Svante Arrhenius

How to use this course

Svante Arrhenius is one of the founders of physical chemistry. Chemists know him for two things: the claim that salts, acids and bases split into ions when they dissolve, and the exponential law that governs how reaction rates depend on temperature. Climate scientists know him for something else: in 1896 he made the first quantitative estimate of how much the Earth would warm or cool if the carbon dioxide in its air changed. He also wrote on cosmic physics, immunology and the origin of life.

Part I follows his formation, from a provincial Swedish childhood to the thesis his examiners nearly failed and the years of travel that made it famous. Part II treats each of his main scientific contributions with modern derivations. Part III covers his years as a public scientist in Stockholm, the Nobel Institute, and his legacy.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Numerical data are rounded modern values.

Timeline

Vik and Uppsala, 1859–1881

Svante August Arrhenius was born on 19 February 1859 on the estate of Vik, near Uppsala in Sweden. His father, Svante Gustav Arrhenius, was a land surveyor who later worked for Uppsala University as a manager of its properties; his mother was Carolina Thunberg. The family moved into Uppsala when Svante was a small child.

He was quick at arithmetic as a boy and did well at the Uppsala cathedral school, which he left in 1876 as one of the youngest in his class. He entered Uppsala University the same year, studying mathematics, physics and chemistry, and took his candidate’s degree after only a few years. Uppsala was then a conservative place for the physical sciences. The physics professor, Tobias Thalén, was a spectroscopist, and the young Arrhenius found little support there for the electrical problem he wanted to study.

In 1881 he left for Stockholm to work under Erik Edlund at the Physical Institute of the Royal Swedish Academy of Sciences, while remaining registered at Uppsala for his doctorate. Edlund was interested in electricity and gave him freedom and apparatus.

Stockholm and the fourth-class thesis, 1881–1884

In Stockholm Arrhenius measured the electrical conductivity of very dilute solutions of salts, acids and bases. The method, using alternating current to avoid polarisation of the electrodes, had been developed by Friedrich Kohlrausch. Arrhenius studied how conductivity per unit of dissolved substance changes with dilution, for many substances, and noticed that it rose toward a limit.

He interpreted this through a distinction between an “active” and an “inactive” part of the electrolyte. Only the active part carried current, and the active fraction grew with dilution. He also found that the more conductive an acid was, the stronger it was as a chemical agent. The thesis, in two parts in French, was submitted to Uppsala in 1884 as Recherches sur la conductibilité galvanique des électrolytes.

The examiners did not know what to make of it. The thesis received the fourth-class grade, the lowest pass, and the defence a slightly better mark. With such a grade an academic career in Sweden was unlikely. In the same work he also put forward definitions of acids and bases based on the ions they produce in water (Lesson 05).

Ostwald’s visit

Arrhenius sent copies of the thesis to leading physical chemists abroad. Wilhelm Ostwald, then a young professor at the Polytechnic in Riga, had been measuring the catalytic strength of acids and saw at once that the conductivity data matched his own rankings. In 1884 he travelled to Uppsala to meet Arrhenius and offered him a position in Riga. Arrhenius stayed in Sweden, but the visit raised his standing at home, and Uppsala appointed him a docent in physical chemistry at the end of 1884.

The travelling years and the Ionists, 1886–1891

A travel grant from the Swedish Academy of Sciences gave Arrhenius five years in the best laboratories in Europe. In 1886 he worked with Ostwald in Riga and with Kohlrausch in Würzburg. In 1887 he was with Ludwig Boltzmann in Graz, and in 1888 with Jacobus van ’t Hoff in Amsterdam. In 1889 he was again with Ostwald, now in Leipzig.

Van ’t Hoff’s theory of dilute solutions, published in 1886, was the key. Van ’t Hoff showed that dissolved substances obey an osmotic analogue of the ideal gas law, except that salts, strong acids and bases behave as though they contained more particles than their formulas suggest. He corrected for this with an empirical factor \(i\). Arrhenius saw that his “active” molecules were molecules dissociated into ions, and that the same degree of dissociation should explain both the conductivity and van ’t Hoff’s factor. In 1887 he published the theory of electrolytic dissociation in the first volume of the Zeitschrift für physikalische Chemie, which Ostwald and van ’t Hoff had just founded.

The three men became known as the Ionists. Their claim that sodium chloride in water exists largely as free sodium and chloride ions seemed absurd to many chemists, who asked why sodium did not react with the water. Critics included Dmitri Mendeleev, who preferred a theory of hydrates, and Henry Armstrong in Britain. The answer, that an ion is a charged species with properties different from the neutral atom, took years to win acceptance. In Leipzig in 1889 Arrhenius also wrote his paper on the rate of inversion of cane sugar by acids, which introduced the temperature law now named after him (Lesson 06).

Electrolytic conductivity

For the general reader. Pure water barely conducts electricity; salty water conducts well. Arrhenius’s thesis measured how well, for many dissolved substances and many concentrations. The pattern in those numbers, that conductivity per dissolved molecule climbs toward a ceiling as a solution is diluted, became the evidence for ions.

Molar conductivity

A cell of electrode area \(A\) and separation \(\ell\) filled with solution has resistance \(R\). The conductivity is \(\kappa=\ell/(RA)\). For an electrolyte at concentration \(c\), the molar conductivity is

If ions moved independently and all molecules were ionised, \(\Lambda_m\) would not depend on \(c\). In fact it rises on dilution toward a limit \(\Lambda_m^\circ\).

Arrhenius’s interpretation and Ostwald’s law

Arrhenius took the ratio \(\alpha=\Lambda_m/\Lambda_m^\circ\) as the fraction of the electrolyte dissociated into ions. For a weak acid \(\mathrm{HA}\rightleftharpoons\mathrm{H^+ + A^-}\) at total concentration \(c\), mass action gives

The law works well for weak acids such as acetic acid. It fails for strong electrolytes such as sodium chloride, whose conductivity falls off as \(\sqrt c\). Those are in fact fully dissociated; the fall is due to interactions between ions, explained by Peter Debye and Erich Hückel in 1923 and Lars Onsager in 1927. Arrhenius’s picture is right for weak electrolytes and needs that correction for strong ones.

Ionic dissociation, osmotic pressure and acids

For the general reader. A dissolved substance lowers the freezing point of water and creates osmotic pressure, in proportion to the number of dissolved particles, not their kind. Salt does this almost twice as strongly as sugar, mole for mole. Arrhenius’s explanation was that each unit of salt becomes two particles, a sodium ion and a chloride ion. The same idea defined acids as sources of hydrogen ions and bases as sources of hydroxide ions.

Van ’t Hoff’s factor

The dissociation hypothesis

Arrhenius compared the two values for dozens of substances and found good agreement, which was the main evidence for his theory. For strong electrolytes the agreement is partly a coincidence: both quantities are reduced by interionic forces, and the modern description uses full dissociation with activity coefficients.

Acids and bases

In the Arrhenius definition, an acid increases the concentration of \(\mathrm{H^+}\) in water and a base increases the concentration of \(\mathrm{OH^-}\). Neutralisation is \(\mathrm{H^+ + OH^-\to H_2O}\), whatever the acid and base. This explains why the heat of neutralisation of any strong acid by any strong base is nearly the same, about 57 kJ per mole of water. The definition was later broadened by Johannes Brønsted and Thomas Lowry (1923), who defined acids as proton donors, and by Gilbert Lewis, who defined them as electron-pair acceptors. The Arrhenius definition covers only aqueous solutions.

The Arrhenius equation

For the general reader. Food spoils faster in summer, and many reactions in the laboratory roughly double in speed for every ten degrees of warming. Arrhenius explained why the effect is so large. Only molecules with an unusually high energy can react, and the share of such molecules rises steeply, exponentially, with temperature.

The law

In 1884 van ’t Hoff showed that an equilibrium constant obeys \(\frac{d\ln K}{dT}=\frac{\Delta H^\circ}{RT^2}\), and suggested a similar form for rate constants. Arrhenius, studying the acid-catalysed inversion of cane sugar in 1889, gave the rule a physical interpretation.

Origin in the Boltzmann factor

Arrhenius supposed that reactant molecules exist in equilibrium between ordinary and “active” forms, with only the active ones reacting, and applied van ’t Hoff’s equation to that equilibrium. In modern language, if \(E_a\) is a barrier, the fraction of molecules or collisions with energy above it carries a Boltzmann factor. For collisions in a gas, counting the energy of relative motion along the line of centres, the fraction exceeding \(E_0\) is exactly \(e^{-E_0/k_BT}\), giving collision-theory rates \(k=\sigma\bar v_{\text{rel}}N_A\,e^{-E_0/RT}\) with a weakly temperature-dependent prefactor.

Transition-state theory (Henry Eyring, Meredith Evans and Michael Polanyi, 1935) gives

which has Arrhenius form with \(E_a=\Delta H^\ddagger+RT\) for a unimolecular reaction in solution. The Arrhenius equation is now best viewed as an empirical law with a clear but approximate physical basis.

Radiation balance and the greenhouse effect

For the general reader. The Earth is warmed by sunlight and cools by giving off infrared radiation to space. Some gases in the air, water vapour and carbon dioxide above all, absorb that infrared and send part of it back down. The surface therefore settles at a higher temperature than it would without them. Arrhenius did not discover this; he was the first to calculate how much it changes when carbon dioxide changes.

Predecessors

Joseph Fourier argued in the 1820s that the atmosphere keeps the Earth warmer than it would otherwise be. Claude Pouillet in 1838 suggested that water vapour and carbon dioxide were responsible. In 1859 John Tyndall measured in the laboratory that nitrogen and oxygen are nearly transparent to infrared while water vapour and carbon dioxide absorb strongly, and suggested that changes in these gases could explain past changes of climate. The Stefan–Boltzmann law, \(j=\sigma T^4\), established by Josef Stefan in 1879 and derived by Boltzmann in 1884, gave Arrhenius the radiation law he needed.

Effective temperature

The Earth intercepts sunlight over its cross-section \(\pi R^2\) and radiates from its whole surface \(4\pi R^2\). With solar constant \(S\) and planetary albedo \(a\),

A single absorbing layer

Derivation. At the surface: \(\tfrac{S(1-a)}4+\varepsilon\sigma T_a^4=\sigma T_s^4\). For the layer: \(\varepsilon\sigma T_s^4=2\varepsilon\sigma T_a^4\). Substituting \(\sigma T_a^4=\tfrac12\sigma T_s^4\) gives \(\sigma T_s^4(1-\varepsilon/2)=\sigma T_e^4\). A larger \(\varepsilon\), meaning more absorbing gas, warms the surface. Arrhenius’s 1896 model has this structure, with an atmospheric absorption that he computed band by band and an atmospheric emissivity of about 0.7–0.8.

The 1896 carbon-dioxide calculation

For the general reader. Arrhenius wanted to explain the ice ages. Could a fall in carbon dioxide have cooled Europe enough to bury it in ice? He spent months computing, by hand, the absorption of infrared radiation by carbon dioxide and water vapour at each latitude and season. He found that cutting carbon dioxide by about 40% would bring ice-age cold, and that doubling it would warm the Earth by several degrees. The method was rough; the order of magnitude has stood.

Method and data

Arrhenius presented the paper to the Swedish Academy of Sciences on 11 December 1895, and an English version appeared in the Philosophical Magazine in April 1896. For absorption he used Samuel Langley’s measurements of infrared radiation from the full Moon, observed through different thicknesses of air and water vapour as the Moon rose and set, divided into 21 wavelength groups. From these he fitted absorption coefficients for carbon dioxide and water vapour. He then computed, for latitude bands from 70° N to 60° S and for each season, the change in surface temperature when the carbon dioxide content \(K\) (with \(K=1\) the present value) was set to 0.67, 1.5, 2, 2.5 and 3. Water vapour was assumed to keep constant relative humidity, so it rose with temperature: an early form of water-vapour feedback. Cloud cover and surface albedo were fixed.

The geologist Arvid Högbom, his colleague in Stockholm, supplied estimates of the natural carbon cycle: volcanic emission, uptake by weathering and carbonate formation, and the then-small contribution of coal burning. They showed that the atmospheric reservoir is small compared with the fluxes passing through it, so large changes over geological time were plausible.

Results

Point 3 is a logarithmic law. The modern fit for the radiative forcing of carbon dioxide, from detailed line-by-line calculations, is

and the equilibrium warming is \(\Delta T=\lambda\,\Delta F\), where the climate sensitivity parameter \(\lambda\) includes feedbacks. Doubling gives \(\Delta F\approx3.7\) W m⁻². The logarithm arises because the centre of the 15 μm band is already saturated, and added gas mainly widens the band wings.

Assessment

Arrhenius’s doubling figure is higher than the modern best estimate of the equilibrium climate sensitivity, about 3 °C with a likely range of 2.5–4 °C (IPCC, 2021). Some errors in his absorption data partly cancelled. In Worlds in the Making (1906) he gave about 4 °C for a doubling. He regarded the warming from coal as slow and probably beneficial for a cold country such as Sweden, writing of future ages with more equable climates and better harvests. His work was criticised in 1900 by Knut Ångström, whose laboratory experiment suggested that carbon dioxide absorption was already saturated; that objection, widely accepted for decades, was mistaken because it ignored the upper atmosphere, where added absorbers raise the effective emission level.

Cosmic physics and panspermia

For the general reader. Arrhenius was interested in everything that physics could say about the Earth and sky: auroras, comets, the Sun, the origin of the planets. He noticed that light pushes on very small particles more strongly than the Sun’s gravity pulls on them. From this he proposed that microscopic spores might be blown from one planetary system to another, seeding life across the universe. He called the idea panspermia.

Cosmic physics

His Lehrbuch der kosmischen Physik (1903) covered astronomy, geophysics, meteorology and atmospheric electricity in one textbook. He used radiation pressure, predicted by Maxwell and measured by Pyotr Lebedev and by Ernest Nichols and Gordon Hull in 1900–1901, to explain why comet tails point away from the Sun and to propose a role for charged solar dust in the aurora.

Radiation pressure on a grain

A spherical grain of radius \(s\) and density \(\rho\) at distance \(r\) from a star of luminosity \(L\) and mass \(M\) feels radiation force \(F_r=\frac{L}{4\pi r^2c}\,Q_{\rm pr}\pi s^2\), where \(Q_{\rm pr}\) is an efficiency of order 1 for grains larger than the wavelength. Gravity is \(F_g=GM\tfrac43\pi s^3\rho/r^2\).

For grains much smaller than the wavelength of sunlight, \(Q_{\rm pr}\) falls steeply, so \(\beta\) peaks near radii of a few tenths of a micrometre. Arrhenius estimated that only particles in a narrow size window, around the size of the smallest bacterial spores, could be driven out. He proposed radiation-driven panspermia in 1903 and named and popularised it in Worlds in the Making. Earlier, in 1871, William Thomson (Kelvin) had suggested that meteorites might carry life.

Assessment

The main objection is radiation damage. Unshielded spores exposed to ultraviolet light in space experiments die within seconds to minutes, and galactic cosmic rays sterilise unshielded material over the long journeys needed between stars. Spores shielded inside rock survive much longer, which keeps alive the meteorite version of the idea (lithopanspermia), at least within one planetary system. Panspermia also moves the problem of the origin of life rather than solving it.

Physical chemistry of toxins and antitoxins

For the general reader. In the early 1900s Arrhenius turned his ideas about chemical equilibrium to immunology. Antitoxins in blood serum neutralise bacterial toxins. Arrhenius argued, against the leading immunologist Paul Ehrlich, that the binding is reversible and follows the same law of mass action as an ordinary reaction in solution. He called the new field immunochemistry.

In the early 1900s Arrhenius worked with the Danish physician Thorvald Madsen of the State Serum Institute in Copenhagen on diphtheria toxin and other toxins and lysins. He lectured on the subject at Berkeley in 1904 and published the lectures as Immunochemistry. Paul Ehrlich had interpreted irregularities in toxin–antitoxin titrations by postulating many distinct toxic components. Arrhenius and Madsen argued that a single reversible equilibrium explains much of the same data. The dispute was sharp and personal. Later immunology showed that antibody–antigen binding is indeed reversible and obeys mass action, while also confirming that antisera and toxins are heterogeneous, so each side was partly right.

Stockholm, the Nobel Prize and the Nobel Institute, 1891–1914

In 1891 Arrhenius declined a chair at Giessen and became a lecturer in physics at Stockholms Högskola, a young private institution that is now Stockholm University. He was made professor of physics in 1895 and later served as the Högskola’s rector. He was a popular teacher and a sociable presence in Stockholm science, active in its physical and chemical societies. In 1894 he married his former student Sofia Rudbeck; their son Olof was born in 1895 and the couple divorced in 1896.

In 1903 he received the Nobel Prize in Chemistry, the first Swede to win a Nobel Prize, for the services he had rendered to chemistry “by his electrolytic theory of dissociation”. He had already received the Davy Medal of the Royal Society in 1902.

In 1905 he left his professorship to become director of the new Nobel Institute for Physical Chemistry, which the Royal Swedish Academy of Sciences set up for him; it moved into its own building in 1909. The same year he married Maria Johansson, with whom he had a son and two daughters. He was a member of the Nobel Committee for Physics and influential in chemistry decisions. Historians, notably Elisabeth Crawford, have documented how he used that influence to support allies such as van ’t Hoff and Ostwald and to oppose candidates he disliked, including Walther Nernst, Paul Ehrlich and Mendeleev.

He wrote widely for the public. Världarnas utveckling (1906; Worlds in the Making, 1908) brought his cosmogony and climate ideas to a large readership, and was followed by books on the fates of the stars and on chemistry in modern life. He also served on the board of the Swedish Society for Racial Hygiene, founded in 1909, and took part in Swedish debates on eugenics, a position common among scientists of his time and now widely condemned.

Last years and legacy, 1914–1927

Arrhenius gave the Faraday Lecture of the Chemical Society in London in 1914, received the first Willard Gibbs Award in 1911 and the Franklin Medal in 1920, and held honorary degrees from many universities, including Cambridge, Oxford, Leipzig and Heidelberg. He was elected a Foreign Member of the Royal Society. His later books included Quantitative Laws in Biological Chemistry (1915), applying reaction kinetics to physiology, and his Silliman lectures at Yale, published as Theories of Solutions.

He remained director of the Nobel Institute until 1927. He fell ill that year and died in Stockholm on 2 October 1927, aged 68. He is buried in Uppsala. His descendants include several scientists: his son Olof Arrhenius worked in agricultural chemistry, and his grandson Gustaf Arrhenius was an oceanographer and geochemist.

Legacy

• Physical chemistry: ionic dissociation, with van ’t Hoff’s solution theory and Ostwald’s kinetics, founded the discipline; ions in solution are now elementary chemistry.
• Kinetics: the Arrhenius equation and the concept of activation energy are used everywhere from industrial chemistry to the shelf life of drugs and the temperature dependence of biological rates.
• Climate science: the 1896 paper is the first quantitative estimate of carbon-dioxide climate sensitivity. Revived by Guy Callendar in 1938 and by the measurements of Charles Keeling from 1958, it is now the starting point of every history of global warming.
• Astrobiology: panspermia remains a live, if minority, hypothesis, and radiation pressure on dust is basic to the study of comets and debris discs.
• Honours: the lunar crater Arrhenius and the Martian crater Arrhenius bear his name, as does the Arrhenius Laboratory at Stockholm University.

Connections

People linked to Svante Arrhenius across this collection. A name without a link has no page here yet. See Svante Arrhenius in the network of all 337 lives →

Teachers and advisors

• Ludwig Boltzmann
• Erik Edlund 1881-1884

Influenced by

• John Tyndall
• Josef Stefan
• Joseph Fourier
• William Thomson, Lord Kelvin
• Friedrich Kohlrausch 1886
• Samuel Pierpont Langley

Influenced

• Peter Debye
• Guy Stewart Callendar

Colleagues and collaborators

• Jacobus Henricus van 't Hoff 1888
• Ludwig Boltzmann 1887
• Wilhelm Ostwald 1884-1890
• Arvid Högbom 1895-1896
• Thorvald Madsen

Rivals

• Dmitri Mendeleev
• Henry Armstrong
• Knut Ångström 1900
• Paul Ehrlich
• Walther Nernst

Family

• Gustaf Arrhenius
• Olof Arrhenius

Also mentioned in this life

• Humphry Davy
• Michael Faraday
• Benjamin Franklin
• Josiah Willard Gibbs
• James Clerk Maxwell

Fields

• physical chemistry 9 lives
• electrochemistry 7 lives

On CoursesHub

• Physical Chemistry

Sources and further reading

Biography

1. Svante Arrhenius, Biographical, NobelPrize.org.
2. The Nobel Prize in Chemistry 1903, NobelPrize.org.
3. Svante Arrhenius, Wikipedia.
4. E. Crawford, Arrhenius: From Ionic Theory to the Greenhouse Effect, Science History Publications, 1996.
5. E. Crawford, The Beginnings of the Nobel Institution: The Science Prizes, 1901–1915, Cambridge University Press, 1984.
6. Science History Institute, Future calculations: the first climate change believer.

Science

1. S. Arrhenius, Recherches sur la conductibilité galvanique des électrolytes, Bihang till K. Svenska Vet.-Akad. Handlingar, 1884.
2. S. Arrhenius, Über die Dissociation der in Wasser gelösten Stoffe, Z. physik. Chem. 1 (1887) 631–648.
3. S. Arrhenius, Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren, Z. physik. Chem. 4 (1889) 226–248.
4. S. Arrhenius, On the influence of carbonic acid in the air upon the temperature of the ground, Phil. Mag. (5) 41 (1896) 237–276.
5. S. Arrhenius, Lehrbuch der kosmischen Physik, Hirzel, Leipzig, 1903.
6. S. Arrhenius, Immunochemistry, Macmillan, New York, 1907.
7. S. Arrhenius, Worlds in the Making, Harper, 1908 (Project Gutenberg).
8. J. H. van ’t Hoff, Études de dynamique chimique, Amsterdam, 1884.
9. Arrhenius equation and Panspermia, Wikipedia.
10. S. Weart, The Discovery of Global Warming: The carbon dioxide greenhouse effect, American Institute of Physics.
11. G. Myhre et al., New estimates of radiative forcing due to well mixed greenhouse gases, Geophys. Res. Lett. 25 (1998) 2715–2718.
12. IPCC, Climate Change 2021: The Physical Science Basis, Chapter 7.