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# Rigid rotor

The rigid rotor is a mechanical model that is used to explain rotating systems. The model consists of two point masses located at fixed distances from their center of mass. The fixed distance between the two masses is an important characteristic of the model. However, for many actual systems this distance is not completely fixed. Fortunately, corrections can be made to compensate for small variations in the distance and therefore the rigid rotor model can still be used to produce fairly accurate results.

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## Rigid Rotor and Quantum Mechanics

The rigid rotor model can be used in quantum mechanics to predict the rotational energy of a diatomic molecule. The rotational energy depends on the moment of inertia for the system, I. In the center of mass reference frame, the moment of inertia is equal to:

I = μR2

where μ is the reduced mass of the molecule and R is the distance between the two atoms.

According to quantum mechanics, the energy (E) of a system can be determined using the Schrödinger equation:

$\hat H Y = E Y$

where Y is the wave function (sometimes the letter ψ is used instead) and $\hat H$ is the energy (Hamiltonian) operator. For the rigid rotor, the energy operator corresponds to the kinetic energy of the system:

$\hat H = - \frac{\hbar^2}{2I} \nabla^2$

where $\hbar$ is Plank's constant divided by and $\nabla^2$ is a symbol known as the Laplacian. The Laplacian is the second derivative with respect to the spatial coordinates. For the rigid rotor model, the energy operator is often written in terms of spherical coordinates:

$\hat H =- \frac{\hbar^2}{2I} \left [ {1 \over \sin \theta} {\partial \over \partial \theta} \left ( \sin \theta {\partial \over \partial \theta} \right ) + {1 \over {\sin^2 \theta}} {\partial^2 \over \partial \phi^2} \right]$

The wave functions for the rigid rotator, Y(θ,φ), can be separated a product of two functions. One of these functions depends only on θ and the other depends only on φ. Multiplied together, these functions result in the complete wave function for the rigid rotor model:

$Y_{l,m}(\theta,\phi) = \left [ \sqrt{ {(2l+1) \over 2} {( l - \left |m \right |)! \over (l + \left |m \right |)!} } P_l^\left |m \right | (\cos \theta) \right ] \left [ \sqrt {{1 \over 2\pi}} \exp (im\phi) \right ]$

where $\sqrt{ {(2l+1) \over 2} {( l - \left |m \right |)! \over (l + \left |m \right |)!} }$ is a normalization constant that depends on the quantum numbers l and m. The symbol $P_l^\left |m \right | (\cos \theta)$ represents a set of functions known as the spherical harmonics, and which also depend on the l and m quantum numbers.

Although the wave functions for the rigid rotor may appear complicated, the equation for the energy of the system is much more compact:

$E_l = {\hbar^2 \over 2I} l \left (l+1\right )$

In units of wave numbers, a unit that is often used for rotational-vibrational spectroscopy, this equation is:

$\bar E_l = \bar {B}l \left (l+1\right )$

where $\bar B$ is known as the rotational constant and:

$\bar B = {h \over 8\pi^2 cI}$

A typical rotational spectrum consists of a series of peaks that correspond to transitions between levels with different values of the angular momentum quantum number (l). Consequently, rotational peaks appear at energies corresponding to an integer multiple of ${2\bar B}$.

## Selection Rules for the Rigid Rotor

Typically, rotational transitions can only be observed when the angular momentum quantum number changes by 1 ($\Delta l = \pm 1$). This selection rule arises from the first-order perturbation theory treatment of rigid rotor using the time-dependent Schrödinger equation . According to this treatment, rotational transitions can only be observed when the z-component of the dipole transition moment:

$\left ( \mu_z \right )_{21} = \int \psi_2^*\mu_z\psi_1, d\tau$

is non-zero. This means that the molecule must have a permanent dipole moment in order to have a spectroscopically observable rotational transition. When this integral is evaluated using the wave function for the rigid rotor and the properties of the spherical harmonics the result is:

$\left ( \mu_z \right )_{l,m;l^',m^'} = \mu \int_0^{2\pi} d\phi \int_0^\pi d\theta \sin \theta \cos \theta Y_{l^1}^{m^1} \left ( \theta , \phi \right )^* Y_l^m \left ( \theta , \phi \right )$

Using the normalization constant for wave function and the orthogonality of the spherical harmonics, it is possible to determine which values of l, m, l', and m' will result in nonzero values for the dipole transition moment integral. This constraint results in the observed selection rules for the rigid rotor:

Δm = 0

$\Delta l = \pm 1$

## Non-Rigid Rotor

The rigid rotor is commonly used to describe the rotational energy of diatomic molecules but it not a completely accurately description of such molecules. This is because molecular bonds (and therefore the interatomic distance R) is not completely fixed; the bond between the atoms stretches out as molecule rotates faster (higher values of the rotational quantum number l). This effect can be accounted for by introducing a correction factor known as the centrifugal distortion constant (D):

$\bar E_l = {E_l \over hc} = \bar {B}l \left (l+1\right ) - \bar {D}l^2 \left (l+1\right )^2$

where

$\bar D = {4 \bar {B}^3 \over \bar {v}^2}$

$\bar v$ is the fundamental vibrational frequency of the bond. This frequency is related to the reduced mass and the force constant (bond strength) of the molecule according to

$\bar v = {1\over 2\pi c} \left ({k \over \mu } \right )^{1 \over 2}$

The non-rigid rotor is an acceptably accurate model for diatomic molecules but is still somewhat imperfect. This is because, although the model does account for bond stretching due to rotation, it ignores any bond stretching due to vibrational energy in the bond.

References: Quantum Chemistry, McQuarrie, Donald A.