Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CY 101- Chemistry Atomic Structure Dr. Priyabrat Dash Email: [email protected] Office: BM-406, Mob: 8895121141 Webpage: http://homepage.usask.ca/~prd822/ Dr. Priyabrat Dash Email: [email protected] Office: BM-406, Mob: 8895121141 Webpage: http://homepage.usask.ca/~prd822/ Dr. Priyabrat Dash Email: [email protected] Office: BM-406, Mob: 8895121141 Webpage: http://homepage.usask.ca/~prd822/ Atoms • • • • The atom is mostly empty space. Two regions. Nucleus- protons and neutrons. Electron cloud- region where you might find an electron. Subatomic particles Name Symbol Charge Relative mass Actual mass (g) Electron e- -1 1/1840 9.11 x 10-28 Proton p+ +1 1 1.67 x 10-24 Neutron n0 0 1 1.67 x 10-24 Bohr’s Model Increasing energy Fifth Fourth Nucleus Orbit Third Second First Nucleus Energy Levels Electron Further away from the nucleus means more energy. There is no “in between” energy •Few difficulty Bohr model had •The spectra of larger atoms. At best, it can make some approximate predictions about the emission spectra for atoms with a single outer-shell electron •The relative intensities of spectral lines •The existence of fine and hyperfine structure in spectral lines. •The Zeeman effect - changes in spectral lines due to external magnetic fields Bohr model: A semiclassical model Wave-particle duality nature of light Classical physics- A system can absorb or emit any amount of energy. Plank’ hypothesis- Energy absorbed or emitted by a black body is restricted to the relation E = hν Consequences: 1. Photoelectricity (Eistein, 1906) 2. Theories of atomic spectra (Bohr, 1913) Photoelectric Effect Emission of electrons from metals when exposed to (ultraviolet) radiation. Explanation (EINSTEIN 1905) Threshold frequency 0 given by = h0 For > 0, the kinetic energy of the emitted electron Ek = h = h( 0). Line Spectrum of Hydrogen atom when subjected to higher temperature or a electric discharge emit electromagnetic radiation R = Rydberg Constant 109677.6 cm-1 Wave particle duality of material particles Louis de Broglie (electron has wave properties) Werner Heisenberg (uncertainty Principle) Erwin Schrodinger (mathematical equations using probability, quantum numbers) de Broglie wavelength of electrons λ= h/p Consequence: Electrons produce diffraction pattern when passed through thin diffraction grating (Davison and Germer) Electron Motion Around Atom Shown as a de Broglie Wave Schrödinger's equation, what is it ? Newton’s law allows you to describe motion of mechanical systems and mathematically predict the outcome of the system. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is a wave equation in terms of the wavefunction which predicts analytically and precisely the probability of events or outcome. The Schrodinger equation gives the quantized energies of the system and gives the form of the wavefunction so that other properties may be calculated. The wave equation developed by Erwin Schrodinger in 1926 (one-dimensional form) About the Wavefunction The wavefunction is assumed to be a single-valued function of position and time, which is sufficient to get a value of probability of finding the particle at a particular position and time. The wavefunction is a complex function, since it is its product with its complex conjugate which specifies the real physical probability of finding the particle in a particular state. WAVEFUNCTION () (PSI) Classically, the state of a system is described by its position and momentum In Quantum theory, the state of a system is described by its wavefunction WAVEFUNCTION () (PSI) 1. A wavefunction is a mathematical function (like sinx, ex). Like any mathematical function it can have large value at some place, small in other and zero elsewhere. It can be real or complex 2. A wavefunction contains all information about the system 3. The wavefunction is a function of Cartesian coordinate and time. ie. (x, y, z, t) 4. If the wavefunction is large at a point in space, the particle has a large probability at that point 5. The more rapidly a wavefunction changes from place to place, higher the K.E. of the particle it describes WAVEFUNCTION () (PSI) A wavefunction describes the state of a system How? The state of a system is described by some measurable quantities such as mass, volume, momentum, position, Energy etc. These quantities are called observables How to determine the observables from wavefunction () By performing a set of well defined mathematically operations on . These mathematical operations are called operators Born interpretation 2 The state of a system (particle) is completely specified by its wavefunction (x,y,z,t), which is a probability amplitude and has the significance that 2 dV (more generally 2dV since may be complex) represents the probability that the particle is located in the infinitesimal element of volume dV about the given point, at time t. NORMALIZATION As per Born interpretation, the probability of existence of the particle in the entire space should be 1. In mathematical term the wave function has to be normalized For one dimension N 2 2 dx 1 or N 2 dx 1 Where N is the normalization constant N 1 dx 2 1 2 In 1913 Niels Bohr came up with a new atomic model in which electrons are restricted to certain energy levels. Schrödinger applied his equation to the hydrogen atom and found that his solutions exactly reproduced the energy levels stipulated by Bohr. The result was amazing and one of the first major achievement of Schrödinger's equation and earned him the 1933 Nobel Prize in physics. Newton’s Law: Conservation of Energy ( Harmonic Oscillator example)) Time independent Schrodinger Equation Quantum conservation of Energy Schrodinger Equation H = E In a wave equation, physical variables takes the form of “operators” (H), Hamiltonian operator In three dimensions, ħ2/2m (2 ψ/x2 + 2 ψ/y2 + 2ψ/z2) + U(x,y,z)ψ(x,y,z) = Eψ(x,y,z) Time dependent Schrodinger Equation Eigen value equations (operator) (function)= (constant factor) (function) d ax ax e ae dx 2 d ( Sin 4 x) 16( Sin 4 x) 2 dx 22 How to proceed further? •Let us test the Schrödinger equation for some simple physical systems! Some model systems at first – easy to test ! Hopefully, some quantum properties of matter will also be understood! •If we can understand these ‘simple systems’ without doing any approximation and can derive ‘exact solutions’, then we shall proceed towards our major target – atoms and molecules •Why we do not go right now to solve the Schrödinger equation for atoms and molecules? Because the mathematical solution for atoms and molecules is complicated. 23 Characteristics of the wavefunctions 1. Wavelength = 2L/n 2. There are n-1 nodes (interior points where the wave function passes through zero) in the wavefunction n 3. The energy increases with increasing number of nodes. The ground state has no nodes. 4. The ground state energy is not 0, but h2/8mL2, the zero point energy. This is a consequence of the uncertainty principle. 24 Applications of this model 1. Calculation of energy of -electrons of conjugated olefins H2 C C H C H CH2 2. Electrons in nano materials 3. Electrons present in cavities or color centers 4. Translational motion of ideal gas molecules 25