What does Boyles law state about pressure and volume at a constant temperature and a constant pressure?

For a fixed mass of gas at constant temperature, the volume is inversely proportional to the pressure. This is mathematically:

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Contributors and Attributions
Relation with kinetic theory and ideal gases

\[pV = constant\]

That means that, for example, if you double the pressure, you will halve the volume. If you increase the pressure 10 times, the volume will decrease 10 times.

Is this consistent with pV = nRT ?

You have a fixed mass of gas, so n (the number of moles) is constant.
R is always constant - it is called the gas constant.
Boyle's Law demands that temperature is constant as well.

That means that everything on the right-hand side of pV = nRT is constant, and so pV is constant - which is what we have just said is a result of Boyle's Law.

This is easiest to see if you think about the effect of decreasing the volume of a fixed mass of gas at constant temperature. Pressure is caused by gas molecules hitting the walls of the container. With a smaller volume, the gas molecules will hit the walls more frequently, and so the pressure increases.

You might argue that this isn't actually what Boyle's Law says - it wants you to increase the pressure first and see what effect that has on the volume. But, in fact, it amounts to the same thing. If you want to increase the pressure of a fixed mass of gas without changing the temperature, the only way you can do it is to squeeze it into a smaller volume. That causes the molecules to hit the walls more often, and so the pressure increases.

But everything in the nR/p part of this is constant. That means that V = constant x T, which is Charles's Law.

Contributors and Attributions

Jim Clark (Chemguide.co.uk)

Each day, hundreds of weather balloons are launched. Made of a synthetic rubber and carrying a box of instruments, each helium-filled balloon rises up into the sky. As a balloon gains altitude, the atmospheric pressure becomes less and the balloon expands. At some point the balloon bursts due to the expansion; the instruments drop (aided by a parachute) to be retrieved and studied for information about the weather.

Robert Boyle (1627-1691), an English chemist, is widely considered to be one of the founders of the modern experimental science of chemistry. He discovered that doubling the pressure of an enclosed sample of gas, while keeping its temperature constant, caused the volume of the gas to be reduced by half. Boyle's law states that the volume of a given mass of gas varies inversely with the pressure when the temperature is kept constant. An inverse relationship is described in this way. As one variable increases in value, the other variable decreases.

Physically, what is happening? The gas molecules are moving and are a certain distance apart from one another. An increase in pressure pushes the molecules closer together, reducing the volume. If the pressure is decreased, the gases are free to move about in a larger volume.

Figure $\PageIndex{1}$: Robert Boyle. (CC BY-NC; CK-12)

Mathematically, Boyle's law can be expressed by the equation:

\[P \times V = k\nonumber \]

The $k$ is a constant for a given sample of gas and depends only on the mass of the gas and the temperature. The table below shows pressure and volume data for a set amount of gas at a constant temperature. The third column represents the value of the constant $\left( k \right)$ for this data and is always equal to the pressure multiplied by the volume. As one of the variables changes, the other changes in such a way that the product of $P \times V$ always remains the same. In this particular case, that constant is $500 \: \text{atm} \cdot \text{mL}$.

Table $\PageIndex{1}$: Pressure-Volume Data

Pressure $\left( \text{atm} \right)$	Volume $\left( \text{mL} \right)$	$P \times V = k$ $\left( \text{atm} \cdot \text{mL} \right)$
0.5	1000	500
0.625	800	500
1.0	500	500
2.0	250	500
5.0	100	500
8.0	62.5	500
10.0	50	500

A graph of the data in the table further illustrates the inverse relationship nature of Boyle's Law (see figure below). Volume is plotted on the $x$-axis, with the corresponding pressure on the $y$-axis.

Figure $\PageIndex{2}$: The pressure of a gas decreases as the volume increases, making Boyle's law an inverse relationship. (CC BY-NC; CK-12)

Boyle's Law can be used to compare changing conditions for a gas. We use $P_1$ and $V_1$ to stand for the initial pressure and initial volume of a gas. After a change has been made, $P_2$ and $V_2$ stand for the final pressure and volume. The mathematical relationship of Boyle's Law becomes:

\[P_1 \times V_1 = P_2 \times V_2\nonumber \]

This equation can be used to calculate any one of the four quantities if the other three are known.

Example $\PageIndex{1}$

A sample of oxygen gas has a volume of $425 \: \text{mL}$ when the pressure is equal to $387 \: \text{kPa}$. The gas is allowed to expand into a $1.75 \: \text{L}$ container. Calculate the new pressure of the gas.

Solution

Step 1: List the known quantities and plan the problem.

Known

$P_1 = 387 \: \text{kPa}$
$V_1 = 425 \: \text{mL}$
$V_2 = 1.75 \: \text{L} = 1750 \: \text{mL}$

Unknown

Use Boyle's Law to solve for the unknown pressure $\left( P_2 \right)$. It is important that the two volumes ($V_1$ and $V_2$) are expressed in the same units, so $V_2$ has been converted to $\text{mL}$.

Step 2: Solve.

First, rearrange the equation algebraically to solve for $P_2$.

\[P_2 = \frac{P_1 \times V_1}{V_2}\nonumber \]

Now substitute the known quantities into the equation and solve.

\[P_2 = \frac{387 \: \text{kPa} \times 425 \: \text{mL}}{1750 \: \text{mL}} = 94.0 \: \text{kPa}\nonumber \]

Step 3: Think about your result.

The volume has increased to slightly over 4 times its original value and so the pressure is decreased by about one fourth. The pressure is in $\text{kPa}$ and the value has three significant figures. Note that any pressure or volume units can be used as long as they are consistent throughout the problem.

Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law (especially in France), is an experimental gas law that describes how the pressure of a gas tends to decrease as the volume of the container increases. A modern statement of Boyle's law is:

An animation showing the relationship between pressure and volume when mass and temperature are held constant

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 gas remain unchanged within a closed system.[1][2]

Mathematically, Boyle's law can be stated as:

P ∝ 1 V {\displaystyle P\propto {\frac {1}{V}}}

Pressure is inversely proportional to the volume

PV = k

Pressure multiplied by volume equals some constant

k

where $P$ is the pressure of the gas, $V$ is the volume of the gas, and $k$ is a constant.

The equation states that the product of pressure and volume is a constant for a given mass of confined gas and this holds as long as the temperature is constant. For comparing the same substance under two different sets of conditions, the law can be usefully expressed as:

P 1 V 1 = P 2 V 2 . {\displaystyle P_{1}V_{1}=P_{2}V_{2}.}

This equation shows that, as volume increases, the pressure of the gas decreases in proportion. Similarly, as volume decreases, the pressure of the gas increases. The law was named after chemist and physicist Robert Boyle, who published the original law in 1662.[3]

Graph of Boyle's original data[citation needed] showing the hyperbolic curve of the relationship between pressure (

P

) and volume (

V

) of the form

P = k/V

This relationship between pressure and volume was first noted by Richard Towneley and Henry Power in the 17th century.[4][5] Robert Boyle confirmed their discovery through experiments and published the results.[6] 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. 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;[7] for example, he published works on the growth of plants without air.[8] Boyle used a closed J-shaped tube and after pouring 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.[9] The French physicist Edme Mariotte (1620–1684) discovered the same law independently of Boyle in 1679,[10] but Boyle had already published it in 1662.[9] Mariotte did, however, discover that air volume changes with temperature.[11] Thus this law is sometimes referred to as Mariotte's law or the Boyle–Mariotte law. Later, in 1687 in the Philosophiæ Naturalis Principia Mathematica, Newton showed mathematically that in 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,[12] 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 Maxwell and Boltzmann.

This law was the first physical law to be expressed in the form of an equation describing the dependence of two variable quantities.[9]

Boyle's law demonstrations

The law itself can be stated as follows:

For a fixed mass of an ideal gas kept at a fixed temperature, pressure and volume are inversely proportional.[2]

Or Boyle's law is a 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

Boyle's law states that at constant temperature the volume of a given mass of a dry gas is inversely proportional to its pressure.

Most gases behave like ideal gases 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.[13] 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 atoms and molecules 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.[14]

The debate between proponents of energetics and atomism led Boltzmann to write a book in 1898, which endured criticism until his suicide in 1906.[14] 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.[14]

Equation

Relationships between Boyle's, Charles's, Gay-Lussac's, Avogadro's, combined and ideal gas laws, with the Boltzmann constant kB = RNA = n RN (in each law, properties circled are variable and properties not circled are held constant)

The mathematical equation for Boyle's law is:

P V = k {\displaystyle PV=k}

where $P$ denotes the pressure of the system, $V$ denotes the volume of the gas, $k$ is a constant value representative of the temperature 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 theory, the application of force to a surface may not be infinitely constant for such values of $V$ , but will have a limit when differentiating 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 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:

P 1 V 1 = P 2 V 2 . {\displaystyle P_{1}V_{1}=P_{2}V_{2}.}

Here $P1$ and $V1$ represent the original pressure and volume, respectively, and $P2$ and $V2$ represent the second pressure and volume.

Boyle's law, Charles's law, and 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.

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.[15]

Related phenomena:

Water thief
Industrial Revolution
Steam engine

Other gas laws:

Dalton's law – Gas law describing pressure contributions of component gases in a mixture
Charles's law – Relationship between volume and temperature of a gas at constant pressure

^ Levine, Ira. N (1978). "Physical Chemistry" University of Brooklyn: McGraw-Hill
^ a b Levine, Ira. N. (1978), p. 12 gives the original definition.
^ 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 J Appl Physiol 98: 31–39, 2005. (Jap.physiology.org Online.)
^
See:
- Henry Power, Experimental Philosophy, in Three Books … (London: Printed by T. Roycroft for John Martin and James Allestry, 1663), pp. 126–130. Available on-line at: 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 V1 ("Ayr") of a given quantity of air at the pressure p1 ("Mercurial standard", i.e., atmospheric pressure at a low altitude), then one can predict the volume V2 ("Ayr dilated") of the same quantity of air at the pressure p2 ("Mercurial complement", i.e., atmospheric pressure at a higher altitude) by means of a proportion (because p1 V1 = p2 V2).
- 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 on-line at: Spain's La Biblioteca Virtual de Patrimonio Bibliográfico. On pages 50, 55–56, and 64, Boyle cited experiments by Towneley and Power showing that air expands as the ambient pressure decreases. On p. 63, Boyle acknowledged 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 acknowledged that Lord Brouncker had also investigated the same subject.)
^ Gerald James Holton (2001). Physics, the Human Adventure: From Copernicus to Einstein and Beyond. Rutgers University Press. pp. 270–. ISBN 978-0-8135-2908-0.
^ R. Boyle, A Defence of the Doctrine Touching the Spring and Weight of the Air, … (London: Thomas Robinson, 1662). Available on-line at: 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."
^ The Boyle Papers BP 9, fol. 75v–76r at BBK.ac.uk Archived 2009-11-22 at the Wayback Machine
^ The Boyle Papers, BP 10, fol. 138v–139r at BBK.ac.uk Archived 2009-11-22 at the Wayback Machine
^ a b c Scientists and Inventors of the Renaissance. Britannica Educational Publishing. 2012. pp. 94–96. ISBN 978-1615308842.
^
See:
- Mariotte, Essais de Physique, ou mémoires pour servir à la science des choses naturelles, … (Paris, France: E. Michallet, 1679); "Second essai. De la nature de l'air".
- (Mariotte, Edmé), Oeuvres de Mr. Mariotte, de l'Académie royale des sciences; … , vol. 1 (Leiden, Netherlands: P. Vander Aa, 1717); see especially pp. 151–153.
- Mariotte's essay "De la nature de l'air" was reviewed by the French Royal Academy of Sciences in 1679. See: (Anon.) (1733) "Sur la nature de l'air," Histoire de l'Académie Royale des Sciences, 1 : 270–278.
- Mariotte's essay "De la nature de l'air" was also reviewed in the Journal des Sçavans (later: Journal des Savants) on 20 November 1679. See: (Anon.) (20 November 1679) "Essais de physique, … ," Journal des Sçavans, pp. 265–269.
^ Ley, Willy (June 1966). "The Re-Designed Solar System". For Your Information. Galaxy Science Fiction. pp. 94–106.
^ Principia, Sec. V, prop. XXI, Theorem XVI
^ Levine, Ira. N. (1978), p. 11 notes that deviations occur with high pressures and temperatures.
^ a b c Levine, Ira. N. (1978), p. 400 – Historical background of Boyle's law relation to Kinetic Theory
^ Gerald J. Tortora, Bryan Dickinson, 'Pulmonary Ventilation' in Principles of Anatomy and Physiology 11th edition, Hoboken: John Wiley & Sons, Inc., 2006, pp. 863–867