User:A.C. Norman/gas

The early gas laws were developed at the end of the eigtheenth 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. The earlier gas laws are now considered as special cases of the ideal gas equation, with one or more of the variables held constant.

This article outlines the historical development of the laws describing ideal gases. Other gas laws are also listed at the bottom of the page. For a detailed description of the ideal gas law itself, and its further development, see Ideal gas, Ideal gas law and Gas

NOTES

gas laws; combined gas law; Avogadro's law; charles's law; Gay-Lussac's law;

Introduction

The laws outlined here were developed empirically from experiment. We now know that gases behave in a similar way over a wide variety of conditions because to a good approximation they all have molecules which are widely spaced. Nowadays the equation of state for an ideal gas is derived from kinetic theory, and these laws are all special cases of that ideal gas equation. They are therefore all limited by the presumptions which are made in that theory.

On this page:

p denotes the pressure of the system.

V is the volume of the gas.

T is the temperature

Boyle's Law (1662)

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

Boyle's law (sometimes referred to as the Boyle-Mariotte law) describes the inversely proportional relationship between the pressure and volume of a gas, if the temperature is kept constant within a closed system.

$\qquad \qquad pV=k$

where k is a constant value representative of the pressure and volume of the system.

The law can be stated in words as follows:

For a fixed amount of an ideal gas kept at a fixed temperature, P [pressure] and V [volume] are inversely proportional (while one increases, the other decreases).

Boyle's law states that at constant temperature, 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.

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 can be 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:

$p_{1}V_{1}=p_{2}V_{2}.\,$

Charles's Law (1787)

An animation demonstrating the relationship between volume and temperature.

Charles' law or the law of volumes relates the volume of a gas to its temperature:

${\frac {V}{T}}=k$

where:

k is a constant.

In words:

At constant pressure, the volume of a given mass of an ideal gas increases or decreases by the same factor as its temperature on the absolute temperature scale.

Put more simply, as temperature increases, the gas expands in proportion.

Since the constant k need not be known to make use of the law in comparison between two volumes of gas at equal pressure, the law can also be usefully expressed as follows:

${\frac {V_{1}}{T_{1}}}={\frac {V_{2}}{T_{2}}}\qquad \mathrm {or} \qquad {\frac {V_{2}}{V_{1}}}={\frac {T_{2}}{T_{1}}}\qquad \mathrm {or} \qquad V_{1}\cdot T_{2}=V_{2}\cdot T_{1}$.

Pressure Law (1809)

The Pressure law or Third gas law (developed by Gay-Lussac in 1809) relates the temperature and pressure of a gas:

$p=kT$

In words:

The pressure of a fixed mass and fixed volume of a gas is directly proportional to the gas's temperature.

Simply put, if a gas's temperature increases then so does its pressure, so long as the mass and volume of the gas are constant.

This law holds true because temperature is a measure of the average kinetic energy of a substance; as the kinetic energy of a gas increases, its particles collide with the container walls more rapidly, thereby exerting increased pressure.

For comparing the same substance under two different sets of conditions, the law can be written as:

${\frac {p_{1}}{T_{1}}}={\frac {p_{2}}{T_{2}}}\qquad \mathrm {or} \qquad {p_{1}}{T_{2}}={p_{2}}{T_{1}}$

Combined gas law

The combined gas law combines Charles's law, Boyle's law, and Gay-Lussac's law, all of which relate just two variables to each other whilst keeping everything else constant. The inter-dependence of pressure, volume and temperature (for a fixed amount of gas) is given by the combined gas law, which states that:

$\qquad {\frac {pV}{T}}=k$

where:

k is a constant (with units of energy divided by temperature).

For comparing the same substance under two different sets of conditions, the law can be written as:

$\qquad {\frac {p_{1}V_{1}}{T_{1}}}={\frac {p_{2}V_{2}}{T_{2}}}$

Avogadro's Law (1811)

Avogadro's law (sometimes referred to as Avogadro's hypothesis or Avogadro's principle) is named after Amedeo Avogadro who, in 1811, hypothesized that:

Equal volumes of ideal or perfect gases, at the same temperature and pressure, contain the same number of particles, or molecules.

Thus, the number of molecules in a specific volume of gas is independent of the size or mass of the gas molecules.

As an example, equal volumes of molecular hydrogen and nitrogen would contain the same number of molecules, as long as are at the same temperature and pressure. In practice, for real gases, the law only holds approximately, but the agreement is close enough for the approximation to be useful.

The law can be stated mathematically as:

${\frac {V}{n}}=k\,$.

where:

n is the amount of substance of the gas.

k is a proportionality constant.

The most significant consequence of Avogadro's law is that the ideal gas constant has the same value for all gases. This means that the constant

${\frac {p_{1}\cdot V_{1}}{T_{1}\cdot n_{1}}}={\frac {p_{2}\cdot V_{2}}{T_{2}\cdot n_{2}}}=constant$

has the same value for all gases, independent of the size or mass of the gas molecules.

One mole of an ideal gas occupies 22.40 litres (dm³) at STP, and occupies 24.45 litres at SATP (Standard Ambient Temperature and Pressure = 298K and 1 atm). This volume is often referred to as the molar volume of an ideal gas. Real gases may deviate from this value.

Relationship to ideal gas equation

(for more information, see ideal gas equation)

The combination of Avogadro's law with the combined gas law yields the ideal gas equation:

$pV=nRT\,$,

where

n is the amount of substance (loosely number of moles of gas)

R is the gas constant, which is the same for all gases (Unit and number used: 0.0821 L·atm·mol^-1·K^-1

(The law works with any consistent set of units, provided that the temperature scale starts at absolute zero, and the gas constant is in the correct units.)

An equivalent formulation of this law is:

$pV=kNT\,$

where

k is the Boltzmann constant

N is the number of molecules.

The ideal gas law encompasses all of the properties highlighted by the individual laws above. In particular:

1. If temperature and pressure are kept constant, then the volume of the gas is directly proportional to the number of molecules of gas. (Charles's Law)
2. If the temperature and volume remain constant, then the pressure of the gas changes is directly proportional to the number of molecules of gas present.
3. If the number of gas molecules and the temperature remain constant, then the pressure is inversely proportional to the volume. (Boyle's Law)
4. If the temperature changes and the number of gas molecules are kept constant, then either pressure or volume (or both) will change in direct proportion to the temperature.

Historical Development

The law was named after chemist and physicist Robert Boyle, who published the original law in 1662. Earlier, in 1660 he had sent his findings in a letter to Charles Boyle, 3rd Viscount Dungarvan, eldest son to Richard Boyle

Boyle's Law is named after the Irish natural philosopher Robert Boyle (Lismore, County Waterford, 1627-1691) who was the first to publish it in 1662. The relationship between pressure and volume was brought to the attention of Boyle by two friends and amateur scientists, Richard Towneley and Henry Power, who discovered it. Boyle confirmed their 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 to be a fluid of particles at rest, with in between small invisible springs. At that time air was still seen as one of the four elements, but Boyle didn't agree. Probably Boyle's interest was to understand air as an essential element of life ; he published e.g. the growth of plants without air . The French physicist Edme Mariotte (1620-1684) discovered the same law independently of Boyle in 1676, so this law may be referred to as Mariotte's or the Boyle-Mariotte law. Later (1687) in the Philosophiæ Naturalis Principia Mathematica Newton 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 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 Maxwell and Boltzmann.

Robert Boyle (and Edme Mariotte) derived Boyle's Law solely on experimental grounds. 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.

Charles' Law was first published by Joseph Louis Gay-Lussac in 1802, but he referenced to unpublished work by Jacques Charles from around 1787. Charles's Law was also known as the Law of Charles and Gay-Lussac for this reason. However, in recent years the term has fallen out of favour since Gay-Lussac has the pressure law presented here and attributed to him. The relationship had been anticipated by the work of Guillaume Amontons in 1702.

Daniel Bernoulli in 1738 derived Boyle's law using Newton's laws of motion 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 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.

The debate between proponents of Energetics and Atomism led Boltzmann to write a book in 1898, which endured criticism up to his suicide in 1901. 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.

Limitations

In modern physics, the historical gas equations above are seen as special cases of the ideal gas equation, in which some of the variables are held constant. The ideal gas equation is usually derived from the kinetic theory of gases (see ideal gas equation), which presumes that molecules are occupy negligible volume, do not attract each other and undergo elastic collisions (no loss of kinetic energy); an imaginary gas with exactly these properties is termed an ideal gas. Real gases are good approximations to an ideal gas under a wide range of circumstances (making the ideal gas law useful), but not all.

Most gases behave like ideal gases at moderate pressures and temperatures. The limited technology of the 1600s 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 would become noticeable, and the relationship between pressure and volume can only be accurately described employing real gas theory. The deviation is expressed as the compressibility factor.

Charles's law of volumes, for example, implies theoretically that as a temperature reaches absolute zero the gas will shrink down to zero volume. This is not physically correct, since in fact all gases turn into liquids at a low enough temperature, and Charles' law is not applicable at low temperatures for this reason.

The fact that the gas will occupy a non-zero volume - even as the temperature approaches absolute zero - arises fundamentally from the uncertainty principle of quantum theory. However, as the temperature is reduced, gases turn into liquids long before the limits of the uncertainty principle come into play due to the attractive forces between molecules which are neglected by these laws.

Other gas laws (See Also)

Other gas laws of historical importance include:

• Graham's law states that the rate at which gas molecules diffuse is inversely proportional to the square root of its density. Combined with Avogadro's law (i.e. since equal volumes have equal number of molecules) this is the same as being inversely proportional to the root of the molecular weight.

• Dalton's law of Partial Pressures states that the pressure of a mixture of gases simply is the sum of the partial pressures of the individual components. Dalton's Law is as follows:

$p_{TotalofGasMixture}=p_{1}+p_{2}+p_{3}+...\,$,

OR

$p_{Total}=p_{Gas}+p_{{H}_{2}{O}}\,$,

Where p_Total is the total pressure of the atmosphere, p_Gas is the Pressure of the gas mixture in the atmosphere, and p_H2O is the water pressure at that temperature.

• Amagat's law

• Henry's law states that:

At a constant temperature, the amount of a given gas dissolved in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid.

$p=k_{\rm {H}}\,c$

References

• Castka, Joseph F.; Metcalfe, H. Clark; Davis, Raymond E.; Williams, John E. (2002). Modern Chemistry. Holt, Rinehart and Winston. ISBN 0-03-056537-5.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Guch, Ian (2003). The Complete Idiot's Guide to Chemistry. Alpha, Penguin Group Inc. ISBN 1-59257-101-8.
• Zumdahl, Steven S (1998). Chemical Principles. Houghton Millfin Company. ISBN 0-395-83995-5.