Diatomic molecules are molecules formed of exactly two atoms, of the same or different chemical elements. The prefix di- means two in Greek. Diatomic elements are those that almost exclusively exist as diatomic molecules, known as homonuclear diatomic molecules in their natural elemental state when they are not chemically bonded with other elements. Examples include H₂ and O₂. Earth's atmosphere is comprised almost completely (99%) of diatomic molecules which are oxygen (O₂) (21%) and nitrogen (N₂) (78%). The remaining 1% is predominantly argon (0.9340%)

Oxygen also exists as the triatomic molecule ozone (O₃).

The diatomic elements are hydrogen, nitrogen, oxygen, and the halogens: fluorine, chlorine, bromine, iodine, and astatine. Astatine is so rare in nature (its most stable isotope has a half-life of only 8.1 hours) that it is usually not considered. Many metals are also diatomic when in their gaseous states.

The bond in a homonuclear diatomic molecule is non polar and fully covalent.

Examples of heteronuclear diatomic molecules include carbon monoxide (CO) and nitric oxide (NO).

Chemical equations

---

In chemical equations, they are important in balancing the equation properly. None of the diatomics can stand alone, they must have a pair or be in a compound. For example in

• Ca + Cl → CaCl₂

However, since chlorine is a diatomic, the actual equation should be:

• Ca + Cl₂→ CaCl₂

The equation thence is already balanced, there is no need for any coefficients, as Cl is in actuality Cl₂ if it is by itself in an equation.

Common analogy

---

A common analogy to remember the diatomic molecules is...

HON and the Halogens:

Hydrogen, Oxygen, Nitrogen (HON) and the Halogens (which are fluorine, chlorine, bromine, iodine, and astatine).

However, since astatine is so rare in nature, it is usually uncounted when remembering the analogy.

Another mnemonic commonly used is Have No Fear Of Ice Cold Beer. This is Hydrogen, Nitrogen, Fluorine, Oxygen, Iodine, Chlorine, Bromine.

Energy levels

---

A common, approximate, model of a diatomic molecule is that of a dumbell - that is, each atom is on one end of a spring or rod.

Now this dumbell molecule can only move in a few specific ways:

• It can vibrate such that the atoms oscillate between getting closer and farther from eachother.
• It can rotate or spin about some axis.

Rotational

Classically, the kinetic energy of rotation is

$E\_\{rot\}\; =\; \backslash frac\{L^2\}\{2\; I\}\; \backslash ,$

where

$L\; \backslash ,$ is the angular momentum

$I\; \backslash ,$ is the moment of inertia of the molecule

Now, for quantum systems like a molecule, angular momentum can only have specific descrete levels. So, angular momentum is given by

$L\; =\; l(l+1)\; \backslash hbar^2\; \backslash ,$

where l is some positive integer and $\backslash hbar$ is Plank's constant.

Also, the moment of inertia of this molecule is

$I\; =\; \backslash mu\; r\_\{0\}^2\; \backslash ,$

where

$\backslash mu\; \backslash ,$ is the reduced mass of the molecule and

$r\_\{0\}\; \backslash ,$ is the average distance between the two atoms in the molecule.

So, plugging in the angular momentum and moment of inertia, the rotational energy levels of a diatomic molecule are given by:

{|cellpadding="2" style="border:2px solid #ccccff" $E\_\{rot\}\; =\; \backslash frac\{l(l+1)\; \backslash hbar^2\}\{\backslash mu\; r\_\{0\}^2\}\; \backslash \; \backslash \; \backslash \; \backslash \; \backslash \; l=0,1,2,...\; \backslash ,$

Vibrational

The other way a diatomic molecule can move is to have to have each atom oscillate - or vibrate - along a line connecting them.

The energy of this vibration is exactly the same as a quantum harmonic oscillator:

{|cellpadding="2" style="border:2px solid #ccccff" $E\_\{vib\}\; =\; \backslash left(n+\backslash frac\{1\}\{2\}\; \backslash right)hf\; \backslash \; \backslash \; \backslash \; \backslash \; \backslash \; n=0,1,2,...\; \backslash ,$

where

n is some integer

h is Plank's constant and

f is the frequency of the vibration.

Comparison between rotation and vibration

The lowest rotational energy level is when $l=0$. So if we were to calculate this lowest energy for a molecule of O₂, it would be about:

{| $E\_\{rot,0\}\; \backslash ,$ $=\; \backslash frac\{\backslash hbar^2\}\{m\_\{O\_\{2\}\}\; r\_\{0\}^2\}\; \backslash ,$ $\backslash approx\; \backslash frac\{\backslash left(10^\{-34\}\; \backslash \; \backslash mathrm\{J\backslash cdot\; s\}\; \backslash right)^2\}\{\backslash left(27\; \backslash times\; 10^\{-27\}\; \backslash \; \backslash mathrm\{kg\}\; \backslash right)\; \backslash left(10^\{-10\}\; \backslash \; \backslash mathrm\{m\}\; \backslash right)^2\}\; \backslash ,$ $\backslash approx\; 4\; \backslash times\; 10^\{-23\}\; \backslash \; \backslash mathrm\{J\}\; \backslash ,$

Thus, transitions between rotational energy levels yield photons in the microwave region.

The lowest vibrational energy level is when $n=0$, and a typical vibration frequency is 5x10¹³ Hz. So, doing a similar calculation as with above gives:

$E\_\{vib,0\}\; \backslash approx\; 3\; \backslash times\; 10^\{-21\}\; \backslash \; \backslash mathrm\{J\}\; \backslash ,$.

So a typical transition between vibrational energy levels is about 100 times greater than a typical transition between rotational energy levels.

References

---

• Hyperphysics - Rotational Spectra of Rigid Rotor Molecules
• Hyperphysics - Quantum Harmonic Oscillator

Molecules

ثنائي الذرة