Introduction to Molecular Spectroscopy

4 Introduction to Molecular Spectroscopy Spectroscopy is the study of interaction of radiation with matter and imparts information regarding molecular structure (molecular symmetry, bond distances, bond angles), chemical properties (electronic distribution, bond strength, intra- and inter-molecular spectra. The methods of analyzing the spectra could be better illustrated by starting with small and simple molecules. However, principles so established may be extended, with some added complexities, to larger molecules. Spectroscopic methods have proved to be very useful for studying the properties of molecules. The principal exploratory work in it may now be complete, but the theory is still of interest because of its applications, because of the way it illustrates many principles of quantum mechanics and group theory, and the way theory is used to solve more complex problems. f course a theory is not only a picture and need not be understood entirely by way of visual images, but by understanding the problem more clearly. 4.1 PARTICLE-WAVE NATURE F RADIATIN AND MATTER A light ray consists of oscillating electric, (e) and magnetic (B) field. The direction of both the fields is perpendicular to the direction in which light is propagated. The wave field has no wave component in its direction of propagation. A monochromatic beam of light can be regarded as the resultant wave of a number of rays (waves) of light but the electrical (or magnetic) vectors are in different planes. A Nicol prism or a piece of crystal, filters the light beam with its electrical (or magnetic) vectors confined to a particular plane, which do not change direction as the waves move on. This light is said to be planepolarized in which vector orientation remains constant with varying amplitudes. Plane-polarized light has an important property, i.e., when it falls on a second Nicol prism; it will pass with undiminished intensity only if the polarizing axes of the two prisms are parallel to each other. Any other orientation of these axes will further decrease the intensity until the axes are perpendicular when intensity approaches zero. Light rays are electromagnetic in nature and as the name implies, both an electric field (ε) and a magnetic field (B) is associated with it. These fields oscillate in a periodic manner, sinusoidal, at mutually perpendicular directions and to the direction of propagation of the radiation. In Fig. 4.1, a radiation with electric and magnetic fields oscillate in fixed planes in space is shown thereby giving it a wave character. The periodic behavior of the electric and magnetic field at a time t in a beam is given by F ε = A cos 2πν t zi c K J HG 109

110 Fundamentals of Molecular Spectroscopy X Y e = A 0 cos 2 pn ( t z c ) Z l Direction of propagation Fig. 4.1 Electric field ε-component in the X-Z plane and magnetic field B-component in the Y-Z plane of plane polarized radiation ε Β = µ A cos 2πν t zi HG c K J...(4.1) Where ε and B are the x and y component of electric and magnetic fields respectively, A is the amplitude of the wave, c velocity of light, ν frequency of light related to velocity of light as given by Eqn. 4.2 subsequently, while ε and µ are dielectric constant and permeability of the medium respectively. The magnetic field B at any point in the waves is equal in magnitude of the electric field ε, provided ε is expressed in electrostatic unit (esu) and magnetic field B is expressed in electromagnetic units (emu). These radiation waves travel with the velocity of light c having a wavelength λ and frequency ν. The frequency represents the number of wave crests or wave fronts passing a given point in a unit time, and is expressed in cycles per second or simply seconds 1. The reciprocal seconds (s 1 ) is called hertz (Hz). In spectroscopy, wavelengths are expressed in a variety of units, chosen so that in any particular range the wavelength does not involve large power of tens. Thus, in microwave region wavelength is measured in cm or mm, while in infrared it is usually given in micrometer (µm) formerly called micron where 1µm = 10 6 m. In visible and ultra violet region, λ is still often expressed in Angstrom although the proper SI unit for this region is nanometer (nm). 1 nm = 10 9 m = 10Å The relation between the wavelength and frequency of light wave is λν = c...(4.2) Where c is velocity of light and is equal to 3 10 8 ms 1 To illustrate the values that occur for λ and ν, consider the orange radiation with wavelength λ = 6.2 10 7 m. The corresponding frequency is 8 1 3 10 ms 14 ν = c / λ = = 4.838 10 Hz 7 6.2 10 m Such inconvenient large numerical values forced the spectroscopist to introduce a third quantity called wave number ( ν ), which is simply given by the relation F ν = 1/λ...(4.3)

Introduction to Molecular Spectroscopy 111 Units of wave numbers are units inverts of wavelength and are smaller than the frequency by a factor of c, expressed in cm 1. Wave number is a permitted non-si unit. Thus, for orange light, the wave number is 1 1 ν = = 16129.03 cm 7 6.2 10 m Classical wave theory assumes that interaction of radiation with molecules changes the velocity of the molecular electrons and this energy is usually transformed into thermal energy. Absorption or emission of radiation is taken as a continuous process with no limit to the amount of energy-electrons in atom or molecule emitted or absorbed. However, experimental observations were incompatible with this view. Max Planck in 1900 published a revolutionary idea that an atom can change its energy only by discrete amounts and hence atoms can exist only in certain states separated by a finite difference of energy called quantum of energy E. Another way of saying it is that a fixed amount of energy E, can make the atom to go from one state of energy E 1 to another state of energy E 2. nly a transition of the molecule from energy state E 1 to energy state E 2 can lead to absorption or emission of radiation of fixed quantum E = E 1 E 2. This energy is directly related to the frequency of radiation and is called the quantization condition, i.e., Excited molecule = Molecule + Quantum of energy E 2 E 1 E E 2 E 1 = E = hν...(4.4) Where h is a universal constant known as Planck s constant and has a value of 6.62 10 34 J s. The quanta of energy E is emitted or absorbed as the equilibrium, in Eqn. 4.4 is moving to right or left and results are an emission or an absorption spectrum. This also modified the theory of light to include that the energy of radiation can change only by an integral number of quanta. Thus, atoms/molecules and light are so intimately related that a section concerning radiation without referring extensively to atomic structure and vice versa cannot be thought of. This similarity between radiation and matter initially was thought to be a matter of energy relation only. In 1924, Louis de Broglie suggested that there is a wave associated with momentum p of every particle given by the equation: λ = h/p = h/mv...(4.5) Even more indicative of the validity of this relation were the experiments of Davison and Garmer in which an electronic beam was shown to give diffraction effect corresponding to a wave with wavelength given by the de Broglie relation. Earlier Einstein had hypothesized that a quantum of light energy hν is concentrated into corpuscles or photon particles to explain the photochemical effect. A light beam of frequency ν and energy nhν could be regarded as containing n light corpuscles called photons. Each photon of wavelength λ carries an energy hν and a momentum hν/c. When a photon of wavelength λ is absorbed by molecule, it goes from state E 1 to state E 2. Radiation pressure can be understood in terms of the pressure imparted to a surface by a steady steam of photons. It is only photons and not light waves, which are observed, waves are the mathematical expression of the way in which photons move and follow the laws of wave motion. C H A P T E R 4 4.2 SPECTRA AND ITS REGINS An isolated molecule possesses: (i) translational energy by virtue of the motion of the molecule as a whole.

112 Fundamentals of Molecular Spectroscopy (ii) rotational energy due to bodily rotation of the molecule about an axis passing through the center of gravity of molecule. (iii) vibrational energy due to periodic displacement of its atoms from their equilibrium positions. (iv) electronic energy since the electrons associated with each atom and bond are in constant motion. (v) spin energy, energy change due to nuclear or electron spin changes. These various types of energies associated with different motions of the molecule are different and thus independent of one another. As a first approximation, the total energy of a molecule can be expressed as the sum of the constituent energies, that is E total = E trans + E rot + E vib + E elec In other words, molecules have different quantized energy levels for different motion. Consider two energy states of a system E 1 and E 2, subscripts 1 and 2 referred are quantum numbers. A transition from E 1 to E 2 can occur provided an appropriate amount of energy E = E 2 E 1 is absorbed by the system to give corresponding spectra. The time required for various transitions are as follows: Nature of transition Electronic Vibrational Rotational Time required 10 15 s 10 13 s 10 10 s These times are so different that as an approximation, they do not interact with one another. Thus a system will continue to vibrate in exactly same way irrespective of whether it is simultaneously undergoing translational motion uniformly through space in any direction. Thus, translational motion is separable from the orbital and other kinds of motion. It is a reasonably good approximation, called Born- ppenheimer approximation, that leads us to have independent spectra of Rotational motion, Vibrational motion, Electronic motion for molecules. In the interaction between the incident radiation and molecule, there should be a mechanism so that there must be some change in electrical or magnetic effects, which should be influenced by the fluctuation of electric and magnetic fields, associated with the radiation. Some of the possibilities are as follows: 1. Radio frequency region: Nucleus and electrons are tiny charged particles and their spin is associated with a tiny magnetic dipole. The reversal of this dipole consequent upon the spin reversal can interact with magnetic field to give magnetic resonance at the appropriate frequency. Consequently, all such spin reversals produced an absorption or emission spectra. 2. The microwave region: A molecule, such as hydrogen chloride, HCl in which one atom (the hydrogen) carries a permanent net positive charge and other a net negative charge is said to have a permanent electric dipole moment. n the other hand, H 2 and Cl 2 have no such charge separation and thus have zero dipole moment. As is evident from Fig. 4.2 in the rotation of HCl, plus and minus charges change places periodically and component of dipole moment fluctuate regularly. This fluctuation is plotted in the lower half of Fig. 4.2 and it is seen to be exactly similar in form to the fluctuating electric field of radiation. Thus, interaction can occur and energy can be absorbed or emitted and thus rotation gives rise to a spectrum. All molecules having a permanent dipole moment are said to be microwave active. If there is no dipole moment as in H 2, Cl 2 no interaction can take place and molecule is microwave inactive.

Introduction to Molecular Spectroscopy 113 Direction of rotation + + + + + + Direction of dipole Vertical component of dipole Time Fig. 4.2 The rotation of a diatomic molecule,hci, showing the fluctuation in the dipole moment measured in a particular direction 3. The infrared region: In this region, molecular vibration should give rise to a change in dipole moment. Consider the carbon dioxide molecule as an example, in which three atoms are arranged linearly with a small net positive charge on carbon and small negative charges on oxygen atoms. δ 2δ+ δ C Normal During the mode of vibration known as the symmetric stretching the molecules are alternatively stretched and compressed, both C bonds changing simultaneously as follow: δ 2δ+ δ C δ 2δ+ δ C δ 2δ+ δ C Stretched Normal Compressed Plainly the dipole moment remains zero throughout the whole of this motion, and this particular vibration is thus infrared inactive. However, there is another stretching vibration called the anti-symmetrical vibration depicted in Fig. 4.3. Here one bond stretches while the other is compressed, and vice-versa. As the figure shows there is a periodic alteration in the dipole moment and the vibration is thus infrared active. The infrared radiation is often further subdivided, as near infrared (10,000 to 4000 cm 1 ), infrared (4000200 cm 1 ) and far infrared (20010 cm 1 ). These divisions may be justified on the basis of the differences in instrumentation or experimental procedure for obtaining the spectra. 4. Raman spectra is associated with scattering of radiation which is dependent on the polarizability of molecule. There is a rater special requirement for a molecular motion to be Raman active, i.e., the vibration should be associated with a change in polarizability of molecule. The symmetrical stretching vibration is associated with polarizability change and thus Raman active. This will be discussed in Chapter 7. C H A P T E R 4

114 Fundamentals of Molecular Spectroscopy Asymmetric stretching vibration C 2 C C C C C C C Dipole moment Component of dipole Fig. 4.3 The asymmetric stretching vibration of the carbon dioxide molecule showing the fluctuation in the moment 5. The visible and ultraviolet regions: The excitation of a valence electron involves the movement of electronic charges in the molecule. The consequent change in the electric dipole gives rise to a spectrum by its interaction with fluctuating electric field of radiation and that gives rise to electronic spectra in this region. The spectrum can be obtained in three different ways: By Emission spectroscopy, Absorption spectroscopy and Raman spectroscopy. Emission Spectroscopy Atoms or molecules are subjected to intense heat or electric discharge so as to absorb the same to become excited. n returning to their lower energy state atoms or molecules may emit radiation. Such emission is the result of a transition of an atom or molecule from an excited state to one of lower energy, usually the ground state. This excess energy is emitted as a photon and the corresponding frequency are recorded as the emission spectrum. If the transition is from energy state E 2 to E 1, the emission spectrum shows a line at a frequency ν given by E 2 E 1 = E=hν...(4.6) Absorption Spectroscopy The absorbing sample is placed between the source of light in the frequency range being studied and the spectrophotometer, which records the percentage of light absorbed against the range of frequencies to give the absorption spectrum. This is possible if the molecule is in energy state E 1 initially and then excited to energy state E 2 following the relation given by Eqn. 4.6. Raman Spectroscopy It is a technique, which explores energy levels of molecules by the scattering of light. Photons of frequency ν in are scattered by collision with the molecules of the sample, when new frequencies are added, because photons can acquire or lose energy during collision. If the molecules are excited by light

Introduction to Molecular Spectroscopy 115 during the collision, molecules withdraw some energy from the photons and so scattered light emerges with a lower frequency (ν in ν). If the molecules are already excited before the photons collide with them, molecules may be able to give up this energy and the emerging photons will have higher frequency, viz. (ν in + ν). Raman spectroscopy has recently undergone considerable development because of the availability of lasers, which are so intense that those small samples, and shorter exposures give very good spectra. Further, lasers are associated with very strong electric field, which has shown that scattering is non-linear in nature. These non-linear interactions has opened up score of Raman spectras as discussed in Chapter 14. Since it is only photons and not light waves which are observed, the waves are the mathematical expressions of the way in which the photons moves and follow the laws of wave motion. Figure 4.4 illustrates the region into which electromagnetic radiation has been divided. The boundaries between the regions are by no means precise, although the molecular processes associated with each region are. Infra-red Visible 10 2 Microwave Radio frequency Ultraviolet X-rays g-rays 10 5 Cosmic rays Nh n/399 in J mol 10 5 1 10 6 10 8 10 10 Log n in Hz Wave length l 5 10 12 14.5 15 16 18 20 3 km 3 cm 0.3 cm 950 nm 300 nm 30 nm 0.3 nm 3 pm Fig. 4.4 The electromagnetic spectrum. Values given here correspond to values at left hand side line 1. Radio Frequency region 10 6 10 10 Hz or 10 m 1 cm, wavelength: The energy change with the change of spin of a nucleus or electron is of the order of 0.001 to 10 J mol 1 and falls in this region. 2. Microwave region 10 10 10 12 Hz or 1 cm 100 µm wavelength: Energy changes due to molecular rotations are observed in this region, with E of the order of 100 J mol 1. 3. Infra-red region 10 12 10 14.5 Hz or 100 µm 1 µm wavelength: Molecular vibrations give one of the most valuable spectroscopy observed in this region. Energy separation in molecular vibrations is of the order of 10 4 J mol 1. 4. Visible and ultra-violet regions 10 14.5 10 16 Hz or 1 µm 10 nm wavelength: Energy separation between the valence electrons observed in this region is of the order of kj mol 1. 5. X-ray region 10 16 10 18 Hz or 10 nm 100 pm wavelength: Energy changes involving the inner orbital electron excitation of an atom or a molecule, could be of the order of 10 5 kj mol 1. 6. γ-ray region 10 18 110 20 Hz or 100 pm 1 pm wavelength: Energy changes involve the re-arrangement of nuclear particles, having energies of 10 9 10 11 Jg 1. C H A P T E R 4 Typical wavelength, frequencies, wave number and energies of photons of visible light are given in Table 4.1.

116 Fundamentals of Molecular Spectroscopy Table 4.1: Typical Wavelength, Frequencies, Wave Numbers and Energies of Visible Region Light Name of light Wavelength, Frequency, ν Wave number Energies E λ (nm) (Hz) ν (cm 1 ) (kj mol 1 ) Red 660 4.55 10 14 1.51 10 4 1.81 10 2 range 620 4.8 10 14 1.6 10 4 1.9 10 2 Yellow 580 5.2 10 14 1.7 10 4 2.1 10 2 Green 530 5.7 10 14 1.9 10 4 2.3 10 2 Blue 470 6.4 10 14 2.1 10 4 2.5 10 2 Violet 420 7.1 10 14 2.4 10 4 2.8 10 2 Near Ultraviolet 400 200 (7.515.0)10 14 (2.55)10 4 (3.06.0)10 2 4.3 INTENSITY F SPECTRAL LINES A recorded spectra is a study of variation of intensity of radiation absorbed/emitted with frequency. Such a study is always on a collection of molecules. A typical absorption spectra is shown in Fig. 4.5. Intensity, e A B l max Wavelength, nm Fig. 4.5 A typical absorption spectra, showing maximum absorption at wavelength λ max. AB correspond to width of spectra. The intensity of absorption can be expressed as transmittance (T), which is defined as the ratio of the intensity of the radiation transmitted from the sample (I) to that of the radiation incident on the sample (I 0 ) i.e., T = I/I 0...(4.7) Intensity of absorption is more conveniently expressed in terms of absorbance (A) i.e., A = log 1/T = log(i 0 /I)...(4.8) Absorbance of a band is related to the sample thickness (l) and concentration of the absorbing species. The relation is expressed in the form of Beer-Lambert law as below: A = log I 0 /I = εcl...(4.9)

Introduction to Molecular Spectroscopy 117 Absorbance is a dimensionless quantity. Concentration c is usually expressed in mol dm 3 and path length (l) in cm, hence ε extinction coefficient, has the units dm 3 mol 1 cm 1. If we use SI units of mol dm 3 for concentration and m for path length, the units of ε will be m 2 mol 1. We can obtain the values of ε in m 2 mol 1 units from those in dm 3 mol 1 cm 1 units in the following manner: ε = dm 3 mol 1 cm 1 =10 3 m 3 mol 1 (10 2 m 1 ) = 10 1 m 2 mol 1 Values of ε in SI units can therefore, easily be obtained from published values in dm 3 mol 1 cm 1 by dividing the numerical quantity in latter units by 10. λ max value: The value of wavelength at which absorption maximum occurs is called the λ max and is expressed in nanometer (nm). ε max value: It is known as molar absorptive or molar extinction coefficient is a measure of intensity of absorption. The ε value is characteristic of a particular compound at a given wavelength. Usually for the wavelength of maximum absorption λ max molar extinction coefficient ε max is expressed. Absorption bands with ε max value > 10 3 m 2 mol 1 are considered to be high intensity or strong bands where as those with ε max values < 10 2 m 2 mol 1 are known as low intensity or weak bands. Any spectrum will show lines with a variety of intensities. Some lines which could be expected do not appear at all, i.e., at an expected frequency ν = E/h no line is observed. In this connection, two main factors have to be considered. Transition Probability: The intensity of spectral lines arising from a molecular transition between a pair of states i and j, with wave functions Ψ i and Ψ j depends on transition probability. It is defined as R ij Ψ i MΨ j dτ...(4.10) Larger the value of R ij, more intense will be the spectral lines. The conditions under which the integral fails to vanish for the various possible pairs of states of a given system are known as selection rules. A transition between a pair of energy levels for which R ij vanishes is said to be a forbidden transition and may appear as a weak spectral line. It is this integral which give rise to selection rules for various spectra as will be discussed at many other places in the book. In most of cases only those transitions are allowed for which i j = 1. This is the selection rule and it holds well in all emission or absorption spectra. However, magnetic dipoles, electric quadruple interactions and other perturbations make the forbidden lines to appear many times but with very low intensities. Population in States: If transition from two energy levels to a third energy level are equally probable, then the most intense spectral line will arise from the level, which has the greater population. Boltzmann s law gives this population distinction; If n i is number of molecules with energy E i then the number of molecules n j with energy E j is n j = n (E j E i )/kt i bviously, before molecules are excited from state Ψ i, this state must be populated. Also, if the populations of two states are equal, no absorption will be observed. The total intensity of a band I λ is then given by I λ = c I 0 I ν e (E j E i )/kt C H A P T E R 4 where c is a proportionality constant, I 0 is a constant factor within a band system and the factor I ν incorporates the intensity due to the transition moment. It may be pointed out here that when several states have the same energy, the population increases correspondingly. Thus, if g is the number of states with energy E j and the lowest energy level E i is a

118 Fundamentals of Molecular Spectroscopy single states, then the number of molecules with energy E j is n j = gn i e(e j E i )/kt In the atomic spectra, for the transition of an electron from 2s to 2p orbital, g = 3, because p- orbitals are three fold degenerate. 4.4 WIDTH F SPECTRAL LINES Spectral transitions are functions of the measured position in terms of frequency, wavelength or wave numbers. An equally important factor is the profiles of the spectral lines. The profiles of spectral lines are complicated functions of the many perturbations that influence the absolute values of energy levels. A very qualitative treatment of the best understood factors that determine the profiles are given in terms of width of spectral lines. The spread of a spectra over a range of frequency is called width of spectra. The width ( ν) of a spectral band is defined in terms of the half bandwidth, that is width of band at half height. By definition, frequency ν = n/t, i.e., n waves that take t seconds to pass an observer, usually the spectrometer, but it is not possible to measure n and t with more precision than the resolving power of the instrument. This width inherent in the instrument in any transition is minimum, beyond which line sharpening, is not possible. The following other factors contribute to it. Heisenberg Uncertainty Principle: Natural line broadening is closely related to the uncertainty according to which δε lower. δt lower = h/2π...(4.11) where δε lower represent uncertainty in the lower energy level from which transition occurs and δt lower the uncertainty in the time for which the energy state is occupied. An analogous relation holds good for the upper energy state. The uncertainty in frequency associated with lack of precision in defining upper and lower quantum states are taken to be additive and are determined by the expression hδν = (δε upper + δε lower ) = δε δε = (2 π δν) 1...(4.12) Experimentally δν appears as a line width at half the maximum intensity and (2πδν) 1 is the uncertainty of time and may in a sense be considered the time over which absorption or emission takes place. Take the case of electronic spectra where the lifetime of a quantum state is of the order of 10 8, which gives an uncertainty in frequency of the order of 10 8 Hz. As the electronic spectrum is observed in the frequency region 10 14 10 16 Hz, comparatively, uncertainty is negligible. Radio frequency (10 8 10 9 Hz) is the other extreme of the spectrum where spin resonance spectroscopy lies. The energy difference in energy levels is very small in this region. Excited electron spin has a lifetime of 10 7 and thus uncertainty effect in it is always there. Doppler Broadening Doppler effect shifts the wavelength of light emitted by a moving atom. If the gas atom are moving at non-relativistic speeds, and the radiation is viewed in x-direction, the wavelength of light emitted by an atom with a component of velocity v x in the x direction is λ = λ o (1 + v x /c)

Introduction to Molecular Spectroscopy 119 The plus sign corresponds to an atom receding from the observer and minus sign to an atom approaching the observer. Molecules move towards as well as away from the observer and since there is continuous range of velocities, perfectly monochromatic radiators moving in this way would appear to emit a range of frequencies. This is called Doppler broadening of spectral line. The frequency of the light emitted by the moving atom is related to the natural frequency ν o by νo vx ν = ν o 1+ v 1± x c c The width ν of the intensity curve due to Doppler effect is given by 1/2 2νo 2kT log2 ν doppler = c M...(4.13) where k is Boltzmann constant, c velocity of light, M molar mass of gas. The quantity is usually much greater than natural width of line. It is clear that ν doppler is proportional to T 1/2, one should work at low temperatures to decrease the importance of the Doppler broadening. We also note from Eqn. 4.13 that Doppler broadening is proportional to the frequency ν o and inversely proportional to the square root of molar mass of gas molecule. Doppler broadening of peak is Gaussian in character. Collision Broadening The atoms emitting or absorbing radiation in a gas undergo collisions. In each collision, there is certain probability that a molecule initially in an excited state to make a radiationless transition to a lower state, so that the lifetime of excited state will be decreased. If the number of collision per second removing molecules from excited state is W c, the total number of transitions out of the excited state per second, is W c + 1/τ b, where τ b is the natural lifetime of the state b. The total observed breadth of the line will then be Γ = h (W c + 1/τ b ) The number of collision per second depends on the pressure of the gas, and this effect is, therefore, also called pressure broadening. Thus, in order to observe experimentally the natural line width it is essential that pressure in spectral source should be low. By changing the pressure and observing the corresponding change in line width, information about the collisions occurring in a gas can be obtained. If each collision changes the molecular state, the emitted radiation cannot be on the average, any longer than the average time between collision t c. The kinetic theory of gases shows that 1 t c =...(4.14) 2unσ where u is the mean velocity, n is the number of molecule per cm 3, σ is the effective cross sectional area for collision broadening, ν collision for such a model is given by ν collision = 2unσ...(4.15) n = number of molecules cm 3, V = 1 cm 3 so that PV=nRT gives n = P/RT Also from kinetic theory of gases u = 3RT/M so that C H A P T E R 4 3RT P 6 P υ collision = 2. σ = σ...(4.16) M RT R MT

120 Fundamentals of Molecular Spectroscopy Collision broadening is directly proportional to the square root of pressure. Lower the pressure lesser is the broadening. Like Doppler broadening, collision broadening is also inversely proportional to the square root of molar mass. However, it is inversely proportional to the square root of temperature. With this inverse relation to temperature, measurement made at lower temperature decreases Doppler broadening but increases collision broadening and vice versa with change of temperature. So, a spectra is recorded over a range of temperature and where minimum width is obtained, at that temperature, Doppler broadening and collision broadening balance one another. Collision broadening has also been used for obtaining information about collision cross-section of atoms. The frequency spread introduced by collision broadening is Lorenzian in character. Interaction Broadening When the electrons of one particle are influenced by electrons in the neighbouring particles, it results in a change of vibration and rotation frequencies. These interactions become noticeable in gases at high pressures and in liquids. Broadening at low pressure can also occur in gases when there is a tendency for the molecules to undergo slightly sticky collision leading to formation of weak quasimolecules in which inter-atomic distances are not sharply defined. In the case of solids, the motion of particles are more limited in extent and less random in direction, so that solid-phase spectra are often sharp but show splitting of lines into two or more components due to strong interactions between the molecules. Even in the absence of above mentioned broadening one has the inherent line width because of the finite life time of excited state due to spontaneous emission which gives a spread of frequency and it is Lorenzian in character. This type of broadening is called Natural Broadening. In the above we have considered, all the mechanisms present simultaneously. The resultant line shape function can be obtained by performing a convolution of different line shapes. If any one of the broadening mechanisms dominates over the other then the line shape would correspond to the dominant mechanism. ne of the utility of this understanding lies in Lasers and their application to second order or third order Raman spectra as discussed in Chapter 14. 4.5 IMPRTANCE F MLECULAR SPECTRSCPY Quantum mechanical calculations give the energy value of the various molecules energy levels. There is no experimental way by which to verify these absolute energy values. n the other hand, molecular spectra gives transition energies between various energy levels. If the difference between energy levels as calculated by quantum mechanics agrees with the corresponding spectral values implies that calculations are correct. So, molecular spectroscopy is the tool to the verification of quantum mechanical calculations. In other words, the two are so interrelated that without one other cannot be appreciated. Thus, two together give a deep insight to various processes that can take place in a molecule. Molecular spectroscopy helps us to give electronic structure of the molecule and thus explain the various possible reactions that can take place. From the energy difference values with the help of statistical mechanics complete partition function for the molecules involved in a chemical reaction can be written. From these partition function one can calculate reaction rate constant for such reactions. In other words, reaction rate constants and thus rates of various chemical reactions are related to the molecular energy levels. This also implies equilibrium constants and thus various thermodynamic quantities can also be calculated from these energy levels. If a molecule shows absorption spectra at wavelength λ, it implies that this radiation is interacting with this molecule. This implies that a molecule can be photochemically active for a light of frequency

Introduction to Molecular Spectroscopy 121 ν 1, only if it shows an absorption peak at this frequency. In case, this light does not interact with molecule, it may be photochemically inactive. Thus, it is the molecular spectroscopy that decides if a compound will be photochemically active or not. From the above discussion it can be stated conclusively that molecular spectroscopy, an experimental branch, is the heart of entire physical chemistry. Manuals are available that may help you to the identification of molecular structure from molecular spectra. However, without deeper understanding of the molecular processes, molecular interactions, the excitement associated with the subject will be lost and one may never be confident of implied results. So understanding of molecular spectroscopy is a must for understanding of chemical system. Problem 4.1: Certain metal ions show characteristic flame test. Whether the colored flame is a case of emission or absorption spectra. Why other metal ions like Zn 2+, Al 3+ do not show flame test? Solution: n placing metal ions in flame, it gets reduced to metal atoms which are excited by the flame. When excited atom falls back to ground state, it emits a radiation that lies in the visible region and thus can be seen. Since coloured flame results from excited to ground state transition, it is a case of emission spectra. The characteristic flame test shown by certain metal ions show the existence of certain quantized but fixed energy levels and hence a characteristic colours. In metal ions like Zn +2, Al +3, the quantized energy levels must be there but the energy difference between first excited state and ground state is so large that radiation emitted does not lie in visible region hence does not impart colour to the flame. Had the energy levels been continuous all metals should have shown flame test but not a characteristic flame. Take the case of copper imparts green colour (ν = 5.7 10 14 Hz) to flame. Energy associated with a flame is kt. Had the energy levels been continuous for ions/atoms to show a flame test this energy, kt should be equal to energy of green radiation or temperature required for this excitation should have been = h 34 14 ν 6. 626 10 57. 10 = k 23 = 27368 K 138. 10 But the maximum temperature that a burner can give is 1200 C = 1473 K. Since with 1473 K flame, 27368 K cannot be attained, hence something is missing. The reasoning say that all copper atoms get excited and when they fall back they emit green colour, which is not the case. Copper atoms have quantized energy levels. n placing copper in flame, copper atoms distribute to various energy levels. This distribution is given by the Boltzmann distribution from which fraction of molecules in excited state to emit green colour can be calculated i.e., 34 14 6. 626 10 5. 7 10 ni = e 23 1. 38 10 1473 = 8.53 10 9 no Thus, nearly 10 7 per cent of the copper atoms will go to excited state, which when fall back on the ground state emit green color. Atoms redistribute themselves so that 10 7 per cent again go to excited state and making the flame green. Since emission is a continuous process, redistribution is also a continuous process and thus a good green flame is obtained. Problem 4.2: The colour shown by some simple transition metal salts is a case of emission or absorption spectrum, Explain. Solution: The transition metal ions with incomplete filled d-orbitals are split into two types of d-orbitals by the electrical field produced by the anions around cations. The energy difference between these two types of d-orbitals is quite small and visible light excite the electron from ground state to excited state. When electron falls back to ground state it emits a radiation, which gives it a colour that lies in visible region. Again, this is a case of emission spectra. C H A P T E R 4

122 Fundamentals of Molecular Spectroscopy Problem 4.3: What is the uncertainty in momentum if we wish to locate an electron within an atom, say, so that x is approximately 50 pm? Solution: p = 34 ħ 1.05 10 Js = = 2.1 10 kg ms x 12 50 10 m 22 1 because p = mv and mass of electron 9.11 10 31 kg; this value of p correspond to v = 22 1 p 2.1 10 kg ms = = 2.3 10 ms m 31 9.11 10 kg 8 1 This is a very large uncertainty in speed, which has to be at least equal to velocity of electron. This velocity is so large that will break the atom. Hence we do not find electrons in the atomic nuclei. REFERENCES FR FURTHER READING 1. Hoffman B., The Strange Story of the Quantum, Dover, New York, 1959. 2. Feynman Richard P., Robert B. Sands Leighton and Mathew, The Feynman Lectures on Physics, Addison-Wesley, Massachusetts, 1963. 3. Atkins P.W., Molecular Quantum Mechanics, Clarendon Press, xford, 1970. 4. Hanna Melvin W., Quantum Mechanics in Chemistry, W.A. Benjamin Inc., New York, 1965. 5. Coulson C.A., Valence, xford University Press, London, 1961. 6. Kauzmanns W., Quantum Chemistry, Academic Press Inc., New York, 1957. 7. Louisell W.H., Quantum Statistical Properties of Radiation, Wiley, New York, 1973. 8. Carrathers P. and Niejo, M.M., Phase and Angle Variables in Quantum Mechanics, Rev. Mod. Physics 40, 411, 1968. 9. Scanberg Sune, Atomic and Molecular Spectroscopy Basic Aspects and Practical Approach. PRBLEMS 4.1. Calculate the energy per photon, energy per mole of photons, and momenta of photons when their wavelength is (a) 400 nm (blue), (b) 200 nm (UV) and (c) 1 cm (microwave). 4.2. A sample is irradiated with Na-D light (λ = 590 nm) of intensity 0.10 Wcm 2, (a) What is the amplitude of radiation s electric field strength? (b) What is the amplitude of radiation s magnetic induction B? (c) How many photons pass through 10 4 m 2 cross-section per seconds? 4.3. A particle is constrained to move in the XY-plane under potential V=1/2 k (x 2 +y 2 ). Write down the Hamiltonian in polar as well as Cartesian coordinates. 4.4. An electron moves in a one-dimensional box of length a with a positive charge at a/2. Show that the transition φ 1 φ 2 is allowed but φ 1 φ 3 is not.

Introduction to Molecular Spectroscopy 123 4.5. A simple model of conjugated polyenes allow their π-electrons to move freely along the chain of atoms. The molecule is then regarded as a collection of independent particles confined to a box and molecular orbital is taken to be square well wave functions. This is the free electron molecular orbitals (femo) picture of structure. Assuming that the potential energy is zero for the polyenes containing even number of carbon atoms, show that n n + 1, n n + 3, n n + 5, and so on, are the only electronic transitions which are allowed. (Dipole moment = ex) where n is the highest filled molecular orbital. 4.6. An atom of oxygen is expelled from a molecule of hemoglobin in a way that fixes its position parallel to the principal plane of hemoglobin with an uncertainty y of 0.15 nm. What is the uncertainty in momentum p, along the y-coordinate axis? 4.7. What is the Doppler shifted wavelength of a red (λ = 660 nm) traffic light when a body moving at 50 mph approaches it? At what speed of approach would red light appear to be green (λ = 520 nm)? 4.8. When an electron in an atom is excited into a level above the ground state, it remains there for sometime of the order of 10 8 seconds. Use the uncertainty principle to compute the minimum width of the spectral line (λ = 5000 Å) emitted when the electron returns to the ground state (This is called the natural line width of a spectral line). 4.9 Why certain organic compounds are colored and other are white? Explain. C H A P T E R 4