Hydrogen

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Hydrogen   ₁H

Purple glow in its plasma state
Spectral lines of hydrogen
General properties
Name, symbol	hydrogen, H
Pronunciation	/ˈhaɪdrədʒən/[1] HY-drə-jən
Appearance	colorless gas
Hydrogen in the periodic table
- ↑ H ↓ Li - ← hydrogen → helium
Atomic number	1
Standard atomic weight	1.008(1)
Element category	diatomic nonmetal, could be considered metalloid
Group, period, block	group 1, period 1, s-block
Electron configuration	1s1 per shell: 1
Physical properties
Color	colorless
Phase	gas
Melting point	13.99 K ​(−259.16 °C, ​−434.49 °F)
Boiling point	20.271 K ​(−252.879 °C, ​−423.182 °F)
Density	0.08988 g/L (at 0 °C, 101.325 kPa)
Liquid density	at m.p.: 0.07 g·cm−3 (solid: 0.0763 g·cm−3)[2] at b.p.: 0.07099 g·cm−3
Triple point	13.8033 K, ​7.041 kPa
Critical point	32.938 K, 1.2858 MPa
Heat of fusion	(H2) 0.117 kJ·mol−1
Heat of vaporization	(H2) 0.904 kJ·mol−1
Molar heat capacity	(H2) 28.836 J·mol−1·K−1
Vapor pressure P (Pa) 1 10 100 1 k 10 k 100 k at T (K) 15 20
Atomic properties
Oxidation states	1, −1 ​(an amphoteric oxide)
Electronegativity	2.20 (Pauling scale)
Ionization energies	1st: 1312.0 kJ·mol−1
Covalent radius	31±5 pm
Van der Waals radius	120 pm
Miscellanea
Crystal structure	​hexagonal
Speed of sound	1310 m·s−1 (gas, 27 °C)
Thermal conductivity	0.1805 W·m−1·K−1
Magnetic ordering	diamagnetic[3]
CAS Number	1333-74-0
History
Discovery	Henry Cavendish[4][5] (1766)
Named by	Antoine Lavoisier[6] (1783)
Most stable isotopes
Main article: Isotopes of hydrogen
iso NA half-life DM DE (MeV) DP 1H 99.9885% 1H is stable with 0 neutrons 2H 0.0115% 2H is stable with 1 neutron 3H trace 12.32 y β− 0.01861 3He

Hydrogen is a chemical element with chemical symbol H and atomic number 1. With an atomic weight of 1.00794 u, hydrogen is the lightest element on the periodic table. Its monatomic form (H) is the most abundant chemical substance in the universe, constituting roughly 75% of all baryonic mass.^{[7][note 1]} Non-remnant stars are mainly composed of hydrogen in its plasma state. The most common isotope of hydrogen, termed protium (name rarely used, symbol ¹H), has a single proton and zero neutrons.

The universal emergence of atomic hydrogen first occurred during the recombination epoch. At standard temperature and pressure, hydrogen is a colorless, odorless, tasteless, non-toxic, nonmetallic, highly combustible diatomic gas with the molecular formula H₂. Since hydrogen readily forms covalent compounds with most non-metallic elements, most of the hydrogen on Earth exists in molecular forms such as in the form of water or organic compounds. Hydrogen plays a particularly important role in acid–base reactions. In ionic compounds, hydrogen can take the form of a negative charge (i.e., anion) known as a hydride, or as a positively charged (i.e., cation) species denoted by the symbol H⁺. The hydrogen cation is written as though composed of a bare proton, but in reality, hydrogen cations in ionic compounds are always more complex species than that would suggest.

As the simplest atom known, the hydrogen atom has had considerable theoretical application. For example, the hydrogen atom is the only neutral atom with an analytic solution to the Schrödinger equation.

Hydrogen gas was first artificially produced in the early 16th century, via the mixing of metals with acids. In 1766–81, Henry Cavendish was the first to recognize that hydrogen gas was a discrete substance,^[8] and that it produces water when burned, a property which later gave it its name: in Greek, hydrogen means "water-former".

Industrial production is mainly from the steam reforming of natural gas, and less often from more energy-intensive hydrogen production methods like the electrolysis of water.^[9] Most hydrogen is employed near its production site, with the two largest uses being fossil fuel processing (e.g., hydrocracking) and ammonia production, mostly for the fertilizer market.

Hydrogen is a concern in metallurgy as it can embrittle many metals,^[10] complicating the design of pipelines and storage tanks.^[11]

Properties

Combustion

The Space Shuttle Main Engine burnt hydrogen with oxygen, producing a nearly invisible flame at full thrust.

Hydrogen gas (dihydrogen or molecular hydrogen)^[12] is highly flammable and will burn in air at a very wide range of concentrations between 4% and 75% by volume.^[13] The enthalpy of combustion for hydrogen is −286 kJ/mol:^[14]

2 H₂(g) + O₂(g) → 2 H₂O(l) + 572 kJ (286 kJ/mol)^{[note 2]}

Hydrogen gas forms explosive mixtures with air if it is 4–74% concentrated and with chlorine if it is 5–95% concentrated. The mixtures may be ignited by spark, heat or sunlight. The hydrogen autoignition temperature, the temperature of spontaneous ignition in air, is 500 °C (932 °F).^[15] Pure hydrogen-oxygen flames emit ultraviolet light and with high oxygen mix are nearly invisible to the naked eye, as illustrated by the faint plume of the Space Shuttle Main Engine compared to the highly visible plume of a Space Shuttle Solid Rocket Booster. The detection of a burning hydrogen leak may require a flame detector; such leaks can be very dangerous. Hydrogen flames in other conditions are blue, resembling blue natural gas flames.^[16] The destruction of the Hindenburg airship was an infamous example of hydrogen combustion; the cause is debated, but the visible orange flames were the result of a rich mixture of hydrogen to oxygen combined with carbon compounds from the airship skin.

H₂ reacts with every oxidizing element. Hydrogen can react spontaneously and violently at room temperature with chlorine and fluorine to form the corresponding hydrogen halides, hydrogen chloride and hydrogen fluoride, which are also potentially dangerous acids.^[17]

Electron energy levels

Depiction of a hydrogen atom with size of central proton shown, and the atomic diameter shown as about twice the Bohr model radius (image not to scale).

The ground state energy level of the electron in a hydrogen atom is −13.6 eV, which is equivalent to an ultraviolet photon of roughly 92 nm wavelength.^[18]

The energy levels of hydrogen can be calculated fairly accurately using the Bohr model of the atom, which conceptualizes the electron as "orbiting" the proton in analogy to the Earth's orbit of the Sun. However, the electromagnetic force attracts electrons and protons to one another, while planets and celestial objects are attracted to each other by gravity. Because of the discretization of angular momentum postulated in early quantum mechanics by Bohr, the electron in the Bohr model can only occupy certain allowed distances from the proton, and therefore only certain allowed energies.^[19]

A more accurate description of the hydrogen atom comes from a purely quantum mechanical treatment that uses the Schrödinger equation or the Feynman path integral formulation to calculate the probability density of the electron around the proton.^[20] The most complicated treatments allow for the small effects of special relativity and vacuum polarization. In the quantum mechanical treatment, the electron in a ground state hydrogen atom has no angular momentum at all— an illustration of how different the "planetary orbit" conception of electron motion differs from reality.

Elemental molecular forms

First tracks observed in liquid hydrogen bubble chamber at the Bevatron

There exist two different spin isomers of hydrogen diatomic molecules that differ by the relative spin of their nuclei.^[21] In the orthohydrogen form, the spins of the two protons are parallel and form a triplet state with a molecular spin quantum number of 1 (¹⁄₂+¹⁄₂); in the parahydrogen form the spins are antiparallel and form a singlet with a molecular spin quantum number of 0 (¹⁄₂–¹⁄₂). At standard temperature and pressure, hydrogen gas contains about 25% of the para form and 75% of the ortho form, also known as the "normal form".^[22] The equilibrium ratio of orthohydrogen to parahydrogen depends on temperature, but because the ortho form is an excited state and has a higher energy than the para form, it is unstable and cannot be purified. At very low temperatures, the equilibrium state is composed almost exclusively of the para form. The liquid and gas phase thermal properties of pure parahydrogen differ significantly from those of the normal form because of differences in rotational heat capacities, as discussed more fully in spin isomers of hydrogen.^[23] The ortho/para distinction also occurs in other hydrogen-containing molecules or functional groups, such as water and methylene, but is of little significance for their thermal properties.^[24]

The uncatalyzed interconversion between para and ortho H₂ increases with increasing temperature; thus rapidly condensed H₂ contains large quantities of the high-energy ortho form that converts to the para form very slowly.^[25] The ortho/para ratio in condensed H₂ is an important consideration in the preparation and storage of liquid hydrogen: the conversion from ortho to para is exothermic and produces enough heat to evaporate some of the hydrogen liquid, leading to loss of liquefied material. Catalysts for the ortho-para interconversion, such as ferric oxide, activated carbon, platinized asbestos, rare earth metals, uranium compounds, chromic oxide, or some nickel^[26] compounds, are used during hydrogen cooling.^[27]

Compounds

Covalent and organic compounds

While H₂ is not very reactive under standard conditions, it does form compounds with most elements. Hydrogen can form compounds with elements that are more electronegative, such as halogens (e.g., F, Cl, Br, I), or oxygen; in these compounds hydrogen takes on a partial positive charge.^[28] When bonded to fluorine, oxygen, or nitrogen, hydrogen can participate in a form of medium-strength noncovalent bonding called hydrogen bonding, which is critical to the stability of many biological molecules.^[29][30] Hydrogen also forms compounds with less electronegative elements, such as the metals and metalloids, in which it takes on a partial negative charge. These compounds are often known as hydrides.^[31]

Hydrogen forms a vast array of compounds with carbon called the hydrocarbons, and an even vaster array with heteroatoms that, because of their general association with living things, are called organic compounds.^[32] The study of their properties is known as organic chemistry^[33] and their study in the context of living organisms is known as biochemistry.^[34] By some definitions, "organic" compounds are only required to contain carbon. However, most of them also contain hydrogen, and because it is the carbon-hydrogen bond which gives this class of compounds most of its particular chemical characteristics, carbon-hydrogen bonds are required in some definitions of the word "organic" in chemistry.^[32] Millions of hydrocarbons are known, and they are usually formed by complicated synthetic pathways, which seldom involve elementary hydrogen.

Hydrides

Compounds of hydrogen are often called hydrides, a term that is used fairly loosely. The term "hydride" suggests that the H atom has acquired a negative or anionic character, denoted H⁻, and is used when hydrogen forms a compound with a more electropositive element. The existence of the hydride anion, suggested by Gilbert N. Lewis in 1916 for group I and II salt-like hydrides, was demonstrated by Moers in 1920 by the electrolysis of molten lithium hydride (LiH), producing a stoichiometry quantity of hydrogen at the anode.^[35] For hydrides other than group I and II metals, the term is quite misleading, considering the low electronegativity of hydrogen. An exception in group II hydrides is BeH
2, which is polymeric. In lithium aluminium hydride, the AlH−
4 anion carries hydridic centers firmly attached to the Al(III).

Although hydrides can be formed with almost all main-group elements, the number and combination of possible compounds varies widely; for example, there are over 100 binary borane hydrides known, but only one binary aluminium hydride.^[36] Binary indium hydride has not yet been identified, although larger complexes exist.^[37]

In inorganic chemistry, hydrides can also serve as bridging ligands that link two metal centers in a coordination complex. This function is particularly common in group 13 elements, especially in boranes (boron hydrides) and aluminium complexes, as well as in clustered carboranes.^[38]

Protons and acids

Oxidation of hydrogen removes its electron and gives H⁺, which contains no electrons and a nucleus which is usually composed of one proton. That is why H+

 is often called a proton. This species is central to discussion of acids. Under the Bronsted-Lowry theory, acids are proton donors, while bases are proton acceptors.

A bare proton, H+, cannot exist in solution or in ionic crystals, because of its unstoppable attraction to other atoms or molecules with electrons. Except at the high temperatures associated with plasmas, such protons cannot be removed from the electron clouds of atoms and molecules, and will remain attached to them. However, the term 'proton' is sometimes used loosely and metaphorically to refer to positively charged or cationic hydrogen attached to other species in this fashion, and as such is denoted "H+" without any implication that any single protons exist freely as a species.

To avoid the implication of the naked "solvated proton" in solution, acidic aqueous solutions are sometimes considered to contain a less unlikely fictitious species, termed the "hydronium ion" (H
3O+). However, even in this case, such solvated hydrogen cations are more realistically conceived as being organized into clusters that form species closer to H
9O+
4.^[39] Other oxonium ions are found when water is in acidic solution with other solvents.^[40]

Although exotic on Earth, one of the most common ions in the universe is the H+
3 ion, known as protonated molecular hydrogen or the trihydrogen cation.^[41]

Isotopes

Hydrogen discharge (spectrum) tube

Deuterium discharge (spectrum) tube

Protium, the most common isotope of hydrogen, has one proton and one electron. Unique among all stable isotopes, it has no neutrons (see diproton for a discussion of why others do not exist).

Hydrogen has three naturally occurring isotopes, denoted 1H, 2H and 3H. Other, highly unstable nuclei (4H to 7H) have been synthesized in the laboratory but not observed in nature.^[42][43]

• 1H is the most common hydrogen isotope with an abundance of more than 99.98%. Because the nucleus of this isotope consists of only a single proton, it is given the descriptive but rarely used formal name protium.^[44]
• 2H, the other stable hydrogen isotope, is known as deuterium and contains one proton and one neutron in its nucleus. Essentially all deuterium in the universe is thought to have been produced at the time of the Big Bang, and has endured since that time. Deuterium is not radioactive, and does not represent a significant toxicity hazard. Water enriched in molecules that include deuterium instead of normal hydrogen is called heavy water. Deuterium and its compounds are used as a non-radioactive label in chemical experiments and in solvents for 1H-NMR spectroscopy.^[45] Heavy water is used as a neutron moderator and coolant for nuclear reactors. Deuterium is also a potential fuel for commercial nuclear fusion.^[46]
• 3H is known as tritium and contains one proton and two neutrons in its nucleus. It is radioactive, decaying into helium-3 through beta decay with a half-life of 12.32 years.^[38] It is so radioactive that it can be used in luminous paint, making it useful in such things as watches. The glass prevents the small amount of radiation from getting out.^[47] Small amounts of tritium occur naturally because of the interaction of cosmic rays with atmospheric gases; tritium has also been released during nuclear weapons tests.^[48] It is used in nuclear fusion reactions,^[49] as a tracer in isotope geochemistry,^[50] and specialized in self-powered lighting devices.^[51] Tritium has also been used in chemical and biological labeling experiments as a radiolabel.^[52]

Hydrogen is the only element that has different names for its isotopes in common use today. During the early study of radioactivity, various heavy radioactive isotopes were given their own names, but such names are no longer used, except for deuterium and tritium. The symbols D and T (instead of 2H and 3H) are sometimes used for deuterium and tritium, but the corresponding symbol for protium, P, is already in use for phosphorus and thus is not available for protium.^[53] In its nomenclatural guidelines, the International Union of Pure and Applied Chemistry allows any of D, T, 2H, and 3H to be used, although 2H and 3H are preferred.^[54]

History

Discovery and use

In 1671, Robert Boyle discovered and described the reaction between iron filings and dilute acids, which results in the production of hydrogen gas.^[55][56] In 1766, Henry Cavendish was the first to recognize hydrogen gas as a discrete substance, by naming the gas from a metal-acid reaction "flammable air". He speculated that "flammable air" was in fact identical to the hypothetical substance called "phlogiston"^[57][58] and further finding in 1781 that the gas produces water when burned. He is usually given credit for its discovery as an element.^[4][5] In 1783, Antoine Lavoisier gave the element the name hydrogen (from the Greek ὑδρο- hydro meaning "water" and -γενής genes meaning "creator")^[6] when he and Laplace reproduced Cavendish's finding that water is produced when hydrogen is burned.^[5]

Antoine-Laurent de Lavoisier

Lavoisier produced hydrogen for his experiments on mass conservation by reacting a flux of steam with metallic iron through an incandescent iron tube heated in a fire. Anaerobic oxidation of iron by the protons of water at high temperature can be schematically represented by the set of following reactions:

   Fe +    H₂O → FeO + H₂

2 Fe + 3 H₂O → Fe₂O₃ + 3 H₂

3 Fe + 4 H₂O → Fe₃O₄ + 4 H₂

Many metals such as zirconium undergo a similar reaction with water leading to the production of hydrogen.

Hydrogen was liquefied for the first time by James Dewar in 1898 by using regenerative cooling and his invention, the vacuum flask.^[5] He produced solid hydrogen the next year.^[5] Deuterium was discovered in December 1931 by Harold Urey, and tritium was prepared in 1934 by Ernest Rutherford, Mark Oliphant, and Paul Harteck.^[4] Heavy water, which consists of deuterium in the place of regular hydrogen, was discovered by Urey's group in 1932.^[5] François Isaac de Rivaz built the first de Rivaz engine, an internal combustion engine powered by a mixture of hydrogen and oxygen in 1806. Edward Daniel Clarke invented the hydrogen gas blowpipe in 1819. The Döbereiner's lamp and limelight were invented in 1823.^[5]

The first hydrogen-filled balloon was invented by Jacques Charles in 1783.^[5] Hydrogen provided the lift for the first reliable form of air-travel following the 1852 invention of the first hydrogen-lifted airship by Henri Giffard.^[5] German count Ferdinand von Zeppelin promoted the idea of rigid airships lifted by hydrogen that later were called Zeppelins; the first of which had its maiden flight in 1900.^[5] Regularly scheduled flights started in 1910 and by the outbreak of World War I in August 1914, they had carried 35,000 passengers without a serious incident. Hydrogen-lifted airships were used as observation platforms and bombers during the war.

The first non-stop transatlantic crossing was made by the British airship R34 in 1919. Regular passenger service resumed in the 1920s and the discovery of helium reserves in the United States promised increased safety, but the U.S. government refused to sell the gas for this purpose. Therefore, H₂ was used in the Hindenburg airship, which was destroyed in a midair fire over New Jersey on 6 May 1937.^[5] The incident was broadcast live on radio and filmed. Ignition of leaking hydrogen is widely assumed to be the cause, but later investigations pointed to the ignition of the aluminized fabric coating by static electricity. But the damage to hydrogen's reputation as a lifting gas was already done.

In the same year the first hydrogen-cooled turbogenerator went into service with gaseous hydrogen as a coolant in the rotor and the stator in 1937 at Dayton, Ohio, by the Dayton Power & Light Co,^[59] because of the thermal conductivity of hydrogen gas this is the most common type in its field today.
The nickel hydrogen battery was used for the first time in 1977 aboard the U.S. Navy's Navigation technology satellite-2 (NTS-2).^[60] For example, the ISS,^[61] Mars Odyssey^[62] and the Mars Global Surveyor^[63] are equipped with nickel-hydrogen batteries. In the dark part of its orbit, the Hubble Space Telescope is also powered by nickel-hydrogen batteries, which were finally replaced in May 2009,^[64] more than 19 years after launch, and 13 years over their design life.^[65]

Role in quantum theory

Hydrogen emission spectrum lines in the visible range. These are the four visible lines of the Balmer series

Because of its relatively simple atomic structure, consisting only of a proton and an electron, the hydrogen atom, together with the spectrum of light produced from it or absorbed by it, has been central to the development of the theory of atomic structure.^[66] Furthermore, the corresponding simplicity of the hydrogen molecule and the corresponding cation H+
2 allowed fuller understanding of the nature of the chemical bond, which followed shortly after the quantum mechanical treatment of the hydrogen atom had been developed in the mid-1920s.

One of the first quantum effects to be explicitly noticed (but not understood at the time) was a Maxwell observation involving hydrogen, half a century before full quantum mechanical theory arrived. Maxwell observed that the specific heat capacity of H₂ unaccountably departs from that of a diatomic gas below room temperature and begins to increasingly resemble that of a monatomic gas at cryogenic temperatures. According to quantum theory, this behavior arises from the spacing of the (quantized) rotational energy levels, which are particularly wide-spaced in H₂ because of its low mass. These widely spaced levels inhibit equal partition of heat energy into rotational motion in hydrogen at low temperatures. Diatomic gases composed of heavier atoms do not have such widely spaced levels and do not exhibit the same effect.^[67]

Natural occurrence

NGC 604, a giant region of ionized hydrogen in the Triangulum Galaxy

Hydrogen, as atomic H, is the most abundant chemical element in the universe, making up 75% of normal matter by mass and over 90% by number of atoms (most of the mass of the universe, however, is not in the form of chemical-element type matter, but rather is postulated to occur as yet-undetected forms of mass such as dark matter and dark energy).^[68] This element is found in great abundance in stars and gas giant planets. Molecular clouds of H₂ are associated with star formation. Hydrogen plays a vital role in powering stars through the proton-proton reaction and the CNO cycle nuclear fusion.^[69]

Throughout the universe, hydrogen is mostly found in the atomic and plasma states whose properties are quite different from molecular hydrogen. As a plasma, hydrogen's electron and proton are not bound together, resulting in very high electrical conductivity and high emissivity (producing the light from the Sun and other stars). The charged particles are highly influenced by magnetic and electric fields. For example, in the solar wind they interact with the Earth's magnetosphere giving rise to Birkeland currents and the aurora. Hydrogen is found in the neutral atomic state in the interstellar medium. The large amount of neutral hydrogen found in the damped Lyman-alpha systems is thought to dominate the cosmological baryonic density of the Universe up to redshift z=4.^[70]

Under ordinary conditions on Earth, elemental hydrogen exists as the diatomic gas, H₂ (for data see table^[ambiguous]). However, hydrogen gas is very rare in the Earth's atmosphere (1 ppm by volume) because of its light weight, which enables it to escape from Earth's gravity more easily than heavier gases. However, hydrogen is the third most abundant element on the Earth's surface,^[71] mostly in the form of chemical compounds such as hydrocarbons and water.^[38] Hydrogen gas is produced by some bacteria and algae and is a natural component of flatus, as is methane, itself a hydrogen source of increasing importance.^[72]

A molecular form called protonated molecular hydrogen (H+
3) is found in the interstellar medium, where it is generated by ionization of molecular hydrogen from cosmic rays. This charged ion has also been observed in the upper atmosphere of the planet Jupiter. The ion is relatively stable in the environment of outer space due to the low temperature and density. H+
3 is one of the most abundant ions in the Universe, and it plays a notable role in the chemistry of the interstellar medium.^[73] Neutral triatomic hydrogen H₃ can only exist in an excited form and is unstable.^[74] By contrast, the positive hydrogen molecular ion (H+
2) is a rare molecule in the universe.

Production

H2 is produced in chemistry and biology laboratories, often as a by-product of other reactions; in industry for the hydrogenation of unsaturated substrates; and in nature as a means of expelling reducing equivalents in biochemical reactions.

Metal-acid

In the laboratory, H
2 is usually prepared by the reaction of dilute non-oxidizing acids on some reactive metals such as zinc with Kipp's apparatus.

Zn + 2 H+ → Zn2+ + H
2

Aluminium can also produce H
2 upon treatment with bases:

2 Al + 6 H
2O + 2 OH− → 2 Al(OH)−
4 + 3 H
2

The electrolysis of water is a simple method of producing hydrogen. A low voltage current is run through the water, and gaseous oxygen forms at the anode while gaseous hydrogen forms at the cathode. Typically the cathode is made from platinum or another inert metal when producing hydrogen for storage. If, however, the gas is to be burnt on site, oxygen is desirable to assist the combustion, and so both electrodes would be made from inert metals. (Iron, for instance, would oxidize, and thus decrease the amount of oxygen given off.) The theoretical maximum efficiency (electricity used vs. energetic value of hydrogen produced) is in the range 80–94%.^[75]

2 H
2O(l) → 2 H
2(g) + O
2(g)

In 2007, it was discovered that an alloy of aluminium and gallium in pellet form added to water could be used to generate hydrogen. The process also creates alumina, but the expensive gallium, which prevents the formation of an oxide skin on the pellets, can be re-used. This has important potential implications for a hydrogen economy, as hydrogen can be produced on-site and does not need to be transported.^[76]

Steam reforming

Hydrogen can be prepared in several different ways, but economically the most important processes involve removal of hydrogen from hydrocarbons. Commercial bulk hydrogen is usually produced by the steam reforming of natural gas.^[77] At high temperatures (1000–1400 K, 700–1100 °C or 1300–2000 °F), steam (water vapor) reacts with methane to yield carbon monoxide and H
2.

CH
4 + H
2O → CO + 3 H
2

This reaction is favored at low pressures but is nonetheless conducted at high pressures (2.0  MPa, 20 atm or 600 inHg). This is because high-pressure H
2 is the most marketable product and Pressure Swing Adsorption (PSA) purification systems work better at higher pressures. The product mixture is known as "synthesis gas" because it is often used directly for the production of methanol and related compounds. Hydrocarbons other than methane can be used to produce synthesis gas with varying product ratios. One of the many complications to this highly optimized technology is the formation of coke or carbon:

CH
4 → C + 2 H
2

Consequently, steam reforming typically employs an excess of H
2O. Additional hydrogen can be recovered from the steam by use of carbon monoxide through the water gas shift reaction, especially with an iron oxide catalyst. This reaction is also a common industrial source of carbon dioxide:^[77]

CO + H
2O → CO
2 + H
2

Other important methods for H
2 production include partial oxidation of hydrocarbons:^[78]

2 CH
4 + O
2 → 2 CO + 4 H
2

and the coal reaction, which can serve as a prelude to the shift reaction above:^[77]

C + H
2O → CO + H
2

Hydrogen is sometimes produced and consumed in the same industrial process, without being separated. In the Haber process for the production of ammonia, hydrogen is generated from natural gas.^[79] Electrolysis of brine to yield chlorine also produces hydrogen as a co-product.^[80]

Thermochemical

There are more than 200 thermochemical cycles which can be used for water splitting, around a dozen of these cycles such as the iron oxide cycle, cerium(IV) oxide–cerium(III) oxide cycle, zinc zinc-oxide cycle, sulfur-iodine cycle, copper-chlorine cycle and hybrid sulfur cycle are under research and in testing phase to produce hydrogen and oxygen from water and heat without using electricity.^[81] A number of laboratories (including in France, Germany, Greece, Japan, and the USA) are developing thermochemical methods to produce hydrogen from solar energy and water.^[82]

Anaerobic corrosion

Under anaerobic conditions, iron and steel alloys are slowly oxidized by the protons of water concomitantly reduced in molecular hydrogen (H
2). The anaerobic corrosion of iron leads first to the formation of ferrous hydroxide (green rust) and can be described by the following reaction:

Fe + 2 H
2O → Fe(OH)
2 + H
2

In its turn, under anaerobic conditions, the ferrous hydroxide (Fe(OH)
2 ) can be oxidized by the protons of water to form magnetite and molecular hydrogen. This process is described by the Schikorr reaction:

3 Fe(OH)
2 → Fe
3O
4 + 2 H
2O + H
2

ferrous hydroxide → magnetite + water + hydrogen

The well crystallized magnetite (Fe
3O
4) is thermodynamically more stable than the ferrous hydroxide (Fe(OH)
2 ).

This process occurs during the anaerobic corrosion of iron and steel in oxygen-free groundwater and in reducing soils below the water table.

Geological occurrence: the serpentinization reaction

In the absence of atmospheric oxygen (O
2), in deep geological conditions prevailing far away from Earth atmosphere, hydrogen (H
2) is produced during the process of serpentinization by the anaerobic oxidation by the water protons (H⁺) of the ferrous (Fe²⁺) silicate present in the crystal lattice of the fayalite (Fe
2SiO
4, the olivine iron-endmember). The corresponding reaction leading to the formation of magnetite (Fe
3O
4), quartz (SiO
2) and hydrogen (H
2) is the following:

3Fe
2SiO
4 + 2 H
2O → 2 Fe
3O
4 + 3 SiO
2 + 3 H
2

fayalite + water → magnetite + quartz + hydrogen

This reaction closely resembles the Schikorr reaction observed in the anaerobic oxidation of the ferrous hydroxide in contact with water.

Formation in transformers

From all the fault gases formed in power transformers, hydrogen is the most common and is generated under most fault conditions; thus, formation of hydrogen is an early indication of serious problems in the transformer's life cycle.^[83]

Xylose

In 2014 a low-temperature 50 °C (122 °F), atmospheric-pressure enzyme-driven process to convert xylose into hydrogen with nearly 100% of the theoretical yield was announced. The process employs 13 enzymes, including a novel polyphosphate xylulokinase (XK).^[84][85]

Applications

Consumption in processes

Large quantities of H
2 are needed in the petroleum and chemical industries. The largest application of H
2 is for the processing ("upgrading") of fossil fuels, and in the production of ammonia. The key consumers of H
2 in the petrochemical plant include hydrodealkylation, hydrodesulfurization, and hydrocracking. H
2 has several other important uses. H
2 is used as a hydrogenating agent, particularly in increasing the level of saturation of unsaturated fats and oils (found in items such as margarine), and in the production of methanol. It is similarly the source of hydrogen in the manufacture of hydrochloric acid. H
2 is also used as a reducing agent of metallic ores.^[86]

Hydrogen is highly soluble in many rare earth and transition metals^[87] and is soluble in both nanocrystalline and amorphous metals.^[88] Hydrogen solubility in metals is influenced by local distortions or impurities in the crystal lattice.^[89] These properties may be useful when hydrogen is purified by passage through hot palladium disks, but the gas's high solubility is a metallurgical problem, contributing to the embrittlement of many metals,^[10] complicating the design of pipelines and storage tanks.^[11]

Apart from its use as a reactant, H
2 has wide applications in physics and engineering. It is used as a shielding gas in welding methods such as atomic hydrogen welding.^[90][91] H₂ is used as the rotor coolant in electrical generators at power stations, because it has the highest thermal conductivity of any gas. Liquid H₂ is used in cryogenic research, including superconductivity studies.^[92] Because H
2 is lighter than air, having a little more than ¹⁄₁₄ of the density of air, it was once widely used as a lifting gas in balloons and airships.^[93]

In more recent applications, hydrogen is used pure or mixed with nitrogen (sometimes called forming gas) as a tracer gas for minute leak detection. Applications can be found in the automotive, chemical, power generation, aerospace, and telecommunications industries.^[94] Hydrogen is an authorized food additive (E 949) that allows food package leak testing among other anti-oxidizing properties.^[95]

Hydrogen's rarer isotopes also each have specific applications. Deuterium (hydrogen-2) is used in nuclear fission applications as a moderator to slow neutrons, and in nuclear fusion reactions.^[5] Deuterium compounds have applications in chemistry and biology in studies of reaction isotope effects.^[96] Tritium (hydrogen-3), produced in nuclear reactors, is used in the production of hydrogen bombs,^[97] as an isotopic label in the biosciences,^[52] and as a radiation source in luminous paints.^[98]

The triple point temperature of equilibrium hydrogen is a defining fixed point on the ITS-90 temperature scale at 13.8033 kelvins.^[99]

Coolant

Hydrogen is commonly used in power stations as a coolant in generators due to a number of favorable properties that are a direct result of its light diatomic molecules. These include low density, low viscosity, and the highest specific heat and thermal conductivity of all gases.

Energy carrier

Hydrogen is not an energy resource,^[100] except in the hypothetical context of commercial nuclear fusion power plants using deuterium or tritium, a technology presently far from development.^[101]
The Sun's energy comes from nuclear fusion of hydrogen, but this process is difficult to achieve controllably on Earth.^[102] Elemental hydrogen from solar, biological, or electrical sources require more energy to make it than is obtained by burning it, so in these cases hydrogen functions as an energy carrier, like a battery. Hydrogen may be obtained from fossil sources (such as methane), but these sources are unsustainable.^[100]

The energy density per unit volume of both liquid hydrogen and compressed hydrogen gas at any practicable pressure is significantly less than that of traditional fuel sources, although the energy density per unit fuel mass is higher.^[100] Nevertheless, elemental hydrogen has been widely discussed in the context of energy, as a possible future carrier of energy on an economy-wide scale.^[103] For example, CO
2 sequestration followed by carbon capture and storage could be conducted at the point of H
2 production from fossil fuels.^[104] Hydrogen used in transportation would burn relatively cleanly, with some NO_x emissions,^[105] but without carbon emissions.^[104] However, the infrastructure costs associated with full conversion to a hydrogen economy would be substantial.^[106]

Semiconductor industry

Hydrogen is employed to saturate broken ("dangling") bonds of amorphous silicon and amorphous carbon that helps stabilizing material properties.^[107] It is also a potential electron donor in various oxide materials, including ZnO,^[108][109] SnO₂, CdO, MgO,^[110] ZrO₂, HfO₂, La₂O₃, Y₂O₃, TiO₂, SrTiO₃, LaAlO₃, SiO₂, Al₂O₃, ZrSiO₄, HfSiO₄, and SrZrO₃.^[111]

Biological reactions

H₂ is a product of some types of anaerobic metabolism and is produced by several microorganisms, usually via reactions catalyzed by iron- or nickel-containing enzymes called hydrogenases. These enzymes catalyze the reversible redox reaction between H₂ and its component two protons and two electrons. Creation of hydrogen gas occurs in the transfer of reducing equivalents produced during pyruvate fermentation to water.^[112]
Water splitting, in which water is decomposed into its component protons, electrons, and oxygen, occurs in the light reactions in all photosynthetic organisms. Some such organisms, including the alga Chlamydomonas reinhardtii and cyanobacteria, have evolved a second step in the dark reactions in which protons and electrons are reduced to form H₂ gas by specialized hydrogenases in the chloroplast.^[113] Efforts have been undertaken to genetically modify cyanobacterial hydrogenases to efficiently synthesize H₂ gas even in the presence of oxygen.^[114] Efforts have also been undertaken with genetically modified alga in a bioreactor.^[115]

Safety and precautions

Hydrogen poses a number of hazards to human safety, from potential detonations and fires when mixed with air to being an asphyxiant in its pure, oxygen-free form.^[116] In addition, liquid hydrogen is a cryogen and presents dangers (such as frostbite) associated with very cold liquids.^[117] Hydrogen dissolves in many metals, and, in addition to leaking out, may have adverse effects on them, such as hydrogen embrittlement,^[118] leading to cracks and explosions.^[119] Hydrogen gas leaking into external air may spontaneously ignite. Moreover, hydrogen fire, while being extremely hot, is almost invisible, and thus can lead to accidental burns.^[120]
Even interpreting the hydrogen data (including safety data) is confounded by a number of phenomena. Many physical and chemical properties of hydrogen depend on the parahydrogen/orthohydrogen ratio (it often takes days or weeks at a given temperature to reach the equilibrium ratio, for which the data is usually given). Hydrogen detonation parameters, such as critical detonation pressure and temperature, strongly depend on the container geometry.^[116]

Leonhard Euler

From Wikipedia, the free encyclopedia

Leonhard Euler
Portrait by Jakob Emanuel Handmann (1756)
Born	15 April 1707 Basel, Switzerland
Died	18 September 1783 (aged 76) [OS: 7 September 1783] Saint Petersburg, Russian Empire
Residence	Kingdom of Prussia, Russian Empire Switzerland
Fields	Mathematics and physics
Institutions	Imperial Russian Academy of Sciences Berlin Academy
Alma mater	University of Basel
Doctoral advisor	Johann Bernoulli
Doctoral students	Nicolas Fuss Johann Hennert Stepan Rumovsky
Other notable students	Joseph Louis Lagrange
Known for	See full list
Signature
Notes He is the father of the mathematician Johann Euler. He is listed by an academic genealogy as the equivalent to the doctoral advisor of Joseph Louis Lagrange.[1]

Leonhard Euler (/ˈɔɪlər/ OY-lər;^[2] German pronunciation: [ˈɔʏlɐ] ( listen), local pronunciation: [ˈɔɪlr̩] ( listen); 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.^[3] He is also renowned for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.^[4]

Euler is considered to be the pre-eminent mathematician of the 18th century and one of the greatest mathematicians to have ever lived. He is also one of the most prolific mathematicians; his collected works fill 60–80 quarto volumes.^[5] He spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia.

A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."^[6][7]

Life

Early years

Old Swiss 10 Franc banknote honoring Euler

Euler was born on 15 April 1707, in Basel to Paul Euler, a pastor of the Reformed Church, and Marguerite Brucker, a pastor's daughter. He had two younger sisters named Anna Maria and Maria Magdalena. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli family—Johann Bernoulli, who was then regarded as Europe's foremost mathematician, would eventually be the most important influence on young Leonhard. Euler's early formal education started in Basel, where he was sent to live with his maternal grandmother. At the age of thirteen he enrolled at the University of Basel, and in 1723, received his Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics.^[8]
Euler was at this point studying theology, Greek, and Hebrew at his father's urging, in order to become a pastor, but Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono.^[9] At that time, he was pursuing an (ultimately unsuccessful) attempt to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition; the problem that year was to find the best way to place the masts on a ship. Pierre Bouguer, a man who became known as "the father of naval architecture" won, and Euler took second place. Euler later won this annual prize twelve times.^[10]

St. Petersburg

Around this time Johann Bernoulli's two sons, Daniel and Nicolas, were working at the Imperial Russian Academy of Sciences in St Petersburg. On 10 July 1726, Nicolas died of appendicitis after spending a year in Russia, and when Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.^[11]

1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.

Euler arrived in the Russian capital on 17 May 1727. He was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he often worked in close collaboration. Euler mastered Russian and settled into life in St Petersburg. He also took on an additional job as a medic in the Russian Navy.^[12]

The Academy at St. Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy in order to lessen the faculty's teaching burden, and the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions.^[10]

The Academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. The Russian nobility then gained power upon the ascension of the twelve-year-old Peter II. The nobility were suspicious of the academy's foreign scientists, and thus cut funding and caused other difficulties for Euler and his colleagues.

Conditions improved slightly upon the death of Peter II, and Euler swiftly rose through the ranks in the academy and was made professor of physics in 1731. Two years later, Daniel Bernoulli, who was fed up with the censorship and hostility he faced at St. Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department.^[13]

On 7 January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell, a painter from the Academy Gymnasium.^[14] The young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood.^[15]

Berlin

Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, nowadays written as v − e + f = 2.

Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia. He lived for twenty-five years in Berlin, where he wrote over 380 articles. In Berlin, he published the two works for which he would become most renowned: The Introductio in analysin infinitorum, a text on functions published in 1748, and the Institutiones calculi differentialis,^[16] published in 1755 on differential calculus.^[17] In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences.

In addition, Euler was asked to tutor Friederike Charlotte of Brandenburg-Schwedt, the Princess of Anhalt-Dessau and Frederick's niece. Euler wrote over 200 letters to her in the early 1760s, which were later compiled into a best-selling volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess.^[18] This work contained Euler's exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Euler's personality and religious beliefs. This book became more widely read than any of his mathematical works, and was published across Europe and in the United States. The popularity of the 'Letters' testifies to Euler's ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist.^[17]

Despite Euler's immense contribution to the Academy's prestige, he was eventually forced to leave Berlin. This was partly because of a conflict of personality with Frederick, who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the Academy. Voltaire was among those in Frederick's employ, and the Frenchman enjoyed a prominent position within the king's social circle. Euler, a simple religious man and a hard worker, was very conventional in his beliefs and tastes. He was in many ways the antithesis of Voltaire. Euler had limited training in rhetoric, and tended to debate matters that he knew little about, making him a frequent target of Voltaire's wit.^[17] Frederick also expressed disappointment with Euler's practical engineering abilities:

I wanted to have a water jet in my garden: Euler calculated the force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouci. My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry!^[19]

A 1753 portrait by Emanuel Handmann. This portrayal suggests problems of the right eyelid, and possible strabismus. The left eye, which here appears healthy, was later affected by a cataract.^[20]

Eyesight deterioration

Euler's eyesight worsened throughout his mathematical career. Three years after suffering a near-fatal fever in 1735, he became almost blind in his right eye, but Euler rather blamed the painstaking work on cartography he performed for the St. Petersburg Academy for his condition. Euler's vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as "Cyclops". Euler later developed a cataract in his left eye, which was discovered in 1766. Just a few weeks after its discovery, he was rendered almost totally blind. However, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and exquisite memory. For example, Euler could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last. With the aid of his scribes, Euler's productivity on many areas of study actually increased. He produced on average, one mathematical paper every week in the year 1775.^[5]

Return to Russia

The situation in Russia had improved greatly since the accession to the throne of Catherine the Great, and in 1766 Euler accepted an invitation to return to the St. Petersburg Academy and spent the rest of his life in Russia. However, his second stay in the country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, and almost his life. In 1773, he lost his wife Katharina after 40 years of marriage. Three years after his wife's death, Euler married her half-sister, Salome Abigail Gsell (1723–1794).^[21] This marriage lasted until his death. He was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 1782.^[22]

In St. Petersburg on 18 September 1783, after a lunch with his family, during a conversation with a fellow academician Anders Johan Lexell, about the newly discovered planet Uranus and its orbit, Euler suffered a brain hemorrhage and died a few hours later.^[23] A short obituary for the Russian Academy of Sciences was written by Jacob von Staehlin-Storcksburg and a more detailed eulogy^[24] was written and delivered at a memorial meeting by Russian mathematician Nicolas Fuss, one of Euler's disciples. In the eulogy written for the French Academy by the French mathematician and philosopher Marquis de Condorcet, he commented,

il cessa de calculer et de vivre—... he ceased to calculate and to live.^[25]

He was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island. In 1785, the Russian Academy of Sciences put a marble bust of Leonhard Euler on a pedestal next to the Director's seat and, in 1837, placed a headstone on Euler's grave. To commemorate the 250th anniversary of Euler's birth, the headstone was moved in 1956, together with his remains, to the 18th-century necropolis at the Alexander Nevsky Monastery.^[26]

Euler's grave at the Alexander Nevsky Monastery

Contributions to mathematics and physics

Euler worked in almost all areas of mathematics, such as geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes.^[5] Euler's name is associated with a large number of topics.

Euler is the only mathematician to have two numbers named after him: the important Euler's Number in calculus, e, approximately equal to 2.71828, and the Euler-Mascheroni Constant γ (gamma) sometimes referred to as just "Euler's constant", approximately equal to 0.57721. It is not known whether γ is rational or irrational.^[27]

Mathematical notation

Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function^[3] and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter e for the base of the natural logarithm (now also known as Euler's number), the Greek letter Σ for summations and the letter i to denote the imaginary unit.^[28] The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it did not originate with him.^[29]

Analysis

The development of infinitesimal calculus was at the forefront of 18th Century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigour^[30] (in particular his reliance on the principle of the generality of algebra), his ideas led to many great advances. Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as

Notably, Euler directly proved the power series expansions for e and the inverse tangent function. (Indirect proof via the inverse power series technique was given by Newton and Leibniz between 1670 and 1680.) His daring use of power series enabled him to solve the famous Basel problem in 1735 (he provided a more elaborate argument in 1741):^[30]

A geometric interpretation of Euler's formula

Euler introduced the use of the exponential function and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers, thus greatly expanding the scope of mathematical applications of logarithms.^[28] He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions. For any real number φ (taken to be radians), Euler's formula states that the complex exponential function satisfies

A special case of the above formula is known as Euler's identity,

called "the most remarkable formula in mathematics" by Richard P. Feynman, for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1, e, i and π.^[31] In 1988, readers of the Mathematical Intelligencer voted it "the Most Beautiful Mathematical Formula Ever".^[32] In total, Euler was responsible for three of the top five formulae in that poll.^[32]

De Moivre's formula is a direct consequence of Euler's formula.

In addition, Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. He also found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis. He also invented the calculus of variations including its best-known result, the Euler–Lagrange equation.

Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions. For example, he proved the infinitude of primes using the divergence of the harmonic series, and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to the development of the prime number theorem.^[33]

Number theory

Euler's interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy. A lot of Euler's early work on number theory was based on the works of Pierre de Fermat. Euler developed some of Fermat's ideas, and disproved some of his conjectures.
Euler linked the nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between the Riemann zeta function and the prime numbers; this is known as the Euler product formula for the Riemann zeta function.

Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and he made distinct contributions to Lagrange's four-square theorem. He also invented the totient function φ(n), the number of positive integers less than or equal to the integer n that are coprime to n. Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem. He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid. He proved that the relationship shown between perfect numbers and Mersenne primes earlier proved by Euclid was one-to-one, a result otherwise known as the Euclid–Euler theorem. Euler also conjectured the law of quadratic reciprocity. The concept is regarded as a fundamental theorem of number theory, and his ideas paved the way for the work of Carl Friedrich Gauss.^[34]

By 1772 Euler had proved that 2³¹ − 1 = 2,147,483,647 is a Mersenne prime. It may have remained the largest known prime until 1867.^[35]

Graph theory

Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges.

In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg.^[36] The city of Königsberg, Prussia was set on the Pregel River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no Eulerian circuit. This solution is considered to be the first theorem of graph theory, specifically of planar graph theory.^[36]

Euler also discovered the formula V − E + F = 2 relating the number of vertices, edges and faces of a convex polyhedron,^[37] and hence of a planar graph. The constant in this formula is now known as the Euler characteristic for the graph (or other mathematical object), and is related to the genus of the object.^[38] The study and generalization of this formula, specifically by Cauchy^[39] and L'Huillier,^[40] is at the origin of topology.

Applied mathematics

Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the Bernoulli numbers, Fourier series, Venn diagrams, Euler numbers, the constants e and π, continued fractions and integrals. He integrated Leibniz's differential calculus with Newton's Method of Fluxions, and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation of integrals, inventing what are now known as the Euler approximations. The most notable of these approximations are Euler's method and the Euler–Maclaurin formula. He also facilitated the use of differential equations, in particular introducing the Euler–Mascheroni constant:

One of Euler's more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually incorporate musical theory as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians.^[41]

Physics and astronomy

Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering. Aside from successfully applying his analytic tools to problems in classical mechanics, Euler also applied these techniques to celestial problems. His work in astronomy was recognized by a number of Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the parallax of the sun. His calculations also contributed to the development of accurate longitude tables.^[42]

In addition, Euler made important contributions in optics. He disagreed with Newton's corpuscular theory of light in the Opticks, which was then the prevailing theory. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christiaan Huygens would become the dominant mode of thought, at least until the development of the quantum theory of light.^[43]

In 1757 he published an important set of equations for inviscid flow, that are now known as the Euler equations.^[44] In differential form, the equations are:

where

• ρ is the fluid mass density,
• u is the fluid velocity vector, with components u, v, and w,
• E = ρ e + ½ ρ ( u² + v² + w² ) is the total energy per unit volume, with e being the internal energy per unit mass for the fluid,
• p is the pressure,
• denotes the tensor product, and
• 0 being the zero vector.

Euler is also well known in structural engineering for his formula giving the critical buckling load of an ideal strut, which depends only on its length and flexural stiffness:^[45]

where

= maximum or critical force (vertical load on column),

= modulus of elasticity,

= area moment of inertia,

= unsupported length of column,

= column effective length factor, whose value depends on the conditions of end support of the column, as follows.

For both ends pinned (hinged, free to rotate), = 1.0.

For both ends fixed, = 0.50.

For one end fixed and the other end pinned, = 0.699....

For one end fixed and the other end free to move laterally, = 2.0.

is the effective length of the column.

Logic

Euler is also credited with using closed curves to illustrate syllogistic reasoning (1768). These diagrams have become known as Euler diagrams.^[46]

Euler's Diagram

An Euler diagram is a diagrammatic means of representing sets and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depict sets. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements that are not members of the set. The sizes or shapes of the curves are not important: the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships (intersection, subset and disjointness). Curves whose interior zones do not intersect represent disjoint sets. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection of the sets). A curve that is contained completely within the interior zone of another represents a subset of it. Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement in the 1960s. Since then, they have also been adopted by other curriculum fields such as reading.^[47]

Personal philosophy and religious beliefs

Euler and his friend Daniel Bernoulli were opponents of Leibniz's monadism and the philosophy of Christian Wolff. Euler insisted that knowledge is founded in part on the basis of precise quantitative laws, something that monadism and Wolffian science were unable to provide. Euler's religious leanings might also have had a bearing on his dislike of the doctrine; he went so far as to label Wolff's ideas as "heathen and atheistic".^[48]

Much of what is known of Euler's religious beliefs can be deduced from his Letters to a German Princess and an earlier work, Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister (Defense of the Divine Revelation against the Objections of the Freethinkers). These works show that Euler was a devout Christian who believed the Bible to be inspired; the Rettung was primarily an argument for the divine inspiration of scripture.^[49]

There is a famous legend^[50] inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg academy. The French philosopher Denis Diderot was visiting Russia on Catherine the Great's invitation. However, the Empress was alarmed that the philosopher's arguments for atheism were influencing members of her court, and so Euler was asked to confront the Frenchman. Diderot was informed that a learned mathematician had produced a proof of the existence of God: he agreed to view the proof as it was presented in court. Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced this non-sequitur: "Sir, , hence God exists—reply!" Diderot, to whom (says the story) all mathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court. Embarrassed, he asked to leave Russia, a request that was graciously granted by the Empress. However amusing the anecdote may be, it is apocryphal, given that Diderot himself did research in mathematics.^[51] The legend was apparently first told by Dieudonné Thiébault^[52] with significant embellishment by Augustus De Morgan.^[53][54]

Commemorations

Euler was featured on the sixth series of the Swiss 10-franc banknote and on numerous Swiss, German, and Russian postage stamps. The asteroid 2002 Euler was named in his honor. He is also commemorated by the Lutheran Church on their Calendar of Saints on 24 May—he was a devout Christian (and believer in biblical inerrancy) who wrote apologetics and argued forcefully against the prominent atheists of his time.^[49]

On 15 April 2013, Euler's 306th birthday was celebrated with a Google Doodle.^[55]

Selected bibliography

The title page of Euler's Methodus inveniendi lineas curvas.

Euler has an extensive bibliography. His best-known books include:

• Elements of Algebra. This elementary algebra text starts with a discussion of the nature of numbers and gives a comprehensive introduction to algebra, including formulae for solutions of polynomial equations.
• Introductio in analysin infinitorum (1748). English translation Introduction to Analysis of the Infinite by John Blanton (Book I, ISBN 0-387-96824-5, Springer-Verlag 1988; Book II, ISBN 0-387-97132-7, Springer-Verlag 1989).
• Two influential textbooks on calculus: Institutiones calculi differentialis (1755) and Institutionum calculi integralis (1768–1770).
• Letters to a German Princess (1768–1772).
• Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti (1744). The Latin title translates as a method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense.^[56]

A definitive collection of Euler's works, entitled Opera Omnia, has been published since 1911 by the Euler Commission of the Swiss Academy of Sciences. A complete chronological list of Euler's works is available at the following page: The Eneström Index (PDF).