Diatomic molecule

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A space-filling model of the diatomic molecule dinitrogen, N₂.

Diatomic molecules are molecules formed of 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 composed almost completely (99%) of diatomic molecules which are oxygen (O₂) (21%) and nitrogen (N₂) (78%). The remaining 1% is predominantly argon (0.9340%)

Oxygen can also exist as the triatomic molecule ozone (O₃), for example in the ozone layer.

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).

Other elements exist which form diatomic molecules but with high instability and reactivity. An example is diphosphorus.

1 Energy levels
2 References

[edit] Energy levels

A common approximate model of a diatomic molecule is that of a dumbbell - that is two point masses (the two atoms) connected by a massless spring.

The various motions of the molecule can be broken down into three categories.

The overall (translational) motion of the molecule
The rotational motion of the molecule
The vibration of the molecule along the "spring" connecting them.

[edit] Translational

The translational energy of the molecule is simply given by:

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

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

[edit] Rotational

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

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

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

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

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

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

[edit] 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:

$E_{vib} = \left(n+\frac{1}{2} \right)hf \ \ \ \ \ n=0,1,2,... \,$

where

n is an integer

h is Plank's constant and

f is the frequency of the vibration.

[edit] Comparison between rotation and vibration

The lowest rotational energy level is when $l = 0$ . The next highest energy level ( $l = 1$ ) of O₂, has an energy of roughly:

$E_{rot,1} \,$	$= \frac{\hbar^2}{2 m_{O_{2}} r_{0}^2} \,$
	$\approx \frac{\left(1.05 \times 10^{-34} \ \mathrm{J\cdot s} \right)^2}{2 \left(27 \times 10^{-27} \ \mathrm{kg} \right) \left(10^{-10} \ \mathrm{m} \right)^2} \,$
	$\approx 2 \times 10^{-23} \ \mathrm{J} \,$

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:

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

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

[edit] References

Hyperphysics - Rotational Spectra of Rigid Rotor Molecules
Hyperphysics - Quantum Harmonic Oscillator
Tipler, Paul (1998). Physics For Scientists and Engineers : Vol. 1 (4th ed.). W. H. Freeman. ISBN 1-57259-491-8.

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Category: Molecules