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

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

Contents

[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}m\left(v_x^2+v_y^2+v_z^2\right)

where m is the mass of the molecule and [vx,vy,vz] is its velocity. Each component of the velocity constitutes an independent quadratic degree of freedom so that there are a total of three translational degrees of freedom.

[edit] Rotational

For more details on this topic, see rigid rotor.

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 descrete 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,... \,

??? Show there are two degrees of freedom

[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 may be estimated by assuming that the vibrations are those of a quantum harmonic oscillator:

E_{vib} = \left(n+\frac{1}{2} \right)hf \ \ \ \ \ n=0,1,2,... \,
where
n is some integer
h is Plank's constant and
f is the frequency of the vibration.
\nu=\frac{1}{2\pi}\sqrt{\frac{k}{\mu}}

where k is force constant, μ is reduced mass, ν is frequency.

\mu=\frac{m_1m_2}{m_1+m_2}
I=m_1r_1^2+m_2r_2^2

where r1 and r2 are distances from CM.

??? Show there are two degrees of freedom

[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] Diatomic spectra

For more details on this topic, see Molecular spectra.

[HCl]

??? Show image

  • Rotational
  • Vibrational

Spectrum looks like vibrational peaks, each split into rotational peaks

[edit] Heat capacity

For more details on this topic, see Heat capacity.
Figure 1 - An idealized plot of the dimensionless specific heat of a diatomic gas as a function of temperature. Below Trot, the molecules behave as point particles with three translational degrees of freedom and no internal degrees of freedom. Between Trot and Tvib, the particle behaves as a rigid body with two rotational degrees of freedom. Above Tvib, the two vibrational degrees of freedom become active, yielding a total of seven degrees of freedom.
Figure 1 - An idealized plot of the dimensionless specific heat of a diatomic gas as a function of temperature. Below Trot, the molecules behave as point particles with three translational degrees of freedom and no internal degrees of freedom. Between Trot and Tvib, the particle behaves as a rigid body with two rotational degrees of freedom. Above Tvib, the two vibrational degrees of freedom become active, yielding a total of seven degrees of freedom.
Figure 2 - A plot of the dimensionless specific heat of a number of diatomic gases in the temperature range 200-2000 degrees K. The lightest gas is H2, while the heaviest is I2. (Adapted from Lewis and Randall)
Figure 2 - A plot of the dimensionless specific heat of a number of diatomic gases in the temperature range 200-2000 degrees K. The lightest gas is H2, while the heaviest is I2. (Adapted from Lewis and Randall)

The heat capacity of a diatomic molecule is best conceptualized in terms of the degrees of freedom of the molecule. The different degrees of freedom correspond to the different ways in which the molecule may store energy. As described above, the diatomic molecule has a total of seven degrees of freedom. Three are translational, two are rotational, and two are vibrational.

If the molecule could be described using classical mechanics, then we could use the theorem of equipartition of energy to predict that each degree of freedom would have an average energy in the amount of (1/2)kT . Our calculation of the heat content would be straightforward. With seven degrees of freedom, each molecule would be holding, on average, an energy of (7/2)kT  and the total internal energy of the diatomic gas would be (7/2)NkT  where N  is the total number of molecules. The heat capacity would then be a constant (7/2)Nk , the specific heat capacity would be (7/2)k  and the dimensionless heat capacity would be just 7/2.

In general, the various degrees of freedom cannot be considered to obey classical mechanics. The degrees of freedom will have quantized energy levels and will accept or lose energy only in certain quantized amounts. Every degree of freedom has a ground state and a first excited state, and the energy of this first excited state can be related to a critical temperature via the equipartition theorem: E=(1/2)nRT. (Actually if a number (f) of degrees of freedom are very similar, it is better to use E=(f/2)nRT.) If the temperature is well below this critical temperature, then those degrees of freeom will contain practically no energy and will be said to be "frozen out". The will therefore not contribute to the heat capacity of the substance.

In the case of translational degrees of freedom, the critical temperature is extremely low. For diatomic molecules it is well below the melting point of the substance, so that the translational degrees of freedom in the gas phase may be considered to always be classical, and therefore will contain an energy of (3/2)nRT.

The critical temperature for rotational degrees of freedom is usually a few tens of degrees K. For light gases, such as H2, it will be rather high, since the frequency of vibration is proportional to the mass of the vibrating atoms. In fact, the excitation temperature for the rotational mode of H2 is about 85 deg. K. For heavier molecules, such as I2, the frequency will be much lower. The critical temperature for rotational modes for I2 is about 0.05 deg K. It can be seen that, at room temperature, the rotational modes are always unfrozen.

The vibrational degrees of freedom are the last to "unfreeze". The excitation temperature for the vibrational motion is usually a few thousands of degrees K. H2 gas has a relatively high transition temperature of about 5984 degrees K, while I2 gas has a relatively low transition temperature of about 307 degrees K, while the above mentioned value for HCl of 4156 degrees K. is about average. Generally speaking, at room temperature, the vibrational modes of a diatomic gas are frozen out, except for the heavier gases such as I2.

The specific heat of a diatomic gas may now be written:

\hat{c}_v = f(T)/2

where f is the effective number of unfrozen degrees of freedom at temperature T. With these concepts in mind, we may draw a rather idealized picture of the behavior of the specific heat as a function of temperature, as shown in Figure 1. As each of the degrees of freedom become unfrozen at their particular critical temperature, the dimensionless specific heat rises to a value of f/2. The dimensionless specific heat for most diatomic gases is very close to 5/2 at room temperature, except for the very heavy gases such as iodine and bromine. Figure 2 plots the actual values of the specific heats of a number of diatomic gases in the temperature range 200-2000 deg K. In this range, all rotational modes are active, and the rise in the dimensionless specific heat from 5/2 to 7/2 is evidence of the various vibrational modes becoming active. (Lewis and Randall 1961)

[edit] References