The Planck Hypothesis In order to explain the frequency distribution of radiation from a hot cavity (blackbody radiation) Planck proposed the ad hoc assumption that the radiant energy could exist only in discrete quanta which were proportional to the frequency. This would imply that higher modes would be less populated and avoid the ultraviolet catastrophe of the Rayleigh-Jeans Law. The quantum idea was soon seized to explain the photoelectric effect, became part of the Bohr theory of discrete atomic spectra, and quickly became part of the foundation of modern quantum theory. http://hyperphysics.phy-astr.gsu.edu/hbase/mod2.html#c3 The Photoelectric Effect The remarkable aspects of the photoelectric effect when it was first observed were: 1. The electrons were emitted immediately - no time lag! 2. Increasing the intensity of the light increased the number of photoelectrons, but not their maximum kinetic energy! 3. Red light will not cause the ejection of The details of the photoelectric effect were in electrons, no matter what the intensity! direct contradiction to the expectations of very 4. A weak violet light will eject only a few well developed classical physics. electrons, but their maximum kinetic energies are greater than those for intense The explanation marked one of the major steps light of longer wavelengths! toward quantum theory. http://hyperphysics.phy-astr.gsu.edu/hbase/mod1.html#c2 http://hyperphysics.phy-astr.gsu.edu/hbase/grexp.html#c1 Properties of Molecules The investigation of molecular structure parallels the study of atomic structure in that the methods of quantum mechanics are applied along with the information obtained from molecular spectra. http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/molec.html#c1 Molecular Spectra The most commonly observed molecular spectra involve electronic, vibrational, or rotational transitions. For a diatomic molecule, the electronic states can be represented by plots of potential energy as a function of internuclear distance. Electronic transitions are vertical or almost vertical lines on such a plot since the electronic transition occurs so rapidly that the internuclear distance can't change much in the process. Vibrational transitions occur between different vibrational levels of the same electronic state. Rotational transitions occur mostly between rotational levels of the same vibrational state, although there are many examples of combination vibration-rotation transitions for light molecules. http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/molec.html#c1 Rotational Spectra Incident electromagnetic waves can excite the rotational levels of molecules provided they have an electric dipole moment. The electromagnetic field exerts a torque on the molecule. The spectra for rotational transitions of molecules is typically in the microwaveregion of the electromagnetic spectrum. The rotational energies for rigid molecules can be found with the aid of the Shrodinger equation. The diatomic molecule can serve as an example of how the determined moments of inertia can be used to calculate bond lengths. The illustration at left shows some perspective about the nature of rotational transitions. The diagram shows a portion of the potential diagram for a stable electronic state of a diatomic molecule. That electronic state will have several vibrational states associated with it, so that vibrational spectra can be observed. Most commonly, rotational transitions which are associated with the ground vibrational state are observed. Rotational Energies The classical energy of a freely rotating molecule can be expressed as rotational kinetic energy where x, y, and z are the principal axes of rotation and Ix represents Index the moment of inertia about the x-axis, etc. In terms of the angular momenta about the principal axes, the expression becomes Molecular spectra concepts The formation of the Hamiltonian for a freely rotating molecule is accomplished by simply replacing the angular momenta with the corresponding quantum mechanical operators. Diatomic molecules Go Back R HyperPhysics***** Quantum Physics Nave Diatomic Molecules For a diatomic molecule the rotational energy is obtained from the Schrodinger equation with the Hamiltonian expressed in terms of the angular momentum operator. More detail where J is the rotational angular momentum quantum number and I is the moment of inertia. Index Determining the rotational constant B Molecular spectra concepts enables you to calculate the bond length R. The allowed transitions for the diatomic molecule are regularly spaced at interval 2B. The measurement and Rotational transitions identification of one spectral line allows one to calculate the moment of inertia and then the bond length. Examples HCl CN CH Table of diatomic molecule parameters R Go Back HyperPhysics***** Quantum Physics Nave Rotational Transitions, Diatomic For a rigid rotor diatomic molecule, the selection rules for rotational transitions are ΔJ = +/-1, ΔMJ = 0 . The rotational spectrum of a diatomic molecule consists of a series of equally spaced absorption lines, typically in themicrowave region of Index the electromagnetic spectrum. The energies of the spectral lines are Molecular 2(J+1)B for the transitions J -> J+1. spectra For real molecules like the example of concepts HCl , the successive transitions are a bit lower than predicted because centrifugal distortion lengthens the molecule, increasing its moment of inertia. Moment of inertia for diatomic molecule Vibration/rotation transitions in HCl R Go Back HyperPhysics***** Quantum Physics Nave Moment of Inertia, Diatomic The moment of inertia about thecenter of mass is From the center of mass definition and http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/rotrig.html#c1 Vibrational Spectra of Diatomic Molecules The lowest vibrational transitions of diatomic molecules approximate the quantum harmonic oscillator and can be used to imply the bond force constants for small oscillations. The following is a sampling of transition frequencies from the n=0 to n=1 vibrational level for diatomic molecules and the calculated force constants. Frequency Force constant Molecule x1013 Hz N/m HF 12.4* 970 HCl 8.66 480 HBr 7.68 410 HI 6.69 320 CO 6.42 1860 NO 5.63 1530 * From vibrational transition 4138.52 cm- 1 in Herzberg'stabulation. http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/vibspe.html#c1 Vibration-Rotation Spectrum of HCl Add annotation to spectrum Index Molecular spectra concepts Reference Tipler & Llewellyn Sec. 9-4 A classic among molecular spectra, the infrared absorption spectrum of HCl can be analyzed to gain information about both rotation and vibration of the molecule. The absorption lines shown involve transitions from the ground to first excited vibrational state of HCl, but also involve changes in the rotational state. The rotational angular momentum changes by 1 during such transitions. If you had a transition from j=0 in the ground vibrational state to j=0 in the first excited state, it would produce a line at the vibrational transition energy. As observed, you get a closely spaced series of lines going upward and downward from that vibrational level difference. The splitting of the lines shows the difference in rotational inertia of the two chlorine isotopes Cl-35(75.5%) and Cl-37(24.5%). From the spectrum above, you can examine details about the following: Bond force constant Bond length j for peak intensity Relative intensities Pure rotational transitions in HCl Vibration-rotation spectrum of HBr HyperPhysics***** Quantum Physics Go Back Bond Force Constant for HCl By treating the vibrational transition in the HCl spectrum from its ground to first excited state as a quantum harmonic oscillator, the bond force constant can be calculated. This transition frequency is related to the molecular parameters by: The desired transition frequency does not show up directly in the observed spectrum, because there is no j=0, v=0 to j=0, v=1 transition; the rotational quantum number must change by one unit. It can be approximated by the midpoint between the j=1,v=0->j=0,v=1 transition and the j=0,v=0- Index >j=1,v=1 transition. This assumes that the difference between the j=0 and j=1 levels is the same for the ground Molecula and first excited state, which amounts to assuming that the r spectra first excited vibrational state does not stretch the bond. This concepts "rigid-rotor" model can't be exactly correct, so it introduces some error. Referenc e For the HCl molecule, the needed reduced mass is Tipler & Llewelly n Sec. 9-4 Note that this is almost just the mass of the hydrogen. The chlorine is so massive that it moves very little while the hydrogen bounces back and forth like a ball on a rubber band! Substituting the midpoint frequency into the expression containing the bond force constant gives: Despite the approximations, this value is quite close to the value given in the table. Pure rotational transitions in HCl HyperPhysics***** Quantum Physics Go Back Bond Length of HCl A bond length for the HCl molecule can be calculated from the HCl spectrum by assuming that it is a rigid rotor and solving the Schrodinger equation for that rotor. For a free diatomic molecule the Hamiltonian can be anticipated from the classical rotational kinetic energy and the energy eigenvalues can be anticipated from the nature of angular momentum. Assuming that the bond length is the same for the ground and first excited states, the difference between the j=1,v=0- >j=0,v=1 transition and the j=0,v=0->j=1,v=1 transition frequencies can be used to estimate the bond length. The separation between the two illustrated vibration-rotation transitions is assumed to be twice the rotational energy change from j=0 to j=1. Substitution of numerical values leads to an estimate of the bond length r: This compares reasonably with the value r=.127 nm obtained from pure rotational spectra. Other Heteronuclear Diatomic Molecules Index Equilibrium Separation (nm) Molecule Dissociation Energy(eV) (Bond length) Tables BN 4.0 0.128 Data CO 11.2 0.113 References HBr ..