2011-04-25

Homonuclear diatomic molecule

Diatomic molecule

Diatomic molecules are molecules composed only of two atoms, of either the same or different chemical elements. The prefix di- means two in Greek. Common diatomic molecules are hydrogen (H2), nitrogen (N2), oxygen (O2), and carbon monoxide (CO). Seven elements exist in the diatomic state in the liquid and solid forms: H2, N2, O2, F2, Cl2, Br2, and I2. Most elements (and many chemical compounds) aside from these form diatomic molecules when evaporated, although at very high temperatures, all materials disintegrate into atoms. The noble gases do not form diatomic molecules.

Occurrence

Hundreds of diatomic molecules have been characterized in the terrestrial environment, laboratory, and interstellar medium. About 99% of the Earth's atmosphere is composed of diatomic molecules, specifically oxygen and nitrogen at 21 and 78%, respectively. 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 diatomic molecule seen in nature. The interstellar medium is, indeed, dominated by hydrogen atoms.

Elements that consist of diatomic molecules, under typical laboratory conditions of 1 bar and 25 °C, include hydrogen (H2), nitrogen (N2), oxygen (O2), and the halogens. Again, many other diatomics are possible and form when elements are evaporated, but these diatomic species repolymerize at lower temperatures. For example, heating ("cracking") elemental phosphorus gives diphosphorus.

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 heteronuclear, such as with CO or NO. The bond in a homonuclear diatomic molecule is non-polar and covalent. In most diatomic molecules, the elements are nonidentical. Prominent examples include carbon monoxide, nitric oxide, and hydrogen chloride, but other important examples include gaseous MgO, SiO, and many other species not normally considered diatomic because they polymerize at room temperature.

Molecular geometry

Diatomic molecules cannot have any geometry but linear, as any two points always lie in a line. This is the simplest spatial arrangement of atoms after the sphericity of single atoms.

Historical significance

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.

Energy levels

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.

Translational energies

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

E_{trans}=\frac{1}{2}mv^2

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

Rotational energies

Classically, the kinetic energy of rotation is

E_{rot} = \frac{L^2}{2 I} \,
where
L \, is the angular momentum
I \, 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

L^2 = l(l+1) \hbar^2 \,
where l is a non-negative integer and \hbar is the reduced Planck constant.

Also, for a diatomic molecule the moment of inertia is

I = \mu r_{0}^2 \,
where
\mu \, is the reduced mass of the molecule and
r_{0} \, 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:

E_{rot} = \frac{l(l+1) \hbar^2}{2 \mu r_{0}^2} \ \ \ \ \ l=0,1,2,... \,

Vibrational energies

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:

E_{vib} = \left(n+\frac{1}{2} \right)\hbar \omega \ \ \ \ \ n=0,1,2,.... \,
where
n is an integer
\hbar is the reduced Planck constant and
ω is the angular frequency of the vibration.

Comparison between rotational and vibrational energy spacings

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

E_{rot,1} \,= \frac{\hbar^2}{2 m_{O_{2}} r_{0}^2} \,
\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} \,
\approx 2 \times 10^{-23} \ \mathrm{J}. \,

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 n = 0, and a typical vibration frequency is 5 x 10 Hz. Doing a calculation similar to that above gives:

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

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

See also

Notes and references

External links

  • 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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