Boyle's law is one of many
gas lawsThe early gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between the pressure, volume and temperature of a sample of gas could be obtained which would hold for all gases...
and a special case of the
ideal gas lawThe ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...
. Boyle's law describes the inversely proportional relationship between the absolute
pressurePressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure. Definition :...
and
volumeVolume is the quantity of threedimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
of a gas, if the temperature is kept constant within a
closed systemIn physics:In thermodynamics, a closed system can exchange energy , but not matter, with its surroundings.In contrast, an isolated system cannot exchange any of heat, work, or matter with the surroundings, while an open system can exchange all of heat, work and matter.For a simple system, with...
. The law was named after
chemistChemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
and
physicistA physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from subatomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
Robert BoyleRobert Boyle FRS was a 17th century natural philosopher, chemist, physicist, and inventor, also noted for his writings in theology. He has been variously described as English, Irish, or AngloIrish, his father having come to Ireland from England during the time of the English plantations of...
, who published the original law in 1662. The law itself can be stated as follows:
History
This relationship between pressure and volume was first noted by two amateur scientists,
Richard TowneleyRichard Towneley was an English mathematician and astronomer from Towneley near Burnley, Lancashire. He was one of a group of seventeenth century astronomers in the north of England, which included Jeremiah Horrocks, William Crabtree and William Gascoigne, the pioneer astronomers who laid the...
and
Henry PowerHenry Power was an English physician and experimenter, one of the first elected Fellows of the Royal Society.Life:He matriculated as a pensioner of Christ's College, Cambridge, in 1641, and graduated B.A. in 1644. He became a regular correspondent of Sir Thomas Browne on scientific subjects. He...
. Boyle confirmed their discovery through experiments and published the results. According to
Robert GuntherRobert Theodore Gunther was a historian of science, zoologist, and founder of the Museum of the History of Science, Oxford....
and other authorities, it was Boyle's assistant,
Robert HookeRobert Hooke FRS was an English natural philosopher, architect and polymath.His adult life comprised three distinct periods: as a scientific inquirer lacking money; achieving great wealth and standing through his reputation for hard work and scrupulous honesty following the great fire of 1666, but...
, 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. At that time, air was still seen as one of the four elements, but Boyle disagreed. Boyle's interest was probably to understand air as an essential element of life; he published e.g. the growth of plants without air. The French physicist
Edme MariotteEdme Mariotte was a French physicist and priest. Biography :Edme Mariotte was the youngest son of Simon Mariotte, administrator at the district TilChâtel , and Catherine Denisot . His parents lived in TilChâtel and had 4 other children: Jean, Denise, Claude, and Catharine...
(1620–1684) discovered the same law independently of Boyle in 1676, but Boyle had already published it in 1662. Thus this law may, improperly, be referred to as Mariotte's or the BoyleMariotte law. Later, in 1687 in the Philosophiæ Naturalis Principia Mathematica,
NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
showed mathematically that if an 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, but this mathematical treatise is not the physical explanation for the observed relationship. Instead of a static theory a kinetic theory is needed, which was provided two centuries later by
MaxwellJames Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
and
BoltzmannLudwig Eduard Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics...
.
Relation to kinetic theory and ideal gases
Boyle’s law states that
at constant temperature for a fixed mass, the absolute pressure and the volume of a gas are inversely proportional. The law can also be stated in a slightly different manner, that the product of absolute pressure and volume is always constant.
Most gases behave like
ideal gasAn ideal gas is a theoretical gas composed of a set of randomlymoving, noninteracting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
es at moderate pressures and temperatures. The technology of the 17th century could not produce high pressures or 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 gasReal gases – as opposed to a perfect or ideal gas – exhibit properties that cannot be explained entirely using the ideal gas law. To understand the behaviour of real gases, the following must be taken into account:* compressibility effects;...
theory. The deviation is expressed as the
compressibility factorThe compressibility factor , also known as the compression factor, is a useful thermodynamic property for modifying the ideal gas law to account for the real gas behavior. In general, deviation from ideal behavior becomes more significant the closer a gas is to a phase change, the lower the...
.
Robert Boyle (and Edme Mariotte) derived the law solely on experimental grounds. The law can also be derived theoretically based on the presumed existence of
atomThe atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s and
moleculeA molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
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 BernoulliDaniel Bernoulli was a DutchSwiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...
in 1738 derived Boyle's law using
Newton's laws of motionNewton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
with application on a 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 JouleJames Prescott Joule FRS was an English physicist and brewer, born in Salford, Lancashire. Joule studied the nature of heat, and discovered its relationship to mechanical work . This led to the theory of conservation of energy, which led to the development of the first law of thermodynamics. The...
,
Rudolf ClausiusRudolf Julius Emanuel Clausius , was a German physicist and mathematician and is considered one of the central founders of the science of thermodynamics. By his restatement of Sadi Carnot's principle known as the Carnot cycle, he put the theory of heat on a truer and sounder basis...
and in particular Ludwig Boltzmann firmly established the kinetic theory of gases and brought attention to both the theories of Bernoulli and Waterston.
The debate between proponents of
EnergeticsEnergetics is the study of energy under transformation. Because energy flows at all scales, from the quantum level to the biosphere and cosmos, energetics is a very broad discipline, encompassing for example thermodynamics, chemistry, biological energetics, biochemistry and ecological energetics...
and
AtomismAtomism is a natural philosophy that developed in several ancient traditions. The atomists theorized that the natural world consists of two fundamental parts: indivisible atoms and empty void.According to Aristotle, atoms are indestructible and immutable and there are an infinite variety of shapes...
led Boltzmann to write a book in 1898, which endured criticism up to his suicide in 1906.
Albert EinsteinAlbert Einstein was a Germanborn theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
in 1905 showed how kinetic theory applies to the
Brownian motionBrownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several realworld applications...
of a fluidsuspended particle, which was confirmed in 1908 by Jean Perrin.
Equation
The mathematical equation for Boyle's law is:
where:
 p denotes the pressure of the system.
 V denotes the volume
Volume is the quantity of threedimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
of the gas.
 k is a constant value representative of the pressure and volume of the system.
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
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 theoryCollision theory is a theory proposed by Max Trautz and William Lewis in 1916 and 1918, that qualitatively explains how chemical reactions occur and why reaction rates differ for different reactions. For a reaction to occur the reactant particles must collide. Only a certain fraction of the total...
, the application of force to a surface may not be infinitely constant for such values of
k, but will have a
limitIn mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
when
differentiatingIn mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
such values over a given time.
Forcing the volume
V of the fixed quantity of gas to increase, keeping the gas at the initially measured temperature, the pressure
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 before and after volumes and pressures of the fixed amount of gas, where the before and after temperatures are the same (heating or cooling will be required to meet this condition), are related by the equation:

Boyle's law,
Charles's lawCharles' law is an experimental gas law which describes how gases tend to expand when heated. It was first published by French natural philosopher Joseph Louis GayLussac in 1802, although he credited the discovery to unpublished work from the 1780s by Jacques Charles...
, and
GayLussac's lawThe expression GayLussac's law is used for each of the two relationships named after the French chemist Joseph Louis GayLussac and which concern the properties of gases, though it is more usually applied to his law of combining volumes, the first listed here...
form the
combined gas lawThe combined gas law is a gas law which combines Charles's law, Boyle's law, and GayLussac's law. These laws each relate one thermodynamic variable to another mathematically while holding everything else constant. Charles's law states that volume and temperature are directly proportional to each...
. The three gas laws in combination with
Avogadro's lawAvogadro's law is a gas law named after Amedeo Avogadro who, in 1811, hypothesized that two given samples of an ideal gas, at the same temperature, pressure and volume, contain the same number of molecules...
can be generalized by the
ideal gas lawThe ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...
.
Examples
 Change of pressure in a syringe
 The popping of a balloon
 Increase in size of bubbles as they rise to the surface
 Death of deep sea creatures due to change in pressure
 Popping of ears at high altitude