Diatomic molecule

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## Contents

* 1 Overview
* 2 Description and occurrence in nature
* 3 Molecular Geometry
* 4 Historical significance
* 5 Energy levels
* 5.1 Translational energies
* 5.2 Rotational energies
* 5.3 Vibrational energies
* 5.4 Comparison between rotational and vibrational energy spacings
* 6 Further reading
* 7 See also
* 8 Notes and references
* 9 External links

## Overview

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Diatomic molecules are molecules made only of two atoms, of either the same or different chemical elements. The prefix di- means two in Greek.

## Description and occurrence in nature

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Huber and Herzberg's book, Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules[1], lists hundreds of diatomic molecules, some which have been detected spectroscopically in interstellar space. However, few diatomics are found to occur naturally on Earth outside of laboratories. About 99% of the Earth's atmosphere is composed of diatomic molecules, specifically  oxygen O2 (21%) and nitrogen N2 (78%), with the remaining 1% being mostly argon (0.9340%). The natural abundance of  hydrogen (H2) in the Earth's atmosphere is only on the order of parts per million, but H2 is, in fact, the most abundant molecule seen in nature, dominating the composition of stars.

Elements that consist of diatomic molecules, under typical laboratory conditions of 1 bar and 25 oC, include hydrogen (H2), nitrogen (N2), oxygen (O2), and the halogens: fluorine (F2), chlorine (Cl2), bromine (Br2), iodine (I2), and, perhaps, astatine (At2).[2] Again note that many other diatomics are possible, such as metals heated to their gaseous states. Also, many diatomic molecules are unstable and highly reactive, such as diphosphorus. A few compounds are made of diatomic molecules, including CO and HBr.

If a diatomic molecule consists of two atoms of the same element, such as H2 and O2, then it is said to be homonuclear, but otherwise it is said to heteronuclear, such as with CO or NO. The bond in a homonuclear diatomic molecule is non-polar and fully covalent.

## Molecular Geometry

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The diatomic molecule molecular geometry is the simplest spatial arrangement of atoms.[3]. This configuration is more commonly referred to in the VSEPR theory as the AX1E*. Unlike other systems in VSEPR, there is no central atom.[4][5]

## Historical significance

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Diatomic elements played an important role in the elucidation of the concepts of element, atom, and molecule in the 19th century, because some of the most common elements, such as hydrogen, oxygen, and nitrogen, occur as diatomic molecules. John Dalton's original atomic hypothesis assumed that all elements were monatomic and that the atoms in compounds would normally have the simplest atomic ratios with respect to one another. For example, Dalton assumed that water's formula was HO, giving the atomic weight of oxygen as 8 times that of hydrogen, instead of the modern value of about 16. As a consequence, confusion existed regarding atomic weights and molecular formulas for about half a century.

As early as 1805, Gay-Lussac and von Humboldt showed that water is formed of two volumes of hydrogen and one volume of oxygen, and by 1811 Amedeo Avogadro had arrived at the correct interpretation of water's composition, based on what is now called Avogadro's law and the assumption of diatomic elemental molecules. However, these results were mostly ignored until 1860. Part of this rejection was due to the belief that atoms of one element would have no chemical affinity towards atoms of the same element, and part was due to apparent exceptions to Avogadro's law that were not explained until later in terms of dissociating molecules.

At the 1860 Karlsruhe Congress on atomic weights, Cannizzaro resurrected Avogadro's ideas and used them to produce a consistent table of atomic weights, which mostly agree with modern values. These weights were an important pre-requisite for the discovery of the periodic law by Dmitri Mendeleev and Lothar Meyer.[6]

## Energy levels

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It is convenient, and common, to represent a diatomic molecule as two point masses (the two atoms) connected by a massless spring. The energies involved in the various motions of the molecule can then be broken down into three categories.

* The translational energies
* The rotational energies
* The vibrational energies

### Translational energies

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The translational energy of the molecule is simply given by the kinetic energy expression:

<math>E_{trans}=\frac{1}{2}mv^2</math>

where m is the mass of the molecule and v is its velocity.

### Rotational energies

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Classically, the kinetic energy of rotation is

<math>E_{rot} = \frac{L^2}{2 I} \,</math>
where

<math>L \,</math> is the angular momentum
<math>I \,</math> is the moment of inertia of the molecule

For microscopic, atomic-level systems like a molecule, angular momentum can only have specific discrete values given by

<math>L^2 = l(l+1) \hbar^2 \,</math>
where l is a positive integer and <math>\hbar</math> is Planck's reduced constant.

Also, for a diatomic molecule the moment of inertia is

<math>I = \mu r_{0}^2 \,</math>
where

<math>\mu \,</math> is the reduced mass of the molecule and
<math>r_{0} \,</math> is the average distance between the two atoms in the molecule.

So, substituting the angular momentum and moment of inertia into Erot, the rotational energy levels of a diatomic molecule are given by:

<math style="border:2px solid #ccccff">E_{rot} = \frac{l(l+1) \hbar^2}{2 \mu r_{0}^2} \ \ \ \ \ l=0,1,2,... \,</math>

### Vibrational energies

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Another way a diatomic molecule can move is to have each atom oscillate - or vibrate \- along a line (the bond) connecting the two atoms. The vibrational energy is approximately that of a quantum harmonic oscillator:

<math style="border:2px solid #ccccff">E_{vib} = \left(n+\frac{1}{2} \right)hf \ \ \ \ \ n=0,1,2,... \,</math>
where

n is an integer
h is Planck's constant and
f is the frequency of the vibration.

### Comparison between rotational and vibrational energy spacings

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The lowest rotational energy level of a diatomic molecule occurs for <math>l=0</math> and gives Erot = 0. For O2, the next highest quantum level (<math>l=1</math>) has an energy of roughly:

<math>E_{rot,1} \,</math> | <math>= \frac{\hbar^2}{2 m_{O_{2}} r_{0}^2} \,</math>

|

| <math>\approx \frac{\left(1.05 \times 10^{-34} \ \mathrm{J\cdot s} \right)^2}{2 \left(27 \times 10^{-27} \ \mathrm{kg} \right) \left(10^{-10} \ \mathrm{m} \right)^2} \,</math>
| <math>\approx 2 \times 10^{-23} \ \mathrm{J} \,</math>

This spacing between the lowest two rotational energy levels of O2 is comparable to that of a photon in the microwave region of the electromagnetic spectrum.

The lowest vibrational energy level occurs for <math>n=0</math>, and a typical vibration frequency is 5 x 1013 Hz. Doing a calculation similar to that above gives:

<math>E_{vib,0} \approx 3 \times 10^{-21} \ \mathrm{J} \,</math>.

So the spacing, and the energy of a typical spectroscopic transition, between vibrational energy levels is about 100 times greater than that of a typical transition between rotational energy levels.

## Further reading

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* Huber, K. P. and Herzberg, G. (1979). Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules. New York: Van Nostrand: Reinhold.CS1 maint: Multiple names: authors list (link)
* Tipler, Paul (1998). Physics For Scientists and Engineers : Vol. 1 (4th ed.). W. H. Freeman. ISBN 1-57259-491-8.

## See also

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* AXE method

## Notes and references

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1. ↑ Huber, K. P. and Herzberg, G. (1979). Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules. New York: Van Nostrand: Reinhold.CS1 maint: Multiple names: authors list (link)
2. ↑ Astatine is so rare in nature, with its most stable isotope having a half-life of only 8.3 hours, that it is usually not considered in discussions of this subject. For details on astatine, see Emsley, J. (1989). The Elements. Oxford: Clarendon Press. pp. 22–23.
3. ↑ "VSEPR - A Summary". University of Berkeley College of Chemistry. 20 Jan 2008. http://mc2.cchem.berkeley.edu/VSEPR/
4. ↑ "Shapes of Molecules". 20 Jan 2008. http://www.ausetute.com.au/shapemol.html
5. ↑ "VSEPR - A Summary". University of Berkeley College of Chemistry. 20 Jan 2008. http://mc2.cchem.berkeley.edu/VSEPR/
6. ↑ Ihde, Aaron J. (1961). "The Karlsruhe Congress: A centennial retrospective". Journal of Chemical Education. 38: 83–86. Retrieved 2007-08-24.

## External links

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* Hyperphysics \- Rotational Spectra of Rigid Rotor Molecules
* Hyperphysics \- Quantum Harmonic Oscillator
* 3D Chem \- Chemistry, Structures, and 3D Molecules
* IUMSC \- Indiana University Molecular Structure Center

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