Cycle 2 Exploration: Rotation of Linear Diatomics

Rigid Rotor Model

Molecules are constantly in motion due to energy in the environment. The Rigid Rotor Model represents one way the rotation of a diatomic molecule can be viewed. In this model, it can predict the rotational energy of quantum molecules. In order for this model to be used, several assumptions are made. First, the two masses are assumed to have a volume of a point. Also, these two masses are located at fixed distances from their center of mass, which they rotate about. In order to use the rigid rotor model the diatomic molecule must have a permanent dipole moment, which is a gross selection rule. A specific selection rule is that when calculating the energy of a rotating diatomic molecule, the change in angular momentum cannot be greater than one. It is also assumed that the two atoms in the diatomic have the same angular momentum.

Figure 1: The rigid rotor model can be thought of as a dumbbell rotating around its center of mass.

Solving for Energy for the Rigid Rotor Model

To calculate the change in rotational energy from one energy level to the next, one needs to know several different things about the rotating diatomic. These include the reduced mass, moment of inertia, and the angular momentum. The reduced mass is used in this equation instead of regular mass because using the regular mass complicates the equation too much. Reduced mass is the mass that a diatomic molecule would experience if it was considered to be one object. This is calculated by adding the masses of the two atoms together and dividing that value by the product of the two masses (Figure 2).

Figure 2: The reduced mass equation, where m₁equals the mass of the first particle and m₂equals the mass of the second particle.

The moment of inertia is another mathematical concept that is needed to calculate the energy. This concept describes how much a molecule resists changes in its rotations. For a diatomic molecule, the moment of inertia is the sum of the moments of each of the masses, but by using the reduced mass, the equation can be simplified (Figure 3).

Figure 3: The moment of inertia equation

The moment of inertia describes how hard a molecule resists changes in its rotations, where r₁is the distance from the first particle to the center of mass, r₂is the distance from the second particle to the center of mass, and r₀is the sum of r₁and r₂.

Like translational and electronic energy, rotational energy is quantized. The diatomic molecule in question can only rotate at specific energy levels, labeled J, which creates a characteristic rotational spectrum (more about that later). If energy was not quantized, then rotating molecules could exist at any intermediate energy, similar to macroscopic rotating objects such as a helicopter propeller. If a helicopter propeller could only spin at quantized energy levels, then we would expect it to only rotate at specific speeds and to “jump” between these speeds. Rotational energy transitions typically correspond to microwave radiation (10^-2m), which makes them lower in energy than electronic and vibrational energy transitions.

Figure 4. Relationship between rotational, vibrational, and electronic energy transitions.

Rotational energy levels can be derived from the Schrodinger Equation, yielding:

E_rot= J(J+1)h²/(8π²I)

Where J is the rotational energy level, h is Planck’s constant, and I is the moment of inertia.

Rotational energy is also commonly solved using the rotational constant B, where B=h²/(8π²I). In this form, rotational energy is:

E_rot= BJ(J+1)

A molecule can jump “up” or “down” one rotational energy level through absorption or emission of a photon of energy that matches the energy transition that can be calculated from the equation.

Applications

Rotational Spectra:

Electromagnetic waves can excite the rotational levels of molecules if the molecule has a permanent dipole moment. How does this happen? The electromagnetic field exerts a torque on the electric dipole of the molecule, which can be explained by the graph below. This torque causes the rotation of the molecule. Rotational transitions of molecules are generally found in the microwave region of the electromagnetic spectrum, as mentioned above.

Figure 5: http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/rotrig.html#c1

The rotational spectrum can be used to learn about the diatomic molecule’s structure. With the help of the Schrodinger equation, the rotational energies for “rigid” molecules can be determined. Using the rigid rotor model for a diatomic molecule, calculated moments of inertia can be used to solve for bond lengths. By using the following equations, along with the assumption that the bond length is the same for the ground and first excited states, one can calculate the bond length by solving for r.

Figure 6: http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/vibrot.html#c3

Microwaves:

Although water is not a diatomic molecule, the heating of water molecules in microwave oven is a good example of how the rotation of molecules is useful in everyday life. Microwave heating is more effective on water than on other materials, such as fats and sugars, because water has a greater dipole moment. Since the microwaves easily make the water molecules rotate, the food can be heated quickly.

Conclusion

The rigid rotor model is an effective way to describe the rotational motion of a diatomic molecule. This model allows us to calculate for the rotational energy levels of the molecule, which gives us the ability to solve for bond length. Therefore, the rigid rotor model is useful in learning about a molecule’s structure.

References:

"Energy Calculation for Rigid Rotor Molecules." HyperPhysics. Ed.

Carl R. Nave. Georgia State University, 2010. Web. 9 Nov. 2010.

"Rigid Rotor." Middlebury College: Community Home Page. 27 Aug. 2009 <http://community.middlebury.edu/~chem/chemistry/class/physical/quantum/help /rigid_rotor/rigid_rotor.html>.

Strong, Benjamin. "Rotational Spectroscopy of Diatomic Molecules." ChemWiki: The Complete Chemistry Textbook . UC Davis, 2010. Web. 26 Oct. 2010.