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Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 Atomic Structure and Electronic Configurations Chemistry for Earth Scientists DM Sherman University of Bristol Discovery of the Electron! • • • • A Crookes tube consists of a cathode and anode in a sealed glass vessel in which there is a very small gas pressure. When a voltage is applied, a glowing discharge results. In 1897 JJ Thomson discovered that the discharge observed in a Crookes Tube (“cathode rays”) consisted of charged particles. He proposed that these very small particles (later called electrons), were fundamental components of matter. Page 1 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 Nuclear Model of the Atom! Gold foil Alpha particle (+ve charge) By bombarding gold foil with alpha particles, Ernest Rutherford (1871-1937) established that atoms consist of mostly empty space with a positively charged nucleus at the center. The electrons must surround the nucleus in clouds(?) or orbits (?)… Discovery of the Proton! • It was realized that atoms must contain positively charged particles to counteract the charge of the electrons. • Rutherford (1919) was able to generate positively charged hydrogen nuclei from the collision between alpha particles and nitrogen gas. • It was inferred that positively charged hydrogen nuclei must be present in all atoms. These fundamental particles were called protons. Page 2 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 Discovery of the Neutron! • In 1920, Ernest Rutherford hypothesized the existence of an uncharged particle as a further component of the atomic nucleus in order to explain the variations of isotopic masses. Atomic Spectroscopy! Atoms absorb and (when thermally excited) emit light but at discrete (quantized) wavelengths. Different elements have their own set of absorption/emission lines. The absorption lines observed in the spectrum of the solar heliosphere allow us to work out the elements present in the sun. Solar spectrum Page 3 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 Review of Waves! y = A sin(kx + ωt + φ) where k = 2π/λ The frequency (ν) of a wave is given by ν = c/λ Interference of Waves! The wave nature of light can be seen by interference patterns and diffraction. Constructive Destructive Page 4 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 Photoelectric Effect: Light as particles (photons)! The energy for the ejected electrons depends on the frequency of the light but not on its intensity. This led Einstein to propose that light occurs as packets or particles called photons with energy E = hν. The Bohr Model of the Atom! • • • Electrons orbit the nucleus at fixed radii with quantized energies. Atomic absorption and emission lines result from excitations between orbitals. However, this violates classical physics; the model offers no explanation for quantization and fails to predict the spectra of other elements. Balmer Series (visible light) 5 -2 4 -2 3 -2 Page 5 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 Wave Nature of Electrons! If light could act as a particle, perhaps particles could act like waves…Louis de Broglie (1892-1987) proposed that very small particles may display wave properties with a wavelength of λ = h/mv (where h is Planck’s constant). In 1927, this was verified for electrons by the experiment of Davisson and Germer where they showed that electrons can diffract. Postulates of Quantum Mechanics! • Any system can be described by a wavefunction Ψ • The probability of finding the system at coordinates (r1,r2..,t) is given by |Ψ(r1,r2..,t)|2 • For every observable λ, there corresponds and operator Λ which operates on the wavefunction: ΛΨ = λΨ Page 6 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 Examples of QM Operators! • The operator for momentum p (a vector) is ! ∂ ∂ ∂ $ p = i # i + j+ k & = i ∇ ∂y ∂z % " ∂x • The operator for total energy (a scaler) is E =− ∂ i ∂t = h / 2π = 1.05459 × 10−34 Joule − sec € € The Schrodinger Equation! Since kinetic energy (T) is p2/2m we get T=− 2 2 ∇ 2m Let V be the potential energy. Since total energy E is T + V € TΨ +VΨ = EΨ We get the Schrodinger equation: € − 2 2 ∇ Ψ +VΨ = EΨ 2m € Page 7 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 The Particle in a Box! Consider a particle in a one-dimensional box with length L. Inside the box, the potential energy V = 0; outside the box V = ∞ Inside the box, the S.E. takes the form − 2 d 2 Ψ = EΨ 2m dx 2 € The Particle in a Box (cont).! The solution to the SE must be of the form Ψ (x ) = Asin(kx ) + B cos(kx ) Since Ψ(0) = 0 , we get Ψ(0) = A sin(0) + B cos(0) = 0 so that B = 0 € Since Ψ(L) = 0 Ψ(L) = A sin(kL) = 0 € kL = nπ where n = 1,2,3,... € Page 8 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 The Particle in a Box (cont).! Now, the energy of the particle is given by the SE − 2 d 2 Ψ = EΨ 2m dx 2 Plugging in € Ψ(x) = A sin( nπ x) L gives the result that € h 2n 2 E = 8mL2 with n = 1, 2… € The Particle in a Box (cont).! The quantum spacing gets smaller as the box gets bigger or as the mass gets bigger relative to h. This is why we don’t see quantized motion in our macroscopic world. Page 9 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 The Free Particle! The quantized energies of the particle in a box result from the particle being bound. The wavefunctions of the particle are standing waves. Imagine if we made the box dimensions very large. The spacing between the quantum levels would go to zero. For a free particle (an infinite box), there is no quantization and the wavefunction is that of plane wave with continuous energies. The Hydrogen Atom! Here, the potential energy (V) of the electron is given by Coulombs Law: electron −e2 V= r -e r € +e nucleus (1 proton) Page 10 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 The Hydrogen Atom! It is convenient to transform the problem to spherical coordinates: z x = r sinθ cos φ y = r sinθ sinφ z = r cos θ (r,θ,φ) r θ φ y x € The Hydrogen Atom! The Schrodinger equation becomes: −2 % 1 ∂ % 2 ∂ψ ( 1 ∂ % ∂ψ ( 1 ∂ 2ψ ( r + sin θ + ' * ' * 2m '& r 2 ∂r & ∂r ) r 2 sinθ ∂θ & ∂θ ) r 2 sin2 θ ∂φ 2 *) − Ze2 ψ = Eψ r We can use a separation of variables to write € ψ (r) = ψ (r ,θ ,φ ) = R(r )Θ(θ )Φ(φ ) € Page 11 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 The Hydrogen Atom! We then get separate equations for R, Θ and Φ 1 d 2Φ = −ml2 2 Φ(φ ) dφ € 1 + 1 d $ dΘ ' ml2 . 2 sin θ − 0=m & ) 2 Θ(θ ) , sinθ dθ % dθ ( sin θ / 1 ⎡ d ⎛ 2 dR ⎞ 2 2m ⎛ e2 ⎞ ⎤ ⎢ ⎜ r ⎟ + r 2 ⎜E + ⎟R ⎥ = β R(r ) ⎢⎣ dr ⎝ dr ⎠ r ⎠ ⎥⎦ ⎝ € € The Hydrogen Atom: quantum numbers! The three separate equations for R, Θ and Φ yield three different quantum numbers n, l and ml • Principle quantum number (from r): n =1, 2, 3… • Angular momentum quantum number (from θ): l = 0, 1, 2 ,…, n-1. (s, p, d and f for l = 0,1, 2 and 3) • Magnetic quantum number (from φ): ml = -l,- l +1,…,+l. However, we also have “Spin quantum number” from relativity: ms = +1/2 and -1/2 (“up” and “down”) Page 12 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 The Hydrogen Atom: orbitals! The hydrogenic wavefunctions (orbitals) are spherical waves with shape determined by the l quantum number and orientation determined by the ml quantum number. The orbital energy is determined from the n quantum number: 1 ⎛ e2 ⎞ En = − 2 ⎜ ⎟ n ⎝ 2a0 ⎠ € What about the other atoms..! If we have more than one electron, it gets very complicated. Consider a He atom with two electrons at position r1 and r2. The potential energy takes the form V(r1,r2 ) = € −e2 −e2 e2 + + r1 r2 r1 − r2 Electron-electron repulsion term: This is the problem! Page 13 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 Electronic Configurations: Pauli Exclusion Principle We can approximate the electronic structures of multielectronic atoms as electronic configurations over one-electron (hydrogen-like) orbitals. The Pauli Exclusion principle states that no two electrons in an atom can have the same four quantum numbers. Hence, we can only put two electrons (one spin-up and one spin-down) in each nlm orbital. The Quantum Numbers and the Periodic Table of the Elements Page 14 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 Notation for X-ray Absorption/Emission! Low level High level Name of emission K (1s) L3 (2p3/2) Kα1 L2 (2p1/2) Kα2 M3 (3p3/2) Kβ1 L3 (2p3/2) M5 (3d5/2) Lα1 L2 (2p1/2) M4 (3d3/2) Lβ1 M5 (3d5/2) N7 (5p3/2) Mα1 The Electromagnetic Spectrum! Vibrational Modes Electronic Transitions Nuclear Transitions Page 15 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 Spectroscopic Methods! Method X-Ray Absorption Energy Range E > 0.5 keV Processes Information Obtained Atomic core electron excitations Element abundances; coordination environments via XANES and EXAFS. Distinguish surface complexation and precipitation from solid solution Identify and quantify aqueous complexes via crystal field and LMCT transitions. Optical 0.5 < E < Electronic Absorption and 4.5 eV excitations of Reflectance transition metal d-electrons. Infrared and < 0.5 eV Vibrations Raman and molecular rotations. Mossbauer 14.4 keV Nuclear (for Fe) transitions Identify complexes from vibrational spectroscopy. Site occupancies and oxidation states of Fe; magnetic properties Synchrotron Radiation! This is the light emitted by electrons when moving in curved trajectory. Synchrotron radiation provides a powerful source of light from the IR to X-rays. Diamond Light Source, UK Page 16 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 Beamlines at Diamond Light Source! X-Ray Absorption Spectroscopy O O 4s,4p 3d 3p 3s XANES = X-ray absorption near-edge spectroscopy Ni Photoelectron O O EXAFS = Extended Xray absorption nearedge spectroscopy X-ray photon 2p 2s 1s Page 17 Chemistry for Earth Scientists DM Sherman, University of Bristol 2011/2012 Summary Important Concepts: • • • • • Fundamental particles Atomic spectra Wave-particle duality Quantum mechanical model of the hydrogen atom Electronic configurations for multielectronic atoms in terms of the hydrogenic orbitals but obeying the Pauli Exclusion principle. Page 18