Boyle–Mariotte law

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Animation: constant mass and temperature

Boyle Mariotte's law (Boyle's law) is one of the gas laws that relates the volume and pressure of a certain amount of gas held at constant temperature.

The Boyle-Mariotte law was formulated independently by the Irish physicist and chemist Robert Boyle in 1662 and the French physicist and botanist Edme Mariotte in 1676.

History

Main article: History of thermodynamics

Chart of Boyle's original data showing the hyperbolic curve of the pressure ratio (

P

) and volume (

V

) shape

P = k/V

This relationship between pressure and volume was first noted by Richard Towneley and Henry Power in the 17th century. Robert Boyle confirmed his discovery through experiments and published the results. 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 a fluid of particles at rest between small invisible springs. At the time, air was still considered one of the four elements, but Boyle disagreed. Boyle's interest was probably in understanding air as an essential element of life; for example, he published papers on the growth of plants without air. Boyle used a closed J-shaped tube and, after pouring mercury through a side, forced the air on the other side to contract under the pressure of the mercury. After repeating the experiment several times and using different amounts of mercury, he discovered that, under controlled conditions, the pressure of a gas is inversely proportional to the volume it occupies. The French physicist Edme Mariotte (1620–1684) independently discovered the same law. by Boyle in 1679, but Boyle had already published it in 1662. Mariotte, however, discovered that the volume of air changes with temperature. This law is therefore sometimes called Mariotte's law or Boyle-Boyle's law. Mariotte. Later, in 1687 in the Philosophiæ Naturalis Principia Mathematica, Newton mathematically proved that in an elastic fluid made up of particles at rest, among which are repulsive forces inversely proportional to their distance, the density would be directly proportional to pressure, but this mathematical treatise is not the physical explanation of 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.

Symbols

Symbolology
Symbol	Name	Unit
$k$	Constant	Pa³
$p$	Pressure	Pa
${displaystyle p_{1}}$	Initial pressure	Pa
${displaystyle p_{2}}$	Final pressure	Pa
$V$	Volume	m³
$V_{1}$	Initial volume	m³
$V_2$	Final volume	m³

Definition

The law itself can be stated as follows:

For a gas at constant temperature, the volume is inversely proportional to the pressure on it.

You can explain mathematically with: ${displaystyle p V=k}$

When the pressure increases, the volume decreases, while if the pressure decreases, the volume increases.

Boyle's law is a gas law, stating that the pressure and volume of a gas are inversely related. If the volume increases, then the pressure decreases and vice versa, when the temperature remains constant.

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

It is not necessary to know the exact value of the constant ( $k$ ) in order to make use of the law: if we consider the two situations of the figure, keeping constant the amount of gas and the temperature, the relation must be fulfilled:

${displaystyle p_{1} V_{1}=p_{2} V_{2}}$

In addition, if any unknown is resolved, the following is obtained:

Deduction
	1
Equations	${displaystyle p_{1} V_{1}=p_{2} V_{2}}$
Clearing	${displaystyle p_{1}={frac {p_{2} V_{2}}{V_{1}}}}$	${displaystyle p_{2}={frac {p_{1} V_{1}}{V_{2}}}}$	${displaystyle V_{1}={frac {p_{2} V_{2}}{p_{1}}}}$	${displaystyle V_{2}={frac {p_{1} V_{1}}{p_{2}}}}$

Also known as the rule of three.

This law is a simplification of the Ideal Gas Law particularized for isothermal processes of a certain constant gas mass.

Together with Charles's law, Gay-Lussac's law, Avogadro's law, and Graham's law, Boyle's law forms the gas laws, which describe the behavior of an ideal gas. The first three laws can be generalized into the ideal gas law.

Boyle's experiment

This theory explained that atoms and molecules, in the gaseous state, behave like point centers of mass that are only affected by attractive forces in the range of high pressures and low temperatures. Outside these limits, the physical properties of a gas are mainly due to the independent movement of its molecules.

If a gas is contained in a container, the pressure it exerts is the force per unit area on the walls due to the elastic impacts of the molecules.

Robert Boyle discovered in 1662 the mathematical relationship between the pressure and volume of a fixed amount of gas at constant temperature. According to Boyle's Law, the volume of a given mass of gas varies inversely as the pressure when the temperature is held at a fixed value. The mathematical expression of the law is written:

${displaystyle p V=k}$ (isothermal process) (13.1)

The magnitude of the constant k is a function of the chemical amount of gas and the temperature.

For two different states 1 and 2, the law implies:

${displaystyle p_{1} V_{1}=p_{2} V_{2}}$ (13.2)

That is, if the physical behavior of a gas is explored according to Boyle's law and assuming ideal behaviour, it can be concluded that, at constant temperature:

If the pressure on a given mass of gas is doubled, its volume is halved. If the volume of a given mass of gas is tripled, the pressure is reduced by one third.

It is usual in experiments on the Boyle law obtain a set of pressure and volume data, which can be graphically represented to obtain the value of ( $k$ ). A chart of ( $p$ ) versus ( $V$ ) (Figure 13.1) results in the characteristic hyperbole corresponding to equation 13.1. If the experiment is repeated at different temperatures, a family of hyperboles is generated, and because the temperature is constant along each line, these curves are called isotermas.

In order to verify his theory, Mariotte introduced a gas into a cylinder with a plunger and checked the different pressures by lowering the plunger. Below is a table showing shows some of the results obtained in this phenomenon as follows:

Experimento de Mariotte
P (atm)	V (L)	P · V
0.5	60	30
1.0	30	30
1.5	20	30
2.0	15	30
2.5	12	30
3.0	10	30

If the data in the table are observed, it can be verified that by increasing the pressure, the volume decreases. This is why an isothermal diagonal is used to represent it in a graph. ( $p$ ) increases and that by multiplying ( $p$ ) and ( $V$ ) you get ${displaystyle p V=30}$ ${displaystyle (atmcdot L)}$ .

Human Respiration System

Boyle's law is often used as part of an explanation of how the respiration system works in the human body. This commonly involves explaining how lung volume can increase or decrease and therefore cause relatively lower or higher air pressure within them (according to Boyle's law). This forms a pressure difference between the air inside the lungs and the ambient air pressure, which in turn precipitates inhalation or exhalation as the air moves from high to low pressure.

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