Physical Chemistry 2 nd Edition

Chapter 19 The Vibrational and Rotational Spectroscopy of Diatomic Molecules Physical Chemistry 2nd Edition Thomas Engel, Philip Reid

Objectives • Describe how light interacts with molecules to induce transitions between states • Discuss the absorption of electromagnetic radiation

Outline • An Introduction to Spectroscopy • Absorption, Spontaneous Emission, and Stimulated Emission • An Introduction to Vibrational Spectroscopy • The Origin of Selection Rules • Infrared Absorption Spectroscopy • Rotational Spectroscopy

19.1 An Introduction to Spectroscopy • Spectroscopy are tools chemists have to probe the species at an atomic and molecular level. • The frequency at which energy is absorbed or emitted is related to the energy levels involved in the transitions by

19.1 An Introduction to Spectroscopy • 19.1 Energy Levels and Emission Spectra

19.1 An Introduction to Spectroscopy • During vibration, oscillator will absorb energy in both the stretching and compression. • The molecule can absorb energy from the field during oscillation.

19.2 Absorption, Spontaneous Emission, and Stimulated Emission • The 3 basic processes by which photon-assisted transitions occur are absorption, spontaneous emission and stimulated emission.

19.2 Absorption, Spontaneous Emission, and Stimulated Emission • In absorption, the incident photon induces a transition to a higher level. • In emission, a photon is emitted as an excited state relaxes to one of lower energy. • Spontaneous emission is a random event and its rate is related to the lifetime of the excited state.

19.2 Absorption, Spontaneous Emission, and Stimulated Emission • At equilibrium, where = radiation density at frequency ν = rate coefficient • Einstein concluded that

Example 19.1 Derive the equations using these two pieces of information: (1) the overall rate of transition between levels 1 and 2 is zero at equilibrium, and (2) the ratio of N2 to N1 is governed by the Boltzmann distribution.

Solution The rate of transitions from level 1 to level 2 is equal and opposite to the transitions from level 2 to level 1. This gives the equation . The Boltzmann distribution function states that

Solution In this case . These two equations can be solved for , giving . Planck has showed that For these two expressions to be equal

19.3 An Introduction to Vibrational Spectroscopy • The vibrational frequency depends on two identity vibrating atoms on both end of the bond. • This property generates characteristic frequencies for atoms joined by a bond known as group frequencies.

Example 19.2 A strong absorption of infrared radiation is observed for 1H35Cl at 2991 cm-1. a. Calculate the force constant, k, for this molecule. b. By what factor do you expect this frequency to shift if deuterium is substituted for hydrogen in this molecule? The force constant is unaffected by this substitution.

Solution a. We first write . Solving for k, b. The vibrational frequency for DCl is lower by a substantial amount.

19.3 An Introduction to Vibrational Spectroscopy • 19.2 The Morse Potential • The bond energy D0 is defined with respect to the lowest allowed level, rather than to the bottom of the potential. • The energy level is

19.3 An Introduction to Vibrational Spectroscopy • Parameters for selected model are shown.

19.4 The Origin of Selection Rules • The transition probability from state n to state m is only nonzero if the transition dipole moment satisfies the following condition: where x = spatial variable μx = dipole moment along the electric field direction

19.5 Infrared Absorption Spectroscopy • Atoms and molecules possess a discrete energy spectrum that can only be absorbed or emitted which correspond to the difference between two energy levels. • Beer-Lambert law states that where I(λ) = intensity of light leaving the cell I0(λ) = intensity of light passing dl distancel = path length ε(λ) = molar absorption coefficient

Example 19.4 The molar absorption coefficient for ethane is 40 (cm bar)-1 at a wavelength of 12 μm. Calculate in a 10-cm-long absorption cell if ethane is present at a contamination level of 2.0 ppm in one bar of air. What cell length is required to make ?

Solution Using This result shows that for this cell length, light absorption is difficult to detect. Rearranging the Beer-Lambert equation, we have

19.5 Infrared Absorption Spectroscopy • Coupled system has two vibrational frequencies: the symmetrical and antisymmetric modes. • For symmetrical and asymmetrical, the vibrational frequency is

19.6 Rotational Spectroscopy • 19.3 Normal Modes for H2O • 19.4 Normal Modes for CO2 • 19.5 Normal Modes for NH3 • 19.6 Normal Modes for Formaldehyde

Example 19.5 Using the following total energy eigenfunctions for the three-dimensional rigid rotor, show that the J=0 → J=1 transition is allowed, and that the J=0 → J=2 transition is forbidden: The notation is used for the preceding functions.

Solution Assuming the electromagnetic field to lie along the zaxis, , and the transition dipole moment takes the form For the J=0 → J=1 transition,

Solution For the J=0 → J=2 transition, The preceding calculations show that the J=0 → J=1 transition is allowed and that the J=0 → J=2 transition is forbidden. You can also show that is also zero unless MJ=0 .

19.6 Rotational Spectroscopy • For vibrational spectroscopy, we have to change the symbol for the angular momentum quantum number from l to J. • Thus the dependence of the rotational energy on the quantum number is given by where rotational constant is

19.6 Rotational Spectroscopy • We can calculate the energy corresponding to rotational transitions

Example 19.5 Because of the very high precision of frequency measurements, bond lengths can be determined with a correspondingly high precision, as illustrated in this example. From the rotational microwave spectrum of 1H35Cl, we find that B=10.59342cm-1. Given that the masses of 1H and 35Cl are 1.0078250 and 34.9688527 amu, respectively, determine the bond length of the 1H35Cl molecule.

Solution We have

19.6 Rotational Spectroscopy • To excite various transitions consistent with the selection rule , we have

19_16fig_PChem.jpg

, R: , Q: (vibrational ). P,Q,R branches of rotational spectrum

19.6 Rotational Spectroscopy • 19.7 Rotational Spectroscopy of Diatomic Molecules • 19.8 Rotational-Vibrational Spectroscopy of Diatomic Molecules • The ratio for value of J relative to the number in the ground state (J=0) can be calculated using the Boltzmann distribution:

Rotational Raman Spectra The molecule can be made anisotropically polarized and Raman active. Selection Rules: Linear rotors Symmetrical rotors

Proof of Rotational Raman Selection Rules Selection rules

A Typical Rotational Raman Spectrum (Linear rotors) (Linear rotors) Stokes lines Anti-Stokes lines

19_21fig_PChem.jpg Vibrational Raman effect, Δ n=+1,-1

Physical Chemistry 2 nd Edition