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Energy levels of atoms and molecules Dr. Yin LI Department of Biophysics , Medical School, PTE [email protected] Outline • Interaction of light with matter – – – • Atomic Energy levels – – • Energy levels of hydrogen atom Bohr model Molecular Energy Levels – – – • Light properties Polarization of light wave Interaction of light with matter From atoms to a molecule (Electronic structure of hydrogen gas molecule (H2)) Electronic structure of other molecules (Types of chemical bonds) Electronic Spectroscopy (UV-vis Absorption & fluorescence spectroscopy, Jablonski diagram) Reference book: – Physical Chemistry, Authored by Peter Atkins & Julio De Paula, Oxford University Press Light Properties: Wave-particle duality • Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. • Properties of light: – – – Light travels in straight lines; Light speed in vacuum: ∼ 3 × 108 m/s; Light can be reflected; normal 1 2 𝜃1 = 𝜃2 Horizontal plane – Light can refract. sin 𝜃1 sin 𝜃2 𝑛 = 𝑛2 (Snell's law) 1 𝑛 denotes the refraction index of medium. Light Properties: Wave-particle duality • Wave Properties of light – Interference : two or more waves with the same frequency combine to reinforce or cancel each other, the amplitude of the resulting wave being equal to the sum of the amplitudes of the combining waves. – Diffraction: when waves encounter a small obstacle or opening, they tend to bend around that barrier or pass through it and become spread out. Light Properties: Wave-particle duality • Einstein's photoelectric effect – Many metals emit electrons when light shines upon them. The emission of photoelectrons has nothing to do with intensity of light (amplitude of electromagnetic wave) and irradiation time. As long as the incoming light exceeds a threshold of frequency, electron will be discharged from the surface of metal. Light behaves differently from ordinary waves. ℎ∙𝑣 ≥𝑊 – – – – • • h is Planck’s constant, W denotes the work function of metal, which means the minimum energy for photoelectrons to escape from the surface of metal. Energy of an ordinary wave: 𝐸 ∝ A2 and 𝐸 ∝ 𝑡 But, energy of light: 𝐸 ∝𝑓 The photoelectric effect elucidates particle properties of light. “Particle” here does NOT mean some staffs like beans or bullets. It represents packets of energy that can not be further divided. The particle of light is called “photon” which has a mass of 0 and energy of ℎ ∙ 𝑓. Properties of Photons (a) Velocity The velocity (ms-1) of any wave is given by the frequency, n (or number of waves per second, s-1) multiplied by the wavelength, l in metres. The velocity of light in vacuum is a fundamental constant, given the special symbol, c. c = n.l c takes the value 3 x 108 ms-1, which is pretty fast. The velocity of light (v) in any medium, of refractive index, n, is slowed compared to vacuum to a velocity v = c / n. The value of n is always greater than or equal to unity. (b) Energy Energy is often a property we associate with a particle (kinetic, potential, etc.). The energy associated with a single photon was deduced by Planck, one of the founders of the quantum theory as E = hn in which h is another fundamental constant, the Planck constant, which has the value h = 6.63 x 10-34 Js so E is in Joules. Clearly the energy of a photon depends on frequency (or wavelength) and this leads to the idea of the electromagnetic spectrum Polarization of light waves • Light is an electromagnetic wave, with Electric and Magnetic fields, that are in phase and orthogonal. • Un-polarized light: electric field vectors vibrate in all orientations that are perpendicular with respect to the direction of propagation. For example: sun light. If the electric field vectors are restricted to some orientations by filtration of the beam with specialized materials, then light is polarized. • The Interaction of Light and Matter Spectroscopy is the study of matter (for chemists matter = molecules) by their interaction with light. What is the mechanism of this interaction? Recall the simples picture of an atom, as a positive nucleus surrounded by a negatively charged ‘electron cloud’. Imagine this atom placed in a static electric field. Electrons will be attracted to the positive pole, the nucleus to the negative. This is polarization, and results in a dipole moment in the atom. This is refereed to as an induced dipole, as it is only present when the field is applied. The size of mind depends on the atom’s polarizability. Since light is an oscillating electric field it is not surprising that it interacts with the atom (the magnetic field is of much less significance in spectroscopy). Energy is transferred from the light to the molecule, and in so doing an induced (oscillating) dipole is generated. This process is referred to as absorption. The reverse process is also possible. When the light field is off the induced dipole relaxes back, and gives out energy as radiation. This process is called emission. Atomic Energy levels • • Hydrogen atom is the simplest model to study atomic energy levels. Emission spectrum of atom hydrogen was detected by the following device. • It was observed that the spectrum of atom hydrogen appears in a series of lines. This observation manifests that the energy levels of hydrogen atom is discrete (quantized). continuous discrete Energy levels of hydrogen atom • In 1885, a swiss mathematician-Balmer analyzed the hydrogen spectrum and proposed an empirical formula for the visible spectral lines of hydrogen atom. 1 1 1 / l R 2 2 2 n – • Where 𝑛 is an integer larger than 2, and 𝑅 is a constant, 0.010972 nm−1 . Wavelength (nm) n 656.2 3 486.1 4 434.0 5 410.1 6 364.6 ∞ Rydberg expanded Blamer’s formula to all series of hydrogen spectrum. 1 1 1 / l RH 2 2 n1 n2 – – 𝑅𝐻 is called Rydberg constant. 𝑅𝐻 ≈ 13.6 eV, or 1.097 × 105 cm−1 𝑛1 Name of Series 1 Lyman 2 Balmer 3 Paschen 4 Brackett 5 Pfund Energy levels of hydrogen atom • In 1913, Bohr developed a model for hydrogen atom. – – – – • Electrons move around the nucleus with high velocity in some certain allowed “stationary” orbits. Each stationary orbit corresponds to a definite energy. The orbit that is closer to nucleus has a less energy compared to the orbit that is far away from nucleus. Electron can jump between orbits by absorption or emission of radiation whose energy is equal to the energy difference between two orbits. Angular momenta for electrons are quantized. Limitations of Bohr model – – According to classical mechanics, any charged particle that is moving is going to emit electromagnetic radiation. Therefore, electron can’t circle around nucleus in “stationary” orbits. The final result is that electron spirals into nucleus. Due to smooth change of electron orbits, it would result in a smooth energy level instead of discrete energy levels. Bohr model just can give a good description of hydrogen atom. It failed in explanation of the multielectron atoms. • Uncertainty principle – Heisenberg proposed that you can be certain about the position or momentum of a subatomic particle but never both at the same time. This gave us a “electron cloud” model in which electrons do not have precisely described orbits. In classical mechanics, you can be certain about both momentum and position of this moving car running on road. In quantum mechanics, you can only know the probability that electron would appear in a certain position. Here comes quantum mechanics! • To correctly describe the quantum states of a physical system, the classical wave expression must be abandoned. In 1926, Schrödinger published his equation which was later known as Schrödinger equation to describe the quantum states of a system. – Time-independent Schrödinger equation 𝐸Ψ = 𝐻 Ψ – – • • Where, Ψ is wave function, 𝐸 is the eigenvalue of a quantum state, 𝐻 is an operator. To calculate the whole energy of the system, we sum both kinetic energy operator and potential energy operator 𝐻 = 𝐾 + 𝑉 , 𝐻 is called Hamiltonian. The requirements for Ψ: – – Ψ must be continuous; and its derivative must also be continuous. Ψ must be single-valued. – Ψ 2 must be integrable, and an electron should be 1. +∞ 2 Ψ d𝜏 −∞ = 1meaning that in whole space the total probability of finding Which function below is a qualified wavefunction? – Figure of wavefunction for 1s electron in hydrogen atom plotted as a function of radial distance between electron and nucleus . ∞ Ψ 𝑟 0 2 ∙ 4π𝑟 2 d𝑟 • Electron orbitals in atom • Electron configuration of atom • Formation of H2 molecule – When two hydrogen atoms approach each other, the electron of one atom begins to feel the attraction from both nuclei of these two atom. Electron cloud of each atom gets distorted and the electron density around each nucleus shifts toward the region between two atoms. As the distance between two atoms decreases, one electron becomes be able to enter the orbital of another electron. Finally, these two electrons are shared by both nuclei and become indistinguishable. At this moment, the covalent bond between these two atoms is formed, H2 molecule is generated. – H atom has only one electron in 1s orbital. The covalent bond formed from the combination of these two 1s electrons is called 𝜎 bonding orbital whose energy is lower than the sum of two hydrogen atoms. Simultaneously, there is also a 𝜎 ∗ antibonding orbital formed at a higher energy level. The energy level of antibonding orbital is higher than bonding orbitals and it raises the energy of molecule and destabilizes the molecule. Therefore, without any extra stimulation, electrons always stay at the bonding orbital. • The essential of molecular orbitals (MOs) – – Molecular orbitals (MOs) is the linear combination of atomic orbitals (LCAOs). Take H2 molecule as an example. Ψ𝐻2 = 𝑐1 ∙ 𝜓1𝐻 +𝑐2 ∙ 𝜓2𝐻 – Where 𝑐1 and 𝑐2 are coefficients, 𝜓1𝐻 and 𝜓2𝐻 are atomic orbitals of each hydrogen atom. 𝜎 bonds (head to head) 𝜋 bonds (back to back) • Potential energy surface The energy is plotted as a function of position of the atoms, which for H2 is just the internuclear separation, r. – – – – – • At infinite separation the energy is defined as zero. As the atoms approach the AOs overlap to form MOs. The bonding MO is more stable than the separated atoms. The stabilization energy reaches a minimum, when the favorable overlap is balanced by the repulsion between the two nuclei- this is the bond length. At smaller r the repulsion increases dramatically. Anti-bonding orbital Bonding orbital Electronic transition – Analogous to electronic transition in atoms, in molecule electron is also able to jump to a higher energy level after absorption of photons with a given wavelength which corresponds to energy difference between two energy levels. 𝜋 → 𝜋 ∗ transition 𝜎 → 𝜎 ∗ transition • Hybridization of s and p electrons in carbon atom E E sp3 sp3 sp3 sp3 Sp3 hybridization Sp3 hybridization Formation of CH4 + sp3 hybridized carbon atom • Hybridization of s and p electrons in carbon atom E 2pz E sp2 sp2 sp2 Sp2 hybridization Formation of CH2=CH2 • Formation of formaldehyde (CH2O) sp2 hybridized carbon atom 2pz E sp2 sp2 sp2 sp2 hybridized oxygen atom 2pz E sp2 sp2 sp2 • Energy levels of formaldehyde (CH2O) The order of energy required for transition : S0 → S2 S0 → S1 300 Collecting together data on molecular electronic transitions we can say that np* are weak np* are of long wavelength (low energy) ss* are strong, but of high energy pp* are strong and at intermediate energies. Such rules can be useful in assigning the UV vis spectra measured in the lab. In addition we have to remember that long conjugation length decreases the energy of pp* transitions. Some of these ideas are evident in the table below. Chromophore transition lmax / nm strong / weak C—C, C—H s s* ~150 strong O n s* ~185 weak N n s* ~195 weak S n s* ~195 weak O n p* ~300 weak O n s* ~190 weak p p* ~190 strong p p* 250 strong C C H2C C H H C C H H C CH2 • Jablonski Energy Diagram • Problems: 1. Work function of metal sodium is equal to 2.29 eV, please calculate whether a radiation with a wavelength of 400 nm can drive electrons to escape from the surface of metal sodium. (1 eV ≈ 1.62 × 10−19 J). 2. Calculate the wavelength of the light emitted by hydrogen from level 4 to level 3 using Rydberg equation. What is the nature of emitted light?(Infrared, visible, ultraviolet or x-Ray) 3. The oxygen atom in water molecule is sp3 hybridized. Please draw the energy levels of this oxygen atom before and after sp3 hybridization. Please also draw all the chemical bonds within water molecule and mark their types (𝜎 or 𝜋 bond?). Explain why water molecule has a permanent dipole.