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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:

When the pressure on a sample of a dry gas is held constant, the Kelvin temperature and the volume will be directly related.[1]

this directly proportional relationship can be written as:

$$V \propto T\,$$

or

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

where:

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

This law describes 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:

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

The equation shows that, as absolute temperature increases, the volume of the gas also increases in proportion.

Discovery and naming of the law

The law was named after scientist Jacques Charles, who formulated the original law in his unpublished work from the 1780s. In two of a series of four essays presented between 2 and 30 October 1801,[2] John Dalton demonstrated by experiment that all the gases and vapours that he studied, expanded by the same amount between two fixed points of temperature. The French natural philosopher Joseph Louis Gay-Lussac confirmed the discovery in a presentation to the French National Institute on 31 Jan 1802,[3] although he credited the discovery to unpublished work from the 1780s by Jacques Charles. The basic principles had already been described a century earlier by Guillaume Amontons[4] and Francis Hauksbee.[5]

Dalton was the first to demonstrate that the law applied generally to all gases, and to the vapours of volatile liquids if the temperature was well above the boiling point. Gay-Lussac concurred.[6] With measurements only at the two thermometric fixed points of water, Gay-Lussac was unable to show that the equation relating volume to temperature was a linear function. On mathematical grounds alone, Gay-Lussac's paper does not permit the assignment of any law stating the linear relation. Both Dalton's and Gay-Lussac's main conclusions can be expressed mathematically as:

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

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. This equation does not contain the temperature and so has nothing to do with what became known as Charles' Law. Gay-Lussac's value for “k” (1⁄2.6666), was identical to Dalton's earlier value for vapours and remarkably close to the present-day value of 1⁄2.7315. Gay-Lussac gave credit for this equation to unpublished statements by his fellow Republican citizen J. Charles in 1787. In the absence of a firm record, the gas law relating volume to temperature cannot be named after Charles. Dalton's measurements had much more scope regarding temperature than Gay-Lussac, not only measuring the volume at the fixed points of water, but also at two intermediate points. Unaware of the inaccuracies of mercury thermometers at the time, which were divided into equal portions between the fixed points, Dalton, after concluding in Essay II that in the case of vapours, “any elastic fluid expands nearly in a uniform manner into 1370 or 1380 parts by 180 degrees (Fahrenheit) of heat”, was unable to confirm it for gases. His conclusion for vapours is a clear statement of what became become known wrongly as Charles' Law, then even more wrongly as Gay-Lussaac's law, but never correctly as Dalton's 2nd law. His 1st law was that of partial pressures.

Relation to absolute zero

Charles' 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 −273.15 °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.[3]

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' 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.[3]

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' law, merely that Charles' 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).

However, Charles also stated:

The volume of a fixed mass of dry gas increases or decreases by 1/273 times the volume at 0oC for every 1o rise or fall in temperature. Thus-

$$V_t=V_0+(1/273\times V_0 )\times t$$

$$V_t=V_0 (1+t/273)$$

Where Vt is the volume of gas at temperature t, V0 is the volume at 0oC.
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' 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:

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

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

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

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Boyle's law
Combined gas law
Gay-Lussac's law
Ideal gas law
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References

Fullick, P. (1994), Physics, Heinemann, pp. 141–42, ISBN 0-435-57078-1.
J. Dalton (1802) "Essay II. On the force of steam or vapour from water and various other liquids, both in vacuum and in air" and Essay IV. "On the expansion of elastic fluids by heat," Memoirs of the Literary and Philosophical Society of Manchester, vol. 5, pt. 2, pages 550-574 and pages 595–602 .
Gay-Lussac, J. L. (1802), "Recherches sur la dilatation des gaz et des vapeurs" [Researches on the expansion of gases and vapors], Annales de chimie 43: 137–175. English translation (extract).
On page 157, Gay-Lussac mentions the unpublished findings of Charles: "Avant d'aller plus loin, je dois prévenir que quoique j'eusse reconnu un grand nombre de fois que les gaz oxigène, azote, hydrogène et acide carbonique, et l'air atmosphérique se dilatent également depuis 0° jusqu'a 80°, le cit. Charles avait remarqué depuis 15 ans la même propriété dans ces gaz ; mais n'avant jamais publié ses résultats, c'est par le plus grand hasard que je les ai connus." (Before going further, I should inform [you] that although I had recognized many times that the gases oxygen, nitrogen, hydrogen, and carbonic acid [i.e., carbon dioxide], and atmospheric air also expand from 0° to 80°, citizen Charles had noticed 15 years ago the same property in these gases; but having never published his results, it is by the merest chance that I knew of them.)
See:

Amontons, G. (presented 1699, published 1732) "Moyens de substituer commodément l'action du feu à la force des hommes et des chevaux pour mouvoir les machines" (Ways to conveniently substitute the action of fire for the force of men and horses in order to power machines), Mémoires de l’Académie des sciences de Paris (presented 1699, published 1732), 112–126; see especially pages 113–117.
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.
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).

Englishman Francis Hauksbee (1660–1713) independently also discovered Charles' law: Francis Hauksbee (1708) "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.

Gay-Lussac (1802), from page 166:
"Si l'on divise l'augmentation totale de volume par le nombre de degrés qui l'ont produite ou par 80, on trouvera, en faisant le volume à la température 0 égal à l'unité, que l'augmentation de volume pour chaque degré est de 1 / 223.33 ou bien de 1 / 266.66 pour chaque degré du thermomètre centrigrade."
If one divides the total increase in volume by the number of degrees that produce it or by 80, one will find, by making the volume at the temperature 0 equal to unity (1), that the increase in volume for each degree is 1 / 223.33 or 1 / 266.66 for each degree of the centigrade thermometer.
From page 174:
" … elle nous porte, par conséquent, à conclure que tous les gaz et toutes les vapeurs se dilatent également par les mêmes degrés de chaleur."
… it leads us, consequently, to conclude that all gases and all vapors expand equally [when subjected to] the same degrees of heat.
Thomson, William (1848), "On an Absolute Thermometric Scale founded on Carnot's Theory of the Motive Power of Heat, and calculated from Regnault's Observations", Philosophical Magazine: 100–6.

Thomson, William (1852), "On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam", Philosophical Magazine 4. Extract.