law relationships.svg}}
The mathematical equation for Boyle's law is:

<math display="block"> PV = k </math>

where {{mvar|P}} denotes the [[pressure]] of the system, {{mvar|V}} denotes the [[volume]] of the gas, {{mvar|k}} is a constant value representative of the temperature of the system and [[Amount of substance|amount]] of gas.

So long as [[temperature]] remains constant the same amount of energy given to the system persists throughout its operation and therefore, theoretically, the value of {{mvar|k}} will remain constant. However, due to the derivation of pressure as perpendicular applied force and the probabilistic likelihood of collisions with other particles through [[collision theory]], the application of force to a surface may not be infinitely constant for such values of {{mvar|V}}, but will have a [[limit (mathematics)|limit]] when [[differential calculus|differentiating]] such values over a given time. Forcing the volume {{mvar|V}} of the fixed quantity of gas to increase, keeping the gas at the initially measured temperature, the pressure {{mvar|P}} must decrease proportionally. Conversely, reducing the volume of the gas increases the pressure. Boyle's law is used to predict the result of introducing a change, in volume and pressure only, to the initial state of a fixed quantity of gas.

The initial and final volumes and pressures of the fixed amount of gas, where the initial and final temperatures are the same (heating or cooling will be required to meet this condition), are related by the equation:

<math display="block">P_1 V_1 = P_2 V_2. </math>

Here {{math|''P''<sub>1</sub>}} and {{math|''V''<sub>1</sub>}} represent the original pressure and volume, respectively, and {{math|''P''<sub>2</sub>}} and {{math|''V''<sub>2</sub>}} represent the second pressure and volume.

Boyle's law, [[Charles's law]], and [[Gay-Lussac's law#Pressure-temperature law|Gay-Lussac's law]] form the [[combined gas law]]. The three gas laws in combination with [[Avogadro's law]] can be generalized by the [[ideal gas law]].

==Human breathing system==

Boyle's law is often used as part of an explanation on how the [[breathing]] system works in the human body. This commonly involves explaining how the lung volume may be increased or decreased and thereby cause a relatively lower or higher air pressure within them (in keeping with Boyle's law). This forms a pressure difference between the air inside the lungs and the environmental air pressure, which in turn precipitates either inhalation or exhalation as air moves from high to low pressure.<ref>Gerald J. Tortora, Bryan Dickinson, 'Pulmonary Ventilation' in ''Principles of Anatomy and Physiology'' 11th edition, Hoboken: John Wiley & Sons, Inc., 2006, pp. 863–867</ref>

==See also==
Related phenomena:
* [[Water thief]]
* [[Industrial Revolution]]
* [[Steam engine]]

Other [[gas laws]]:
* {{annotated link|Dalton's law}}
* {{annotated link|Charles's law}}
* {{annotated link|Ideal gas law}}

==Citations==
{{Reflist}}

==External links==
* {{Commonscatinline|Boyle's Law}}

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