Molecular vibration

From Wikipedia, the free encyclopedia

A molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion. The frequency of the periodic motion is known as a vibration frequency, and the typical frequencies of molecular vibrations range from less than 10¹² to approximately 10¹⁴ Hz.

In general, a molecule with N atoms has 3N – 6 normal modes of vibration, but a linear molecule has 3N – 5 such modes, as rotation about its molecular axis cannot be observed.^[1] A diatomic molecule has one normal mode of vibration. The normal modes of vibration of polyatomic molecules are independent of each other but each normal mode will involve simultaneous vibrations of different parts of the molecule such as different chemical bonds.

A molecular vibration is excited when the molecule absorbs a quantum of energy, E, corresponding to the vibration's frequency, ν, according to the relation E = hν (where h is Planck's constant). A fundamental vibration is excited when one such quantum of energy is absorbed by the molecule in its ground state. When two quanta are absorbed the first overtone is excited, and so on to higher overtones.
To a first approximation, the motion in a normal vibration can be described as a kind of simple harmonic motion. In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first overtone has twice the frequency of the fundamental. In reality, vibrations are anharmonic and the first overtone has a frequency that is slightly lower than twice that of the fundamental. Excitation of the higher overtones involves progressively less and less additional energy and eventually leads to dissociation of the molecule, as the potential energy of the molecule is more like a Morse potential.
The vibrational states of a molecule can be probed in a variety of ways. The most direct way is through infrared spectroscopy, as vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum. Raman spectroscopy, which typically uses visible light, can also be used to measure vibration frequencies directly. The two techniques are complementary and comparison between the two can provide useful structural information such as in the case of the rule of mutual exclusion for centrosymmetric molecules.

Vibrational excitation can occur in conjunction with electronic excitation (vibronic transition), giving vibrational fine structure to electronic transitions, particularly with molecules in the gas state.
Simultaneous excitation of a vibration and rotations gives rise to vibration-rotation spectra.

Vibrational coordinates

The coordinate of a normal vibration is a combination of changes in the positions of atoms in the molecule. When the vibration is excited the coordinate changes sinusoidally with a frequency ν, the frequency of the vibration.

Internal coordinates

Internal coordinates are of the following types, illustrated with reference to the planar molecule ethylene,

• Stretching: a change in the length of a bond, such as C-H or C-C
• Bending: a change in the angle between two bonds, such as the HCH angle in a methylene group
• Rocking: a change in angle between a group of atoms, such as a methylene group and the rest of the molecule.
• Wagging: a change in angle between the plane of a group of atoms, such as a methylene group and a plane through the rest of the molecule,
• Twisting: a change in the angle between the planes of two groups of atoms, such as a change in the angle between the two methylene groups.
• Out-of-plane: a change in the angle between any one of the C-H bonds and the plane defined by the remaining atoms of the ethylene molecule. Another example is in BF₃ when the boron atom moves in and out of the plane of the three fluorine atoms.

In a rocking, wagging or twisting coordinate the bond lengths within the groups involved do not change. The angles do. Rocking is distinguished from wagging by the fact that the atoms in the group stay in the same plane.

In ethene there are 12 internal coordinates: 4 C-H stretching, 1 C-C stretching, 2 H-C-H bending, 2 CH₂ rocking, 2 CH₂ wagging, 1 twisting. Note that the H-C-C angles cannot be used as internal coordinates as the angles at each carbon atom cannot all increase at the same time.

Vibrations of a methylene group (-CH₂-) in a molecule for illustration

The atoms in a CH₂ group, commonly found in organic compounds, can vibrate in six different ways: symmetric and asymmetric stretching, scissoring, rocking, wagging and twisting as shown here:

Symmetrical stretching	Asymmetrical stretching	Scissoring (Bending)
Rocking	Wagging	Twisting

(These figures do not represent the "recoil" of the C atoms, which, though necessarily present to balance the overall movements of the molecule, are much smaller than the movements of the lighter H atoms).

Symmetry-adapted coordinates

Symmetry-adapted coordinates may be created by applying a projection operator to a set of internal coordinates.^[2] The projection operator is constructed with the aid of the character table of the molecular point group. For example, the four(un-normalised) C-H stretching coordinates of the molecule ethene are given by

where are the internal coordinates for stretching of each of the four C-H bonds.
Illustrations of symmetry-adapted coordinates for most small molecules can be found in Nakamoto.^[3]

Normal coordinates

The normal coordinates, denoted as Q, refer to the positions of atoms away from their equilibrium positions, with respect to a normal mode of vibration. Each normal mode is assigned a single normal coordinate, and so the normal coordinate refers to the "progress" along that normal mode at any given time. Formally, normal modes are determined by solving a secular determinant, and then the normal coordinates (over the normal modes) can be expressed as a summation over the cartesian coordinates (over the atom positions). The advantage of working in normal modes is that they diagonalize the matrix governing the molecular vibrations, so each normal mode is an independent molecular vibration, associated with its own spectrum of quantum mechanical states. If the molecule possesses symmetries, it will belong to a point group, and the normal modes will "transform as" an irreducible representation under that group. The normal modes can then be qualitatively determined by applying group theory and projecting the irreducible representation onto the cartesian coordinates. For example, when this treatment is applied to CO₂, it is found that the C=O stretches are not independent, but rather there is an O=C=O symmetric stretch and an O=C=O asymmetric stretch.

• symmetric stretching: the sum of the two C-O stretching coordinates; the two C-O bond lengths change by the same amount and the carbon atom is stationary. Q = q₁ + q₂
• asymmetric stretching: the difference of the two C-O stretching coordinates; one C-O bond length increases while the other decreases. Q = q₁ - q₂

When two or more normal coordinates belong to the same irreducible representation of the molecular point group (colloquially, have the same symmetry) there is "mixing" and the coefficients of the combination cannot be determined a priori. For example, in the linear molecule hydrogen cyanide, HCN, The two stretching vibrations are

1. principally C-H stretching with a little C-N stretching; Q₁ = q₁ + a q₂ (a << 1)
2. principally C-N stretching with a little C-H stretching; Q₂ = b q₁ + q₂ (b << 1)

The coefficients a and b are found by performing a full normal coordinate analysis by means of the Wilson GF method.^[4]

Newtonian mechanics

The HCl molecule as an anharmonic oscillator vibrating at energy level E₃. D₀ is dissociation energy here, r₀ bond length, U potential energy. Energy is expressed in wavenumbers. The hydrogen chloride molecule is attached to the coordinate system to show bond length changes on the curve.

Perhaps surprisingly, molecular vibrations can be treated using Newtonian mechanics to calculate the correct vibration frequencies. The basic assumption is that each vibration can be treated as though it corresponds to a spring. In the harmonic approximation the spring obeys Hooke's law: the force required to extend the spring is proportional to the extension. The proportionality constant is known as a force constant, k. The anharmonic oscillator is considered elsewhere.^[5]

By Newton’s second law of motion this force is also equal to a reduced mass, μ, times acceleration.

Since this is one and the same force the ordinary differential equation follows.

The solution to this equation of simple harmonic motion is

A is the maximum amplitude of the vibration coordinate Q. It remains to define the reduced mass, μ. In general, the reduced mass of a diatomic molecule, AB, is expressed in terms of the atomic masses, m_A and m_B, as

The use of the reduced mass ensures that the centre of mass of the molecule is not affected by the vibration. In the harmonic approximation the potential energy of the molecule is a quadratic function of the normal coordinate. It follows that the force-constant is equal to the second derivative of the potential energy.

When two or more normal vibrations have the same symmetry a full normal coordinate analysis must be performed (see GF method). The vibration frequencies,ν_i are obtained from the eigenvalues,λ_i, of the matrix product GF. G is a matrix of numbers derived from the masses of the atoms and the geometry of the molecule.^[4] F is a matrix derived from force-constant values. Details concerning the determination of the eigenvalues can be found in.^[6]

Quantum mechanics

In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave equation, the energy states for each normal coordinate are given by

,

where n is a quantum number that can take values of 0, 1, 2 ... In molecular spectroscopy where several types of molecular energy are studied and several quantum numbers are used, this vibrational quantum number is often designated as v.^[7][8]

The difference in energy when n (or v) changes by 1 is therefore equal to , the product of the Planck constant and the vibration frequency derived using classical mechanics. For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency (in the harmonic oscillator approximation).

See quantum harmonic oscillator for graphs of the first 5 wave functions, which allow certain selection rules to be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes by one,

but this does not apply to an anharmonic oscillator; the observation of overtones is only possible because vibrations are anharmonic. Another consequence of anharmonicity is that transitions such as between states n=2 and n=1 have slightly less energy than transitions between the ground state and first excited state. Such a transition gives rise to a hot band.

Intensities

In an infrared spectrum the intensity of an absorption band is proportional to the derivative of the molecular dipole moment with respect to the normal coordinate.^[9] The intensity of Raman bands depends on polarizability.

Beer–Lambert law

From Wikipedia, the free encyclopedia

An example of Beer–Lambert law: green laser light in a solution of Rhodamine 6B. The beam intensity becomes weaker as it passes through solution.

The Beer–Lambert law, also known as Beer's law, the Lambert–Beer law, or the Beer–Lambert–Bouguer law relates the attenuation of light to the properties of the material through which the light is traveling. The law is commonly applied to chemical analysis measurements and used in understanding attenuation in physical optics, for photons, neutrons or rarefied gases. In mathematical physics, this law arises as a solution of the BGK equation.

Equations

The law states that there is a logarithmic dependence between the transmission (or transmissivity or transmittance), T, of light through a substance and the product of the attenuation coefficient of the substance, Σ, and the distance the light travels through the material (i.e., the path length), ℓ. The attenuation coefficient can, in turn, be written as a product of either an absorptivity of the attenuator, ε, and the concentration c of attenuating species in the material, or a total (absorption and scattering) cross section, σ, and the (number) density N' of attenuators. In some chemistry applications for liquids these relations are usually written with the notation:

whereas in biology and physics, they are normally written as:

where and are the intensity (power per unit area) of the incident radiation and the transmitted radiation, respectively; σ is attenuation cross section and N is the concentration (number per unit volume) of attenuating medium. The base 10 and base e conventions must not be confused because they have different numerical values for the attenuation coefficient: . However, it is easy to convert one to the other, using

and to express the decimal logarithm quantity with the decibel unit of measure.

The transmissivity (ability to transmit) is expressed in terms of an absorbance which is defined as

whereas it can be expressed in decibels as:

This implies that the absorbance becomes linear with the concentration (or number density of attenuators) according to

and

for the two cases, respectively. Thus, if the path length and the attenuation coefficient (or the total cross section) are known and the absorbance is measured, the concentration of the substance (or the number density of attenuators) can be deduced. Although several of the expressions above often are used as Beer–Lambert law, the name should strictly speaking only be associated with the latter two. The reason is that historically, the Lambert law states that attenuation is proportional to the light path length, whereas the Beer law states that attenuation is proportional to the concentration of attenuating species in the material.^[1] If the concentration is expressed as a mole fraction i.e., a dimensionless fraction, the absorptivity of the attenuator (ε) takes the same dimension as the attenuation coefficient, i.e., reciprocal length (e.g., m⁻¹). However, if the concentration is expressed in moles per unit volume, the attenuation coefficient (ε) is used in L·mol⁻¹·cm⁻¹, or sometimes in converted SI units of m²·mol⁻¹. The attenuation coefficient Σ' is one of many ways to describe the attenuation of electromagnetic waves. For the others, and their interrelationships, see the article: Mathematical descriptions of opacity. For example, Σ' can be expressed in terms of the imaginary part of the refractive index, κ, and the wavelength of the radiation(in free space), λ₀, according to

In molecular attenuation spectrometry, the attenuation cross section σ is expressed in terms of a linestrength, S, and an (area-normalized) lineshape function, Φ. The frequency scale in molecular spectroscopy is often in cm⁻¹, where the lineshape function is expressed in units of 1/cm⁻¹. Since N is given as a number density in units of 1/cm³, the linestrength is often given in units of cm²cm⁻¹/molecule. A typical linestrength in one of the vibrational overtone bands of smaller molecules, e.g., around 1.5 μm in CO or CO₂, is around 10⁻²³ cm²cm⁻¹, although it can be larger for species with strong transitions, e.g., C₂H₂. The linestrengths of various transitions can be found in large databases, e.g., HITRAN. The lineshape function often takes a value around a few 1/cm⁻¹, up to around 10/cm⁻¹ under low pressure conditions, when the transition is Doppler broadened, and below this under atmospheric pressure conditions, when the transition is collision broadened. It has also become commonplace to express the linestrength in units of cm⁻²/atm since then the concentration is given in terms of a pressure in units of atm. A typical linestrength is then often in the order of 10⁻³ cm⁻²/atm. Under these conditions, the detectability of a given technique is often quoted in terms of ppm•m. The fact that there are two commensurate definitions of attenuation (in base 10 or e) implies that the absorbance and the attenuation coefficient for the cases with gases, A' and Σ', are ln 10 (approximately 2.3) times as large as the corresponding values for liquids, i.e., A and Σ, respectively. Therefore, care must be taken when interpreting data that the correct form of the law is used. The law tends to break down at very high concentrations, especially if the material is highly scattering. If the radiation is especially intense, nonlinear optical processes can also cause variances. The main reason, however, is the following. At high concentrations, the molecules are closer to each other and begin to interact with each other. This interaction will change several properties of the molecule, and thus will change the attenuation. If the attenuation is different at higher concentrations than at lower ones, then the plot of the attenuation coefficient will not be linear, as is suggested by the equation, so you can only use it when all the concentrations you are working with are low enough that the absorbtivity is the same for all of them.

Derivation

Classically, the Beer–Lambert law was first devised independently where Lambert's law stated that absorbance is directly proportional to the thickness of the sample, and Beer's law stated that absorbance is proportional to the concentration of the sample. The modern derivation of the Beer–Lambert law combines the two laws and correlate the absorbance to both, the concentration as well as the thickness (path length) of the sample.^[2] In concept, the derivation of the Beer–Lambert law is straightforward. Divide the attenuating sample into thin slices that are perpendicular to the beam of light. The light that emerges from a slice is slightly less intense than the light that entered because some of the photons have run into molecules in the sample and did not make it to the other side. For most cases where measurements of attenuation are needed, a vast majority of the light entering the slice leaves without being attenuated. Because the physical description of the problem is in terms of differences—intensity before and after light passes through the slice—we can easily write an ordinary differential equation model for attenuation. The difference in intensity due to the slice of attenuating material is reduced; leaving the slice, it is a fraction of the light entering the slice . The thickness of the slice is , which scales the amount of attenuation (thin slice does not attenuates much light but a thick slice attenuates a lot). In symbols, , or . This conceptual overview uses to describe how much light is attenuated. All we can say about the value of this constant is that it will be different for each material. Also, its values should be constrained between −1 and 0. The following paragraphs cover the meaning of this constant and the whole derivation in much greater detail. Assume that particles may be described as having an attenuation cross section (i.e., area), σ, perpendicular to the path of light through a solution, such that a photon of light is attenuated if it strikes the particle, and is transmitted if it does not. Define z as an axis parallel to the direction that photons of light are moving, and A and dz as the area and thickness (along the z axis) of a 3-dimensional slab of space through which light is passing. We assume that dz is sufficiently small that one particle in the slab cannot obscure another particle in the slab when viewed along the z direction. The concentration of particles in the slab is represented by N. It follows that the fraction of photons attenuated (absorbed and scattered away) when passing through this slab is equal to the total opaque area of the particles in the slab, σAN dz, divided by the area of the slab A, which yields σN dz. Expressing the number of photons attenuated by the slab as dI_z, and the total number of photons incident on the slab as I_z, the number of photons attenuated by the slab is given by

Note that because there are fewer photons which pass through the slab than are incident on it, dI_z is actually negative (It is proportional in magnitude to the number of photons attenuated). The solution to this simple differential equation is obtained by integrating both sides to obtain I_z as a function of z

The difference of intensity for a slab of real thickness ℓ is I₀ at z = 0, and I_l at z = ℓ. Using the previous equation, the difference in intensity can be written as,

rearranging and exponentiating yields,

This implies that

and

The quantity Σ is called the total macroscopic cross section or attenuation coefficient, depending on the topic (for example in respectively the first term is used transport theory and the second one in shielding and radiation protection). The derivation assumes that every attenuating particle behaves independently with respect to the light and is not affected by other particles. While it is commonly thought that error is introduced when particles are lying along the same optical path such that some particles are in the shadow of others, this is actually a key part of the derivation and why integration is used.
When the path taken is long enough to make the medium attenuation coefficient not uniform, the original equation must be modified as follows:

where z is the distance along the path through the medium, all other symbols are as defined above.^[3] This is taken into account in each in the atmospheric equation above.

Deviations from Beer–Lambert law

Under certain conditions Beer–Lambert law fails to maintain a linear relationship between attenuation and concentration of analyte.^[4] These deviations are classified into three categories:

1. Real – fundamental deviations due to the limitations of the law itself.
2. Chemical – deviations observed due to specific chemical species of the sample which is being analyzed.
3. Instrument – deviations which occur due to how the attenuation measurements are made.

Prerequisites

There are at least six conditions that need to be fulfilled in order for Beer’s law to be valid. These are:

1. The attenuators must act independently of each other;
2. The attenuating medium must be homogeneous in the interaction volume
3. The attenuating medium must not scatter the radiation – no turbidity - unless this is accounted for as in DOAS;
4. The incident radiation must consist of parallel rays, each traversing the same length in the absorbing medium;
5. The incident radiation should preferably be monochromatic, or have at least a width that is narrower than that of the attenuating transition. Otherwise a spectrometer as detector for the intensity is needed instead of a photodiode which has not a selective wavelength dependence; and
6. The incident flux must not influence the atoms or molecules; it should only act as a non-invasive probe of the species under study. In particular, this implies that the light should not cause optical saturation or optical pumping, since such effects will deplete the lower level and possibly give rise to stimulated emission.

If any of these conditions are not fulfilled, there will be deviations from Beer’s law.

Chemical analysis

Beer's law can be applied to the analysis of a mixture by spectrophotometry, without the need for extensive pre-processing of the sample. An example is the determination of bilirubin in blood plasma samples. The spectrum of pure bilirubin is known, so the molar attenuation coefficient is known. Measurements are made at one wavelength that is nearly unique for bilirubin and at a second wavelength in order to correct for possible interferences.The concentration is given by c = A^corrected / ε. For a more complicated example, consider a mixture in solution containing two components at concentrations c₁ and c₂. The absorbance at any wavelength, λ is, for unit path length, given by

Therefore, measurements at two wavelengths yields two equations in two unknowns and will suffice to determine the concentrations c₁ and c₂ as long as the molar absorbances of the two components, ε₁ and ε₂ are known at both wavelengths. This two system equation can be solved using Cramer's rule. In practice it is better to use linear least squares to determine the two concentrations from measurements made at more than two wavelengths. Mixtures containing more than two components can be analyzed in the same way, using a minimum of n wavelengths for a mixture containing n components. The law is used widely in infra-red spectroscopy and near-infrared spectroscopy for analysis of polymer degradation and oxidation (also in biological tissue). The carbonyl group attenuation at about 6 micrometres can be detected quite easily, and degree of oxidation of the polymer calculated.

Beer–Lambert law in the atmosphere

This law is also applied to describe the attenuation of solar or stellar radiation as it travels through the atmosphere. In this case, there is scattering of radiation as well as absorption. The Beer–Lambert law for the atmosphere is usually written

where each is the optical depth whose subscript identifies the source of the absorption or scattering it describes:

• refers to aerosols (that absorb and scatter)
• are uniformly mixed gases (mainly carbon dioxide () and molecular oxygen () which only absorb)
• is nitrogen dioxide, mainly due to urban pollution (absorption only)
• are effects due to Raman scattering in the atmosphere
• is water vapour absorption
• is ozone (absorption only)
• is Rayleigh scattering from molecular oxygen () and nitrogen () (responsible for the blue color of the sky).
• The selection of the attenuators which have to be considered depends on the wavelength range and can include various other compounds. This can include Tetraoxygen, HONO, Formaldehyde, Glyoxal, a series of halogen radicals and others.

is the optical mass or airmass factor, a term approximately equal (for small and moderate values of ) to , where is the observed object's zenith angle (the angle measured from the direction perpendicular to the Earth's surface at the observation site). This equation can be used to retrieve , the aerosol optical thickness, which is necessary for the correction of satellite images and also important in accounting for the role of aerosols in climate.

History

The law was discovered by Pierre Bouguer before 1729.^[5] It is often attributed to Johann Heinrich Lambert, who cited Bouguer's Essai d'Optique sur la Gradation de la Lumiere (Claude Jombert, Paris, 1729) — and even quoted from it — in his Photometria in 1760.^[6] Much later, August Beer extended the exponential attenuation law in 1852 to include the concentration of solutions in the attenuation coefficient.^[7]