LECTURE 17 ATOMIC STRUCTURE & SPECTRA Instructor: Kazumi Tolich Lecture 17 2 ¨ Reading chapter 36-1 to 36-3 & 36-6 ¤ Atoms ¤ The Bohr model of the hydrogen atom ¤ Quantum theory of atoms ¤ The periodic table Atoms 3 ¨ 118 chemical elements have been discovered as of today. ¨ 94 chemical elements are naturally occurring on Earth. ¨ Each element is characterized by a number of protons, atomic number, Z. ¨ A neutral atom contains an equal number of protons and electrons. Atomic emission of light 4 ¨ The spacing and relative intensities of spectral lines in a spectrograph are Hydrogen emission spectrum characteristic of the element. Mercury emission spectrum Gas discharge tube Demo 1 5 ¨ Line Spectra ¤ Hydrogen, neon, mercury, and helium emission tubes are examined with transmission gratings. Thomson’s plum-pudding model of the atom 6 ¨ In 1904, J. J. Thomson, who discovered electrons, proposed the “plum-pudding model.” ¤ Atoms consist of a “cloud” of positive charge roughly 10-10 m in diameter. It was known that electrons are much smaller and less massive than atoms. ¤ Electrons are embedded like plums in a plum pudding. Rutherford’s model of the atom 7 ¨ Around 1911 Ernest Rutherford, with his students, showed that the positive charge and most of the mass of an atom is concentrated in a very small region (~1 fm), now called the nucleus. ¨ The majority of alpha particles fired at gold atoms were deflected through only a small angle, but a small number were deflected by a large angle. ¨ “It was quite the most incredible event that ever happened to me in my life. It was almost as incredible as if you had fired a 15-inch shell at a piece of tissue paper, and it came back and hit you.” Classical planetary model of H 8 ¨ The electron (−�) of the hydrogen atom moves in a circular orbit around the positive nucleus (+��) according to Coulomb’s law. ¨ The energy of the system is the sum of the kinetic energy of the electron plus the potential energy stored between the electron and the nucleus: 1 ��, � = − 8��+ � ¨ Classical EM theory says that an electron in a circular orbit is accelerating, so it would radiate an EM wave and loses its energy. ¨ This atom would quickly collapse as the electron spirals into the nucleus and radiates away the energy. Bohr’s postulate 1 9 ¨ As a solution for the collapsing atom problem, Bohr postulated that ¤ only certain “stationary states” are allowed. ¤ an atom with an electron in a stationary state does not radiate. ¤ an atom radiates only when the electron makes a transition from one stationary state to another. The electron in the hydrogen atom can move only in certain non- radiating, circular orbits called stationary states. Bohr’s postulate 2 10 ¨ The 2nd postulate relates the frequency of radiation, �, to the energies of the stationary states, E. � − � ��, 1 1 � = / 0 = − ℎ 8��+ℎ �0 �/ ¤ where Ei and Ef are the initial and final energies of the atom, and ri and rf are the radii of the initial and final orbits, respectively. Bohr’s postulate 3 11 4 4 ¨ From the Rydberg-Ritz formula, � = �� 6 − 6 , and Bohr’s 2nd postulate, 56 57 896 4 4 � = − , it is evident that r of stable orbits must be proportional :;<=> ?@ ?A to n2. ¨ Bohr postulated that the magnitude of the angular momentum of the electron in a stable orbit is given by 5> � ≡ �� � = �ℏ = , where � = 1,2,3, … 5 5 5 ,; Radii of Bohr orbits 12 ¨ The centripetal force on the electron is the Coulomb’s force. ¨ The orbital electron has the angular momentum given by Bohr’ 3rd postulate. ¨ The radii of the stationary state orbits: , 4��+ℏ �+ 0.0529 nm � = �, ≡ �, = �, 5 ���, � � �+: first Bohr radius Energy Levels in Bohr H 13 ¨ There is an energy level in H corresponding to each stationary state radius. �, ��U � = − 5 , , , , � 32� �+ ℏ �, �, � = − � = −13.6 eV 5 �, + �, Ionization 14 ¨ Removing an electron from an atom is called ionization. ¨ The minimum energy required to ionize the atom is the ionization energy. ¨ In a H atom in its ground state, electron is bound to the atom by the binding energy of 13.6 eV. ¨ The ionization energy of the ground-state hydrogen atom is 13.6 eV. Ground state Example 1 15 a) What is the wavelength of the least energetic photon emitted in the Lyman series of the hydrogen atom spectrum lines? b) What is the wavelength of the most energetic photon emitted in the Lyman series? Paschen Balmer Series Series (IR) (Visible) Lyman Series (UV) Success and failure 16 ¨ Bohr explained the stability and discrete spectra of H by introducing stationary states. ¨ However, there were problems. ¤ There was no justification for the stationary states or for the quantization of angular momentum. ¤ The Bohr’s method failed when two or more electrons were in orbit and could not predict the spectrum of helium. ¨ The quantum-mechanical model resolves these difficulties. Quantum theory of atoms 17 ¨ The stationary states of the Bohr model are replaced by the standing- wave solutions of the Schrödinger equation. ¨ Energy quantization is a consequence of the standing-wave solutions of the Schrödinger equation. ¨ For H, QM and the Bohr’s model give the same energy levels. ¨ The quantization of angular momentum postulated by Bohr is also predicted by QM. Schrödinger equation for an electron in an atom 18 ¨ The wave function for an electron in an atom is three dimensional. ¨ The requirement that wave function be continuous and normalizable introduces three quantum numbers for three dimensional wave function. ¨ Since the potential energy stored in the electron-nucleus system is spherically symmetric, it is convenient to use spherical coordinates. ℏ, − �,� + � � � = 0 2� Quantum numbers in spherical coordinates 19 - ¨ �: principal quantum number is associated with the probability of finding e at various r. ¨ �: orbital quantum number is related to the magnitude of the orbital angular momentum � = ℏ] � � + 1 . ¨ �^: magnetic quantum number is related to the component of the � along z- axis, �_ = �^ℏ. For l = 2 Quantum numbers in spherical coordinates 2 20 ¨ The possible values of these quantum numbers are ¤ n: principal quantum number can be any positive integer. ¤ l: orbital quantum number can be 0 or any positive integer up to n - 1. ¤ ml: magnetic quantum number has 2l + 1 possible values, ranging from -l to l. Energy and quantum numbers 21 ¨ General trend of energy vs. quantum numbers are: ¤ The lower the values of n, the lower the energy. ¤ For a given value of n, the lower the value of l, the lower the energy. ¨ The dependence of the energy on l is due to the interaction of the electrons in the atom with each other. Electron configuration 22 ¨ An electron has an intrinsic spin, so there is an additional quantum number, ms, which can be either +1/2 or -1/2. ¨ The stationary states of each electron in an atom is described by 4 quantum numbers: n, l, ml, and ms. ¨ Electron configuration can be specified by these 4 quantum numbers. ¨ No two electrons in an atom can be in the same quantum state. Quiz: 1 23 Example 2 24 ¨ An electron in an atom has principal quantum number n = 3. a) What are the possible values of l? b) What are the possible combinations of l and ml? c) Using the fact that there are two quantum states for each combination of l and ml because of electron spin, find the total number of electron states for n = 3. Quantum weirdness ¨ Bohr used to say “if you aren’t confused by quantum mechanics, then you don’t really understand it.” ¨ Based on the QM of a two-particle system, Einstein, Podolsky, and Rosen predicted results of a thought-experiment that were too weird for Einstein to accept QM. EPR thought experiment ¨ Suppose two photons A and B are simultaneously emitted in opposite direction by a source S. ¨ Each photon has a polarization X that may have two values, X1 and X2. ¨ It is always true that if A has X1, the B has X2, and vice versa. ¨ There are two quantum states: (AX1, BX2) and (AX2, BX1). ¨ Before any measurements is made, QM predicts that the actual state of this two-photon system is an equal-parts mixture of both states. ¨ By making a measurement on A, an experimentalist can reveal either X1 or X2 for A. ¨ Once the experimentalist reveals that A has X1, the system is no longer a mixture of states, but (AX1, BX2) alone. ¨ No matter how far apart A and B are, a measurement made on A automatically removes any choice for the state of B. ¨ This prediction was verified in 1980’s by Aspect et al. B S A Quantum weirdness: Schrödinger’s cat 27 ¨ Suppose a cat is placed in a sealed box containing a device that has a 50% chance of killing the cat. ¨ The wave function of the cat is ¨ The cat is dead AND alive until you open the box and check the status of the cat. ¨ When you make the observation, the cat is either dead OR alive. Quantum weirdness: e- double slit experiment ¨ When nobody is looking, an electron goes through both slits and produces interference patterns on the screen. ¨ If you place a detector to see which slit an electron goes through, interference patterns disappear. Quantum weirdness: non-locality ¨ Wave function collapses instantaneously when a detection is made somewhere. ¨ It is as if one throws a beer bottle into Boston Harbor. Its wave spreads all over the Atlantic. ¨ Then in Copenhagen, someone finds the beer bottle suddenly, and the ripples disappear everywhere at that moment.