Frequency simple harmonic motion

The hydrogen atom attached to an alkane molecule vibrates along the bond axis at a frequency of about 3000 cm. What wavelength of electromagnetic radiation is resonant with this vibration What is the frequency in hertz What is the force constant of the C II bond if the alkane is taken to be a stationary mass because of its size and the H atom is assumed to execute simple harmonic motion ... 

There are many different solutions for X1 and X2 to this pair of coupled equations, but it proves possible to find two particularly simple ones called normal modes of vibration. These have the property that both particles execute simple harmonic motion at the same angular frequency. Not only that, every possible vibrational motion of the two particles can be described in terms of the normal modes, so they are obviously very important. 

There are thus two frequencies at which the two particles will show simple harmonic motion at the same frequency. 

A molecule composed of A atoms has in general 3N degrees of freedom, which include three each for translational and rotational motions, and (3N — 6) for the normal vibrations. During a normal vibration, all atoms execute simple harmonic motion at a characteristic frequency about their equilibrium positions. For a linear molecule, there are only two rotational degrees of freedom, and hence (3N — 5) vibrations. Note that normal vibrations that have the same symmetry and frequency constitute the equivalent components of a degenerate normal mode hence the number of normal modes is always equal to or less than the number of normal vibrations. In the following discussion, we shall demonstrate how to determine the symmetries and activities of the normal modes of a molecule, using NH3 as an example. 

The Vibration of Diatomic Molecules.—In addition to their rotation, we have seen that diatomic molecules can vibrate with simple harmonic motion if the amplitude is small enough. We shall use only this approximation of small amplitude, and our first stop will be to calculate the frequency of vibration. To do this, we must first find the linear restoring force when the interatomic distance is displaced slightly from its equilibrium value / ,. We can get this from Eq. (1.2) by expanding the force in Taylor s series in (r — rt). We have... 

In a diatomic molecule, the masses mv and m2 vibrate back and forth relative to their centre of mass in opposite directions, as shown in the following figure. The two masses reach the extremes of their respective motions at the same time. The diatomic molecule has only one vibrational degree of freedom, i.e., it has only one frequency, called the fundamental vibrational frequency. During vibrational motion, the bond of the molecule behave like a spring and the molecule exhibits a simple harmonic motion provided the displacement of the nuclei from the equilibrium configuration is not too much. At the two extremes of motion which correspond to extension and compression of the chemical bond between the two atoms, the potential energy is maximum. On... 

Normal modes In a normal vibration of a molecule, all atoms carry out simple harmonic motion at the same frequency and the same phase. Any vibration of the molecule can be represented as a superposition of these normal modes. Low-frequency modes are bending modes, while high-frequency modes are stretching modes. 

Classically, in a polyatomic molecule, a normal mode of vibration occurs when each nucleus undergoes simple harmonic motion with the same frequency as and in phase with the other nuclei. The possible normal modes of vibrational of a non-linear polyatomic molecule with A atoms are 3A— 6 and are categorized as arising from stretching, bending, torsional and non-bonded interactions. Whether a particular mode is IR active depends on whether the symmetry of the vibration corresponds to one of the components of the dipole moment. 

Under the conditions of the experimental method, the sample will exhibit simple harmonic motion, and thus the angular frequency (co) may be related to the shear modulus (G) ... 

A normal mode or normal vibration of a polyatomic system is defined as a vibrational state in which each atom moves in simple harmonic motion about its equilibrium position, each atom having the same frequency of oscillation with, generally, all atoms moving in phase. 

We can ascribe these frequency differences to the effect of the different reduced masses, fi, and force constants, / From considerations of simple harmonic motion [3]. 

The energy levels of a molecule vibrating with simple harmonic motion are given by sv-lb — hv(v + ), where v is the vibrational frequency and v the vibrational quantum number. To obtain the vibrational partition function we sum the energy levels measuring the energy from the lowest available level, e0 %hv (the zero-point energy), to obtain... 

Figure 1-7 indicates the normal modes of vibration in CO2 and H2O molecules. In each normal vibration, the individual nuclei carry out a simple harmonic motion in the direction indicated by the arrow, and all the nuclei have the same frequency of oscillation (i.e., the frequency of the normal vibration) and are moving in the same phase. Furthermore, the relative lengths of the arrows indicate the relative velocities and the amplitudes for each nucleus. The 2 vibrations in CO2 are worth comment, since they differ from the others in that two vibrations (i>2a and p2 ) I ave exactly the same frequency. Apparently, there are an infinite number of normal vibrations of this type, which differ only in their directions perpendicular to the molecular axis. Any of them, however, can be resolved into two vibrations such as P2o and p2fe>... 

The cross products in T and V may be eliminated by a linear transformation to a set of new (normal) co-ordinates Q in which each nucleus executes a simple harmonic motion about its equilibrium position with frequency vt and in phase with all other nuclei, v is determined from ... 

A qualitative illustration of this proportionality is easily proAdded. The frequency of a simple harmonic motion is given by the formula... 

When two masses joined by a spring execute a simple harmonic motion about their equilibrium position, the frequency is given by the equation i f... 

As the molecule vibrates (undergoes atom displacements)) the electronic charge distribution and, hence, the polarizability (a) varies in time. The polarizability is related to the electron density of the molecule and is often visualized in three dimensions as an ellipsoid and represented mathematically as a symmetric second-rank tensor. The time-dependent amplitude (Q ) of a normal vibrational mode executing simple harmonic motion is written in terms of the equilibrium amplitude Q , the normal mode frequency o), and time t). 

Figure 1 illustrates the simplest theoretical treatment for the energy absorption behavior of covalent bonds. Hooke s law curve corresponds to bonds vibrating in simple harmonic motion. In practice, covalent bond vibrations are anharmonic and follow the potential energies depicted in the Morse curve, also illustrated in Figure 1. It is this phenomenon of anharmonicity that allows the overtones or harmonics to occur in the NIR region. Thus, the frequencies of the overtones are slightly less than whole number...

It is important to appreciate that an electron oscillating in a sphere of positive electrification will execute a simple harmonic motion and emit radiation of definite frequency, giving a sharp spectrum line. Thomson investigated mathematically the stable distribution of electrons in rings in his atom model, and found that the results agreed with some experiments on floating magnets made by A. M. Mayer, which Thomson used to show in his lectures at the Royal Institution. 

Properties. It is of considerable importance to examine the nature of die solutions obtained above. It is evident from Eq. (9), Sec. 2-2, that each atom is oscillating about its equilibrium position with a simple harmonic motion of amplitude Aik — Kkhk, frequency x /27t, and phase e. Ihirthermore, corresponding to a given solution X of the secular equation, i he frequency and phase of the motion of each coordinate is the same, but I lie amplitudes may be, and usually are, different for each coordinate. On account of the equality of phase and frequency, each atom reaches its position of maximum displacement at the same time, and each atom pa.sscs through its equilibrium position at the same time. A mode of ibration having all these characteristics is called a normal mode of vibra-iion, and its frequency is known as a normal, or fundamental, frequency of (he molecule. 

In a diatomic molecule, Ar will be small and this will result in a simple harmonic motion this type of system is referred to as a harmonic oscillator. Assuming that this vibrational motion is equivalent to a harmonic oscillator allows the frequency of the oscillation to be calculated by... 

Vibrations are considered in terms of the classical expressions governing motion of nuclei vibrating about their equilibrium positions with a simple harmonic motion (40). The potential and kinetic potential energies of molecules are defined in terms of the coordinates most appropriate to the molecular structures. All relative motions of atoms about the center of mass (vibrations) are linear combinations of a set of coordinates, known as normal coordinates. For every normal mode of vibration, all coordinates vary periodically with the same frequency and pass through equilibrium simultaneously. 

The frequency at which two masses, joined together by a spring, vibrate, is described by the equations of simple harmonic motion ... 

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