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Chapter 12 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 12 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 12 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 12 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 12 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula
CHAPTER 12 Rotational and vibrational spectra The origin of spectral lines in molecular spectroscopy is the absorption, emission, or scattering of a photon when the energy of a molecule changes The difference from atomic spectroscopy (Topic 9C) is that the energy of a molecule can change not only as a result of electronic transitions but also because it can undergo changes of rotational and vibrational state Molecular spectra are therefore more complex than atomic spectra However, they also contain information relating to more properties, and their analysis leads to values of bond strengths, lengths, and angles They also provide a way of determining a variety of molecular properties, such as dipole moments The general strategy we adopt in this chapter is to set up expressions for the energy levels of molecules and then infer the form of rotational and vibrational spectra Electronic spectra are considered in Chapter 13 12A General features of molecular spectroscopy This Topic begins with a discussion of the theory of absorption and emission of radiation, leading to the factors that determine the intensities and widths of spectral lines Then we describe features of instrumentation used to monitor the absorption, emission, and scattering of radiation spanning a wide range of frequencies 12B Molecular rotation In this Topic we see how to derive expressions for the values of the rotational energy levels of diatomic and polyatomic molecules The most direct procedure, which we adopt, is to identify the expressions for the energy and angular momentum obtained in classical physics, and then to transform these expressions into their quantum mechanical counterparts 12C Rotational spectroscopy This Topic focuses on the interpretation of pure rotational and rotational Raman spectra, in which only the rotational state of a molecule changes We explain in terms of nuclear spin and the Pauli principle the observation that not all molecules can occupy all rotational states 12D Vibrational spectroscopy of diatomic molecules In this Topic we consider the vibrational energy levels of diatomic molecules and see that we can use the properties of harmonic oscillators developed in Topic 8B, but must also take into account deviations from harmonic oscillation We also see that vibrational spectra of gaseous samples show features that arise from the rotational transitions that accompany the excitation of vibrations 12E Vibrational spectroscopy of polyatomic molecules The vibrational spectra of polyatomic molecules may be discussed as though they consisted of a set of independent harmonic oscillators, so the same approach as employed for diatomic molecules may be used We also see that the symmetry properties of the atomic displacements of polyatomic molecules are helpful for deciding which modes of vibration can be studied spectroscopically What is the impact of this material? Molecular spectroscopy is also useful to astrophysicists and environmental scientists In Impact I12.1 we see how the 12 Rotational and vibrational spectra identities of molecules found in interstellar space can be inferred from their rotational and vibrational spectra In Impact I12.2 we turn our attention back towards the Earth and see how the vibrational properties of its atmospheric constituents can affect its climate 475 To read more about the impact of this material, scan the QR code, or go to bcs.whfreeman.com/webpub/chemistry/ pchem10e/impact/pchem-12-1.html 12A General features of molecular spectroscopy Contents 12A.1 The absorption and emission of radiation Stimulated and spontaneous radiative processes Brief illustration 12A.1: The Einstein coefficients (b) Selection rules and transition moments (c) The Beer–Lambert law Example 12A.1: Determining a molar absorption coefficient ➤➤ What you need to know already? 477 (a) 12A.2 Spectral linewidths Doppler broadening Brief illustration 12A.2: Doppler broadening (b) Lifetime broadening Brief illustration 12A.3: Lifetime broadening (a) 12A.3 Experimental techniques Sources of radiation (b) Spectral analysis Example 12A.2: Calculating a Fourier transform (c) Detectors (d) Examples of spectrometers (a) Checklist of concepts Checklist of equations 477 478 478 479 480 480 481 481 482 482 482 482 483 484 485 485 486 487 ➤➤ Why you need to know this material? To interpret data from the wide range of varieties of molecular spectroscopy you need to understand the experimental and theoretical features that all types of spectra share ➤➤ What is the key idea? A transition from a low energy state to one of higher energy can be stimulated by absorption of electromagnetic radiation; a transition from a higher to a lower state may be either spontaneous (resulting in emission of radiation) or stimulated by radiation You need to be familiar with quantization of energy in molecules (Topics 8A–8C), and the concept of selection rules in spectroscopy (Topic 9C) In emission spectroscopy, a molecule undergoes a transition from a state of high energy E1 to a state of lower energy E2 and emits the excess energy as a photon In absorption spectroscopy, the net absorption of incident radiation is monitored as its frequency is varied We say net absorption, because, when a sample is irradiated, both absorption and emission at a given frequency are stimulated, and the detector measures the difference, the net absorption In Raman spectroscopy, changes in molecular state are explored by examining the frequencies present in the radiation scattered by molecules The energy, hν, of the photon emitted or absorbed, and therefore the frequency ν of the radiation emitted or absorbed, is given by the Bohr frequency condition (eqn 7A.12 of Topic 7A, hν = |E1 − E2|) Emission and absorption spectroscopy give the same information about electronic, vibrational, or rotational energy level separations, but practical considerations generally determine which technique is employed In Raman spectroscopy the difference between the frequencies of the scattered and incident radiation is determined by the transitions that take place within the molecule as a result of the impact of the incoming photon; this technique is used to study molecular vibrations and rotations About in 107 of the incident photons that collide with the molecules, give up some of their energy, and emerge with a lower energy These scattered photons constitute the lower-frequency Stokes radiation from the sample (Fig 12A.1) Other incident photons may collect energy from the molecules (if they are already excited), and emerge as higher-frequency anti-Stokes radiation The component scattered without change of frequency is called Rayleigh radiation Atomic spectroscopy is discussed in Topic 9C Here we set the stage for detailed discussion of rotational (Topics 12B 12A General features of molecular spectroscopy Energy Anti-Stokes Rayleigh Incident radiation Stokes energy state to one of higher energy that is driven by the electromagnetic field oscillating at the transition frequency This process is called stimulated absorption The rate of this type of transition is proportional to the intensity of the incident radiation: the more intense the incident radiation, the greater is the rate of the transition and the stronger is the absorption by the sample Einstein wrote this transition rate as w f ←i = Bfi ρ Figure 12A.1 In Raman spectroscopy, an incident photon is scattered from a molecule with either an increase in frequency (if the radiation collects energy from the molecule) or with a lower frequency (if it loses energy to the molecule) to give the anti-Stokes and Stokes lines, respectively Scattering without change of frequency results in the Rayleigh lines The process can be regarded as taking place by an excitation of the molecule to a wide range of states (represented by the shaded band), and the subsequent return of the molecule to a lower state; the net energy change is then carried away by the photon and 12C), vibrational (Topics 12D and 12E), and electronic (the several Topics of Chapter 13) transitions in molecules Techniques that probe transitions between spin states of electrons and nuclei are also useful They rely on special experimental approaches and theoretical considerations described in Chapter 14 12A.1 The absorption and emission of radiation As mentioned in Foundations B, the separation of rotational energy levels (in small molecules, ΔE ≈ 0.01 zJ, corresponding to about 0.01 kJ mol−1) is smaller than that of vibrational energy levels (ΔE ≈ 10 zJ, corresponding to 10 kJ mol−1), which itself is smaller than that of electronic energy levels (ΔE ≈ 0.1–1 aJ, corresponding to about 102–103 kJ mol−1) From ν = ΔE/h, it follows that rotational, vibrational, and electronic transitions result from the absorption of emission of microwave, infrared, and ultraviolet/visible/far infrared radiation, respectively (see also Chapter 8) Here we turn our attention to the origins of spectroscopic transitions, focusing on concepts that apply generally to all varieties of spectroscopy (a) Stimulated and spontaneous radiative processes Albert Einstein identified three contributions to the transitions between states First, he recognized the transition from a low 477 Stimulated absorption Transition rate (12A.1) The constant Bfi is the Einstein coefficient of stimulated absorption and ρdν is the energy density of radiation in the frequency range from ν to ν + dν, where ν is the frequency of the transition For instance, when the atom or molecule is exposed to black-body radiation from a source of temperature T, ρ is given by the Planck distribution (eqn 7A.6 of Topic 7A): ρ= 8πh3 /c eh/kT −1 Planck distribution (12A.2) At this stage Bfi is an empirical parameter that characterizes the transition: if it is large, then a given intensity of incident radiation will induce transitions strongly and the sample will be strongly absorbing The total rate of absorption, Wf←i, is the transition rate of a single molecule multiplied by the number of molecules Ni in the lower state: Wf ←i = N i w f ←i = N i Bfi ρ Total absorption rate (12A.3) Einstein considered that the radiation was also able to induce the molecule in the upper state to undergo a transition to the lower state, and hence to generate a photon of frequency ν Thus, he wrote the rate of this stimulated emission as w f →i = Bif ρ Stimulated emission Transition rate (12A.4) where Bif is the Einstein coefficient of stimulated emission This coefficient is in fact equal to the coefficient of stimulated absorption as we shall see below Moreover, only radiation of the same frequency as the transition can stimulate an excited state to fall to a lower state At this point, it is tempting to suppose that the total rate of emission is this individual rate multiplied by the number of molecules in the upper state, Nf, and therefore to write Wf→i = NfBif ρ But here we encounter a problem: at equilibrium (as in a black-body container), the rate of emission is equal to the rate of absorption, so NiBfiρ = NfBif ρ and therefore, since Bif= Bfi, Ni = Nf The conclusion that the populations must be equal at equilibrium is in conflict with another very fundamental conclusion, that the ratio of populations is given by the Boltzmann distribution (Foundations B and Topic 15A) which implies that Ni ≠ Nf Einstein realized that to bring the analysis of transition rates into alignment with the Boltzmann distribution there must be 478 12 Rotational and vibrational spectra another route for the upper state to decay into the lower state, and wrote w f →i = A + Bif ρ Emission rate (12A.5) The constant A is the Einstein coefficient of spontaneous emission The total rate of emission, Wf→i, is therefore Wf →i = N f w f →i = N f ( A + Bif ρ) Total emission rate (12A.6) At thermal equilibrium, Ni and Nf not change over time This condition is reached when the total rates of emission and absorption are equal: N i Bfi ρ = N f ( A + Bif ρ) Thermal equilibrium (12A.7) and therefore ρ= Nf A N i Bfi − N f Bif divide by Nf Bfi = A/Bfi A/Bfi = N i /N f − Bif /Bfi eh/kT − Bif /Bfi 6.00 × 1011 s −1 To assess the relative significance of spontaneous emission, with rate A, and stimulated emission, with rate Bρ, at 298 K, we rearrange eqn 12A.8, with B = Bfi = Bif, when it becomes ρ= to form the ratio A −34 11 −1 −23 −1 = eh/kT − = e(6.626 ×10 J s) × (6.00 ×10 s )/(1.381×10 J K ) × (298 K ) −1 Bρ = 0.101 and both spontaneous and stimulated emission are significant at this wavelength Self-test 12A.1 Calculate the ratio A/Bρ at 298 K for a transition in the infrared region of the electromagnetic spectrum, corresponding to excitation of a molecular vibration, with wavenumber 2000 cm−1 What conclusion can you draw? Answer: A/Bρ = 1.6 × 104; for vibrational transitions spontaneous emission is more significant than stimulated emission (12A.8) We have used the Boltzmann expression (Foundations B and Topic 15A) for the ratio of populations of the upper state (of energy Ef ) and lower state (of energy Ei): h Nf = e −( E −E )/kT Ni f i This result has the same form as the Planck distribution (eqn 12A.2), which describes the radiation density at thermal equilibrium Indeed when we compare eqns 12A.2 and 12A.8, we can conclude that Bif = Bfi (as we promised to show) and that 8πh3 A= B c (12A.9) The important point about eqn 12A.9 is that it shows that the relative importance of spontaneous emission increases as the cube of the transition frequency and therefore that it is therefore potentially of great importance at very high frequencies Conversely, spontaneous emission can be ignored at low transition frequencies, in which case intensities of those transitions can be discussed in terms of stimulated emission and absorption alone Brief illustration 12A.1 The Einstein coefficients For a transition in the microwave region of the electromagnetic spectrum (corresponding to an excitation of a molecular rotation), a typical frequency is 600 GHz (1 GHz = 109 Hz), or A/B eh/kT −1 (b) Selection rules and transition moments We first met the concept of a ‘selection rule’ in Topic 9C as a statement about whether a transition is forbidden or allowed Selection rules also apply to molecular spectra, and the form they take depends on the type of transition The underlying classical idea is that, for the molecule to be able to interact with the electromagnetic field and absorb or create a photon of frequency ν, it must possess, at least transiently, a dipole oscillating at that frequency In Topic 9C it is shown that this transient dipole is expressed quantum mechanically in terms of the transition dipole moment, μfi, between states ψi and ψf: ∫ μfi = ψ f* μψ i dτ Definition Transition dipole moment (12A.10) where μˆ is the electric dipole moment operator The size of the transition dipole can be regarded as a measure of the charge redistribution that accompanies a transition: a transition is active (and generates or absorbs photons) only if the accomp anying charge redistribution is dipolar (Fig 12A.2) Only if the transition dipole moment is nonzero does the transition contribute to the spectrum It follows that, to identify the selection rules, we must establish the conditions for which μfi ≠ 0 A gross selection rule specifies the general features that a molecule must have if it is to have a spectrum of a given kind For instance, in Topic 12C it is shown that a molecule gives a rotational spectrum only if it has a permanent electric dipole moment This rule and others like it for other types of transition 12A General features of molecular spectroscopy x 479 x + dx Intensity, I – dI Intensity, I (a) Length, L (b) Figure 12A.2 (a) When a 1s electron becomes a 2s electron, there is a spherical migration of charge There is no dipole moment associated with this migration of charge, so this transition is electric-dipole forbidden (b) In contrast, when a 1s electron becomes a 2p electron, there is a dipole associated with the charge migration; this transition is allowed are explained in relevant Topics A detailed study of the transition moment leads to the specific selection rules that express the allowed transitions in terms of the changes in quantum numbers (c) The Beer–Lambert law Consider the absorption of radiation by a sample It is found empirically that the transmitted intensity I varies with the length, L, of the sample and the molar concentration, [J], of the absorbing species J in accord with the Beer–Lambert law: I = I 10− ε[J]L Beer–Lambert law (12A.11) where I0 is the incident intensity The quantity ε (epsilon) is called the molar absorption coefficient (formerly, and still widely, the ‘extinction coefficient’) The molar absorption coefficient depends on the frequency of the incident radiation and is greatest where the absorption is most intense Its dimensions are 1/(concentration × length), and it is normally convenient to express it in cubic decimetres per mole per centimetre (dm3 mol−1 cm−1); in SI base units it is expressed in metressquared per mole (m2 mol−1) The latter units imply that ε may be regarded as a (molar) cross-section for absorption and that the greater the cross-sectional area of the molecule for absorption, the greater is its ability to block the passage of the incident radiation at a given frequency The Beer–Lambert law is an empirical result However, it is simple to account for its form as we show in the following Justification Justification 12A.1 The Beer–Lambert law We think of the sample as consisting of a stack of infinitesimal slices, like sliced bread (Fig 12A.3) The thickness of each layer is dx The change in intensity, dI, that occurs when electromagnetic radiation passes through one particular slice is Figure 12A.3 To establish the Beer–Lambert law, the sample is supposed to be sliced into a large number of planes The reduction in intensity caused by one plane is proportional to the intensity incident on it (after passing through the preceding planes), the thickness of the plane, and the concentration of absorbing species proportional to the thickness of the slice, the concentration of the absorber J, and the intensity of the incident radiation at that slice of the sample, so dI ∝[J]Idx Because dI is negative (the intensity is reduced by absorption), we can write dI = –κ [J]Idx where κ (kappa) is the proportionality coefficient Division of both sides by I gives dI = −κ [J]dx I This expression applies to each successive slice To obtain the intensity that emerges from a sample of thickness L when the intensity incident on one face of the sample is I0, we sum all the successive changes Because a sum over infinitesimally small increments is an integral, we write: Integral A.2 ln( I /I0 ) I dI = −κ I0 I ∫ ∫ L [J]dx [J] uniform = Integral A.1 L −κ [J] ∫ L dx in the second step we have supposed that the concentration is uniform, so [J] is independent of x and can be taken outside the integral Therefore ln I = −κ [J]L I0 Because ln x = (ln 10)log x, we can write ε = κ/ln 10 and obtain log I = −ε[J]L I0 which, on taking (common) antilogarithms, is the Beer– Lambert law (eqn 12A.11) 480 12 Rotational and vibrational spectra The spectral characteristics of a sample are commonly reported as the transmittance, T, of the sample at a given frequency: I I0 Definition Transmittance (12A.12) and the absorbance, A, of the sample: A = log I0 I Definition Answer: 1.1 × 104 dm3 mol−1 cm−1, A = 0.17, T = 0.68 Absorbance (12A.13) The two quantities are related by A = −log T (note the common logarithm) and the Beer–Lambert law becomes A = ε[J]L (12A.14) The product ε[J]L was known formerly as the optical density of the sample Example 12A.1 Determining a molar absorption coefficient Radiation of wavelength 280 nm passed through 1.0 mm of a solution that contained an aqueous solution of the amino acid tryptophan at a concentration of 0.50 mmol dm−3 The light intensity is reduced to 54 per cent of its initial value (so T = 0.54) Calculate the absorbance and the molar absorption coefficient of tryptophan at 280 nm What would be the transmittance through a cell of thickness 2.0 mm? Method From A = −log T = ε [J]L , it fol lows t hat ε = − (log T)/[J]L For the transmittance through the thicker cell, we use T = 10−A and the value of ε calculated here Solution The molar absorption coefficient is ε= − log 0.54 = 5.4 ×102 dm mol −1 mm −1 (5.0 ×10−4 mol dm −3 ) × (1.0 mm) These units are convenient for the rest of the calculation (but the outcome could be reported as 5.4 × 103 dm3 mol−1 cm−1 if desired) The absorbance is A = – log 0.54 = 0.27 contained the amino acid tyrosine at a molar concentration of 0.10 mmol dm−3 was measured as 0.14 at 240 nm in a cell of length 5.0 mm Calculate the molar absorption coefficient of tyrosine at that wavelength and the absorbance of the solution What would be the transmittance through a cell of length 1.0 mm? The absorbance of a sample of length 2.0 mm is A = (5.4 ×102 dm mol −1 mm −1) × (5.0 ×10−4 mol dm −3 ) × (2.0 mm) = 0.54 It follows that the transmittance is now T = 10 – A = 10 –0.54 = 0.29 That is, the emergent light is reduced to 29 per cent of its incident intensity The maximum value of the molar absorption coefficient, εmax, is an indication of the intensity of a transition However, as absorption bands generally spread over a range of wavenumbers, quoting the absorption coefficient at a single wavenumber might not give a true indication of the intensity of a transition The integrated absorption coefficient, A, is the sum of the absorption coefficients over the entire band (Fig 12A.4), and corresponds to the area under the plot of the molar absorption coefficient against wavenumber: A= ∫ ε( )d Definition Integrated absorption coefficient (12A.15) band For lines of similar widths, the integrated absorption coefficients are proportional to the heights of the lines 12A.2 Spectral linewidths A number of effects contribute to the widths of spectroscopic lines Some contributions to linewidths can be modified by changing the conditions, and to achieve high resolutions we need to know how to minimize these contributions Other contributions cannot be changed, and represent an inherent limitation on resolution Molar absorption coefficient, ε T= Self-test 12A.2 The transmittance of an aqueous solution that Area = integrated g absorption coefficient, coeff ficient, icient A ~ Wavenumber, ν Figure 12A.4 The integrated absorption coefficient of a transition is the area under a plot of the molar absorption coefficient against the wavenumber of the incident radiation 12A General features of molecular spectroscopy (a) Doppler broadening One important broadening process in gaseous samples is the Doppler effect, in which radiation is shifted in frequency when the source is moving towards or away from the observer When a source emitting electromagnetic radiation of frequency ν moves with a speed s relative to an observer, the observer detects radiation of frequency − s/c receding = + s/c 1/2 1+ s/c approaching = 1− s/c approaching ≈ 1− s/c (12A.16b) Atoms and molecules reach high speeds in all directions in a gas, and a stationary observer detects the corresponding Doppler-shifted range of frequencies Some molecules approach the observer, some move away; some move quickly, others slowly The detected spectral ‘line’ is the absorption or emission profile arising from all the resulting Doppler shifts As shown in the following Justification, the profile reflects the distribution of velocities parallel to the line of sight, which is a bellshaped Gaussian curve The Doppler line shape is therefore also a Gaussian (Fig 12A.5), and we show in the Justification that, when the temperature is T and the mass of the atom or molecule is m, then the observed width of the line at half-height (in terms of frequency or wavelength) is 2 2kT ln c m 1/2 δλobs = 2λ 2kT ln c m 1/2 Doppler broadening (12A.17) Absorption intensity δobs = Brief illustration 12A.2 Doppler broadening where c is the speed of light For nonrelativistic speeds (s ≪ c), these expressions simplify to 1+ s/c Doppler broadening increases with temperature because the molecules acquire a wider range of speeds Therefore, to obtain spectra of maximum sharpness, it is best to work with cool samples For a molecule like N2 at T = 300 K, 1/2 Doppler shifts (12A.16a) receding ≈ 481 T/3 δobs 2kT ln = c mN2 1/2 = 2.998 ×108 m s −1 kg g m2 s−2 23 − × 1.380 ×10 J K −1 × (300 K) × ln × −26 4.653 × 10 kg 1/2 = 2.34 × 10−6 For a transition wavenumber of 2331 cm−1 (from the Raman spectrum of N2), corresponding to a frequency of 69.9 THz (1 THz = 1012 Hz), the linewidth is 164 MHz Self-test 12A.3 What is the Doppler-broadened linewidth of the transition at 821 nm in atomic hydrogen at 300 K? Answer: 4.38 GHz Justification 12A.2 Doppler broadening It follows from the Boltzmann distribution (Foundations B and Topic 15A) that the probability that an atom or molecule of mass m and speed s in a gas phase sample at a temperature T has kinetic energy E k = 12 ms is proportional to e −ms /2 kT The observed frequencies, ν obs , emitted or absorbed by the molecule are related to its speed by eqn 12A.16b When s ≪ c, the Doppler shift in the frequency is obs − ≈ ± s / c T 3T Frequency Figure 12A.5 The Gaussian shape of a Doppler-broadened spectral line reflects the Maxwell distribution of speeds in the sample at the temperature of the experiment Notice that the line broadens as the temperature is increased More specifically, the intensity I of a transition at νobs is proportional to the probability of there being an atom that emits or absorbs at ν obs , so it follows from the Boltzmann distribution and the expression for the Doppler shift in the form s = (νobs − ν)c/ν that I (obs ) ∝ e − mc (obs − )2 /2 2 kT (12A.18) which has the form of a Gaussian function Because the width 2 at half-height of a Gaussian function ae −( x −b) /2σ (where a, b, and σ are constants) is δx = 2σ(2 ln 2)1/2, δνobs can be inferred directly from the exponent of eqn 12A.18 to give eqn 12A.17 482 12 Rotational and vibrational spectra (b) Lifetime broadening It is found that spectroscopic lines from gas-phase samples are not infinitely sharp even when Doppler broadening has been largely eliminated by working at low temperatures This residual broadening is due to quantum mechanical effects Specifically, when the Schrödinger equation is solved for a system that is changing with time, it is found that it is impossible to specify the energy levels exactly If on average a system survives in a state for a time τ, the lifetime of the state, then its energy levels are blurred to an extent of order δE ≈ ħ/τ With the energy spread expressed as a wavenumber through δE = hcδ , and the values of the fundamental constants introduced, this relation becomes δ ≈ 5.3 cm −1 τ /ps Lifetime broadening (12A.19) and given an indication of lifetime broadening of spectral lines No excited state has an infinite lifetime; therefore, all states are subject to some lifetime broadening and the shorter the lifetimes of the states involved in a transition the broader are the corresponding spectral lines Brief illustration 12A.3 Lifetime broadening A typical electronic excited state natural lifetime is about τ = 10−8 s = 1.0 × 104 ps, corresponding to a linewidth of δ ≈ 5.3 cm −1 = 5.3 ×10−4 cm −1 1.0 ×104 which corresponds to 16 MHz Self-test 12A.4 Consider a molecular rotation with a lifetime of about 103 s What is the linewidth of the spectral line? Answer: 5 × 10 −15 cm−1 (of the order of 10 −4 Hz) Two processes are responsible for the finite lifetimes of excited states The dominant one for low frequency transitions is collisional deactivation, which arises from collisions between atoms or with the walls of the container If the collisional lifetime, the mean time between collisions, is τcol, the resulting collisional linewidth is δEcol ≈ ħ/τcol Because τcol = 1/z, where z is the collision frequency, and from the kinetic model of gases (Topic 1B), which implies that z is proportional to the pressure, we conclude that the collisional linewidth is proportional to the pressure The collisional linewidth can therefore be minimized by working at low pressures The rate of spontaneous emission cannot be changed It is a natural limit to the lifetime of an excited state which cannot be changed by modifying the conditions, and the resulting lifetime broadening is the natural linewidth of the transition Because the rate of spontaneous emission increases as ν3, the lifetime of the excited state decreases as ν3, and the natural linewidth increases with the transition frequency Thus, rotational (microwave) transitions occur at much lower frequencies than vibrational (infrared) transitions and consequently have much longer lifetimes and hence much smaller natural linewidths: at low pressures rotational linewidths are due principally to Doppler broadening 12A.3 Experimental techniques We now turn to practical aspects of molecular spectroscopy Common to all spectroscopic techniques is a spectrometer, an instrument that detects the characteristics of radiation scattered, emitted, or absorbed by atoms and molecules As an example, Fig 12A.6 shows the general layout of an absorption spectrometer Radiation from an appropriate source is directed toward a sample and the radiation transmitted strikes a device that separates it into different frequencies The intensity of radiation at each frequency is then analysed by a suitable detector (a) Sources of radiation Sources of radiation are either monochromatic, those spanning a very narrow range of frequencies around a central value, or polychromatic, those spanning a wide range of frequencies Monochromatic sources that can be tuned over a range of frequencies include the klystron and the Gunn diode, which operate in the microwave range, and lasers (Topic 13C) Polychromatic sources that take advantage of black-body radiation from hot materials (Topic 7A) can be used from the infrared to the ultraviolet regions of the electromagnetic spectrum Examples include mercury arcs inside a quartz envelope (35 cm −1 < < 200 cm −1 ), Nernst filaments and globars (200 cm −1 < < 4000 cm −1 ), and quartz–tungsten–halogen lamps (320 nm