The infrared spectrum and other things

I wanted to use the ammonia infrared spectrum as an example of the application of group theory to physics, but along the way I assembled some ammoniacal lore that may be interesting. The group theory follows this.

Ammonia Lore

Ammonia, NH3, is a colorless gas under standard conditions, with a sharp, acrid odor. Its molecular weight of 17 makes it a good deal lighter than air, M = 29, so it tends to rise when released. Under 1 atm, it melts at -77.7°C, and boils at -33.4°C. Its critical temperature is 132.4°C, so it can be liquefied at room temperature by pressure alone. On evaporating, it absorbs 327 cal/g, which is rather a lot, so it makes a good refrigerant. It was widely used for this purpose until the introduction of Freon (chlorinated hydrocarbon) refrigerants which have done so much for the ozone layer.

Ammonia is remarkably soluble in water. Under standard conditions, the saturated solution is about 30% ammonia. By heating the solution, ammonia is expelled, and this is a good source of small amounts of ammonia for experiments. The solution feels soapy (because it dissolves skin) and is a powerful cleaning agent. A little added to window-cleaning solution prevents streaking. Most of the dissolved ammonia is still NH3, but a little combines with the water to form NH4OH, ammonium hydroxide, which partially dissociates into NH4+ and OH-, so the solution becomes quite basic. Liquid ammonia, not the water solution, dissolves sodium, potassium, calcium and such to make bluish, metallic solutions that are shiny and electrically conductive, but not very stable. Liquid ammonia dissociates slightly into NH2-, amide ion, and NH4+, ammonium ion, just as water dissociates into OH- and H3O+.

Nitrogen is an essential element for plant growth. Since intensive agriculture depletes this element in the soil, it must be added as a fertilizer, or otherwise supplied, to maintain fertility. Plants cannot use elemental nitrogen, N2, which makes up 78% of the atmosphere, but require it as nitrate, NO3-, or similar combined form. Bacteria in nodules on the roots of legumes can "fix" atmospheric nitrogen to nitrates. Nitrogen is often added in the form of ammonia itself ("anhydrous ammonia"), or its ammonium compounds. Soil bacteria then convert this to nitrite, NO2-, which is then oxidized to nitrate. Ammonium nitrate, NH4NO3, contains nitrogen in both forms.

If you remove one electron from nitrogen, the N+ ion has the electron configuration of C, and its valency can be satisfied by four protons in a similar way, making NH4+, the ammonium ion. This ion forms many salts, mostly white powders, all of which are soluble in water. Smelling salts is ammonium carbonate, (NH4)2CO3, which decomposes readily into ammonia, water and carbon dioxide. It is kept in a vial, a whiff of which opens the eyes. Ammonium salts all evolve ammonia on heating, more or less readily. Ammonium chloride, NH4Cl is the white powder that seemed to cover everything with a thin film in old Chemistry buildings. This is sal ammoniac, formed from the vapors of the qualitative analysis laboratory with its ammonia and hydrochloric acid.

The name ammonia comes from Jupiter Ammon, whose temple in Libya was famous for producing sal ammoniac from camel dung, noted from the 8th century. A more famous temple of Amon-Ra at Thebes is sometimes mentioned, but they probably did not make sal ammoniac there. Ammonites are the fossils that look like ram's horns, which Amon-Ra wore. Ammonites are also the people from trans-Jordan whose capital was at present-day Amman. Neither of these seem to have anything to do with ammonia.

A product of protein metabolism is urea, CO(NH2)2. This was the first "organic" compound artificially synthesized, by Wöhler in 1828. Prior to that, organic compounds were regarded as requiring the principle of life for their synthesis. It probably occurs significantly in camel dung, since camels are noted water conservers. The rest of us excrete urea in urine. Heated, it gives off ammonia by pyrolysis. In fact, the reaction of ammonia and carbon dioxide to form urea and water is reversible, pressure favoring the production of urea, heat the production of ammonia. This reaction is used commercially to make urea fertilizer from ammonia. An early source of ammonia was pyrolysis of hoofs and horns, giving spirits of hartshorn. When coal is distilled to make coke and town gas, considerable ammonia results, which was captured by sulphuric acid to make ammonium sulphate, (NH4)2SO4, a nitrogenous fertilizer.

Most nitrogen is "fixed" by the direct combination of atmospheric nitrogen with hydrogen by the Haber process. Low temperatures and high pressures increase the yield of ammonia. The low temperatures (500°C) mean that a catalyst is necessary to increase the reaction rate. Finely divided iron, with some activators, is used. In the Cyanamide process, lime and coke are cooked to form calcium carbide, which reacts with elemental nitrogen to form calcum cyanimide, which then is hydrolyzed (combined with water) to form calcium carbonate and ammonia. The electric arc process combines atmospheric nitrogen and oxygen directly to nitrogen oxides. Finally, the Serpek process for the purification of aluminum ores has ammonia as a by-product.

In addition to NH3, there is also an HN3, hydrazoic acid. (Nitrogen is called Azote, "lifeless," in French because of its reluctance to combine chemically when in the form N2.) Lead azide, Pb(N3)2, explodes when shocked, and is used as a detonator. The fertilizer ammonium nitrate is also an explosive, especially when mixed with hydrocarbonaceous material. It is not very sensitive, so it is relatively safe. A shipload of it blew up in Texas City some time ago.

Infrared Spectrum

Molecules absorb and emit electromagnetic radiation in wide areas of the spectrum. If electrons change state, the radiation may be in the visible region. Molecular ultraviolet spectra are rather rare, since molecules fall apart at these high energies. Changes in vibrational states are associated with infrared wavelengths, and changes in rotational states with the far infrared. There are even finer energy differences that cause spectra even in the radiofrequency region. All of these generally consist of a great number of lines, sometimes not resolved individually, forming bands and such.

Infrared spectra are a valuable tool for determining the structure of molecules. The wavelengths range from the end of the visible at about 1 μ wavelength to the limits of the dispersing elements in the spectroscopes. An infrared band is simpler than the band spectra in the visible, but still rather complex, consisting of several series of lines corresponding to transitions between different rotational states. Two methods are generally used, absorption spectra that study the transitions from the ground state to excited states, and Raman spectra that studies the changes in wavelength in scattered radiation. Raman spectroscopy can be done in the visible region with its more convenient experimental conditions, and with the powerful beams of lasers.

Quantum mechanics is necessary for the understanding of molecular spectra, which it perfectly explains. Another paper explains the relation of group theory to quantum mechanics. Symmetry is a powerful tool in the quantum mechanics of molecules, and the ammonia molecule furnishes a good example. Let us consider what infrared and Raman spectra are to be expected if the molecule is a symmetrical pyramid, which is indeed the case. The other possibilities, which have been eliminated by experiment, can be considered in the same way. You will have to know character analysis to appreciate the discussion.

C3v E 2C basis functions
A1 1 1 1 Tz, x2+y2,z2
A2 1 1 -1 Rz
E 2 -1 0 (Tx,Ty) (Rx,Ry) (x2 - y2, xy) (xz,yz)
The symmetrical pyramid has the symmetry group C3v, whose character table is shown at the left. T and R show the representations to which components of the translation and rotation displacements belong; these are vectors and axial vectors, respectively. T also shows the representations of the dipole moment operator which produces the infrared spectrum. The quadratic functions shown transform like the molecular polarizability, the operator which produces the Raman spectrum.

We assign three displacement coordinates to each atom, 12 in all for the four atoms. The first thing to do is to find the characters of this representation. The character for E is 12, since the identity transforms each coordinate into itself. The rotations about the axis leave only the displacements on the nitrogen in the same place, and the character is the same as that of the three T components, or 1 - 1 = 0. Reflections in a vertical plane leave the nitrogen and one hydrogen unmoved, and the character is easily seen to be 2 - 1 = 1 for each atom. Therefore, the characters of the reducible representation of the displacements is 12, 0, 2. This must include the representations of the translation and rotation of the molecule as a whole, A1 + A2 + 2E. Therefore, we subtract the characters 6, 0, 0 to find the character of the vibrations, 6, 0, 2. By character analysis, we find that this gives 2A1 + 2E. Ammonia, therefore, should exhibit four fundamentals, all active in both infrared and Raman spectra. This is exactly what is observed. The Raman spectra of the E fundamentals ought to be faint, and they were not observed (or were not until lasers came in). If the ammonia molecule were planar, two more fundamentals would be expected, and they are not observed.

Herzfeld gives the four modes as follows. There is a very strong band at 1627.5 cm-1 (infrared spectroscopists use the reciprocal of the wavelength, since it is proportional to the frequency and the quantum energy), about 6.1 μ, and is a so-called perpendicular band, which would be expected from the x and y components of the dipole moment. This is one of the doubly-degenerate E fundamentals, a symmetric bending of two of the hydrogens to or away from each other. The asymmetric bending is of higher frequency, 3414 cm-1, and difficult to observe. These are the two E modes. There is a strong parallel band at 931.58 and 968.08 cm-1, about 10.6 μ corresponding to an A1 representation. This band is double, and the reason is curious. The ammonia molecule can turn itself inside-out; that is, the nitrogen can pass through the plane of the hydrogens. This isn't easy, but the nitrogen can tunnel through, and the doubling is the result. The states divide into those symmetrical with respect to this inversion, and those that are antisymmetrical (change sign). The selection rules on the rotational transitions make the band separations the sum of the inversion splittings in the two cases. In the Raman spectrum, the separation is the difference of the splittings. The Raman bands are observed at 934.0 and 964.3 cm-1. Finally, there is a strong band at 3335.9 and 3337.5 cm-1, and a Raman shift at 3334.2 cm-1 (about 3.0 μ) corresponding to the other A1 fundamental. In this mode, the bond lengths lengthen and shorten symmetrically. The two A modes can be called bending and stretching, respectively.

ND3, with the heavier deuterium substituted for the protons, gives somewhat different (lower) frequencies, and the shifts can be used to nail down the identification of the vibrational frequencies, confirming the conclusion that ammonia is a symmetrical pyramid. The inversion doubling is a very interesting phenomenon. It turns out to be possible to separate molecules in even and odd inversion states, and this led to the ammonia maser, the first of its kind. Although we can form a good picture of ammonia as if it were a macroscopic object, try to picture it with the nitrogen partly on both sides of the hydrogens!


  1. D. Schonland, Molecular Symmetry (London: D. Van Nostrand and Co., 1965), Chapter 8.
  2. G. Herzberg, Infrared and Raman Spectra (New York: D. Van Nostrand and Co., 1945), pp. 256-257 and pp. 294-297.

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Composed by J. B. Calvert
Created 23 November 2000
Last revised 15 March 2004