Avogadro's Law
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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 Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
of a gas to the amount of substance of gas present. The law is a specific case of the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
. A modern statement is:
Avogadro's law states that "equal volumes of all gases, at the same
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
and
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, have the same number of
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
s." For a given mass of an
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
, the volume and amount (moles) of the gas are
directly proportional In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
if the temperature and pressure are constant.
The law is named after
Amedeo Avogadro Lorenzo Romano Amedeo Carlo Avogadro, Count of Quaregna and Cerreto (, also , ; 9 August 17769 July 1856) was an Italian scientist, most noted for his contribution to molecular theory now known as Avogadro's law, which states that equal volume ...
who, in 1812, 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 Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
and
nitrogen Nitrogen is the chemical element with the symbol N and atomic number 7. Nitrogen is a nonmetal and the lightest member of group 15 of the periodic table, often called the pnictogens. It is a common element in the universe, estimated at se ...
contain the same number of atoms when they are at the same temperature and pressure, and observe
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
behavior. In practice,
real gas Real gases are nonideal gases whose molecules occupy space and have interactions; consequently, they do not adhere to the ideal gas law. To understand the behaviour of real gases, the following must be taken into account: *compressibility effect ...
es 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: :V \propto n or :\frac = k where *''V'' is the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
of the gas; *''n'' is the amount of substance of the gas (measured in
moles Moles can refer to: * Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain * The Moles (Australian band) *The Moles, alter ego of Scottish band Simon Dupree and the Big Sound People *Abraham Moles, French engin ...
); *''k'' is a constant for a given temperature and pressure. This law describes how, under the same condition of
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
and
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, equal
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
s of all
gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
es contain the same number of
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
s. For comparing the same substance under two different sets of conditions, the law can be usefully expressed as follows: :\frac = \frac 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 In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, ...
of the gas.


Derivation from the ideal gas law

The derivation of Avogadro's law follows directly from the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
, i.e. :PV = nRT , where ''R'' is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
, ''T'' is the Kelvin temperature, and ''P'' is the pressure (in
pascals The pascal (symbol: Pa) is the unit of pressure in the International System of Units (SI), and is also used to quantify internal pressure, stress, Young's modulus, and ultimate tensile strength. The unit, named after Blaise Pascal, is defined ...
). Solving for ''V/n'', we thus obtain :\frac = \frac. Compare that to :k=\frac 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 The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
''k''B, as :PV = N k_\text T, where ''N'' is the number of particles in the gas, and the ratio of ''R'' over ''k''B is equal to the
Avogadro constant The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining c ...
. In this form, for ''V/N'' is a constant, we have :\frac = k' = \frac. If ''T'' and ''P'' are taken at standard conditions for temperature and pressure (STP), then ''k''′ = 1/''n''0, where ''n''0 is the
Loschmidt constant The ''Loschmidt constant'' or Loschmidt's number (symbol: ''n''0) is the number of particles (atoms or molecules) of an ideal gas in a given volume (the number density), and usually quoted at standard temperature and pressure. The 2014 CODATA recom ...
.


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 Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law (especially in France), is an experimental gas law that describes the relationship between pressure and volume of a confined gas. Boyle's law has been stated as: The ...
(1662),
Charles's law Charles's law (also known as the law of volumes) is an experimental gas law that describes how gases tend to expand when heated. A modern statement of Charles's law is: When the pressure on a sample of a dry gas is held constant, the Kelvin t ...
(1787) and
Gay-Lussac's law Gay-Lussac's law usually refers to Joseph-Louis Gay-Lussac's law of combining volumes of gases, discovered in 1808 and published in 1809. It sometimes refers to the proportionality of the volume of a gas to its absolute temperature at constant pr ...
(1808). The hypothesis was first published by Amadeo Avogadro in 1811, and it reconciled
Dalton atomic theory John Dalton (; 5 or 6 September 1766 – 27 July 1844) was an English chemist, physicist and meteorologist. He is best known for introducing the atomic theory into chemistry, and for his research into Color blindness, colour blindness, which ...
with the "incompatible" idea of
Joseph Louis Gay-Lussac Joseph Louis Gay-Lussac (, , ; 6 December 1778 – 9 May 1850) was a French chemist and physicist. He is known mostly for his discovery that water is made of two parts hydrogen and one part oxygen (with Alexander von Humboldt), for two laws ...
that some gases were composite of different fundamental substances (molecules) in integer proportions. In 1814, independently from Avogadro, André-Marie Ampère published the same law with similar conclusions. As Ampère was more well known in France, the hypothesis was usually referred there as Ampère's hypothesis,First used by Jean-Baptiste Dumas in 1826. and later also as Avogadro–Ampère hypothesisFirst used by
Stanislao Cannizzaro Stanislao Cannizzaro ( , also , ; 13 July 1826 – 10 May 1910) was an Italian chemist. He is famous for the Cannizzaro reaction and for his influential role in the atomic-weight deliberations of the Karlsruhe Congress in 1860. Biograph ...
in 1858.
or even Ampère–Avogadro hypothesis. Experimental studies carried out by Charles Frédéric Gerhardt and Auguste Laurent on
organic chemistry Organic chemistry is a subdiscipline within chemistry involving the scientific study of the structure, properties, and reactions of organic compounds and organic materials, i.e., matter in its various forms that contain carbon atoms.Clayden, J.; ...
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 Stanislao Cannizzaro ( , also , ; 13 July 1826 – 10 May 1910) was an Italian chemist. He is famous for the Cannizzaro reaction and for his influential role in the atomic-weight deliberations of the Karlsruhe Congress in 1860. Biograph ...
, as announced at
Karlsruhe Congress The Karlsruhe Congress was an international meeting of chemists held in Karlsruhe, Germany from 3 to 5 September 1860. It was the first international conference of chemistry worldwide. The meeting The Karlsruhe Congress was called so that Euro ...
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, giving rise to the ideal gas law. At the end of the 19th century, later developments from scientists like
August Krönig August Karl Krönig (; 20 September 1822 – 5 June 1879) was a German chemist and physicist who published an account of the kinetic theory of gases in 1856, probably after reading a paper by John James Waterston. Biography Krönig was born i ...
,
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Sadi Carnot's principle ...
,
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and li ...
and
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of ther ...
, 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 constant

Avogadro's law provides a way to calculate the quantity of gas in a receptacle. Thanks to this discovery,
Johann Josef Loschmidt Johann Josef Loschmidt (15 March 1821 – 8 July 1895), who referred to himself mostly as Josef Loschmidt (omitting his first name), was a notable Austrian scientist who performed ground-breaking work in chemistry, physics (thermodynamics, optics, ...
, in 1865, was able for the first time to estimate the size of a molecule.English translation
His calculation gave rise to the concept of the
Loschmidt constant The ''Loschmidt constant'' or Loschmidt's number (symbol: ''n''0) is the number of particles (atoms or molecules) of an ideal gas in a given volume (the number density), and usually quoted at standard temperature and pressure. The 2014 CODATA recom ...
, a ratio between macroscopic and atomic quantities. In 1910, Millikan's
oil drop experiment The oil drop experiment was performed by Robert A. Millikan and Harvey Fletcher in 1909 to measure the elementary electric charge (the charge of the electron). The experiment took place in the Ryerson Physical Laboratory at the University of C ...
determined the
charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...
of the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
; using it with the
Faraday constant In physical chemistry, the Faraday constant, denoted by the symbol and sometimes stylized as ℱ, is the electric charge per mole of elementary charges. It is named after the English scientist Michael Faraday. Since the 2019 redefinition of ...
(derived by
Michael Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic inducti ...
in 1834), one is able to determine the number of particles in a
mole Mole (or Molé) may refer to: Animals * Mole (animal) or "true mole", mammals in the family Talpidae, found in Eurasia and North America * Golden moles, southern African mammals in the family Chrysochloridae, similar to but unrelated to Talpida ...
of substance. At the same time, precision experiments by
Jean Baptiste Perrin Jean Baptiste Perrin (30 September 1870 – 17 April 1942) was a French physicist who, in his studies of the Brownian motion of minute particles suspended in liquids (sedimentation equilibrium), verified Albert Einstein’s explanation of this p ...
led to the definition of the Avogadro number as the number of molecules in one gram-molecule of
oxygen Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as wel ...
. 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 The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining c ...
.


Molar volume

At standard temperature and pressure (100 
kPa KPA may refer to: * Keele Postgraduate Association, Keele University, UK, formerly Keele Research Association (KRA) * Kensington (Olympia) station, London, England, National Rail station code * Kenya Ports Authority * ''Kiln phosphoric acid'', a ...
and 273.15  K) , we can use Avogadro's law find the molar volume of an ideal gas: : V_\text = \frac = \frac \approx \frac \approx \mathrm = \mathrm


Notes


References

{{Authority control Gas laws Amount of substance it:Volume molare#Legge di Avogadro