Short description: Relationship between volume and number of moles of a gas at constant temperature and pressure.

Avogadro's law (sometimes referred to as Avogadro's hypothesis or Avogadro's principle) or Avogadro-Ampère's hypothesis is an experimental gas law relating the volume of a gas to the amount of substance of gas present.[1] The law is a specific case of the ideal gas law. A modern statement is:

Avogadro's law states that "equal volumes of all gases, at the same temperature and pressure, have the same number of molecules."[1]

For a given mass of an ideal gas, the volume and amount (moles) of the gas are directly proportional if the temperature and pressure are constant.

The law is named after Amedeo Avogadro who, in 1812,[2][3] hypothesized that two given samples of an ideal gas, of the same volume and at the same temperature and pressure, contain the same number of molecules. As an example, equal volumes of gaseous hydrogen and nitrogen contain the same number of atoms when they are at the same temperature and pressure, and observe ideal gas behavior. In practice, real gases show small deviations from the ideal behavior and the law holds only approximately, but is still a useful approximation for scientists.

## Mathematical definition

The law can be written as:

$\displaystyle{ V \propto n }$

or

$\displaystyle{ \frac{V}{n} = k }$

where

This law describes how, under the same condition of temperature and pressure, equal volumes of all gases contain the same number of molecules. For comparing the same substance under two different sets of conditions, the law can be usefully expressed as follows:

$\displaystyle{ \frac{V_1}{n_1} = \frac{V_2}{n_2} }$

The equation shows that, as the number of moles of gas increases, the volume of the gas also increases in proportion. Similarly, if the number of moles of gas is decreased, then the volume also decreases. Thus, the number of molecules or atoms in a specific volume of ideal gas is independent of their size or the molar mass of the gas.

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

### Derivation from the ideal gas law

The derivation of Avogadro's law follows directly from the ideal gas law, i.e.

$\displaystyle{ PV = nRT , }$

where R is the gas constant, T is the Kelvin temperature, and P is the pressure (in pascals).

Solving for V/n, we thus obtain

$\displaystyle{ \frac{V}{n} = \frac{RT}{P}. }$

Compare that to

$\displaystyle{ k=\frac{RT}{P} }$

which is a constant for a fixed pressure and a fixed temperature.

An equivalent formulation of the ideal gas law can be written using Boltzmann constant kB, as

$\displaystyle{ PV = N k_\text{B} T, }$

where N is the number of particles in the gas, and the ratio of R over kB is equal to the Avogadro constant.

In this form, for V/N is a constant, we have

$\displaystyle{ \frac{V}{N} = k' = \frac{k_{\text{B}}T}{P}. }$

If T and P are taken at standard conditions for temperature and pressure (STP), then k′ = 1/n0, where n0 is the Loschmidt constant.

## Historical account and influence

Avogadro's hypothesis (as it was known originally) was formulated in the same spirit of earlier empirical gas laws like Boyle's law (1662), Charles's law (1787) and Gay-Lussac's law (1808). The hypothesis was first published by Amadeo Avogadro in 1811,[4] and it reconciled Dalton atomic theory with the "incompatible" idea of Joseph Louis Gay-Lussac that some gases were composite of different fundamental substances (molecules) in integer proportions.[5] In 1814, independently from Avogadro, André-Marie Ampère published the same law with similar conclusions.[6] As Ampère was more well known in France, the hypothesis was usually referred there as Ampère's hypothesis,[note 1] and later also as Avogadro–Ampère hypothesis[note 2] or even Ampère–Avogadro hypothesis.[7]

Experimental studies carried out by Charles Frédéric Gerhardt and Auguste Laurent on organic chemistry demonstrated that Avogadro's law explained why the same quantities of molecules in a gas have the same volume. Nevertheless, related experiments with some inorganic substances showed seeming exceptions to the law. This apparent contradiction was finally resolved by Stanislao Cannizzaro, as announced at Karlsruhe Congress in 1860, four years after Avogadro's death. He explained that these exceptions were due to molecular dissociations at certain temperatures, and that Avogadro's law determined not only molecular masses, but atomic masses as well.

### Ideal gas law

Boyle, Charles and Gay-Lussac laws, together with Avogadro's law, were combined by Émile Clapeyron in 1834,[8] giving rise to the ideal gas law. At the end of the 19th century, later developments from scientists like August Krönig, Rudolf Clausius, James Clerk Maxwell and Ludwig Boltzmann, gave rise to the kinetic theory of gases, a microscopic theory from which the ideal gas law can be derived as an statistical result from the movement of atoms/molecules in a gas.

Avogadro's law provides a way to calculate the quantity of gas in a receptacle. Thanks to this discovery, Johann Josef Loschmidt, in 1865, was able for the first time to estimate the size of a molecule.[9] His calculation gave rise to the concept of the Loschmidt constant, a ratio between macroscopic and atomic quantities. In 1910, Millikan's oil drop experiment determined the charge of the electron; using it with the Faraday constant (derived by Michael Faraday in 1834), one is able to determine the number of particles in a mole of substance. At the same time, precision experiments by Jean Baptiste Perrin led to the definition of Avogadro's number as the number of molecules in one gram-molecule of oxygen. Perrin named the number to honor Avogadro for his discovery of the namesake law. Later standardization of the International System of Units led to the modern definition of the Avogadro constant.

## Molar volume

Main page: Molar volume

Taking STP to be 101.325 kPa and 273.15 K, we can find the volume of one mole of gas:

$\displaystyle{ V_{\rm m} = \frac{V}{n} = \frac{RT}{P} \approx \frac{8.314 \text{ J}\text{ mol}^{-1} \text{ K}^{-1} \times 273.15 \text{ K}}{101.325 \text{ kPa}} \approx 22.41 \text{ dm}^3 \text{ mol}^{-1} = 22.41 \text{ litres/mol} }$

For 101.325 kPa and 273.15 K, the molar volume of an ideal gas is 22.4127 dm3mol−1.

• Boyle's law – Relationship between pressure and volume in a gas at constant temperature
• Physics:Charles's law – Relationship between volume and temperature of a gas at constant pressure
• Physics:Gay-Lussac's law – Relationship between pressure and temperature of a gas at constant volume.
• Physics:Ideal gas – Mathematical model which approximates the behavior of real gases

## Notes

1. First used by Jean-Baptiste Dumas in 1826.
2. First used by Stanislao Cannizzaro in 1858.

## References

1. Editors of the Encyclopædia Britannica. "Avogadro's law".
2. Avogadro, Amedeo (1810). "Essai d'une manière de déterminer les masses relatives des molécules élémentaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons". Journal de Physique 73: 58–76.  English translation
3. Avogadro, Amadeo (July 1811). "Essai d'une maniere de determiner les masses relatives des molecules elementaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons" (in fr). Journal de Physique, de Chimie, et d'Histoire Naturelle 73: 58–76.
4. Ampère, André-Marie (1814). "Lettre de M. Ampère à M. le comte Berthollet sur la détermination des proportions dans lesquelles les corps se combinent d'après le nombre et la disposition respective des molécules dont les parties intégrantes sont composées" (in fr). Annales de Chimie 90 (1): 43–86.
5. Scheidecker-Chevallier, Myriam (1997). "L'hypothèse d'Avogadro (1811) et d'Ampère (1814): la distinction atome/molécule et la théorie de la combinaison chimique" (in fr). Revue d'Histoire des Sciences 50 (1/2): 159–194. doi:10.3406/rhs.1997.1277.
6. Clapeyron, Émile (1834). "Mémoire sur la puissance motrice de la chaleur" (in fr). Journal de l'École Polytechnique XIV: 153–190.
7. Loschmidt, J. (1865). "Zur Grösse der Luftmoleküle". Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien 52 (2): 395–413.  English translation.