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Charles' law

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 Title: Charles' law Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

Charles' law

Charles' law (also known as the law of volumes) is an experimental gas law which describes how gases tend to expand when heated. A modern statement of Charles' law is:

The volume of a given mass of an ideal gas is directly proportional to its temperature on the absolute temperature scale (in Kelvin) if pressure and the amount of gas remain constant; that is, the volume of the gas increases or decreases by the same factor as its temperature.[1]

this directly proportional relationship can be written as:

$V \propto T\,$

or

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

where:

V is the volume of the gas
T is the temperature of the gas (measured in Kelvin).
k is a constant.

This law explains how a gas expands as the temperature increases; conversely, a decrease in temperature will lead to a decrease in volume. For comparing the same substance under two different sets of conditions, the law can be written as:

$\frac\left\{V_1\right\}\left\{T_1\right\} = \frac\left\{V_2\right\}\left\{T_2\right\} \qquad \mathrm\left\{or\right\} \qquad \frac \left\{V_2\right\}\left\{V_1\right\} = \frac\left\{T_2\right\}\left\{T_1\right\} \qquad \mathrm\left\{or\right\} \qquad V_1 T_2 = V_2 T_1.$

The equation shows that, as absolute temperature increases, the volume of the gas also increases in proportion. The law was named after scientist Jacques Charles, who formulated the original law in his unpublished work from the 1780s.

History

It was first published by French natural philosopher Joseph Louis Gay-Lussac in 1802,[2] although he credited the discovery to unpublished work from the 1780s by Jacques Charles. The law was independently discovered by British natural philosopher John Dalton by 1801, although Dalton's description was less thorough than Gay-Lussac's.[3][4] The basic principles had already been described a century earlier by Guillaume Amontons.[5]

Gay-Lussac was the first to demonstrate that the law applied generally to all gases, and to the vapours of volatile liquids if the temperature was more than a few degrees above the boiling point.[6] His statement of the law can be expressed mathematically as:

$V_\left\{100\right\} - V_0 = kV_0\,$

where V100 is the volume occupied by a given sample of gas at 100 °C; V0 is the volume occupied by the same sample of gas at 0 °C; and k is a constant which is the same for all gases at constant pressure. Gay-Lussac's value for k was 12.6666, remarkably close to the present-day value of 12.7315. This law was first given by J. Charles in 1787.

Relation to absolute zero

Charles's law appears to imply that the volume of a gas will descend to zero at a certain temperature (−266.66 °C according to Gay-Lussac's figures) or −273°C. Gay-Lussac was clear in his description that the law was not applicable at low temperatures:

but I may mention that this last conclusion cannot be true except so long as the compressed vapors remain entirely in the elastic state; and this requires that their temperature shall be sufficiently elevated to enable them to resist the pressure which tends to make them assume the liquid state.[2]

Gay-Lussac had no experience of liquid air (first prepared in 1877), although he appears to believe (as did Dalton) that the "permanent gases" such as air and hydrogen could be liquified. Gay-Lussac had also worked with the vapours of volatile liquids in demonstrating Charles's law, and was aware that the law does not apply just above the boiling point of the liquid:

I may however remark that when the temperature of the ether is only a little above its boiling point, its condensation is a little more rapid than that of atmospheric air. This fact is related to a phenomenon which is exhibited by a great many bodies when passing from the liquid to the solid state, but which is no longer sensible at temperatures a few degrees above that at which the transition occurs.[2]

The first mention of a temperature at which the volume of a gas might descend to zero was by William Thomson (later known as Lord Kelvin) in 1848:[7]

This is what we might anticipate, when we reflect that infinite cold must correspond to a finite number of degrees of the air-thermometer below zero; since if we push the strict principle of graduation, stated above, sufficiently far, we should arrive at a point corresponding to the volume of air being reduced to nothing, which would be marked as −273° of the scale (−100/.366, if .366 be the coefficient of expansion); and therefore −273° of the air-thermometer is a point which cannot be reached at any finite temperature, however low.

However, the "absolute zero" on the Kelvin temperature scale was originally defined in terms of the second law of thermodynamics, which Thomson himself described in 1852.[8] Thomson did not assume that this was equal to the "zero-volume point" of Charles's law, merely that Charles's law provided the minimum temperature which could be attained. The two can be shown to be equivalent by Ludwig Boltzmann's statistical view of entropy (1870).

Relation to kinetic theory

The kinetic theory of gases relates the macroscopic properties of gases, such as pressure and volume, to the microscopic properties of the molecules which make up the gas, particularly the mass and speed of the molecules. In order to derive Charles's law from kinetic theory, it is necessary to have a microscopic definition of temperature: this can be conveniently taken as the temperature being proportional to the average kinetic energy of the gas molecules, Ek:

$T \propto \bar\left\{E_\left\{\rm k\right\}\right\}.\,$

Under this definition, the demonstration of Charles's law is almost trivial. The kinetic theory equivalent of the ideal gas law relates pV to the average kinetic energy:

$pV = \frac\left\{2\right\}\left\{3\right\} N \bar\left\{E_\left\{\rm k\right\}\right\}\,$

where N is the number of molecules in the gas sample. If the pressure is constant, the volume is directly proportional to the average kinetic energy (and hence to the temperature) for any given gas sample.

References

1. Amontons, G. (presented 1702, published 1743) "Discours sur quelques propriétés de l'Air, & le moyen d'en connoître la température dans tous les climats de la Terre" (Discourse on some properties of air and on the means of knowing the temperature in all climates of the Earth), Mémoires de l’Académie des sciences de Paris, 155–174.
2. Review of Amontons' findings: "Sur une nouvelle proprieté de l'air, et une nouvelle construction de Thermométre" (On a new property of the air and a new construction of thermometer), Histoire de l'Academie royale des sciences, 1–8 (submitted: 1702 ; published: 1743).
3. Englishman "An account of an experiment touching the different densities of air, from the greatest natural heat to the greatest natural cold in this climate," Philosophical Transactions of the Royal Society of London 26(315): 93–96.

Further reading

• . Facsimile at the Bibliothèque nationale de France (pp. 315–22).
• . Facsimile at the Bibliothèque nationale de France (pp. 353–79).
• . (French)

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