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Physics of the Atom Physics of the Atom Ernest The Nobel Prize in Chemistry 1908 Rutherford "for his investigations into the disintegration of the elements, and the chemistry of radioactive substances" Niels Bohr The Nobel Prize in Physics 1922 "for his services in the investigation of the structure of atoms and of the radiation emanating from them" W. Ubachs – Lectures MNW-1 Early models of the atom atoms : electrically neutral they can become charged positive and negative charges are around and some can be removed. popular atomic model “plum-pudding” model: W. Ubachs – Lectures MNW-1 Rutherford scattering Rutherford did an experiment that showed that the positively charged nucleus must be extremely small compared to the rest of the atom. Result from Rutherford scattering 2 dσ ⎛ 1 Zze2 ⎞ 1 = ⎜ ⎟ ⎜ ⎟ 4 dΩ ⎝ 4πε0 4K ⎠ sin ()θ / 2 Applet for doing the experiment: http://www.physics.upenn.edu/courses/gladney/phys351/classes/Scattering/Rutherford_Scattering.html W. Ubachs – Lectures MNW-1 Rutherford scattering the smallness of the nucleus the radius of the nucleus is 1/10,000 that of the atom. the atom is mostly empty space Rutherford’s atomic model W. Ubachs – Lectures MNW-1 Atomic Spectra: Key to the Structure of the Atom A very thin gas heated in a discharge tube emits light only at characteristic frequencies. W. Ubachs – Lectures MNW-1 Atomic Spectra: Key to the Structure of the Atom Line spectra: absorption and emission W. Ubachs – Lectures MNW-1 The Balmer series in atomic hydrogen The wavelengths of electrons emitted from hydrogen have a regular pattern: Johann Jakob Balmer W. Ubachs – Lectures MNW-1 Lyman, Paschen and Rydberg series the Lyman series: the Paschen series: W. Ubachs – Lectures MNW-1Janne Rydberg The Spectrum of the hydrogen Atom A portion of the complete spectrum of hydrogen is shown here. The lines cannot be explained by classical atomic theory. W. Ubachs – Lectures MNW-1 The Bohr Model Solution to radiative instability of the atom: 1. atom exists in a discrete set of stationary states no radiation when atom is in such state 2. radiative transitions Æ quantum jumps between levels hc hν = = E − E λ i f 3. angular momentum is quantized . These are ad hoc hypotheses by Bohr, against intuitions of classical physics W. Ubachs – Lectures MNW-1 The Bohr Model: derivation An electron is held in orbit by the Coulomb force: (equals centripetal force) Fcentr = FCoulomb mv2 Ze2 = 2 rn 4πε0rn Ze2 r = Solve this equations for r: n 2 4πε0mv Impose the quantisation condition: L = mvrn = nh W. Ubachs – Lectures MNW-1 The Bohr Model: derivation Solve: the size of the orbit is quantized (and we know the size of an atom !) . Size of an atom: 0.5 Å W. Ubachs – Lectures MNW-1 The Bohr Model: derivation The energy of a particle in orbit 2 2 1 2 Ze ⎛ 1 ⎞ Ze En = Ekin + E pot = mv − = ⎜ −1⎟ 2 4πε0rn ⎝ 2 ⎠ 4πε0rn So 2 1 Ze2 ⎛ Z 2 ⎞⎛ e2 ⎞ m E = − = −⎜ ⎟⎜ ⎟ n ⎜ 2 ⎟⎜ ⎟ 2 2 4πε0rn ⎝ n ⎠⎝ 4πε0 ⎠ 2h Define ⎛ Z 2 ⎞ E = −⎜ ⎟R n ⎜ 2 ⎟ ∞ ⎝ n ⎠ 2 ⎛ e2 ⎞ m With the Rydberg constant R = ⎜ ⎟ e ∞ ⎜ ⎟ 2 ⎝ 4πε0 ⎠ 2h W. Ubachs – Lectures MNW-1 EXTRA Reduced mass in the old Bohr model Æ the one particle problem Relative coordinates: Velocity vectors: r r r M r = r1 − r2 vr = vr 1 m + M Centre of Mass r M r v2 = − v r r m + M mr1 + Mr2 = 0 Relative velocity Position vectors: r r dr M v = rr = rr dt 1 m + M m rr = − rr 2 m + M W. Ubachs – Lectures MNW-1 EXTRA Reduced mass mM μ = m + M Centripetal force m v2 m v2 μv2 F = 1 1 = 2 2 = r1 r2 r Kinetic energy 1 1 1 K = m v2 + m v2 = μv2 Quantisation of angular momentum: 2 1 1 2 2 2 2 h L = μvr = n = nh Angular momentum 2π L = m v r + m v r = μvr These are the equations for a 1 1 1 2 2 2 single (reduced) particle W. Ubachs – Lectures MNW-1 EXTRA Result for the one-particle problem Quantisation of radius in orbit: Rewrite: 2 2 2 2 Z ⎛ μ ⎞ 1 2 2 n 4πε0h n me En = − ⎜ ⎟ α mec rn = = a0 2 ⎜ m ⎟ 2 Z e2μ Z μ n ⎝ e ⎠ dimensionless energy Energy levels in the Bohr model: 1 α ≈ 137 Z 2 μ E = − R n 2 ∞ n me mM μ = For all atoms: μ ≈ me Isotope effect m + M W. Ubachs – Lectures MNW-1 Calculate the isotope shift on Lyman-α: Optical transitions in The Bohr Model The lowest energy level is called the ground state; the others are excited states. Calculate wavelengths and frequencies c hc λ = = f E2 − E1 W. Ubachs – Lectures MNW-1 Quantum states and kinetics Hydrogen at 20°C. Estimate the average kinetic energy of whole hydrogen atoms (not just the electrons) at room temperature, and use the result to explain why nearly all H atoms are in the ground state at room temperature, and hence emit no light. 3 K = k T = 6.2×10−21 J = 0.04eV 2 B Gas at elevated temperature – probability exited state is populated: −En / kT Pn (T ) = e Energy in gas: −En / kT ∑ Ene E = n ∑e−En / kT n W. Ubachs – Lectures MNW-1 Another way of looking at the quantisation in the atom de Broglie’s Hypothesis Applied to Atoms Electron of mass m has a wave nature Electron in orbit: a standing wave 2πrn = nλ Substitution gives the quantum condition nh L = mvr = n 2π W. Ubachs – Lectures MNW-1 de Broglie’s Hypothesis Applied to Atoms These are circular standing waves for n = 2, 3, and 5. Standing waves do not radiate; Interpretation: electron does not move (no acceleration) W. Ubachs – Lectures MNW-1.
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